quadrature of the circle_.” Seeing therefore you were resolved to show
where I erred, you should have proved either that the supposition was
true, and the conclusion falsely inferred, or contrarily, that though
the supposition be false, yet the conclusion is true; for else you
object nothing to my geometry, but only to my judgment, in thinking fit
to publish it; which nevertheless you cannot justly do, seeing it was
likely to give occasion to ingenious men (the practice of it being so
accurate to sense) to inquire wherein the fallacy did consist. And for
the problem as it was first printed, but never published, and
consequently ought to have passed for a private paper stolen out of my
study, your public objecting against it (in the opinion of all men that
have conversed so much with honest company as to know what belongs to
civil conversation), was sufficiently barbarous in divines. And seeing
you knew I had rejected that proposition, it was but a poor ambition to
take wing as you thought to do, like beetles from my egestions. But let
that be as it will, you will think strange now I should resume, and make
good, at least against your objection, that very same proposition. So
much of the figure as is needful you will find noted with the same
letters, and placed at the end of this fifth lesson. Wherein let B I, be
an arch not greater than the radius of the circle, and divided into four
equal parts, in L, N, O. Draw S N, the sine of the arch B N, and produce
it to T, so as S T be double to S N, that is, equal to the chord B I.
Draw likewise _a_ L, the sine of the arch B L, and produce it to _c_, so
as _a c_ be quadruple to _a_ L, that is, equal to the two chords B N, N
I. Upon the centre N with the radius N I, draw the arch I _d_, cutting B
U the tangent in _d_. Then will B N produced cut the arch I _d_, in the
midst at _o_. In the line B S produced take S _b_, equal to B S; then
draw and produce _b_ N, and it will fall on the point _d_. And B _d_, S
T, will be equal; and _d_ T joined and produced will fall upon _o_, the
midst of the arch I _d_. Join I T, and produce it to the tangent B U in
U. I say, that the straight line I T U shall pass through _c_. For
seeing B S, S _b_, are equal, and the angle at S a right angle, the
straight lines B N, and _b_ N, are also equal, and the triangles B N
_b_, _d_ N _o_ like and equal; and the lines _d_ T, T _o_ equal. Draw _o
i_ parallel to _d_ U, cutting I U in _i_; and the triangles _d_ T U, _o_
T _i_ will also be like and equal. Produce S T to the arch _d o_ I in
_e_, and produce it further to _f_, so that the line _e f_ be equal to T
_e_; and then S _f_ will be equal to _a c_. Therefore _f c_ joined will
be parallel to B S. In _c f_ produced take _f g_ equal to _c f_; and
draw _g m_ parallel to _d_ U, cutting I U in _m_, and _d o_ in _n_; and
let the intersection of the two lines _a c_ and _d o_ be in _r_; which
being done, the triangles _m n_ T, _r c_ T will be like and equal.
Therefore _m n_ and _r c_ are equal; and consequently the straight line
I _m_ T U shall pass through _c_. Dividing therefore _a c_ in the midst
at _t_, and S N in the midst at _l_, and joining _t_ N, L _l_, the lines
L _l_, _t_ N, and _c_ T produced, will all meet in one and the same
point of B S produced; suppose at _q_. Therefore the point _q_ being
given by the two known points T and I, the lines drawn from _q_ through
equal parts of the sine of the arch B I, (for example through the points
P, Q, R, of the sine M I), shall cut off equal arches, as B L, L N, N O,
O I. And this is enough to make good that problem, as to your objection.
The straight line therefore B U, for any thing you have said, is proved
equal to the arch B I, and the division of any angle given into any
proportion given, the quadrature of any sector, and the construction of
any equilateral polygon is also given. And though in this also I should
have erred, yet it cannot be denied but that I have used a more natural,
a more geometrical, and a more perspicuous method in the search of this
so difficult a problem, than you have done in your _Arithmetica
Infinitorum_. For though it be true that the aggregate of all the mean
proportionals between the radius, together with an infinitely little
part of the same, and the radius wanting an infinitely little part of
the same; and again, between the radius, together with two infinitely
little parts, and the radius wanting two infinitely little parts, and so
on eternally, will be equal to the quadrant (a thing which every mean
geometrician knew before); yet it was absurd to think those means could
be calculated in numbers by interpoling of a symbol; especially when you
make that symbol to stand for a number neither true nor surd; as if
there were a number that could neither be uttered in words, nor not be
uttered in words. For what else is surd, but that which cannot be
spoken?
To the fifth article, though your discourse be long, you object but two
things. One is, that “_Whereas the spiral of Archimedes is made of two
motions, one straight, the other circular, both uniform, I taking the
motion compounded of them both for one of those that are compounded,
conclude falsely, that the generation of the spiral is like to the
generation of the parabola_.” What heed you use to take in your
reprehensions, appears most manifestly in this objection. For I say in
that demonstration of mine, that _the velocity of the point A in
describing the spiral increaseth continually in proportion to the
times_. For seeing it goes on uniformly in the semidiameter, it is
impossible it should not pass into greater and greater circles,
proportionally to the times, and consequently it must have a swifter and
swifter motion circular, to be compounded with the uniform motion in
every point of the radius as it turneth about. This objection therefore
is nothing but an effect of a will, without cause, to contradict.
The other objection is, that “_Granting all to be true hitherto, yet
because it depends upon the finding of a straight line equal to a
parabolical line in the eighteenth chapter, where I was deceived, I am
also deceived here_.” True. But because in the eighteenth chapter of
this English edition I have found a straight line equal to the spiral
line of Archimedes. I must here put you in mind that by these words in
your objections to the fifth article at your number two, _Quatenus verum
est, etc._, _we have demonstrated prop. 10, 11, 13_, _Arithmetica
Infinitorum_; you make it appear that you thought your spiral (made of
arches or circles) was the true spiral of Archimedes; which is fully as
absurd as the quadrature of Joseph Scaliger, whose geometry you so much
despise.
To the sixth article, which is a digression concerning the analytics of
geometricians, you deny _that the efficient cause of the construction
ought to be contained in the demonstration_. As if any problem could be
known to be truly done, otherwise than by knowing first how, that is to
say, by what efficient cause, and in what manner, it is to be done.
Whatsoever is done without that knowledge, cannot be demonstrated to be
done; as you see in your computation of the parabola, and paraboloeides,
in your _Arithmetica Infinitorum_.
And whereas I said that _the ends of all straight lines drawn from a
straight line, and passing through one and the same point, if their
parts be proportional, shall be in a straight line_; is true and
accurate; as also, _if they begin in the circumference of a circle, they
shall also be in the circumference of another circle_. And so is this:
_if the proportion be duplicate, they shall be in a parabola_. All this
I say is true and accurately spoken. But this was no place for the
demonstration of it. Others have done it. And I perceive by that you put
in by parenthesis (“_Intelligis credo inter duas peripherias
concentricas_”) that you understand not what I mean.
Hitherto reach your objections to my geometry: for the rest of your
book, it containeth nothing but a collection of lies, wherewith you do
what you can, to extenuate as vulgar, and disgrace as false, that which
followeth, and to which you have made no special objection.
I shall therefore only add in this place concerning your _Analytica per
Potestates_, that it is no art. For the rule, both in Mr. Ougthred, and
in Des Cartes, is this: “_When a problem or question is propounded,
suppose the thing required done, and then using a fit ratiocination, put
A or some other vowel for the magnitude sought_.” How is a man the
better for this rule without another rule, how to know when the
ratiocination is fit? There may therefore be in this kind of analysis
more or less natural prudence, according as the analyst is more or less
wise, or as one man in choosing of the unknown quantity with which he
will begin, or in choosing the way of the consequences which he will
draw from the hypothesis, may have better luck than another. But this is
nothing to art. A man may sometimes spend a whole day in deriving of
consequences in vain, and perhaps another time solve the same problem in
a few minutes.
I shall also add, that symbols, though they shorten the writing, yet
they do not make the reader understand it sooner than if it were written
in words. For the conception of the lines and figures (without which a
man learneth nothing) must proceed from words either spoken or thought
upon. So that there is a double labour of the mind, one to reduce your
symbols to words, which are also symbols, another to attend to the ideas
which they signify. Besides, if you but consider how none of the
ancients ever used any of them in their published demonstrations of
geometry, nor in their books of arithmetic, more than for the roots and
potestates themselves; and how bad success you have had yourself in the
unskilful using of them, you will not, I think, for the future be so
much in love with them as to demonstrate by them that first part you
promise of your _Opera Mathematica_. In which, if you make not amends
for that which you have already published, you will much disgrace those
mathematicians you address your epistles to, or otherwise have
commended; as also the Universities, as to this kind of learning, in the
sight of learned men beyond sea. And thus having examined your pannier
of Mathematics, and finding in it no knowledge, neither of quantity, nor
of measure, nor of proportion, nor of time, nor of motion, nor of any
thing, but only of certain characters, as if a hen had been scraping
there; I take out my hand again, to put it into your other pannier of
theology, and good manners. In the mean time I will trust the objections
made by you the astronomer (wherein there is neither close reasoning,
nor good style, nor sharpness of wit, to impose upon any man) to the
discretion of all sorts of readers.
==========
OF MANNERS.
TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN
THE UNIVERSITY OF OXFORD.
LESSON VI.
Having in the precedent lessons maintained the truth of my geometry, and
sufficiently made appear that your objections against it are but so many
errors of your own, proceeding from misunderstanding of the propositions
you have read in Euclid, and other masters of geometry; I leave it to
your consideration to whom belong, according to your own sentence, the
unhandsome attributes you so often give me upon supposition, that you
yourselves are in the right, and I mistaken; and come now to purge
myself of those greater accusations which concern my manners. It cannot
be expected that there should be much science of any kind in a man that
wanteth judgment; nor judgment in a man that knoweth not the manners due
to a public disputation in writing; wherein the scope of either party
ought to be no other than the examination and manifestation of the
truth. For whatsoever is added of contumely, either directly or
_scommatically_, is want of charity and uncivil, unless it be done by
way of reddition from him that is first provoked to it. I say unless it
be by way of reddition; for so was the judgment given by the emperor
Vespasian in a quarrel between a senator and a knight of Rome which had
given him ill language. For when the knight had proved that the first
ill language proceeded from the senator, the emperor acquitted him in
these words: “_Maledici senatoribus non oportere; remaledicere, fas et
civile esse_.” Nevertheless, now-a-days, uncivil words are commonly and
bitterly used by all that write in matter of controversy, especially in
divinity, excepting now and then such writers as have been more than
ordinarily well bred, and have observed how heinous and hazardous a
thing such contumely is amongst some sorts of men, whether that which is
said in disgrace be true or false. For evil words by all men of
understanding are taken for a defiance, and a challenge to open war. But
that you should have observed so much, who are yet in your mother’s
belly, was not a thing to be much expected.
The faults in manners you lay to my charge are these: 1. _Self-conceit._
2. _That I will be very angry with all men that do not presently submit
to my dictates._ 3. _That I had my doctrine concerning Vision, out of
papers which I had in my hands of Mr. Warner’s._ 4. _That I have injured
the universities._ 5. _That I am an enemy to religion._ These are great
faults; but such as I cannot yet confess. And therefore I must, as well
as I can, seek out the grounds upon which you build your accusation.
Which grounds (seeing you are not acquainted with my conversation) must
be either in my published writings, or reported to you by honest men,
and without suspicion of interest in reporting it. As for my
self-conceit and ostentation, you shall find no such matter in my
writings. That which you allege from thence is first, that in the
epistle dedicatory I say of my book _De Corpore_, “_though it be little,
yet it is full; and if good may go for great, great enough._” When a man
presenting a gift great or small to his betters, adorneth it the best he
can to make it the more acceptable; he that thinks this to be
ostentation and self-conceit, is little versed in the common actions of
human life. And in the same epistle, where I say of civil philosophy:
“_It is no ancienter than my book De Cive_;” these words are added: “_I
say it provoked, and that my detractors may see they lose their
labour_.” But that which is truly said, and upon provocation, is not
boasting, but defence. A short sum of that book of mine, now publicly in
French, done by a gentleman I never saw, carrieth the title of _Ethics
Demonstrated_ . The book itself translated into French, hath not only a
great testimony from the translator Sorberius, but also from Gassendus,
and Mersennus, who being both of the Roman religion had no cause to
praise it, or the divines of England have no cause to find fault with
it. Besides, you know that the doctrine therein contained is generally
received by all but those of the clergy, who think their interest
concerned in being made subordinate to the civil power; whose
testimonies therefore are invalid. Why therefore, if I commend it also
against them that dispraise it publicly, do you call it boasting? “_You
have heard_,” you say, “_that I had promised the quadrature of the
circle, &c._” You heard then that which was not true. I have been asked
sometimes, by such as saw the figure before me, what I was doing, and I
was not afraid to say I was seeking for the solution of that problem;
but not that I had done it. And afterwards being asked of the success, I
have said, I thought it done. This is not boasting; and yet it was
enough, when told again, to make a fool believe it was boasting. But
you, the astronomer, in the epistle before your philosophical essay, say
“_You had a great expectation of my philosophical and mathematical
works, before they were published_.” It may be so. Is that my fault? Can
a man raise a great expectation of himself by boasting? If he could,
neither of you would be long before you raised it of yourselves; saving
that what you have already published, has made it now too late. For I
verily believe there was never seen worse reasoning than in that
philosophical essay; which any judicious reader would believe proceeded
from a prevaricator, rather than from a man that believed himself; nor
worse principles, than those in your books of Geometry. The expectation
of that which should be written by me, was raised partly by the
_Cogitata Physica-Mathematica_ of Mersennus, wherein I am often named
with honour; and partly by others with whom I then conversed in Paris,
without any ostentation. That no man has a great expectation of any
thing that shall proceed from either of you two, I am content to let it
be your praise.
Another argument of my self-conceit, you take from my contempt of the
writers of metaphysics and school-divinity. If that be a sign of
self-conceit, I must confess I am guilty; and if your geometry had then
been published, I had contemned that as much. But yet I cannot see the
consequence (unless you lend me your better logic) from despising
insignificant and absurd language, to self-conceit.
And again, in your _Vindiciæ Academiarum_, you put for boasting, that in
my _Leviathan_ , page 331, I would have _that book by entire sovereignty
imposed upon the Universities_; and in my _Review_ , p. 713, that I say
of my _Leviathan_ , “_I think it may be profitably printed, and more
profitably taught in the University_.” The cause of my writing that
book, was the consideration of what the ministers before, and in the
beginning of, the civil war, by their preaching and writing did
contribute thereunto. Which I saw not only to tend to the abatement of
the then civil power, but also to the gaining of as much thereof as they
could (as did afterwards more plainly appear) unto themselves. I saw
also that those ministers, and many other gentlemen who were of their
opinion, brought their doctrines against the civil power from their
studies in the Universities. Seeing therefore that so much as could be
contributed to the peace of our country, and the settlement of sovereign
power without any army, must proceed from teaching; I had reason to
wish, that civil doctrine were truly taught in the Universities. And if
I had not thought that mine was such, I had never written it. And having
written it, if I had not recommended it to such as had the power to
cause it to be taught, I had written it to no purpose. To me therefore
that never did write anything in philosophy to show my wit, but, as I
thought at least, to benefit some part or other of mankind, it was very
necessary to commend my doctrine to such men as should have the power
and right to regulate the Universities. I say my doctrine; I say not my
_Leviathan_ . For wiser men may so digest the same doctrine as to fit it
better for a public teaching. But as it is, I believe it hath framed the
minds of a thousand gentlemen to a conscientious obedience to present
government, which otherwise would have wavered in that point. This
therefore was no vaunting, but a necessary part of the business I took
in hand. You ought also to have considered, that this was said in the
close of that part of my book which concerneth policy merely civil.
Which part, if you, the astronomer, that now think the doctrine unworthy
to be taught, were pleased once to honour with praises printed before
it, you are not very constant nor ingenuous. But whether you did so or
not, I am not certain, though it was told me for certain. If it were not
you, it was somebody else whose judgment has as much weight at least as
yours.
And for anything you have to say from your own knowledge, I remember not
that I ever saw either of your faces. Yet you, the professor of
geometry, go about obliquely to make me believe that Vindex hath
discoursed with me, once at least, though I remember it not. I suppose
it therefore true; but this I am sure is false, that either he or any
man living did ever hear me brag of my science, or praise myself, but
when my defence required it. Perhaps some of our philosophers that were
at Paris at the same time, and acquainted with the same learned men that
I was acquainted with, might take for bragging the maintaining of my
opinions, and the not yielding to the reasons alledged against them. If
that be ostentation, they tell you the truth. But you that are so wise
should have considered, that even such men as profess philosophy are
carried away with the passions of emulation and envy (the sole ground of
this your accusation) as well as other men, and instanced in yourselves.
And this is sufficient to shake off your aspersions of ostentation and
self-conceit. For if I added, that my acquaintance know that I am
naturally of modest rather than of boasting speech, you will not believe
it; because you distinguish not between that which is said upon
provocation, and that which is said without provocation, from vain
glory.
The next accusation is: “_That I will be very angry with all men that do
not presently submit to my dictates; and that for advancing the
reputation of my own skill, I care not what unworthy reflections I cast
on others_.” This is in the epistle placed before the _Vindiciæ
Academiarum_, subscribed by N S, as the plain song for H D in the rest
of the book to descant upon. I know well enough the authors’ names; and
am sorry that N S has lent his name to be abused to so ill a purpose.
But how does this appear? What argument, what witness is there of it?
You offer none; nor am I conscious of any. I begin to suspect since you,
the professor of geometry, have in your objections to the twentieth
chapter these words concerning “_Vindex, ocularis ille testis de quo hic
agitur, erat, ni fallor, ille ipse_,”--that Vindex himself, in other
company, has bestowed a visit on me. Seeing you will have me believe it,
let it be so; and, as it is likely, not long after my return into
England. At which time (for the reputation, it seems, I had gotten by my
boasting) divers persons that professed to love philosophy and
mathematics, came to see me; and some of them to let me see them, and
hear and applaud what they applauded in themselves. I see now it hath
happened to me with Vindex, as it happened to Dr. Harvey with Moranus.
Moranus, a jesuit, came out of Flanders hither, especially, as he says,
to see what learned men in divinity, ethics, physics, and geometry, were
here yet alive, to the end that by discoursing with them in these
sciences, he might correct either his own, or their errors. Amongst
others he was brought, he says, to that most civil and renowned old man
Dr. Harvey. That is very well. And in good earnest if he had made good
use of the time which was very patiently afforded him, he might have
learned of him (or of no man living) very much knowledge concerning the
circulation of the blood, the generation of living creatures, and many
other difficult points of natural philosophy. And if he had had anything
in him but common and childish learning, he could have showed it nowhere
more to his advantage, than before him that was so great a judge of such
matters. But what did he? That precious time (which was but little,
because he was to depart again presently for Flanders) he bestowed
wholly in venting his own childish opinions, not suffering the Doctor
scarce to speak; losing thereby the benefit he came for, and discovering
that he came not to hear what others could say, but to show to others
how learned he was himself already. Why else did he take so little time,
and so misspend it? Or why returned he not again? But when he had talked
away his time, and found (though patiently and civilly heard) he was not
much admired, he took occasion, writing against me, to be revenged of
Dr. Harvey, by slighting his learning publicly; and tells me that his
learning was only experiments; which he says I say have no more
certainty than civil histories. Which is false. My words are: “_Ante hos
nihil certi in physica erat præter experimenta cuique sua, et historias
naturales, si tamen et hæ dicendæ certæ sint, quæ civilibus historiis
certiores non sunt_.” Where I except expressly from uncertainty the
experiments that every man maketh to himself. But you see the near cut,
by which vain glory joined with ignorance passeth quickly over to envy
and contumely.
Thus it seems by your own confession I was used by Vindex. He comes with
some of my acquaintance in a visit. What he said I know not, but if he
discoursed then, as in his philosophical essay he writeth, I will be
bold to say of myself, I was so far from morosity, or, to use his
phrase, from being tetrical, as I may very well have a good opinion of
my own patience. And if there passed between us the discourse you
mention in your _Elenchus_, page 116, it was an incivility in him so
great, that without great civility I could not have abstained from
bidding him be gone. That which passed between us you say was this: “_I
complained that whereas I made sense, nothing but a perception of motion
in the organ, nevertheless, the philosophy schools through all Europe,
led by the text of Aristotle, teach another doctrine, namely, that
sensation is performed by species_.” This is a little mistaken. For I do
glory, not complain, that whereas all the Universities of Europe hold
sensation to proceed from species, I hold it to be a perception of
motion in the organ. The answer of Vindex, you say, was: “_That the
other hypothesis, whereby sense was explicated by the principles of
motion, was commonly admitted here before my book came out, as having
been sufficiently delivered by Des Cartes, Gassendus, and Sir Kenelm
Digby, before I had published anything in this kind_.” This then, it
seems, was it that made me angry. Truly I remember not an angry word
that ever I uttered in all my life to any man that came to see me,
though some of them have troubled me with very impertinent discourse;
and with those that argued with me, how impertinently soever, I always
thought it more civility to be somewhat earnest in the defence of my
opinion, than by obstinate and affected silence to let them see I
contemned them, or hearkened not to what they said. If I were earnest in
making good, that the manner of sensation by such motion as I had
explicated in my _Leviathan_ , is in none of the authors by him named,
it was not anger, but a care of not offending him, with any sign of the
contempt which his discourse deserved. But it was incivility in him to
make use of a visit, which all men take for a profession of friendship,
to tell me that that which I had already published for my own, was found
before by Des Cartes, Gassendus, and Sir Kenelm Digby. But let any man
read Des Cartes; he shall find, that he attributeth no motion at all to
the object of sense, but an inclination to action, which inclination no
man can imagine what it meaneth. And for Gassendus, and Sir Kenelm
Digby, it is manifest by their writings, that their opinions are not
different from that of Epicurus, which is very different from mine. Or
if these two, or any of those I conversed with at Paris, had prevented
me in publishing my own doctrine, yet since it was there known and
declared for mine by Mersennus in the preface to his _Ballistica_ (of
which the three first leaves are employed wholly in the setting forth of
my opinion concerning sense, and the rest of the faculties of the soul)
they ought not therefore to be said to have found it out before me. And
consequently this answer which you say was given me by Vindex was
nothing else but untruth and envy; and, because it was done by way of
visit, incivility. But you have not alleged, nor can allege, any words
of mine, from which can be drawn that I am so angry as you say I am with
those that submit not to my dictates. Though the discipline of the
University be never so good; yet certainly this behaviour of yours and
his are no good arguments to make it thought so. But you the professor
of geometry, that out of my words spoken against Vindex in my twentieth
chapter, argue my angry humour, do just as well, as when (in your
_Arithmetica Infinitorum_) from the continual increase of the excess of
the row of squares above the third part of the aggregate of the
greatest, you conclude they shall at last be equal to it. For though you
knew that Vindex had given me first the worst words that possibly can be
given, yet you would have that return of mine to be a demonstration of
an angry humour; not then knowing what I told you even now in the
beginning of this lesson, of the sentence given by Vespasian. But to
this point I shall speak again hereafter.
Your third accusation is: “_That I had my doctrine of vision, which I
pretended to be my own, out of papers which I had a long time in my
hands of Mr. Warner’s_.” I never had sight of Mr. Warner’s papers in all
my life, but that of _Vision by Refraction_ (which by his approbation I
carried with me to Park, and caused it to be printed under his own name,
at the end of Mersennus his _Cogitata Physico-Mathematica_, which you
may have there seen, and another treatise of the proportions of alloy in
gold and silver coin; which is nothing to the present purpose). In all
my conversation with him, I never heard him speak of anything he had
written, or was writing, _De penicillo optico_. And it was from me that
he first heard it mentioned that light and colour were but fancy. Which
he embraced presently as a truth, and told me it would remove a rub he
was then come to in the discovery of the place of the image. If after my
going hence he made any use of it (though he had it from me, and not I
from him), it was well done. But wheresoever you find my principles,
make use of them, if you can, to demonstrate all the symptoms of vision;
and I will do (or rather have done and mean to publish) the same; and
let it be judged by that, whether those principles be of mine, or other
men’s invention. I give you time enough, and this advantage besides,
that much of my optics hath been privately read by others. For I never
refused to lend my papers to my friends, as knowing it to be a thing of
no prejudice to the advancement of philosophy, though it be, as I have
found it since, some prejudice to the advancement of my own reputation
in those sciences; which reputation I have always postposed to the
common benefit of the studious.
You say further (you the geometrician) that I had the proposition of the
spiral line equal to a parabolical line from Mr. Robervall: true. And if
I had remembered it, I would have taken also his demonstration; though
if I had published his, I would have suppressed mine. I was comparing in
my thoughts those two lines, spiral and parabolical, by the motions
wherewith they were described; and considering those motions as uniform,
and the lines from the centre to the circumference, not to be little
parallelograms, but little sectors, I saw that to compound the true
motion of that point which described the spiral, I must have one line
equal to half the perimeter, the other equal to half the diameter. But
of all this I had not one word written. But being with Mersennus and Mr.
Robervall in the cloister of the convent, I drew a figure on the wall,
and Mr. Robervall perceiving the deduction I made, told me that since
the motions which make the parabolical line, are one uniform, the other
accelerated, the motions that make the spiral must be so also; which I
presently acknowledged; and he the next day, from this very method,
brought to Mersennus the demonstration of their equality. And this is
the story mentioned by Mersennus, prop. 25, corol. 2, of his
_Hydraulica_; which I know not who hath most magnanimously interpreted
to you in my disgrace.
The fourth accusation is: “_That I have injured the Universities_.”
Wherein? First, “_In that I would have the doctrine of my Leviathan by
entire sovereignty be imposed on them_.” You often upbraid me with
thinking well of my own doctrine; and grant by consequence, that I
thought this doctrine good; I desired not therefore that anything should
be imposed upon them, but what (at least in my opinion) was good both
for the Commonwealth and them. Nay more, I would have the state make use
of them to uphold the civil power, as the Pope did to uphold the
ecclesiastical. Is it not absurdly done to call this an injury? But to
question, you will say, whether the civil doctrine there taught be such
as it ought to be, or not, is a disgrace to the Universities. If that be
certain, it is certain also that those sermons and books, which have
been preached and published, both against the former and the present
government, directly or obliquely, were not made by such ministers and
others as had their breeding in the Universities; though all men know
the contrary. But the doctrine which I would have to be taught there,
what is it? It is this: “_That all men that live in a Commonwealth, and
receive protection of their lives and fortunes from the supreme governor
thereof, are reciprocally bound, as far as they are able, and shall be
required, to protect that governor_.” Is it, think you, an unreasonable
thing to impose the teaching of such doctrine upon the Universities? Or
will you say they taught it before, when you know that so many men which
came from the Universities to preach to the people, and so many others
that were not ministers, did stir the people up to resist the then
supreme civil power? And was it not truly therefore said, that the
Universities receiving their discipline from the authority of the pope,
were the shops and operatories of the clergy? Though the competition of
the papal and civil power be taken now away, yet the competition between
the ecclesiastical and the civil power hath manifestly enough appeared
very lately. But neither is this an upbraiding of an University (which
is a corporation or body artificial), but of particular men, that desire
to uphold the authority of a Church, as of a distinct thing from the
Commonwealth. How would you have exclaimed, if, instead of recommending
my _Leviathan_ to be taught in the Universities, I had recommended the
erecting of a new and lay-university, wherein lay-men should have the
reading of physics, mathematics, moral philosophy, and politics, as the
clergy have now the sole teaching of divinity? Yet the thing would be
profitable, and tend much to the polishing of man’s nature, without much
public charge. There will need but one house, and the endowment of a few
professions. And to make some learn the better, it would do very well
that none should come thither sent by their parents, as to a trade to
get their living by, but that it should be a place for such ingenuous
men, as being free to dispose of their own time, love truth for itself.
In the mean time divinity may go on in Oxford and Cambridge to furnish
the pulpit with men to cry down the civil power, if they continue to do
as they did. If I had, I say, made such a motion in my _Leviathan_ ,
though it would have offended the divines, yet it had been no injury.
But it is an injury, you will say, to deny in general the utility of the
ancient schools, and to deny that we have received from them our
geometry. True, if I had not spoken distinctly of the schools of
philosophy, and said expressly, that the geometricians passed not then
under the name of philosophers; and that in the school of Plato (the
best of the ancient philosophers) none were received that were not
already in some measure geometricians. Euclid taught geometry; but I
never heard of a sect of philosophers called Euclidians, or
Alexandrians, or ranged with any of the other sects, as Peripatetics,
Stoics, Academics, Epicureans, Pyrrhonians, &c. But what is this to the
Universities of Christendom? Or why are we beholden for geometry to our
universities, more than to Gresham College, or to private men in London,
Paris, and other places, which never taught or learned it in a public
school? For even those men that living in our Universities have most
advanced the mathematics, attained their knowledge by other means than
that of public lectures, where few auditors, and those of unequal
proficiency, cannot make benefit by one and the same lesson. And the
true use of public professors, especially in the mathematics, being to
resolve the doubts, and problems, as far as they can, of such as come
unto them with desire to be informed.
That the Universities now are not regulated by the Pope, but by the
civil power, is true, and well. But where say I the contrary? And thus
much for the first injury.
Another, you say, is this, that in my _Leviathan_ , p. 670, I say: “_The
principal schools were ordained for the three professions of Roman
religion, Roman law, and the art of medicine_.” Thirdly, that I say:
“_Philosophy had no otherwise place there than as a hand-maid to Roman
religion_.” Fourthly: “_Since the authority of Aristotle was received
there, that study is not properly philosophy, but Aristotelity_.”
Fifthly: “_That for geometry, till of late times it had no place there
at all_.” As for the second, it is too evident to be denied; the
fellowships having been all ordained for those professions; and (saving
the change of religion) being so yet. Nor hath this any reflection upon
the Universities, either as they now are, or as they then were, seeing
it was not in their own power to endow themselves, or to receive other
laws and discipline than their founder and the state was pleased to
ordain. For the third, it is also evident. For all men know that none
but the Roman religion had any stipend or preferment in any university,
where that religion was established? No, nor for a great while, in their
commonwealths; but were everywhere persecuted as heretics. But you will
say, the words of my _Leviathan_ are not, philosophy “_had no place_,”
but “_hath no place_.” Are you not ashamed to lay to my charge a mistake
of the word _hath_ for _had_? which was either a mistake of the printer,
or if it were so in the copy, it could be no other than the mistake of a
letter in the writing, unless you think you can make men believe that
after fifty years being acquainted with what was publicly professed and
practised in Oxford and Cambridge, I knew not what religion they were
of. This taking of advantage from the mistake of a word, or of a letter,
I find also in the _Elenchus_, where for _prætendit se scire_, there is
_prætendit scire_, which you the geometrician sufficiently mumble,
mistaking it I think for an anglicism, not for a fault of the
impression.
To the fourth, you pretend, that men are not now so tied to Aristotle as
not to _enjoy a liberty of philosophising, though it were otherwise when
I was conversant in Magdalen Hall_. Was it so then? Then am I absolved,
unless you can show some public act of the university made since that
time to alter it. For it is not enough to name some few particular
ingenuous men that usurp that liberty in their private discourses, or,
with connivance, in their public disputations. And your doctrine, that
even here you avow, of _abstracted essences_, _immaterial substances_,
and of _nunc-stans_; and your improper language in using the word (not
as mine, for I have it nowhere) _successive eternity_; as also your
doctrine of _condensation_, and your arguing from natural reason the
incomprehensible mysteries of religion, and your malicious writing, are
very shrewd signs that you yourselves are none of those which you say do
_freely philosophise_; but that both your philosophy and your language
are under the servitude, not of the Roman religion, but of the ambition
of some other doctors, that seek, as the Roman clergy did, to draw all
human learning to the upholding of their power ecclesiastical. Hitherto
therefore there is no injury done to the universities. For the fifth,
you grant it, namely, “_that till of late there was no allowance for the
teaching of geometry_.” But lest you should be thought to grant me
anything, you say, you the astronomer, “_geometry hath now so much place
in the universities, that when Mr. Hobbes shall have published his
philosophical and geometrical pieces, you assure yourself you shall be
able to find a greater number in the university who will understand as
much, or more, of them than he desired they should_,” &c. But though
this be true of the _now_, yet it maketh nothing against my _then_. I
know well enough that Sir Henry Savile’s lectures were founded and
endowed since. Did I deny _then_ that there were in Oxford many good
geometricians? But I deny _now_, that either of you is of the number.
For my philosophical and geometrical pieces are published, and you have
understood only so much in them, as all men will easily see by your
objections to them, and by your own published geometry, that neither of
you understand anything either in philosophy or in geometry. And yet you
would have those books of yours to stand for an argument, and to be an
index of the philosophy and geometry to be found in the universities.
Which is a greater injury and disgrace to them, than any words of mine,
though interpreted by yourselves.
Your last and greatest accusation, or rather railing (for an accusation
should contain, whether true or false, some particular fact, or certain
words, out of which it might seem at least to be inferred), is, that I
am an enemy to religion. Your words are: “_It is said that Mr. Hobbes is
no otherwise an enemy to the Roman religion, saving only as it hath the
name of religion_.” This is said by Vindex. You, the geometrician, in
your epistle dedicatory, say thus: “_With what pride and imperiousness
he tramples on all things both human and divine, uttering fearful and
horrible words of God_, (peace), _of sin, of the holy Scripture, of all
incorporeal substances in general, of the immortal soul of man, and of
the rest of the weighty points of religion_ (down), _it is not so much
to be doubted as lamented_.” And at the end of your objections to the
eighteenth chapter, “_Perhaps you take the whole history of the fall of
Adam for a fable, which is no wonder, when you say the rules of
honouring and worshipping of God are to be taken from the laws_.” Down,
I say; you bark now at the supreme legislative power. Therefore it is
not I, but the laws which must rate you off. But do not many other men,
as well as you, read my _Leviathan_ , and my other books? And yet they
all find not such enmity in them against religion. Take heed of calling
them all atheists that have read and approved my _Leviathan_ . Do you
think I can be an atheist and not know it? Or knowing it, durst have
offered my atheism to the press? Or do you think him an atheist, or a
contemner of the Holy Scripture, that sayeth nothing of the Deity but
what he proveth by the Scripture? You that take so heinously that I
would have the rules of God’s worship in a Christian commonwealth taken
from the laws, tell me, from whom you would have them taken? From
yourselves? Why so, more than from me? From the bishops? Right, if the
supreme power of the commonwealth will have it so; if not, why from them
rather than from me? From a consistory of presbyters by themselves, or
joined with lay-elders, whom they may sway as they please? Good, if the
supreme governor of the commonwealth will have it so; if not, why from
them, rather than from me, or from any man else? They are wiser and
learneder than I. It may be so; but it has not yet appeared. Howsoever,
let that be granted. Is there any man so very a fool as to subject
himself to the rules of other men in those things which so nearly
concern himself, for the title they assume of being wise and learned,
unless they also have the sword which must protect them. But it seems
you understand the sword as comprehended. If so, do you not then receive
the rules of God’s worship from the civil power? Yes, doubtless; and you
would expect, if your consistory had that sword, that no man should dare
to exercise or teach any rules concerning God’s worship which were not
by you allowed. See therefore how much you have been transported by your
malice towards me, to injure the civil power by which you live. If you
were not despised, you would in some places and times, where and when
the laws are more severely executed, be shipped away for this your
madness to America, I would say, to Anticyra. What luck have I, when
this, of the laws being the rules of God’s public worship, was by me
said and applied to the vindication of the Church of England from the
power of the Roman clergy, it should be followed with such a storm from
the ministers, presbyterian and episcopal, of the Church of England?
Again, for those other points, namely, that I approve not of incorporeal
bodies, nor of other immortality of the soul, than that which the
Scripture calleth eternal life, I do but as the Scripture leads me. To
the texts whereof by me alleged, you should either have answered, or
else forborne to revile me for the conclusions I derived from them.
Lastly, what an absurd question is it to ask me whether it be in the
power of the magistrate, whether the world be eternal or not? It were
fit you knew it is in the power of the supreme magistrate to make a law
for the punishment of them that shall pronounce publicly of that
question anything contrary to that which the law hath once pronounced.
The truth is, you are content that the papal power be cut off, and
declaimed against as much as any man will; but the ecclesiastical power,
which of late was aimed at by the clergy here, being a part thereof,
every violence done to the papal power is sensible to them yet; like
that which I have heard say of a man, whose leg being cut off for the
prevention of a gangrene that began in his toe, would nevertheless
complain of a pain in his toe, when his leg was cut off.
Thus much in my defence; which I believe if you had foreseen, this
accusation of yours had been left out. I come now to examine (though it
be done in part already) what manners those are which I find everywhere
in your writings.
And first, how came it into your minds that a man can be an atheist, I
mean an atheist in his conscience? I know that David confesseth of
himself, upon sight of the prosperity of the wicked, that his feet had
almost slipped, that is, that he had slipped into a short doubtfulness
of the Divine Providence. And if anything else can cause a man to slip
in the same kind, it is the seeing such as you (who though you write
nothing but what is dictated to each of you by a doctor of divinity) do
break the greatest of God’s commandments, which is charity, in every
line before his face. And though such forgettings of God be somewhat
more than short doubtings, and sudden transportations incident to human
passion, yet I do not for that cause think you atheists and enemies of
religion, but only ignorant and imprudent Christians. But how, I say,
could you think me an atheist, unless it were because finding your
doubts of the Deity more frequent than other men do, you are thereby the
apter to fall upon that kind of reproach? Wherein you are like women of
poor and evil education when they scold; amongst whom the readiest
disgraceful word is whore: why not thief, or any other ill name, but
because, when they remember themselves, they think that reproach the
likeliest to be true?
Secondly, tell me what crime it was which the Latins called by the name
of _scelus_? You think not, unless you be Stoics, that all crimes are
equal. _Scelus_ was never used but for a crime of greatest mischief, as
the taking away of life and honour; and besides, basely acted, as by
some clandestine way, or by such a way as might be covered with a lie.
But when you insinuate in a writing published that I am an atheist, you
make yourselves authors to the multitude, and do all you can to stir
them up to attempt upon my life; and if it succeed, then to sneak out of
it by leaving the fault on them that are but actors. This is to
endeavour great mischief basely, and therefore _scelus_. Again, to
deprive a man of the honour he hath merited, is no little wickedness;
and this you endeavour to do by publishing falsely that I challenge as
my own the inventions of other men. This is therefore _scelus_ publicly
to tell all the world that I will be angry with all men that do not
presently submit to my dictates; to deprive me of the friendship of all
the world; great damage, and a lie, and yours. For to publish any
untruth of another man to his disgrace, on hearsay from his enemy, is
the same fault as if he published it on his own credit. If I should say
I have heard that Dr. Wallis was esteemed at Oxford for a simple fellow,
and much inferior to his fellow-professor Dr. Ward (as indeed I have
heard, but do not believe it), though this be no great disgrace to Dr.
Wallis, yet he would think I did him injury. Therefore public accusation
upon hearsay is _scelus_. And whosoever does any of these things does
_sceleratè_. But you the professors of the mathematics at Oxford, by the
advice of two doctors of divinity have dealt thus with me. Therefore you
have done, I say not foolishly, though no wickedness be without folly,
but _sceleratè_, ὅπερ ἔδει δεῖξαι.
Thirdly, it is ill manners, in reprehending truth, to send a man in a
boasting way to your own errors; as you the professor of geometry have
often sent me to your two tractates of the _Angle of Contact_ and
_Arithmetica Infinitorum_.
Fourthly, it is ill manners, to diminish the just reputation of worthy
men after they be dead, as you the professor of geometry have done in
the case of Joseph Scaliger.
Fifthly, when I had in my _Leviathan_ suffered the clergy of the Church
of England to escape, you did imprudently in bringing any of them in
again. An Ulysses upon so light an occasion would not have ventured to
return again into the cave of Polyphemus.
Lastly, how ill does such levity and scurrility, which both of you have
shown so often in your writings, become the gravity and sanctity
requisite to the calling of the ministry? They are too many to be
repeated. Do but consider, you the geometrician, how unhandsome it is to
play upon my name, when both yours and mine are plebeian names; though
from Willis by Wallis, you go from yours in Wallisius. The jest of using
at every word _mi Hobbi_, is lost to them beyond sea. But this is not so
ill as some of the rest. I will write out one of them, as it is in the
fourth page of your _Elenchus_: “_Whence it appears that your Empusa was
of the number of those fairies which you call in English hob-goblins.
The word is made of_ ἕν and πους; _and thence comes the children’s play
called the play of Empusa, Anglicè_ (hitherto in Latin all but
_hob-goblins_, then follows in English) _fox, fox, come out of your
hole_ (then in Latin again), _in which the boy that is called the fox,
holds up one foot, and jumps with the other, which in English is to
hop_.” When a stranger shall read this, and hoping to find therein some
witty conceit, shall with much ado have gotten it interpreted and
explained to him, what will he think of our doctors of divinity at
Oxford, that will take so much pains as to go out of the language they
set forth in, for so ridiculous a purpose? You will say it is a pretty
_paranomasia_. How you call it there I know not, but it is commonly
called here a _clinch_; and such a one as is too insipid for a boy of
twelve years old, and very unfit for the sanctity of a minister, and
gravity of a doctor of divinity. But I pray you tell me where it was you
read the word _empusa_ for the boy’s play you speak of, or for any other
play amongst the Greeks? In this (as you have done throughout all your
other writings) you presume too much upon your first cogitations. There
be a hundred other scoffing passages, and ill-favoured attributes given
me in both your writings, which the reader will observe without my
pointing to them, as easily as you would have him; and which perhaps
some young students, finding them full of gall, will mistake for salt.
Therefore to disabuse those young men, and to the end they may not
admire such kind of wit, I have here and there been a little sharper
with you than else I would have been. If you think I did not spare you,
but that I had not wit enough to give you as scornful names as you give
me, are you content I should try? Yes (you the geometrician will say)
give me what names you please, so you call me not _Arithmetica
Infinitorum_. I will not. Nor _Angle of Contact_ ; nor _Arch Spiral_ ;
nor _Quotient_ . I will not. But I here dismiss you both together. So go
your ways, you _Uncivil Ecclesiastics, Inhuman Divines, Dedoctors of
morality, Unasinous Colleagues, Egregious pair of Issachars, most
wretched Vindices and Indices Academiarum_; and remember Vespasian’s
law, that it is uncivil to give ill language first, but civil and lawful
to return it. But much more remember the law of God, to obey your
sovereigns in all things; and not only not to derogate from them, but
also to pray for them, and as far as you can to maintain their
authority, and therein your own protection. And, do you hear? take heed
of speaking your mind so clearly in answering my _Leviathan_, as I have
done in writing it. You should do best not to meddle with it at all,
because it is undertaken, and in part published already, and will be
better performed, from term to term, by one Christopher Pike.
ΣΤΙΓΜΑΙ
Αγεωμετρίας, Αγροικίας, Αντίπολιτείας, Αμαθείας,
OR
MARKS
OF THE
ABSURD GEOMETRY, RURAL LANGUAGE, SCOTTISH
CHURCH POLITICS, AND BARBARISMS
OF
JOHN WALLIS,
PROFESSOR OF GEOMETRY AND DOCTOR OF DIVINITY.
BY
THOMAS HOBBES,
OF MALMESBURY.
TO THE RIGHT HONOURABLE
HENRY, LORD PIERREPONT,
VISCOUNT NEWARK, EARL OF KINGSTON, AND
MARQUIS OF DORCHESTER.
==========
MY MOST NOBLE LORD,
I did not intend to trouble your Lordship twice with this contention
between me and Dr. Wallis. But your Lordship sees how I am constrained
to it; which, whatsoever reply the Doctor makes, I shall be constrained
to no more. That which I have now said of his Geometry, Manners,
Divinity, and Grammar, altogether is not much, though enough. As for
that which I here have written concerning his Geometry, which you will
look for first, is so clear, that not only your Lordship, and such as
have proceeded far in that science, but also any man else that doth but
know how to add and subtract proportions, (which is taught at the
twenty-third proposition of the sixth of Euclid), may see the Doctor is
in the wrong. That which I say of his ill language and politics is yet
shorter. The rest, which concerneth grammar, is almost all another
man’s, but so full of learning of that kind, as no man that taketh
delight in knowing the proprieties of the Greek and Latin tongues, will
think his time ill bestowed in the reading it. I give the Doctor no more
ill words, but am returned from his manners to my own. Your Lordship may
perhaps say, my compliment in my title-page is somewhat coarse; and it
is true. But, my Lord, it is since the writing of the title-page, that I
am returned from the Doctor’s manners to my own; which are such as I
hope you will not be ashamed to own me, my Lord, for one of
Your Lordship’s most humble
and obedient servants,
THOMAS HOBBES.
==========
TO
DOCTOR WALLIS,
IN ANSWER TO HIS
SCHOOL DISCIPLINE
---
SIR,
When unprovoked you addressed unto me, in your _Elenchus_, your harsh
compliment with great security, wantonly to show your wit, I confess you
made me angry, and willing to put you into a better way of considering
your own forces, and to move you a little as you had moved me, which I
perceive my lessons to you have in some measure done; but here you shall
see how easily I can bear your reproaches, now they proceed from anger,
and how calmly I can argue with you about your geometry and other parts
of learning.
I shall in the first part confer with you about your _Arithmetica
Infinitorum_, and afterwards compare our manner of elocution; then your
politics; and last of all your grammar and critics. Your spiral line is
condemned by him whose authority you use to prove me a plagiary, (that
is, a man that stealeth other men’s inventions, and arrogates them to
himself), whether it be Roberval or not that writ that paper, I am not
certain. But I think I shall be shortly; but whosoever it be, his
authority will serve no less to show that your doctrine of the spiral
line, from the fifth to the eighteenth proposition of your _Arithmetica
Infinitorum_, is all false; and that the principal fault therein (if all
faults be not principal in geometry, when they proceed from ignorance of
the science) is the same that I objected to you in my _Lessons_. And for
the author of that paper, when I am certain who it is, it will be then
time enough to vindicate myself concerning that name of plagiary. And
whereas he challenges the invention of your method delivered in your
_Arithmetica Infinitorum_, to have been his before it was yours, I
shall, I think, by and by say that which shall make him ashamed to own
it; and those that writ those encomiastic epistles to you ashamed of the
honour they meant to you. I pass therefore to the nineteenth
proposition, which in Latin is this: your geometry!
“_Si proponatur series quantitatum in duplicata ratione arithmetice
proportionalium (sive juxta seriem numerorum quadraticorum) continue
crescentium, a puncto vel 0 inchoatarum, (puta ut 0. 1. 4. 9. 16. etc.),
propositum sit, inquirere quam habeat illa rationem ad seriem totidem
maximæ æqualium._
“_Fiat investigatio per modum inductionis ut_ (_in prop. 1_)
_Eritque_,
(0 + 1 = 1)/(1 + 1 = 2) = (1)/(3) + (1)/(6)
(0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (1)/(3) + (1)/(12)
(0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) = (1)/(3) + (1)/(18) _et sic
deinceps_.
“_Ratio proveniens est ubique major quam subtripla seu (1)/(3); excessus
autem perpetuo decrescit prout numerus terminorum augetur (puta (1)/(6)
(1)/(12) (1)/(18) (1)/(24) etc.) aucto nimirum fractionis denominatore
sive consequente rationis in singulis locis numero senario (ut patet) ut
sit rationis provenientis excessus supra subtriplam, ea quam habet
unitas ad sextuplum numeri terminorum post 0; adeoque._”
That is, if there be propounded a row of quantities in duplicate
proportion of the quantities arithmetically proportional (or proceeding
in the order of the square numbers) continually increasing; and
beginning at a point or 0; let it be propounded to find what proportion
the row hath; to as many quantities equal to the greatest;
Let it be sought by induction (as in the first proposition).
The proportion arising is everywhere greater than subtriple, or (1)/(3),
and the excess perpetually decreaseth as the number of terms is
augmented, as here, (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c.
denominator of the fraction being in every place augmented by the number
six, as is manifest; so that the excess of the rising proportion above
subtriple is the same which unity hath to six times the number of terms
after 0; and so.
Sir, in these your characters I understand by the cross + that the
quantities on each side of it are to be added together and make one
aggregate; and I understand by the two parallel lines = that the
quantities between which they are placed are one to another equal; this
is your meaning, or you should have told us what you meant else; I
understand also, that in the first row 0 + 1 is equal to 1, and 1 + 1
equal to 2; and that in the second row 0 + 1 + 4 is equal to 5; and 4 +
4 + 4 equal to 12; but (which you are too apt to grant) I understand
your symbols no further; but must confer with yourself about the rest.
And first I ask you (because fractions are commonly written in that
manner) whether in the uppermost row (which is (0 + 1 = 1)/(1 + 1 = 2) =
(1)/(3) + (1)/(6))(0)/(1) be a fraction, (1)/(1) be a fraction, (1)/(2)
be a fraction, that is to say, a part of an unit, and if you will, for
the cypher’s sake, whether (0)/(1), be an infinitely little part of 1;
and whether (1)/(1) or 1 divided by 1 signify an unity? if that be your
meaning, then the fraction (0)/(1) added to the fraction (1)/(1) is
equal to the fraction (1)/(2): But the fraction (0)/(1) is equal to O;
therefore the fraction (0)/(1) + (1)/(1) is equal to the fraction
(1)/(1); and (1)/(1) equal to (1)/(2) which you will confess to be an
absurd conclusion, and cannot own that meaning.
I ask you therefore again, if by (0)/(1) you mean the proportion of 0 to
1; and consequently by (1)/(1) the proportion of 1 to 1, and by (1)/(2)
the proportion of 1 to 2: if so, then it will follow, that if the
proportions of 0 to 1 and of 1 to 1 be compounded by addition, the
proportion arising will be the proportion of 1 to 2. But the proportion
of 0 to 1 is infinitely little, that is, none. Therefore the proposition
arising by composition will be that of 1 to 1, and equal (because of the
symbol =) to the proportion of 1 to 2, and so 1 = 2. This also is so
absurd that I dare say that you will not own it.
There may be another meaning yet: perhaps you mean that the uppermost
quantity 0 + 1 is equal to the uppermost quantity 1; and the lowermost
quantity 1 + 1 equal to the lowermost quantity 2: which is true. But how
then in this equation (1)/(2) = (1)/(3) + (1)/(6)? Is the uppermost
quantity 1 equal to the uppermost quantity 1 + 1; or the lowermost
quantity 2 equal to the lowermost quantity 3 + 6? Therefore neither can
this be your meaning. Unless you make your symbols more significant, you
must not blame me for want of understanding them.
Let us now try what better success we shall have where the places are
three, as here:
(0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (5)/(12) = (1)/(3) + (1)/(12):
If your symbols be fractions, the compound of them by addition is
(5)/(4), for 0(1)/(4) and (4)/(4) make (5)/(4); and consequently
(because of the symbol = ) (5)/(4) equal to (5)/(12), which is not to be
allowed, and therefore that was not your meaning. If you meant that the
proportions of 0 to 4 and of 1 to 4 and of 4 to 4 compounded, is equal
to the proportion of 5 to 12, you will fall again into no less an
inconvenience. For the proportion arising out of that composition will
be the proportion of 1 to 4. For the proportion of 0 to 4 is infinitely
little. Then to compound the other two, set them in this order 1. 4. 4.
and you have a proportion compounded of 1 to 4 and of 4 to 4, namely,
the proportion of the first to the last, which is of 1 to 4, which must
be equal, by this your meaning, to the proportion of 5 to 12, and
consequently as 5 to 12, so is 1 to 4, which you must not own. Lastly,
if you mean that the uppermost quantities to the uppermost, and the
lowermost to the lowermost in the first equation are equal, it is
granted, but then again in the second equation it is false. It concerns
your fame in the mathematics to look about how to justify these
equations which are the premises to your conclusion following, namely,
that the proportion arising is every where greater than sub-triple, or a
third; and that the excess (that is, the excess above subtriple)
perpetually decreaseth as the number of terms is augmented, as here
(1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. which I will show you
plainly is false.
But first I wonder why you were so angry with me for saying you made
proportion to consist in the quotient, as to tell me it was abominably
false, and to justify it, cite your own words _penes quotientem_; do not
you say here, the proportion is everywhere greater than subtriple, or
(1)/(3)? And is not (1)/(3) the quotient of 1 divided by 3? You cannot
say in this place that _penes_ is understood; for if it were expressed
you would not be able to proceed.
But I return to your conclusion, that the excess of the proportion of
the increasing quantities above the third part of so many times the
greatest, decreaseth, as (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30),
&c. For by this account in this row (0 + 1)/(1 + 1) = (1)/(2) where the
quantity above exceeds the third part of the quantities below by
(1)/(3), you make (1)/(3) equal to (1)/(6), which you do not mean. It
may be said your meaning is, that the proportion of 1 to the subtriple
of 2 which is (2)/(3), exceedeth what? I cannot imagine what, nor
proceed further where the terms be but two. Let us therefore take the
second row, that is, (0 + 1 + 4)/(4 + 4 + 4) = (5)/(12). The sum above
is 5, the sum below is 12, the third part whereof is 4; if you mean,
that the proportion of 5 to 4 exceeds the proportion of 4 to 12 (which
is subtriple) by (1)/(12), you are out again. For 5 exceeds 4 by unity,
which is (12)/(12). I do not think you will own such an equation as
(12)/(12) = (1)/(12) Therefore I believe you mean (and your next
proposition assures me of it), that the proportion of 5 to 4 exceeds
subtriple proportion by the proportion of 1 to 12; if you do so, you are
yet deceived.
For if the proportion of 5 to 4 exceeds subtriple proportion by the
proportion of 1 to 12, then subtriple proportion, that is, of 4 to 12
added to the proportion of 1 to 12 must make the proportion of 5 to 4.
But if you look on these quantities, 4, 12, 144, you will see, and must
not dissemble, that the proportion of 4 to 12 is subtriple, and the
proportion of 12 to 144 is the same with that of 1 to 12. Therefore by
your assertion it must be as 5 to 4 so 4 to 144, which you must not own.
And yet this is manifestly your meaning, as appeareth in these words:
“_Ut sit rationis provenientis excessus supra subtriplam ea quam habet
unitas ad sextuplum numeri terminorum post 0, adeoque_,” which cannot be
rendered in English, nor need to be. For you express yourself in the
twentieth proposition very clearly; I noted it only that you may be more
merciful hereafter to the stumblings of a hasty pen. For _excessus ea
quam_ does not well, nor is to be well excused by _subauditur ratio_.
Your twentieth proposition is this:
“_Si proponatur series quantitatum in duplicata ratione arithmetice
proportionalium (sive juxta seriem numerorum quadraticorum) continue
crescentium, a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem
totidem maximæ æqualium subtriplam superabit; eritque excessus ea ratio
quam habet unitas ad sextuplum numeri terminorum post 0, sive quam habet
radix quadratica termini primi post 0 ad sextuplum radicis quadraticæ
termini maximi._”
That is, if there be propounded a row of quantities in duplicate
proportion of arithmetically-proportionals (or according to the row of
square numbers) continually increasing, and beginning with a point or O.
The proportion of that row to a row of so many equals to the greatest,
shall be greater than subtriple proportion, and the excess shall be that
proportion which unity hath to the sextuple of the number of terms after
0, or the same which the square root of the first number after 0, hath
to the sextuple of the square root of the greatest.
For proof whereof you have no more here than _patet ex præcedentibus_;
and no more before but _adeoque_. You do not well to pass over such
curious propositions so slightly; none of the ancients did so, nor, that
I remember, any man before yourself. The proposition is false, as you
shall presently see.
Take, for example, any one of your rows: as (0 + 1 + 4)/(4 + 4 + 4). By
this proportion of yours 1 + 4, which makes 5, is to 12 in more than
subtriple proportion; by the proportion of 1 to the sextuple of 2 which
is 12. Put in order these three quantities 5, 4, 12, and you must see
the proportion of 5 to 12 is greater than the proportion of 4 to 12,
that is, subtriple proportion, by the proportion of 5 to 4. But by your
account the proportion of 5 to 4 is greater than that of 4 to 12 by the
proportion of 1 to 12. Therefore, as 5 to 4 so is 1 to 12, which is a
very strange paradox.
After this you bring in this consectary: “_Cum autem crescente numero
terminorum excessus ille supra rationem subtriplam continue minuatur, ut
tandem quovis assignabili minor evadat (ut patet) si in infinitum
producatur, prorsus evaniturus est. Adeoque._”
That is, seeing as the number of terms increaseth, that excess above
subtriple proportion continually decreaseth, so as at length it becomes
less than any assignable (as is manifest) if it be produced infinitely,
it shall utterly vanish, and so. And so what?
Sir, this consequence of yours is false. For two quantities being given,
and the excess of the greater above the less, that excess may
continually be decreased, and yet never quite vanish. Suppose any two
unequal quantities differing by more than an unit, as 3 and 6, the
excess 3, let 3 be diminished, first by an unit, and the excess will be
2, and the quantities will be 3 and 5; 5 is greater than 4, the excess
1. Again, let 1 be diminished and made (1)/(2), the excess 4 and the
quantities 3 and 4(1)/(2), 4(1)/(2) is yet greater than 4. Again
diminish the excess to (1)/(4), the quantities will be 3 and 4(1)/(4),
yet still 4(1)/(4) is greater than 4. In the same manner you may proceed
to (1)/(8) (1)/(16) (1)/(32), &c. infinitely; and yet you shall never
come within an unit (though your unit stand for 100 miles) of the lesser
quantity propounded 3, if that 3 stands for 300 miles. The excesses
above subtriple proportion do not decrease in the manner you say it
does, but in the manner which I now shall show you.
In the first row (0 + 1)/(1 + 1) a third of the quantities below is
(2)/(3), set in order these three quantities 1 (2)/(9) (2)/(3). The
first is 1, equal to the sum above, the last is (2)/(3), equal to the
subtriple of the sum below. The middlemost is (2)/(9) subtriple to the
last quantity (2)/(3). The excess of the proportion of 1 to (2)/(3)
above the subtriple proportion of (2)/(9) to (2)/(3) is the proportion
of 1 to (2)/(9) that is of 9 to 2, that is, of 18 to 4.
Secondly, in the second row, which is (0 + 1 + 4)/(4 + 4 + 4), a third
of the sum below is 4, the sum above is 5. Set in order these
quantities, 1, 5, 4, 12. There the proportion of 15 to 12 is the
proportion of 5 to 4. The proportion of 4 to 12 is subtriple; the excess
is the proportion of 15 to 4, which is less than the proportion of 18 to
4, as it ought to be; but not less by the proportion of (1)/(6) to
(1)/(12) as you would have it.
Thirdly, in the third row, which is (0 + 1 + 4 + 9)/(9 + 9 + 9 + 9). A
third of the sum below is 12, the sum above is 14. Set in order these
quantities, 42, 4, 12. There the proportion of 42 to 12 is the same with
that of 14 to 4. And the proportion of 4 to 12 subtriple, less than the
former excess of 15 to 4. And so it goes on decreasing all the way in
this manner, 18 to 4, 15 to 4, 14 to 4, &c. which differs very much from
your 1 to 6, 1 to 12, 1 to 18, &c. and the cause of your mistake is
this: you call the twelfth part of twelve (1)/(12), and the eighteenth
part of thirty-six you call (1)/(18), and so of the rest. But what need
of all those equations in symbols, to show that the proportion
decreases; is there any man can doubt, but that the proportion of 1 to 2
is greater than that of 5 to 12, or that of 5 to 12 greater than that of
14 to 36, and so on continually forwards; or could you have fallen into
this error, unless you had taken, as you have done in very many places
of your _Elenchus_, the fractions (1)/(6) and (1)/(12), &c. which are
the quotients of 1 divided by 6 and 12, for the very proportions of 1 to
6 and 1 to 12. But notwithstanding the excess of the proportions of the
increasing quantities, to subtriple proportion decrease, still, as the
number of terms increaseth, and that what proportions soever I shall
assign, the decrement will in time (in time, I say, without proceeding
_in infinitum_) produce a less, yet it does not follow that the row of
increasing quantities shall ever be equal to the third part of the row
of so many equals to the last or greatest. For it is not, I hope, a
paradox to you, that in two rows of quantities the proportion of the
excesses may decrease, and yet the excesses themselves increase, and do
perpetually.
For in the second and third rows, which are (0 + 1 + 4 = 5)/(4 + 4 + 4
= 12) and (0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) 5 exceeds the third
part of 12 by a quarter of the square of 4, and 14 exceeds the third
part of 36 by 2 quarters of the square of 4, and proceeding on, the sum
of the increasing quantities where the terms are 5 (which sum is 30)
exceedeth the third part of those below, (those below are 80, and their
third part 26(2)/(3)) by 3 quarters and (1)/(2) a quarter of the square
of 4, and when the terms are 6, the quantities above will exceed the
third part of them below by 5 quarters of the square of 4. Would you
have men believe, that the further they go, the excess of the increasing
quantities above the third part of those below shall be so much the
less? And yet the proportions of those above, to the thirds of those
below, shall decrease eternally; and therefore your twenty-first
proposition is false, namely this:
“_Si proponatur series infinita quantitatum in duplicata ratione
arithmetice proportionalium (sive juxta seriem numerorum quadraticorum),
continue crescentium a puncto sive 0 inchoatarum; erit illa ad seriem
totidem maximæ æqualium, ut 1 ad 3._”
That is, if an infinite row of quantities be propounded in duplicate
proportion of arithmetically-proportionals (or according to the row of
quadratic numbers), continually increasing and beginning from a point or
0; that row shall be to the row of as many equals to the greatest, as 1
to 3. This is false, _ut patet ex præcedentibus_; and, consequently, all
that you say in proof of the proportion of your _parabola_ to a
_parallelogram_, or of the _spiral_ (the true _spiral_) to a _circle_ is
in vain.
But your spiral puts me in mind of what you have under-written to the
diagram of your proposition 5. _The spiral, in both figures, was to be
continued whole to the middle, but, by the carelessness of the graver,
it is in one figure_ manca, _in the other_ intercisa.
Truly, Sir, you will hardly make your reader believe that a graver could
commit those faults without the help of your own copy, nor that it had
been in your copy, if you had known how to describe a spiral line then
as now. This I had not said, though truth, but that you are pleased to
say, though not truth, that I attributed to the printer some faults of
mine.
I come now to the thirty-ninth proposition, which is this:
“_Si proponatur series quantitatum in triplicata ratione arithmetice
proportionalium (sive juxta seriem numerorum cubicorum), continue
crescentium a puncto sive 0 inchoatarum (puta ut 0, 1, 8, 27, etc.),
propositum sit inquirere quam habeat series illa rationem ad seriem
totidem maximæ æqualium_:
“_Fiat investigatio per modum inductionis_ (_ut in prop. 1, et prop.
19_):
_Eritque_
(0 + 1 = 1)/(1 + 1 = 2) = (2)/(4) = (1)/(4) + (1)/(4)
(0 + 1 + 8 = 9)/(8 + 8 + 8 = 24) = (1)/(4) + (1)/(8)
(0 + 1 + 8 + 27 = 36)/(27 + 27 + 27 + 27 = 108) = (4)/(12)
= (1)/(4) + (1)/(12)
_Et sic deinceps._
“_Ratio proveniens est ubique major quam subquadrupla, sive (1)/(4).
Excessus autem perpetuo decrescit, pro ut numerus terminorum augetur,
puta (1)/(4) (1)/(8) (1)/(12) (1)/(16) etc. Aucto nimirum fractionis
denominatore sive consequente rationis in singulis locis numero
quaternatio, ut patet, ut sit rationis provenientis excessus supra
subquadruplam ea quam habet unitas ad quadruplum numeri terminorum post
0 adeoque._”
That is, if a row of quantities be propounded in triplicate proportion
of arithmetically proportionals (or according to the row of cubic
numbers), continually increasing, and beginning from a point or 0, as 0,
1, 8, 27, 64, &c., let it be propounded to inquire, what proportion that
row hath to a row of as many equals to the greatest.
Be it sought by way of induction, as in proposition 1 and 19.
The proposition arising is everywhere greater than subquadruple, or
(1)/(4), and the excess perpetually decreaseth as the number of terms
increaseth, as (1)/(4) (1)/(8) (1)/(12) (1)/(16) (1)/(20) &c. The
denominator of the fraction, or consequent of the proportion, being in
every place augmented by the number 4, as is manifest, so that the
excess of the arising proportion above subquadruple is the same with
that which an unit hath to the quadruple of the number of the terms
after 0, and so. Here are just the same faults which are in proposition
19.
For, if (0)/(1) be a fraction, and (1)/(1) be a fraction, and (1)/(2) be
another fraction, then this equation (0 + 1 = 1)/(1 + 1 = 2) is false.
For this fraction (0)/(1) is equal to 0; and, therefore, we have (1)/(1)
= (1)/(2), that is, the whole equal to half. But perhaps you do not mean
them fractions, but proportions; and, consequently, that the proportion
of 0 to 1, and of 1 to 1, compounded by addition (I say by addition, not
that I, but that you think there is a composition of proportions by
multiplication, which I shall show you anon is false), must be equal to
the proportion of 1 to 2, which cannot be. For the proportion of 0 to 1
is infinitely little, that is, none at all; and, consequently, the
proportion of 1 to 1 is equal to the proportion of 1 to 2, which is
again absurd. There is no doubt but the whole number of 0 + 1 is equal
to 1, and the whole number of 1 + 1 equal to 2. But, reckoning them as
you do, not for whole numbers, but for fractions or proportions, the
equations are false.
Again, your second equation, (2)/(4) = (1)/(4) + (1)/(4), though meant
of fractions, that is, of quotients, it be true, and serve nothing to
your purpose, yet, if it be meant of proportions, it is false. For the
proportion of 1 to 4, and of 1 to 4 being compounded, are equal to the
proportion of 1 to 16, and so you make the proportion of 2 to 4 equal to
the proportion of 1 to 16, where, as it is but subquaduplicate, as you
call it, or the quarter of it, as I call it. And, in the same manner,
you may demonstrate to yourself the same fault in all the other rows of
how many terms soever they consist. Therefore, you may give for lost
this thirty-ninth proposition, as well as all the other thirty-eight
that went before. As for the conclusion of it, which is, _that the
excess of the arising proportion_, &c. They are the words of your
fortieth proposition, where you express yourself better, and make your
error more easy to be detected.
The proposition is this:
“_Si proponatur series quantitatum in triplicata ratione arithmetice
proportionalium (sive juxta seriem numerorum cubicorum) continue
crescentium a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem
totidem maximæ æqualium subquadruplam superabit; eritque excessus ea
ratio quam habet unitas ad quadruplum numeri terminorum post 0; sive
quam habet radix cubica termini primi post 0 ad quadruplum radicis
cubicæ termini maximi. Patet ex præcedente._
“_Quum autem crescente numero terminorum excessus ille supra rationem
subquadruplam ita continuo minuatur, ut tandem quolibet assignabili
minor evadat, ut patet, si in infinitum procedatur, prorsus evaniturus
est, adeoque._
“_Patet ex propositione_ _præcedente._”
That is, if a row of quantities be propounded in triplicate proportion
of arithmetically proportionals (or according to the row of cubic
numbers), continually increasing, and beginning at a point or 0; the
proportion which that row hath to a row of as many equals to the
greatest, is greater than subquadruple proportion; and the excess is
that proportion which one unit hath to the quadruple of the number of
terms after 0; or, which the cubic root of the first term after 0 hath
to the quadruple of the root of the greatest term.
It is manifest by the precedent propositions.
And, seeing the number of terms increasing, that excess above quadruple
proportion doth so continually decrease, as that, at length, it becomes
less than any proportion that can be assigned, as is manifest, if the
proceeding be infinite, it shall quite vanish. And so
This conclusion was annexed to the end of your thirty-ninth proposition,
as there proved. What cause you had to make a new proposition of it,
without other proof than _patet ex præcedente_, I cannot imagine. But,
howsoever, the proposition is false.
For example, set forth any of your rows, as this of fewer terms:
(0 + 1 + 8 + 27 = 36)/((27 + 27 + 27 + 27 =
108)
The row above is 36, the fourth part of the row below is 27. The
quadruple of the number of terms after 0 is 12. Then, by your account,
the proportion of 36 to 108 is greater than subquadruple proportion by
the proportion of 1 to 12. For trial whereof, set in order these three
quantities, 36, 27, 108. The proportion of 36 (the uppermost row) to 108
(the lowermost row) is compounded by addition of the proportions 36 to
27, and 27 to 108. And the proportion of 36 to 108, exceedeth the
proportion of 27 to 108, by the proportion of 36 to 27. But the
proportion of 27 to 108 is subquadruple proportion. Therefore, the
proportion of 36 to 108 exceedeth subquadruple proportion, by the
proportion of 36 to 27. And, by your account, by the proportion of 1 to
12; and, consequently, as 36 to 27, so is 1 to 12. Did you think such
demonstrations as these should always pass?
Then, for your inference from the decrease of the proportions of the
excess, to the vanishing of the excess itself, I have already showed it
to be false; and by consequence that your next proposition, namely, the
fortieth, is also false.
The proposition is this:
“_Si proponatur series infinita quantitatum in triplicata ratione
arithmetice proportionalium (sive juxta seriem numerorum cubicorum),
continue crescentium a puncto sive 0 inchoatarum, erit illa ad seriem
totidem maximæ æqualium, ut 1 ad 4, patet ex præcedente._”
That is, if there be propounded an infinite row of quantities in
triplicate proportion of arithmetically proportionals (or according to
the row of cubic numbers), continually increasing, and beginning at a
point or 0; it shall be to the row of as many equals to the greatest as
1 to 4. Manifest out of the precedent proposition.
Even as manifest as that 36, 27, 1, 12, are proportionals. Seeing,
therefore, your doctrine of the spiral lines and the spaces is given by
yourself for lost, and a vain attempt, your first forty-one propositions
are undemonstrated, and the grounds of your demonstrations all false.
The cause whereof is partly your taking quotient for proportion, and a
point for 0, as you do in the first, sixteenth, and fortieth
propositions, and in other places where you say, _beginning at a point
or 0_, though now you deny you ever said either. There be very many
places in your _Elenchus_, where you say both; and have no excuse for
it, but that, in one of the places, you say the proportion is _penes
quotientem_, which is to the same or no sense.
Your forty-second proposition is grounded on the fortieth; and
therefore, though true, and demonstrated by others, is not demonstrated
by you.
Your forty-third is this:
“_Pari methodo invenietur ratio seriei infinitæ quantitatum arithmetice
proportionalium in ratione quadruplicata, quintuplicata, sextuplicata,
etc., arithmetice proportionalium a puncto seu 0 inchoatarum, ad seriem
totidem maximæ æqualium. Nempe in quadruplicata erit, ut 1 ad 5; in
quintuplicata, ut 1 ad 6; in sextuplicata, ut 1 ad 7. Et sic deinceps._”
That is, by the same method will be found, the proportion of an infinite
row of arithmetically proportionals, in proportion quadruplicate,
quintuplicate, sextuplicate, &c., of arithmetically proportionals,
beginning at a point or 0, to the row of as many equals to the greatest;
namely, in quadruplicate, it shall be as 1 to 5; in quintuplicate, as 1
to 6; in sextuplicate, as 1 to 7; and so forth.
But by the same method that I have demonstrated, that the propositions
19, 20, 21, 39, 40, and 41, are false: any man else, that will examine
the forty-third may find it false also. And, because all the rest of the
propositions of your _Arithmetica Infinitorum_ depend on these, they may
safely conclude, that there is nothing demonstrated in all that book,
though it consist of 194 propositions. The proportions of your
parabolocides to their parallelograms are true, but the demonstrations
false, and infer the contrary. Nor were they ever demonstrated (at least
the demonstrations are not extant) but by me; nor can they be
demonstrated, but upon the same grounds, concerning the nature of
proportion, which I have clearly laid, and you not understood. For, if
you had, you could never have fallen into so gross an error as is this
your book of _Arithmetica Infinitorum_, or that of the angle of contact.
You may see by this, that your symbolic method is not only not at all
inventive of new theorems, but also dangerous in expressing the old. If
the best masters of symbolics think for all this you are in the right,
let them declare it. I know how far the analysis by the powers of the
lines extendeth, as well as the best of your half-learnt epistlers, that
approve so easily of such analogisms as those, 5, 4, 1, 12, and 36, 27,
1, 12, &c.
It is well for you that they who have the disposing of the professors’
places take not upon them to be judges of geometry. For, if they did,
seeing you confess you have read these doctrines in your school, you had
been in danger of being put out of your place.
When the author of the paper wherein I am called Plagiary, and wherein
the honour is taken from you of being the first inventor of these fine
theorems, shall read this that I have here written, he will look to get
no credit by it; especially if it be Roberval, which methinks it should
not be. For he understands what proportion is, better than to make 5 to
4 the same with 1 to 12. Or to make, again, the proportion of 36 to 27
the same with that of 1 to 12; and innumerable _disproportionalites_
that may be inferred from the grounds you go on. But if it be Roberval
indeed, that snatches this invention from you, when he shall see this
burning coal hanging at it, he will let it fall again, for fear of
spoiling his reputation.
But what shall I answer to the authority of the three great
mathematicians that sent you those encomiastic letters. For the first,
whom you say I use to praise, I shall take better heed hereafter of
praising any man for his learning whilst he is young, further than that
he is in a good way. But it seems he was in too ready a way of thinking
very well of himself, as you do of yourself. For the muddiness of my
brain I must confess it; but, Sir, ought not you to confess the same of
yours? No, men of your tenets use not to do so. He wonders, say you, you
thought it worth the while to foul your fingers about such a piece. It
is well; every man abounds in his own sense. If you and I were to be
compared by the compliments that are given us in private letters, both
you and your complimentors would be out of countenance; which
compliments, besides that which has been printed and published in the
commendations of my writings, if it were put together, would make a
greater volume than either of your libels. And truly, Sir, I had never
answered your Elenchus as proceeding from Dr. Wallis, if I had not
considered you also as the minister to execute the malice of that sort
of people that are offended with my _Leviathan_.
As for the judgment of that public Professor that makes himself a
witness of the goodness of your geometry, a man may easily see by the
letter itself that he is a dunce. And for the English person of quality
whom I know not, I can say no more yet than I can say of all three, that
he is so ill a geometrician, as not to detect those gross paralogisms as
infer that 5 to 4 and 1 to 12 are the same proportion. He came into the
cry of those whom your title had deceived.
And now I shall let you see that the composition of proportion by
multiplication, as it is in the fifth definition of the sixth element,
is but another way of adding proportions one to another. Let the
proportions be of 2 to 3, and of 4 to 5. Multiply 2 into 4 and 3 into 5,
the proportion arising is of 8 to 15. Put in order these three
quantities, 8, 12, 15. The proportion therefore of 8 to 15, compounded
of the proportions of 8 to 12, (that is, of 2 to 3) and of 12 to 15,
that is, of 4 to 5 by addition. Again, let the proportion be of 2 to 3,
and of 4 to 5, multiply 2 into 5 and 3 into 4, the proportions arising
is of 10 to 12. Put in order these three numbers, 10, 8, 12. The
proportion 10 to 12 is compounded of the proportions of 10 to 8, that is
of 5 to 4, and of 8 to 12, that is, of 2 to 3 by addition. I wonder you
know not this.
I find not any more clamour against me for saying the proportion of 1 to
2 is double to that of 1 to 4.
Your book, you speak of, concerning proportion against _Meibomius_ is
like to be very useful when neither of you both do understand what
proportion is.
You take exceptions, as that I say, that _Euclid_ has but one word for
_double_ and _duplicate_; which nevertheless was said very truly, and
that word is sometimes διπλάσιος and sometimes διπλάσιων. And you think
you have come off handsomely with asking me whether διπλάσιος and
διπλασίων be one word.
Nor are you now of the mind you were, that a point is not _quantity
unconsidered_, but that in an infinite series it may be safely
neglected. What is _neglected_ but unconsidered.
Nor do you any more stand to it, that the _quotient_ is the
_proportion_. And yet were these the main grounds of your _Elenchus_.
But you will say, perhaps, I do answer to the defence you have now made
in this your _School Discipline_: ’tis true. But ’tis not because you
answer never a word to my former objections against these propositions
19, 89; but because you do so shift and wriggle, and throw out ink, that
I cannot perceive which way you go, nor need I, especially in your
vindication of your _Arithmetica Infinitorum_. Only I must take notice
that in the end of it, you have these words, “Well, _Arithmetica
Infinitorum_ _is come off clear_” You see the contrary. For sprawling is
no defence.
It is enough to me that I have clearly demonstrated both before
sufficiently, and now again abundantly, that your book of _Arithmetica
Infinitorum_ is all nought from the beginning to the end, and that
thereby I have effected that your authority shall never hereafter be
taken for a prejudice. And, therefore, they that have a desire to know
the truth in the questions between us, will henceforth, if they be wise,
examine my geometry, by attentive reading me in my own writings, and
then examine, whether this writing of yours confute or enervate mine.
There is in my fifth lesson a proposition, with a diagram to it, to make
good, I dare say, at least against you, my twentieth chapter concerning
the dimension of a circle. If that demonstration be not shown to be
false, your objections to that chapter, though by me rejected, come to
nothing. I wonder why you pass it over in silence. But you are not, you
say, bound to answer it. True, nor yet to defend what you have written
against me.
Before I give over the examination of your geometry, I must tell you
that your words, (p. 101 of your _School Discipline_), against the first
corollary are untrue.
Your words are these: “_you affirm that the proportion of the parabola A
B I to the parabola A F K is triplicate to the proportion of the time A
B to A F, as it is in the English_.” This is not so. Let the reader turn
to the place and judge. And going on you say, “_or of the impetus B I to
F K as it is in the Latin_.” Nay, as it is in the English, and the other
in the Latin. It is but your mistake; but a mistake is not easily
excused in a false accusation.
Your exception to my saying, “_that the differences of two quantities is
their proportion_,” (when they differ, as the no difference, when they
be equal), might have been put in amongst other marks of your not
sufficiently understanding the Latin tongue. _Differre_ and
_differentia_ differ no more than _vivere_ and _vita_, which is nothing
at all, but as the other words require that go with them, which other
words you do not much use to consider. But _differre_ and _the quantity
by which they differ_, are quite of another kind. _Differre_ (τὸ
διαφέρειν, τὸ ὑπερέχειν) _differing_, _exceeding_, is not quantity, but
relation. But the quantity by which they differ is always a certain and
determined quantity, yet the word _differentia_ serves for both, and is
to be understood by the coherence with that which went before. But I had
said before, and expressly to prevent cavil, that relation is nothing
but a comparison, and that proportion is nothing but relation of
quantities, and so defined them, and therefore I did there use the word
_differentia_ for _differing_, and not for the quantity which was left
by subtraction. For a quantity is not a differing. This I thought the
intelligent reader would of himself understand without putting me,
instead of _differentia_, to use (as some do, and I shall never do) the
mongrel word τὸ _differre_. And whereas in one only place for _differre
ternario_ I have writ _ternarius_, if you had understood what was
clearly expressed before, you might have been sure it was not my
meaning, and therefore the excepting against it was either want of
understanding, or want of candour, choose which you will.
You do not yet clear your doctrine of _condensation_ and _rarefaction_.
But I believe you will by degrees become satisfied that they who say the
same numerical body may be sometimes greater, sometimes less, speak
absurdly, and that _condensation_ and _rarefaction_ here, and
_definitive_ and _circumscriptive_, and some other of your distinctions
elsewhere are but snares, such as school divines have invented
——ᾥσπερ άράχνης
Ὀυλόμενος χέζει ἀλύσεις μυίαις ἀθαρέσσι,
to entangle shallow wits.
And that that distinction which you bring here, “_that it is of the same
quantity while it is in the same place, but it may be of a different
quantity when it goes out of its place_,” (as if the place added to, or
took any quantity from the body placed), is nothing but mere words. It
is true that the body which swells changeth place, but it is not by
becoming itself a greater body, but by admixtion of air or other body,
as when water riseth up in boiling, it taketh in some parts of air. But
seeing the first place of the body is to the body equal, and the second
place equal to the same body, the places must also be equal to one
another, and consequently the dimensions of the body remain equal in
both places.
Sir, when I said that such doctrine was taught in the Universities, I
did not speak against the Universities, but against such as you. I have
done with your geometry, which is one στιγμὴ.
RURAL LANGUAGE.
As for your eloquence, let the reader judge whether yours or mine be the
more _muddy_, though I in plain scolding should have outdone you, yet I
have this excuse which you have not, that I did but answer your
challenge at that weapon which you thought fit to choose. The catalogue
of the hard language which you put in at pages 3 and 4 of your _School
Discipline_, I acknowledge to be mine, and would have been content you
had put in all. The titles you say I give you of _fools_, _beasts_, and
_asses_, I do not give you, but drive back upon you, which is no more
than not to own them; for the rest of the catalogue, I like it so well
as you could not have pleased me better than by setting those passages
together to make them more conspicuous; that is all the defence I will
make to your accusations of that kind.
And now I would have you to consider whether you will make the like
defence against the faults that I shall find in the language of your
_School Discipline_.
I observe, first, the facetiousness of your title-page, “_Due correction
for Mr. Hobbes, or School Discipline, for not saying his Lessons
right_.” What a quibble is this upon the word lesson; besides, you know
it has taken wind; for you vented it amongst your acquaintance at Oxford
then when my _Lessons_ were but upon the press. Do you think if you had
pretermitted that piece of wit, the opinion of your judgment would have
been ere the less? But you were not content with this, but must make
this metaphor from the rod to take up a considerable part of your book,
in which there is scarce anything that yourself can think wittily said
besides it. Consider also these words of yours: “_It is to be hoped that
in time you may come to learn the language, for you be come to great_ A
_already_.” And presently after, “_were I great_ A, _before I would be
willing to be so used, I should wish myself little_ a _a hundred
times_.” Sir, you are a doctor of divinity and a professor of geometry,
but do not deceive yourself, this does not pass for wit in these parts,
no, nor generally at Oxford; I have acquaintance there that will blush
at the reading it.
Again, in another place you have these words: “_Then you catechize us_,
‘_what is your name? Are you geometricians? Who gave you that name_,’”
&c. Besides in other places such abundance of the like insipid conceits,
as would make men think, if they were no otherwise acquainted with the
University but by reading your books, that the dearth there of salt were
very great. If you have any passage more like to salt than these are
(excepting _now and anon_) you may do well to show it to your
acquaintance, lest they despise you; for, since the detection of your
geometry, you have nothing left you else to defend you from contempt.
But I pass over this kind of eloquence, and come to somewhat yet more
rural.
Page 27, line 1, you say I have given Euclid his _lurry_. And again,
page 129, line 11, “_and now he is left to learn his lurry_.” I
understand not the word _lurry_. I never read it before, nor heard it,
as I remember, but once, and that was when a clown threatening another
clown said he would give him _such a lurry come poop_, &c. Such words as
these do not become a learned mouth, much less are fit to be registered
in the public writings of a doctor of divinity. In another place you
have these words, “_just the same to a cow’s thumb_,” a pretty adage.
Page 2, “_But prithee tell me_.” And again, page 95, “_prithee tell me,
why dost thou ask me such a question_,” and the like in many other
places.
You cannot but know how easy it is and was for me to have spoken to you
in the same language. Why did I not? Because I thought that amongst men
that were civilly bred it would have redounded to my shame, as you have
cause to fear that this will redound to yours. But what moved you to
speak in that manner? Were you angry? If I thought that the cause, I
could pardon it the sooner, but it must be very great anger that can put
a man, that professeth to teach good manners, so much out of his wits as
to fall into such a language as this of yours. It was perhaps an
imagination that you were talking to your inferior, which I will not
grant you, nor will the heralds, I believe, trouble themselves to decide
the question. But, howsoever, I do not find that civil men use to speak
so to their inferiors. If you grant my learning but to be equal to
yours, (which you may certainly do without very much disparaging of
yourself abroad in the world), you may think it less insolence in me to
speak so to you in respect of my age, than for you to speak so to me in
respect of your young doctorship. You will find that for all your
doctorship, your elders, if otherwise of as good repute as you, will be
respected before you. But I am not sure that this language of yours
proceeded from that cause; I am rather inclined to think you have not
been enough in good company, and that there is still somewhat left in
your manners for which the honest youths of Hedington and Hincsey may
compare with you for good language, as great a doctor as you are.
For my verses of the Peak, though they be as ill in my opinion as I
believe they are in yours, and made long since, yet they are not so
obscene as that they ought to be blamed by Dr. Wallis. I pray you, sir,
whereas you have these words in your _School Discipline_, page 96,
“_unless you will say that one and the same motion may be now and anon
too_.” What was the reason you put these words, _now and anon too_, in a
different character, that makes them to be more taken notice of? Do you
think that the story of the minister that uttered his affection (if it
be not a slander) not unlawfully but unseasonably, is not known to
others as well as to you? What needed you then, when there was nothing
that I had said could give the occasion, to use those words; there is
nothing in my verses that do _olere hircum_ so much as this of yours. I
know what good you can receive by ruminating on such ideas, or
cherishing of such thoughts. But I go on to other words of mine by you
reproached, “_you may as well seek the focus of the parabola of Dives
and Lazarus_,” which you say is mocking the Scripture; to which I answer
only, that I intended not to mock the Scripture, but you, and that which
was not meant for mocking was none. And thus you have a second στιγμὴ.
GRAMMAR AND CRITIQUES.
I come now to the comparison of our Grammar and Critiques. You object
first against the signification I give of στιγμὴ, and say thus: “_What
should come into your cap_ (that, if you mark it, in a man that wears a
square cap to one that wears a hat, is very witty) _to make you think
that_ στιγμὴ _signifies a mark or brand with a hot iron? I perceive
where the business lies, it was_ στίγμα _run in your mind when you
talked of_ στιγμὴ; _and because the words are somewhat alike you jumble
them both together_.” Sir, I told you once before, you presume too much
upon your first cogitations. Aristophanes, in _Ranis_, Act. V. Scen. 5,
Κἄν μὴ ταχέως ἥκωσι
Νὴ τεν Ἀπόλλο στίξας ἀυτοὺς.
The old commentator upon the word στίξας saith thus, ϛίξας ἀντὶ τοῦ
ϛιγματίσας, ἠν γάρ ξένος. That is, στίξας for ϛιγματίσας, for he
(Adimantus) was not a citizen. I hope the commentator does not here mock
Aristophanes for jumbling ϛίξας and ϛιγματίσας together, for want of
understanding Greek. No, ϛίξας and στιγματίσας signify the same, save
that for branding I seldom read ϛιγματίσας but ϛίξας. For ϛίγμα does no
more signify a brand with a hot iron, than ϛιγμὴ a point made also with
a hot iron. They have both one common theme ϛίζω, which does not signify
_pungo_, nor _interpungo_, nor _inuro_, for all your Lexicon, but _notam
imprimere_, or _pungendo notare_, without any restriction to burning or
punching. It is therefore no less proper to say that ϛιγμὴ is a mark
with a hot iron, than to say the same of στίγμα. The difference is only
this, that when they marked a slave, or a rascal, as you are not
ignorant is usually done here at the assizes in the hand or shoulder
with a hot iron, they called that ϛίγμα, not for the burning, but for
the mark. And as it would have been called ϛίγμα that was imprinted on a
slave, though made by staining or incision, so it is ϛιγμὴ, though done
with a hot iron. And therefore there was no jumbling of those two words
together, as for want of reading Greek authors, and by trusting too much
to your dictionaries, which you say are proofs good enough for such a
business, you were made to imagine. The use I have made thereof was to
show that a point, both by the word Σημεῖον in Euclid, and by the word
στιγμὴ in some others, was not _nothing_, but a _visible_ mark, the
ignorance whereof hath thrown you into so many paralogisms in geometry.
But do you think you can defend your _Adducis Malleum_ as well as I have
now defended my ϛιγμὴ? You have brought, I confess, above a hundred
places of authors, where there is the word _duco_, or some of its
compounds, but none of them will justify _Adducis Malleum_, and,
excepting two of those places, you yourself seem to condemn them all,
comparing yours with none of the rest but with these two only, both out
of Plautus, by you not well understood. The first is in _Casina_, Act.
V. Scen. 2, “_Ubi intro hanc novam nuptam deduxi, via recta, clavem
abduxi_;” which you, presently presuming of your first thoughts, a
peculiar fault to men of your principles, assure yourself is right. But
if you look on the place as Scaliger reads it, cited by the commentator,
you will find it should be _obduxi_, and that _clavis_ is there used for
the bolt of the lock. Besides, he bolted it within. Whither then could
he carry away the key? The place is to be rendered thus, _when I had
brought in this new bride I presently locked the door_, and is this _as
bad every whit_ as _Adducis Malleum_? The second place is in
_Amphytruo_, Act. I. Scen. 1, “_Eam_ (cirneam), _ut a matre fuerat
natum, plenam vini eduxi meri_,” which you interpret _I brought out a
flagon of wine_, unlearnedly. They are the words of Mercury transformed
into Sosia. And to try whether Mercury were Sosia or not, Sosia asked
him where he was and what he did during the battle; to which Mercury
answered, who knew where Sosia then was and what he did, _I was in the
cellar, where I filled a cirnea, and brought it up full of wine, pure as
it came from its mother_. By the mother of the wine meaning the vine,
and alluding to the education of children, for _ebibi_ said _eduxi_, and
with an _emphasis_ in _meri_, because _cirnea_ (from Κφνάω, _misceo_)
was a vessel wherein they put water to temper to their wine. Intimating
that though the vessel was _cirnea_, yet the wine was _merum_. This is
the true sense of the place; but you will have _eduxi_ to be, _I brought
out_, though he came not out himself. You see, sir, that neither this is
so bad as _Adducis Malleum_.
But suppose out of some one place in some one blind author you had
paralleled your _Adducis Malleum_, do you think it must therefore
presently be held for good Latin? Why more than _learn his lurry_ must
be therefore thought good English a thousand years hence, because it
will be read in Dr. Wallis’s long-lived works. But how do you construe
this passage (1 Tim. ii. 15) of the Greek Testament: Σωθήσεται δὲ διὰ
τῆς τεκνογονίας, ἐὰν μείνωσιν ἐν πίστει? You construe it thus: _she
shall be saved notwithstanding child-bearing, if (the woman) remain in
the faith_. Is child-bearing any obstacle to the salvation of women? You
might as well have translated the first verse of the fifth of Romans in
this manner, _Being then justified by faith, we have peace with God
notwithstanding our Lord Jesus Christ_. I let pass your not finding in
τεκνογονίας, as good a grammarian as you are, a nominative case to
μείνωσιν. If you had remembered the place, 1 Pet. iii. 20, εσώθησαν δὶ
ὑδατος, that is, _they were saved in the waters_, you would have thought
your construction justified then very well; but you had been deceived,
for διὰ does not there signify _causam, ablationem impedimenti_, but
_transitum_; not _cause or removing an impediment_, but _passage_. Being
come thus far, I found a friend that hath eased me of this dispute; for
he showed me a letter written to himself from a learned man, that hath
out of very good authors collected enough to decide all the grammatical
questions between you and me, both Greek and Latin. He would not let me
know his name, nor anything of him but only this, that he had better
ornaments than to be willing to go clad abroad in the habit of a
grammarian. But he gave me leave to make use of so much of the letter as
I thought fit in this dispute, which I have done, and have added it to
the end of this writing. But before I come to that, you must not take it
ill, though I have done with your _School Discipline_, if I examine a
little some other of your printed writings as you have examined mine;
for neither you in geometry, nor such as you in church politics, cannot
expect to publish any unwholesome doctrine without some antidotes from
me, as long as I can hold a pen. But why did you answer nothing to my
sixth _Lesson_? Because, you say, it concerned your colleague only. No,
sir, it concerned you also, and chiefly, for I have not heard that your
colleague holdeth those dangerous principles which I take notice of in
you, in my sixth _Lesson_, page 350, upon the occasion of these words,
not his but yours: “_Perhaps you take the whole history of the fall of
Adam for a fable, which is no wonder, seeing you say the rules of
honouring and worshipping of God are to be taken from the laws_.” In
answer to which I said thus: “_You that take so heinously, that I would
have the rule of God’s worship in a Christian commonwealth to be taken
from the laws, tell me from whom you would have them taken? From
yourself? Why so, more than from me? From the bishops? Right, if the
supreme power of the commonwealth will have it so; if not, why from them
rather than from me? From a consistory of presbyters themselves, or
joined with lay elders, whom they may sway as they please? Good, if the
supreme governor of the commonwealth will have it so. If not, why from
them rather than from me, or from any man else? They are wiser and
learneder than I; it may be so, but it has not yet appeared. Howsoever,
let that be granted. Is there any man so very a fool as to subject
himself to the rules of other men in those things which do so nearly
concern himself, for the title they assume of being wise and learned,
unless they also have the sword which must protect them? But it seems
you understand the sword as comprehended. If so, do not you then receive
the rules of God’s worship from the civil power? Yes, doubtless; and you
would expect, if your consistory had that sword, that no man should dare
to exercise or teach any rules concerning God’s worship which were not
by you allowed._”
This will be thought strong arguing, if you do not answer it. But the
truth is, you could say nothing against it without too plainly
discovering your disaffection to the government. And yet you have
discovered it pretty well in your second _Thesis_, maintained in the Act
at Oxford, 1654, and since by yourself published. This _Thesis_ I shall
speak briefly to.
SCOTCH CHURCH POLITICS.
You define ministers of the Gospel to be those _to whom the preaching of
the Gospel by their office is enjoined by Christ_. Pray you, first, what
do you mean by saying preaching _ex officio is enjoined by Christ_? Are
they preachers _ex officio_, and afterwards enjoined to preach? _Ex
officio_ adds nothing to the definition; but a man may easily see your
purpose to disjoin yourself from the state by inserting it.
Secondly, I desire to know in what manner you will be able out of this
definition to prove yourself a minister? Did Christ himself immediately
enjoin you to preach, or give you orders? No. Who then, some bishop, or
minister, or ministers? Yes; by what authority? Are you sure they had
authority immediately from Christ? No. How then are you sure but that
they might have none? At least, some of them through whom your authority
is derived might have none. And therefore if you run back for your
authority towards the Apostles’ times but a matter of sixscore years,
you will find your authority derived from the Pope, which words have a
sound very unlike to the voice of the laws of England. And yet the Pope
will not own you. There is no man doubts but that you hold that your
office comes to you by successive imposition of hands from the time of
the Apostles; which opinion in those gentle terms passeth well enough;
but to say you derive your authority from thence, not through the
authority of the sovereign power civil, is too rude to be endured in a
state that would live in peace. In a word, you can never prove you are a
minister, but by the supreme authority of the commonwealth. Why then do
you not put some such clause into your definition? As thus, _ministers
of the Gospel are those to whom the preaching of the Gospel is enjoined
by the sovereign power in the name of Christ_. What harm is there in
this definition, saving only it crosses the ambition of many men that
hold your principles? Then you define the power of a minister thus:
“_The power of a minister is that which belongeth to a minister of the
Gospel in virtue of the office he holds, inasmuch as he holds a public
station, and is distinguished from private Christians. Such as is the
power of preaching the Gospel, administering the sacrament, the use of
ecclesiastical censures, and ordaining of ministers_,” _&c._
Again, how will you prove out of this definition that you, or any man
else, hath the power of a minister, if it be not given him by him that
is the sovereign of the commonwealth? For seeing, as I have now proved,
it is from him that you must derive your ministry, you can have no other
power than that which is limited in your orders, nor that neither longer
than he thinks fit. For if he give it you for the instruction of his
subjects in their duty, he may take it from you again whensoever he
shall see you instruct them with undutiful and seditious principles. And
if the sovereign power give me command, though without the ceremony of
imposition of hands, to teach the doctrine of my _Leviathan_ in the
pulpit, why am not I, if my doctrine and life be as good as yours, a
minister as well as you, and as public a person as you are? For _public
person_, primarily, is none but the civil sovereign, and so secondarily,
all that are employed in the execution of any part of the public charge.
For all are his ministers, and therefore also Christ’s ministers because
he is so; and other ministers are but his vicars, and ought not to do or
say anything to his people contrary to the intention of the sovereign in
giving them their commission.
Again, if you have in your commission a power to excommunicate, how can
you think that your sovereign who gave you that commission, intended it
for a commission to excommunicate himself? that is, as long as he stand
excommunicate, to deprive him of his kingdom. If all subjects were of
your mind, as I hope they will never be, they will have a very unquiet
life. And yet this has, as I have often heard, been practised in
Scotland, when ministers holding your principles had power enough,
though no right, to do it.
And for administration of the sacraments, if by the supreme power of the
commonwealth it were committed to such of the laity as know how it ought
to be done as well as you, they would _ipso facto_ be ministers as good
as you. Likewise the right of ordination of ministers depends not now on
the imposition of hands of a minister or presbytery, but on the
authority of the Christian sovereign, Christ’s immediate vicar and
supreme governor of all persons and judge of all causes, both spiritual
and temporal, in his own dominions, which I believe you will not deny.
This being evident, what acts are those of yours which you call
_authoritative_, and receive not from the authority of the civil power?
A constable does the acts of a constable _authoritatively_ in that
sense. Therefore you can no otherwise claim your power than a constable
claimeth his, who does not exercise his office in the constabulary of
another. But you forget that the Scribes and the Pharisees sit no more
in Moses’ chair.
You would have every minister to be a minister of the universal Church,
and that it be lawful for you to preach your doctrine at Rome; if you
would be pleased to try, you would find the contrary. You bring no
argument for it that looks like reason. Examples prove nothing, where
persons, times, and other circumstances differ; as they differ very much
now when kings are Christians, from what they were then when kings
persecuted Christians. It is easy to perceive what you aim at.
You would fain have market-day lectures set up by authority, (not by the
authority of the civil power, but by the authority of example of the
Apostles in the emission of preachers to the infidels), not knowing that
any Christian may lawfully preach to the infidels; that is to say,
proclaim unto them that Jesus is the Messiah, without need of being
otherways made a minister, as the deacons did in the Apostles’ time; nor
that many teachers, unless they can agree better, do anything else but
prepare men for faction, nay, rather you know it well enough, but it
conduces to your end upon the market-days to dispose at once both town
and country, under a false pretence of obedience to God, to a neglecting
of the commandments of the civil sovereign, and make the subject to be
wholly ruled by yourselves, wherein you have already found yourselves
deceived. You know how to trouble and sometimes undo a slack government,
and had need to be warily looked to, but are not fit to hold the reins.
And how should you, being men of so little judgment as not to see the
necessity of unity in the governor, and of absolute obedience in the
governed, as is manifest out of the place of your _Elenchus_ above
recited. The doctrine of the duty of private men in a commonwealth is
much more difficult, not only than the knowledge of your symbols, but
also than the knowledge of geometry itself. How then do you think, when
you err so grossly in a few equations, and in the use of most common
words, you should be fit to govern so great nations as England, Ireland,
and Scotland, or so much as to teach them? For it is not reading but
judgment that enables one man to teach another.