To the sixteenth article you bring no argument, but fall into a loud

_oncethmus_ (the special figure wherewith you grace your oratory),

offended with my unexpected crossing of the doctrine you teach, that

proportion consisteth in a quotient. For that being denied you, your

_a/b - c/d + e/f - g/h + i/k_ comes to nothing, that is, to just as much

as it is worth. But are not you very simple men, to say that all

mathematicians speak so, when it is not speaking? When did you see any

man but yourselves publish his demonstrations by signs not generally

received, except it were not with intention to demonstrate, but to teach

the use of signs? Had Pappus no analytics? or wanted he the wit to

shorten his reckoning by signs? Or has he not proceeded analytically in

a hundred problems (especially in his seventh book), and never used

symbols? Symbols are poor unhandsome, though necessary, scaffolds of

demonstration; and ought no more to appear in public, than the most

deformed necessary business which you do in your chambers. “_But why_,”

say you, “_is this limitation to the proportion of the greater to the

less?_” I will tell you; because iterating of the proportion of the less

to the greater, is a making of the proportion less, and the defect

greater. And it is absurd to say that the taking of the same quantity

twice should make it less. And thence it is, that in quantities which

begin with the less, as one, two, four, the proportion of one to two is

greater than that of one to four, as is demonstrated by Euclid, Elem. 5,

prop. 8; and by consequent the proportion of one to four, is a

proportion of greater littleness than that of one to two. And who is

there, that when he knoweth that the respective greatness of four to

one, is double to that of the respective greatness of four to two, or of

two to one, will not presently acknowledge that the respective greatness

of one to two, or two to four, is double to the respective greatness of

one to four? But this was too deep for such men as take their opinions,

not from weighing, but from reading.

Lastly you object against the corollary of art. 28; which you make

absurd enough by rehearsing it thus: _Si quantitas aliqua divisa

supponatur in partes aliquot æquales numero infinitas_, &c. Do you think

that of _partes aliquot_, or of _partes aliquotæ_, it can be said

without absurdity, that they are _numero infinitæ_? And then you say I

seem to mean, that if of the quantity A B, there be supposed a part C B,

infinitely little; and that between A C and A B be taken two means, one

arithmetical, A E, the other geometrical, A D, the difference between A

D and A E, will be infinitely little. My meaning is, and is sufficiently

expressed, that the said means taken everywhere (not in one place only)

will be the same throughout: and you that say there needed not so much

pains to prove it, and think you do it shorter, prove it not at all. For

why may not I pretend against your demonstration, that B E, the

arithmetical difference, is greater than B D, the geometrical

difference. You bring nothing to prove it; and if you suppose it, you

suppose the thing you are to prove. Hitherto you have proceeded in such

manner with your _Elenchus_, as that so many objections as you have

made, so many false propositions you have advanced. Which is a peculiar

excellence of yours, that for so great a stipend as you receive, you

will give place to no man living for the number and grossness of errors

you teach your scholars.

At the fourteenth chapter your first exception is to the second article;

where I define a plane in this manner: _A plane superficies is that

which is described by a straight line so moved, as that every point

thereof describe a several straight line_. In which you require, first,

that instead of _describe_, I should have said _can describe_. Why do

you not require of Euclid, in the definition of a cone, instead of

_continetur_, _is contained_, he say _contineri potest_, _can be

contained_ ? If I tell you how one plane is generated, cannot you apply

the same generation to any other plane? But you object, that the plane

of a circle may be generated by the motion of the _radius_, whose every

point describeth, not a straight, but a crooked line, wherein you are

deceived; for you cannot draw a circle (though you can draw the

perimeter of a circle) but in a plane already generated. For the motion

of a straight line, whose one point resting, describeth with the other

points several perimeters of circles, may as well describe a conic

superficies, as a plane. The question, therefore, is, how you will, in

your definition, take in the plane which must be generated before you

begin to describe your circle, and before you know what point to make

your centre. This objection, therefore, is to no purpose; and besides,

that it reflecteth upon the perfect definitions of Euclid before the

eleventh Element, it cannot make good his definition (which is nothing

worth) of a plane superficies, before his first Element.

In the next place, you reprehend briefly this _corollary, that two

planes cannot enclose a solid_. I should, indeed, have added, _with the

base on whose extremes they insist_: but this is not a fault to be

ashamed of; for any man, by his own understanding, might have mended my

expression without departing from my meaning. But from your doctrine,

_that a superficies has no thickness_, it is impossible to include a

solid, with any number of planes whatsoever, unless you say that solid

is included which nothing at all includes.

At the third article, where I say _of crooked lines, some are everywhere

crooked, and some have parts not crooked_. You ask me what crooked line

has parts not crooked; and I answer, it is that line which with a

straight line makes a rectilineal triangle. But this, you say, cannot

stand with what I said before, namely, that a straight and crooked line

cannot be coincident; which is true, nor is there any contradiction; for

that part of a crooked line which is straight, may with a straight line

be coincident.

To the fourth article, where I define _the centre of a circle to be that

point of the radius, which in the description of the circle is unmoved_;

you object as a contradiction, that I had before defined a point to be

the body which is moved in the description of a line: foolishly, as I

have already shown at your objection to Chap. VIII. art. 12.

But at the sixth article, where I say, that _crooked and incongruous

lines touch one another but in one point_, you make a cavil from this,

that _a circle may touch a parabola in two points_. Tell me truly, did

you read and understand these words that followed? “_A crooked line

cannot be congruent with a straight line; because if it could, one and

the same line should be both straight and crooked._” If you did, you

could not but understand the sense of my words to be this: _when two

crooked lines which are incongruous, or a crooked and a straight line

touch one another, the contact is not in a line, but only in one point_;

and then your instance of a circle and a parabola was a wilful cavil,

not befitting a doctor. If you either read them not, or understood them

not, it is your own fault. In the rest that followeth upon this article,

with your diagram, there is nothing against me, nor anything of use,

novelty, subtlety, or learning.

At the seventh article, where I define both an _angle_, simply so

called, and an _angle of contingence_, by their several generations;

namely, that the former is generated _when two straight lines are

coincident, and one of them is moved, and distracted from the other by

circular motion upon one common point resting, &c._; you ask me “_to

which of these kinds of angle I refer the angle made by a straight line

when it cuts a crooked line_?” I answer easily and truly, To that kind

of angle which is called simply an angle. This you understand not. “For

how”, will you say, “can that angle which is generated by the divergence

of two straight lines, be other than rectilineal? or how can that angle

which is not comprehended by two straight lines, be other than

curvilineal?” I see what it is that troubles you; namely, the same which

made you say before, that if the body which describes a line be a point,

then there is nothing which is not moved that can be called a point. So

you say here, “If an angle be generated by the motion of a straight

line, then no angle so generated can be curvilineal;” which is as well

argued, as if a man should say, the house was built by the carriage and

motion of stone and timber, therefore, when the carriage and that motion

is ended, it is no more a house. Rectilineal and curvilineal hath

nothing to do with the nature of an angle simply so called, though it be

essential to an angle of contact. The measure of an angle, simply so

called, is a circumference of a circle; and the measure is always the

same kind of quantity with the thing measured. The rectitude or curvity

of the lines, which drawn from the centre, intercept the arch, is

accidentary to the angle, which is the same, whether it be drawn by the

motion circular of a straight line or of a crooked. The diameter and the

circumference of a circle make a right angle, and the same which is made

by the diameter and the tangent. And because the point of contact is

not, as you think, nothing, but a line unreckoned, and common both to

the tangent and the circumference; the same angle computed in the

tangent is rectilineal, but computed in the circumference, not

rectilineal, but mixed: or, if two circles cut one another, curvilineal.

For every chord maketh the same angle with the circumference which it

maketh with the line that toucheth the circumference at the end of the

chord. And, therefore, when I divide an angle, simply so called, into

rectilineal and curvilineal, I respect no more the generation of it,

than when I divide it into right and oblique. I then respect the

generation, when I divide an angle into an angle simply so called, and

an angle of contact. This that I have now said, if the reader remember

when he reads your objections to this, and to the ninth article, he will

need no more to make him see that you are utterly ignorant of the nature

of an angle; and that if ignorance be madness, not I, but you, are mad:

and when an angle is comprehended between a straight and a crooked line

(if I may compute the same angle as comprehended between the same

straight line and the point of contact), that it is consonant to my

definition of a point by a _magnitude not considered_. But when you, in

your treatise, _De Angulo Contactus_ (chap. III. p. 6, l. 8) have these

words: “_Though the whole concurrent lines incline to one another, yet

they form no angle anywhere but in the very point of concourse_:” you,

that deny a point to be anything, tell me how two nothings can form an

angle; or if the angle be not formed, neither before the concurrent

lines meet, nor in the point of concourse, how can you apprehend that

any angle can possibly be framed? But I wonder not at this absurdity;

because this whole treatise of yours is but one absurdity, continued

from the beginning to the end, as shall then appear when I come to

answer your objections to that which I have briefly and fully said of

that subject in my 14th chapter.

At the twelfth article, I confess your exception to my universal

definition of parallels to be just, though insolently set down. For it

is no fault of ignorance (though it also infect the demonstration next

it), but of too much security. The definition is this: _Parallels are

those lines or superficies, upon which two straight lines falling, and

wheresoever they fall, making equal angles with them both, are equal_;

which is not, as it stands, universally true. But inserting these words

_the same way_, and making it stand thus: _parallel lines or

superficies, are those upon which two straight lines falling the same

way, and wheresoever they fall, making equal angles, are equal_, it is

both true and universal; and the following consectary, with very little

change, as you may see in the translation, perspicuously demonstrated.

The same fault occurreth once or twice more; and you triumph

unreasonably, as if you had given therein a very great proof of your

geometry.

The same was observed also upon this place by one of the prime

geometricians of Paris, and noted in a letter to his friend in these

words (Chap. XIV. art. 12): “_The definition of parallels wanteth

somewhat to be supplied_.” And of the consectary he says, “_It

concludeth not, because it is grounded on the definition of parallels_.”

Truly and severely enough, though without any such words as savour of

arrogance, or of malice, or of the clown.

At the thirteenth article you recite the demonstration by which I prove

the perimeters of two circles to be proportional to their semidiameters;

and with _esto_, _fortasse_, _recte_, _omnino_, noddying to the several

parts thereof, you come at length to my last inference: _Therefore, by_

Chap. XIII. art. 6, _the perimeters and semidiameters of circles are

proportional_; which you deny; and therefore deny, because you say it

followeth by the same ratiocination, that _circles also and spheres are

proportional to their semidiameters_. “_For the same distance_, you say,

_of the perimeter from the centre which determines the magnitude of the

semidiameter, determines also the magnitude both of the circle and of

the sphere_.” You acknowledge that perimeters and semidiameters have the

cause of their determination such as in equal times make equal spaces.

Suppose now a sphere generated by the semidiameters, whilst the

semicircle is turned about. There is but one _radius_ of an infinite

number of _radii_, which describes a great circle; all the rest describe

lesser circles parallel to it, in one and the same time of revolution.

Would you have men believe, that describing greater and lesser circles,

is according to the supposition (_temporibus æqualibus æqualia facere_)

to make equal spaces in equal times? Or, when by the turning about of

the semidiameter is described the plane of a circle, does it, think you,

in equal times make the planes of the interior circles equal to the

planes of the exterior? Or is the _radius_ that describes the inner

circles equal to the _radius_ that describes the exterior? It does not,

therefore, follow from anything I have said in this demonstration, that

either spheres or planes of circles, are proportional to their _radii_;

and consequently, all that you have said, triumphing in your own

incapacity, is said imprudently by yourselves to your own disgrace. They

that have applauded you, have reason by this time to doubt of all the

rest that follows, and if they can, to dissemble the opinion they had

before of your geometry. But they shall see before I have done, that not

only your whole _Elenchus_ , but also your other books of the _Angle of

Contact_ , &c. are mere ignorance and gibberish.

To the fourteenth article you object, that (in the sixth figure) I

assume gratis, that F G, D E, B C, are proportional to A F, A D, A B;

and you refer it to be judged by the reader: and to the reader I refer

it also. The not exact drawing of the figure (which is now amended) is

it that deceived you. For A F, F D, D B, are equal by construction.

Also, A G, G E, E C, are equal by construction. And F G, D K, B H, K E,

H I, I C, are equal by parallelism. And because A F, F G, are as the

velocities wherewith they are described; also 2 A F (that is A D) and 2

F G (that is D E) are as the same velocities. And finally, 3 A F (that

is A B) and 3 F G (that is B C) are as the same velocities. It is not

therefore assumed gratis, that F G, D E, B C are proportional to A F, A

D, A B, but grounded upon the sixth article of the thirteenth chapter;

and consequently your objection is nothing worth. You might better have

excepted to the placing of D E, first at adventure, and then making A D

two-thirds of A B; for that was a fault, though not great enough to

trouble a candid reader; yet great enough to be a ground, to a malicious

reader, of a cavil.

That which you object to the third _corollary_ of art. 15, was certainly

a dream. There is no assuming of an angle C D E, for an angle H D E, or

B D E, neither in the demonstration, nor in any of the corollaries. It

may be you dreamt of somewhat in the twentieth article of chapter XVI.

But because that article, though once printed, was afterwards left out,

as not serving to the use I had designed it for, I cannot guess what it

is: for I have no copy of that article, neither printed nor written; but

am very sure, though it were not useful, it was true.

Article the sixteenth. Here we come to the controversy concerning the

_angle of contact, which_, you say, _you have handled, in a special

treatise published; and that you have clearly demonstrated, in your

public lectures, that Peletarius was in the right. But that I agree not

sufficiently, neither with Peletarius nor with Clavius._ I confess I

agree not in all points with Peletarius, nor in all points with Clavius.

It does not thence follow that I agree not with the truth. I am not, as

you, of any faction, neither in geometry nor in politics. If I think

that you, or Peletarius, or Clavius, or Euclid, have erred, or been too

obscure, I see no cause for which I ought to dissemble it. And in this

same question I am of opinion that Peletarius did not well in denying

the _angle of contingence_ to be _an angle_. And that Clavius did not

well to say, _the angle of a semicircle_ was less than _a right-lined

right angle_. And that Euclid did not well to leave it so obscure what

he meant by _inclination_ in the definition of a _plane angle_, seeing

elsewhere he attributeth inclination only to acute angles; and scarce

any man ever acknowledged inclination in a straight line, to any other

line to which it was perpendicular. But you, in this question of what is

inclination, though you pretend not to depart from Euclid, are,

nevertheless, more obscure than he; and also are contrary to him. For

Euclid by inclination meaneth the inclination of one line _to_ another;

and you understand it of the inclination of one line _from_ another;

which is not inclination, but declination. For you make two straight

lines, when they lie one on another, to lie ἁκλινῶς, that is, without

any inclination (because it serves your turn); not observing that it

followeth thence, that inclination is a digression of one line _from_

another. This is in your first argument in the behalf of Peletarius (p.

10, l. 22), and destroys his opinion. For, according to Euclid, the

greatest angle is the greatest inclination; and an angle equal to two

right angles by this ἀκλισία, should not be the greatest inclination, as

it is, but the least that can be. But if by the inclination of two

lines, we understand that proceeding of them to a common point, which is

caused by their generation, which, I believe, was Euclid’s meaning; then

will the _angle of contact_ be no less an _angle_ than a _rectilineal_

angle, but only (as Clavius truly says it is) heterogeneous to it; and

the doctrine of Clavius more conformable to Euclid than that of

Peletarius. Besides, if it be granted you, that there is no inclination

of the circumference to the tangent, yet it does not follow that their

concourse doth not form some kind of angle; for Euclid defineth there

but one of the kinds of a plane angle. And then you may as much in vain

seek for the proportion of such angle to the angle of contact, as seek

for the _focus_ or _parameter of the parabola of Dives and Lazarus_.

Your first argument therefore is nothing worth, except you make good

that which in your second argument you affirm, namely, that all plane

angles, not excepting the angle of contact, are (_homogeneous_) of the

same kind. You prove it well enough of other curvilineal angles; but

when you should prove the same of an angle of contact, you have nothing

to say but (p. 17, l. 15), “_Unde autem illa quam somniet heterogenia

oriatur, neque potest ille ullatenus ostendere, neque ego vel

somniare_:” “_Whence should arise that diversity of kind which he dreams

of, neither can he at all show, nor I dream_;” as if you knew what he

could do if he were to answer you; or all were false which you cannot

dream of. So that besides your customary vanity, here is nothing

hitherto proved, neither for the opinion of Peletarius, nor against that

of Clavius. I have, I think, sufficiently explicated, in the first

lesson, that the angle of contact is quantity, namely, that it is the

quantity of that crookedness or flexion, by which a straight line is

bent into an arch of a circle equal to it; and that because the

crookedness of one arch may be greater than the crookedness of another

arch of another circle equal to it; therefore the question _quanta est

curvitas_, how much is the crookedness, is pertinent, and to be answered

by _quantity_. And I have also shown you in the same lesson, that the

quantity of one angle of contact is compared with that of another angle

of contact by a line drawn from the point of contact, and intercepted by

their circumferences; and that it cannot be compared by any measure with

a rectilineal angle.

[Illustration]

But let us see how you answer to that which Clavius has objected

already. “_They are heterogeneous_,” says he, “_because the angle of

contact, how oft soever multiplied, can never exceed a rectilineal

angle_.” To answer which, you allege _it is no angle at all; and that

therefore, it is no angle at all, because the lines have no inclination

one to another_. How can lines that have no inclination one to another,

ever come together? But you answer, _at least they have no inclination

in the point of contact_. And why have two straight lines inclination

before they come to touch, more than a straight line and an arch of a

circle? And in the point of contact itself, how can it be that there is

less inclination of the two points of a straight line and an arch of a

circle, than of the points of two straight lines? But the straight

lines, you say, will cut; which is nothing to the question; and yet this

also is not so evident, but that it may receive an objection. Suppose

two circles, A G B and C F B, to touch in B, and have a common tangent

through B. Is not the line C F B G A a crooked line? and is it not cut

by the common tangent D B E? What is the quantity of the two angles F B

E and G B D, seeing you say neither D B G nor E B F is an angle? It is

not, therefore, the cutting of a crooked line, and the touching of it,

that distinguisheth an angle simply, from an angle of contact. That

which makes them differ, and in kind, is, that the one is the quantity

of a _revolution_, and the other, the quantity of _flexion_.

In the seventh chapter of the same treatise, you think you prove the

angle of contact, if it be an angle, and a rectilineal angle to be

(_homogeneous_) of the same kind; when you prove nothing but that you

understand not what you say. Those quantities which can be added

together, or subtracted one from another, are of the same kind; but an

angle of contact may be subtracted from a right angle, and the remainder

will be the angle of a semicircle, &c. So you say, but prove it not,

unless you think a man must grant you that the superficies contained

between the tangent and the arch, which is it you subtract, is the angle

of contact; and that the plane of the semicircle is the angle of the

semicircle, which is absurd; though, as absurd as it is, you say it

directly in your _Elenchus_ , p. 35, l. 14, in these words: “_When

Euclid defines a plane angle to be the inclination of two lines, he

meaneth not their aggregate, but that which lies between them_.” It is

true, he meaneth not the aggregate of the two lines; but that he means

that which lies between them, which is nothing else but an indeterminate

superficies, is false, or Euclid was as foolish a geometrician as either

of you two.

Again, you would prove the angle of contact, if it be an angle, to be of

the same kind with a rectilineal angle, out of Euclid (III. 16); where

he says, _it is less than any acute angle_. And it follows well, that if

it be an angle, and less than any rectilineal angle, it is also of the

same kind with it. But, to my understanding, Euclid meant no more, but

that it was neither greater nor equal; which is as truly said of

heterogeneous, as of homogeneous quantities. If he meant otherwise, he

confirms the opinion of Clavius against you, or makes the quantity of an

angle to be a superficies, and indefinite. But I wonder how you dare

venture to determine whether two quantities be homogeneous or not,

without some definition of homogeneous (which is a hard word), that men

may understand what it meaneth.

In your eighth chapter you have nothing but Sir H. Savile’s authority,

who had not then resolved what to hold; but esteeming the angle of

contact, first, as others falsely did, by the superficies that lies

between the tangent and the arch, makes the angle of contact and a

rectilineal angle homogeneous; and afterwards, because no multiplication

of the angle of contact can make it equal to the least rectilineal

angle, with great ingenuity returneth to his former uncertainty.

In your ninth and tenth chapters you prove with much ado, that the

angles of like segments are equal; as if that might not have been taken

gratis by Peletarius, without demonstration. And yet your argument,

contained in the ninth chapter, is not a demonstration, but a

conjectural discourse upon the word _similitude_. And in the eleventh

chapter, wherein you answer to an objection, which might be made to your

argument in the precedent page, taken from the parallelism of two

concentric circles, though objection be of no moment, yet you have in

the same treatise of yours that which is much more foolish, which is

this, (p. 38, l. 12): “_Non enim magnitudo anguli_,” _&c._ _“_The_

magnitude of an angle is not to be estimated by that straddling of the

legs, which it hath without the point of concourse, but by that

straddling which it hath in the point of the concourse itself._” I pray

you tell me what straddling there is of two coincident points,

especially such points as you say are nothing? When did you ever see two

nothings straddle?

The arguments in your twelfth and thirteenth chapters are grounded all

on this untruth, that an angle is that which is contained between the

lines that make it; that is to say, is a plane superficies, which is

manifestly false; because the measure of an angle is an arch of a

circle, that is to say, a line; which is no measure of a superficies.

Besides this gross ignorance, your way of demonstration, by putting N

for a great number of sides of an equilateral polygon, is not to be

admitted; for, though you understand something by it, you demonstrate

nothing to anybody but those who understand your symbolic tongue, which

is a very narrow language. If you had demonstrated it in Irish or Welsh,

though I had not read it, yet I should not have blamed you, because you

had written to a considerable number of mankind, which now you do not.

In your last chapters you defend Vitellio without need; for there is no

doubt but that whatsoever crooked line be touched by a straight line,

the angle of contingence will neither add anything to, nor take anything

from, a rectilineal right angle; but that it is because the angle of

contact is no angle, or no quantity, is not true. For it is therefore an

angle, because an angle of contact; and therefore quantity, because one

angle of contact may be greater than another; and therefore

heterogeneal, because the measure of an angle of contact cannot

(_congruere_) be applied to the measure of a rectilineal angle, as they

think it may, who affirm with you that the nature of an angle consisteth

in that which is contained between the lines that comprehend it, viz.,

in a plane superficies. And thus you see in how few lines, and without

brachygraphy, your treatise of the angle of contingence is discovered

for the greatest part to be false, and for the rest, nothing but a

detection of some errors of Clavius grounded on the same false

principles with your own. To return now from your treatise of the angle

of contact back again to your _Elenchus_ .

The fault you find at art. 18, is, that I understand not that Euclid

makes a _plane angle_ to be that which is contained between the two

lines that form it. It is true, that I do not understand that Euclid was

so absurd, as to think the nature of an angle to consist in superficies;

but I understand that you have not had the wit to understand Euclid.

The nineteenth article of mine in this fourteenth chapter, is this:

“_All respect or variety of position of two lines, seemeth to be

comprehended in four kinds_. For they are either _parallel_, or (_being

if need be produced_) _make an angle_; or, (if drawn out far enough)

_touch_; or, lastly, they are _asymptotes_”; in which you are first

offended with the word _It seems_. But I allow you, that never err, to

be more peremptory than I am. For to me it seemed (I say again seemed)

that such a phrase, in case I should leave out something in the

enumeration of the several kinds of position, would save me from being

censured for untruth; and yet your instance of two straight lines in

divers planes, does not make my enumeration insufficient. For those

lines, though not parallels, nor cutting both the planes, yet being

moved parallelly from one plane to another, will fall into one or other

of the kinds of position by me enumerated; and consequently, are as much

that position, as two straight lines in the same plane, not parallel,

make the same angle, though not produced till they meet, which they

would make if they were so produced: for you have nowhere proved, nor

can prove, that two such lines do not make an angle. It is not the

actual concurrence of the lines, but the arch of a circle, drawn upon

that point for centre, in which they would meet if they were produced,

and intercepted between them, that constitutes the angle.

Also your objection concerning asymptotes _in general_ is absurd. You

would have me add, that _their distance shall at last be less than any

distance that can be assigned_; and so make the definition of the

_genus_ the same with that of the _species_. But because you are not

professors of logic, it is not necessary for me to follow your counsel.

In like manner, if we understand one line to be moved towards another

always parallelly to itself, which is, though not actually, yet

potentially the same position, all the rest of your instances will come

to nothing.

At the two-and-twentieth article you object to me the use of the word

_figure_, before I had defined it: wherein also you do absurdly; for I

have nowhere before made such use of the word _figure_, as to argue

anything from it; and therefore your objection is just as wise as if you

had found fault with putting the word figure in the titles of the

chapters placed before the book. If you had known the nature of

demonstration, you had not objected this.

You add further, that by my definition of _figure_, a solid sphere, and

a sphere made hollow within, is the same figure; but you say not why,

nor can you derive any such thing from my definition. That which

deceived your shallowness, is, that you take those points that are in

the concave superficies of a hollowed sphere, not to be contiguous to

anything without it, because that whole concave superficies is within

the whole sphere. Lastly, for the fault you find with the definition of

_like figures in like positions_, I confess there wants the same word

which was wanting in the definition of parallels; namely, _ad easdem

partes_ (_the same way_) which should have been added in the end of the

definition of like figures, &c., and may easily be supplied by any

student of geometry, that is not otherwise a fool.

At the fifteenth chapter, art. 1, number 6, you object as a

contradiction, that _I make motion to be the measure of time; and yet,

in other places, do usually measure motion and the affections thereof by

time_. If your thoughts were your own, and not taken rashly out of

books, you could not but, (with all men else that see time measured by

clocks, dials, hour-glasses, and the like), have conceived sufficiently,

that there cannot be of time any other measure besides motion; and that

the most universal measure of motion, is a line described by some other

motion; which line being once exposed to sense, and the motion whereby

it was described sufficiently explicated, will serve to measure all

other motions and their time: for time and motion (time being but the

mental image or remembrance of the motion) have but one and the same

dimension, which is a line. But you, that would have me measure

_swiftness_ and _slowness_ by longer and shorter motion, what do you

mean by _longer_ and _shorter motion_? Is _longer_ and _shorter_ in the

motion, or in the duration of the motion, which is time? Or is the

motion, or the duration of the motion, that which is exposed, or

designed by a line? Geometricians say often, _let the line A B be the

time_; but never say, _let the line A B be the motion_. There is no

unlearned man that understandeth not what is time, and motion, and

measure; and yet you, that undertake to teach it (most egregious

professors) understand it not.

At the second article you bring another argument (which it seems in its

proper place you had forgotten), to prove that a point is not quantity

not considered, but absolutely nothing; which is this, _That if a point

be not nothing, then the whole is greater than its two halves_. How does

that follow? Is it impossible when a line is divided into two halves,

that the middle point should be divided into two halves also, being

quantity?

At the seventh article, I have sufficiently demonstrated, that all

motion is infinitely propagated, as far as space is filled with body.

You allege no fault in the demonstration, but object from sense, _that

the skipping of a flea is not propagated to the Indies_. If I ask you

how you know it, you may wonder perhaps, but answer you cannot. Are you

philosophers, or geometricians, or logicians, more than are the simplest

of rural people? or are you not rather less, by as much as he that

standeth still in ignorance, is nearer to knowledge, than he that

runneth from it by erroneous learning?

And, lastly, what an absurd objection is it which you make to the eighth

article, where I say that _when two bodies of equal magnitude fall upon

a third body, that which falls with greater velocity, imprints the

greater motion_? You object, _that not so much the magnitude is to be

considered as the weight_; as if the weight made no difference in the

velocity, when notwithstanding weight is nothing else but motion

downward. Tell me, when a weighty body thrown upwards worketh on the

body it meeteth with, do you not then think it worketh the more for the

greatness, and the less for the weight.

==========

OF THE FAULTS THAT OCCUR IN

DEMONSTRATION.

TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN

THE UNIVERSITY OF OXFORD.

LESSON IV.

Of twenty articles which you say (of nineteen which I say) make the

sixteenth chapter, you except but three, and confidently affirm the rest

are false. On the contrary, except three or four faults, such as any

geometrician may see proceed not from ignorance of the subject, or from

want of the art of demonstration, (and such as any man might have mended

of himself) but from security; I affirm that they are all true, and

truly demonstrated; and that all your objections proceed from mere

ignorance of the mathematics.

The first fault you find is this, that I express not (art. 1) what

_impetus_ it is, which I would have to be multiplied into the time.

The last article of my thirteenth chapter was this, “_If there be a

number of quantities propounded, howsoever equal or unequal to one

another; and there be another quantity which so often taken as there be

quantities propounded, is equal to their whole sum; that quantity I call

the mean arithmetical of them all_.” Which definition I did there insert

to serve me in the explication of those propositions of which the

sixteenth chapter consisted, but did not use it here as I intended. My

first proposition therefore as it standeth yet in the Latin, being this,

“_the velocity of any body moved during any time, is so much as is the

product of the impetus in one point of time, multiplied into the whole

time_;” to a man that hath not skill enough to supply what is wanting,

is not intelligible. Therefore I have caused it in the English to go

thus: “_the velocity of any body in whatsoever time moved, hath its

quantity determined by the sum of all the several_ (impetus)

_quicknesses, which it hath in the several points of the time of the

body’s motion_. And added, _that all the_ impetus _together taken

through the whole time is the same thing with the mean_ impetus (which

mean is defined (Chapter XIII. art. 29) _multiplied into the whole

time_.” To this first article, as it is uncorrected in the Latin, you

object, _that meaning by_ impetus _some middle_ impetus, _and assigning

none, I determine nothing_. And it is true. But if you had been

geometricians sufficient to be professors, you would have shewed your

skill much better, by making it appear that this middle _impetus_ could

be none but that, which being taken so often, as there be points in the

line of time, would be equal to the sum of all the several _impetus_

taken in the points of time respectively; which you could not do.

To the _corollary_, you ask first how _impetus_ can be ordinately

applied to a line; absurdly. For does not Archimedes sometimes say, and

with him many other excellent geometricians, _let such a line be the

time_? And do they not mean, that that line, or the motion over it, is

the measure of the time? And may not also a line serve to measure the

swiftness of a motion? _You thought_, you say, _only lines ought to be

said to be ordinately applied to lines_. Which I easily believe; for I

see you understand not that a line, though it be not the time itself,

may be the quantity of a time. You thought also, all you have said in

your _Elenchus_ , in your doctrine of the _angle of contact_, in your

_Arithmetica Infinitorum_, and in your _Conics_ , is true; and yet it is

almost all proved false, and the rest nothing worth.

Secondly, you object, that _I design a parallelogram by one only side_.

It was indeed a great oversight, and argueth somewhat against the man,

but nothing against his art. For he is not worthy to be thought a

geometrician that cannot supply such a fault as that, and correct his

book himself. Though you could not do it, yet another from beyond sea

took notice of the same fault in this manner, “_He maketh a

parallelogram of but one side_; it should be thus: _vel denique per

parallelogrammum cujus unum latus est medium proportionale inter impetum

maximum (sive ultimo acquisitum) et impetus ejusdem maximi semissem;

alterum vero latus, medium proportionale, inter totum tempus, et ejusdem

totius temporis semissem_.” Which I therefore repeat, that you may learn

good manners; and know, that they who reprehend, ought also, when they

can, to add to their reprehension the correction.

At the second article, you are pleased to advise me, instead of _in omni

motu uniformi_, to put in _in omnibus motibus uniformibus_. You have a

strange opinion of your own judgment, to think you know to what end

another man useth any word, better than himself. My intention was only

to consider motions uniform, and motions from rest uniformly, or

regularly accelerated, that I might thereby compute the lengths of

crooked lines, such as are described by any of those motions. And

therefore it was enough to prove this theorem to be true in all uniform

or uniformly accelerated _motion_, not _motions_; though it be true also

in the plural. It seems you think a man must write all he knows, whether

it conduce, or not, to his intended purpose. But that you may know that

I was not (as you think), ignorant how far it might be extended, you may

read it demonstrated at the same article in the English universally.

Against the demonstration itself you run into another article, namely,

the thirteenth, which is this problem: “_the length being given, which

is passed over in a given time by uniform motion, to find the length

which shall be passed over by motion uniformly accelerated in the same

time, so as that the_ impetus _last acquired be equal to the time_.”

Which you recite imperfectly, thereby to make it seem that such a length

is not determined. Whether you did this out of ignorance, or on purpose,

thinking it a piece of wit, as your pretended mystery which goes

immediately before, I cannot tell, for in neither place can any wit be

espied by any but yourselves. To imagine motions with their times and

ways, is a new business, and requires a steady brain, and a man that can

constantly read in his own thoughts, without being diverted by the noise

of words. The want of this ability, made you stumble and fall

unhandsomely in the very first place (that is in Chap. XIII. art. 13),

where you venture to reckon both motion and time at once; and hath made

you in this chapter to stumble in the like manner at every step you go.

As, for example, when I say, _as the product of the time, and impetus,

to the product of the time and impetus, so the space to the space when

the motion is uniform_; you come in with, _nay, rather as the time to

the time_; as if the parallelograms A I, and A H, were not also as the

times A B, and A F. Thus it is, when men venture upon ways they never

had been in before, without a guide.

In the corollary, you are offended with the permutation of the

proportion of times and lines, because you think, (you that have scarce

one right thought of the principles of geometry), that line and time are

heterogeneous quantities. I know time and line are of divers natures;

and more, that neither of them is _quantity_. Yet they may be both of

them _quanta_, that is, they may _have quantity_; but that their

quantities are heterogeneous is false. For they are compared and

measured both of them by straight lines. And to this there is nothing

contrary in the place cited by you out of Clavius; or if there were, it

were not to be valued. And to your question, what is the proportion of

an _hour_ to an _ell_? I answer, it is the same proportion that _two

hours_ have to _two ells_. You see your question is not so subtle as you

thought it. By and bye you confess that in times and lines there is

_quid homogeneum_ (this _quid_ is an infallible sign of not fully

understanding what you say); which is false if you take it of the lines

themselves; though if you take it of their quantities, it is true

without a _quid_. Lastly, you tell m”e how I might have expressed myself

so as it might have been true. But because your expressions please me

not, I have not followed your advice.

To the third article, which is this: “_In motu uniformiter a quiete

accelerato_,” _etc._ “_In motion uniformly accelerated from rest, that

is, when the impetus increaseth in proportion to the times, the length

run over in one time is to the length run over in another time, as the

product of the impetus multiplied by the time, to the product of the

impetus multiplied by the time_;” you object, “_that the lengths run

over are in that proportion which the impetus hath to the impetus; not

that which the impetus hath to the time, because impetus to time has no

proportion, as being heterogeneous_.” First, when you say the impetus,

do you mean some one impetus designed by some one of the unequal

straight lines parallel to the base B I? That is manifestly false. You

mean the aggregate of all those unequal parallels. But that is the same

thing with the time multiplied into the mean impetus. And so you say the

same that I do. Again, I ask, where it is that I say or dream that the

lengths run over are in the proportion of the impetus to the times? Is

it you or I that dream? And for your heterogeneity of the quantities of

time and of swiftness, I have already in divers places showed you your

error. Again, why do you make B I represent the lengths run over, which

I make to be represented by D E, a line taken at pleasure; and you also

a few lines before make the same B I to design the greatest acquired

impetus? These are things which show that you are puzzled and entangled

with the unusual speculation of time and motion, and yet are thrust on

with pride and spite to adventure upon the examination of this chapter.

Secondly, you grant the demonstration to be good, supposing I mean it,

as I seem to speak, of one and the same motion. But why do I not mean it

of one and the same motion, when I say not in _motions_, but in _motion_

uniform? _Because_, say you, _in that which follows, I draw it also to

different motions_. You should have given at least one instance of it;

but there is no such matter. And yet the proposition is in that case

also true; though then it must not be demonstrated by the similitude of

triangles, as in the case present. And therefore the objections you make

from different impetus acquired in the same time, and from other cases

which you mention, are nothing worth.

At the fourth article, you allow the demonstration all the way (except

the faults of the third, which I have already proved to be none) till I

come to say, “_that because the proportion of F K to B I is double to

the proportion of A F to A B, therefore the proportion of A B to A F is

double to the proportion of B I to F K_.” This you deny, and wonder at

as strange, (for it is indeed strange to you), and in many places you

exclaim against it as extreme ignorance in geometry. In this place you

only say, “_no such matter; for though one proportion be double to

another, yet it does not follow that the converse is the double of the

converse_.” So that this is the issue to which the question is reduced,

whether you have any or no geometry. I say, if there be three quantities

in continual proportion, and the first be the least, the proportion of

the first to the second is double to the proportion of the first to the

third; and you deny it. The reason of our dissent consisteth in this,

that you think the doubling of a proportion to be the doubling of the

quantity of the proportion, as well in proportions of defect, as in

proportions of excess; and I think that the doubling of a proportion of

defect, is the doubling of the defect of the quantity of the same. As

for example in these three numbers, 1, 2, 4, which are in continual

proportion, I say the quantity of the proportion of one to two, is

double the quantity of the proportion of one to four. And the quantity

of the proportion of one to four, is half the quantity of the proportion

of one to two. And yet deny not but that the quantity of the defect in

the proportion of one to two is doubled in the proportion of one to

four. But because the doubling of defect makes greater defect, it maketh

the quantity of the proportion less. And as for the part which I hold in

this question, first, there is thus much demonstrated by Euclid, El. v.

prop. 8; that the proportion of one to two, is greater than the

proportion of one to four, though how much it is greater be not there

demonstrated. Secondly, I have demonstrated (Chap, XIII. art. 16); that

it is twice as great, that is to say, (to a man that speaks English),

double. The introducing of _duplicate_, _triplicate_, &c. instead of

_double_, _triple_, &c. (though now they be words well understood by

such as understand what proportion is), proceeded at first from such as

durst not for fear of absurdity, call the half of any thing double to

the whole, though it be manifest that the half of any defect is a double

quantity to the whole defect; for want added to want maketh greater

want, that is, a less positive quantity. This difference between

_double_ and _duplicate_, lighting upon weak understandings, has put men

out of the way of true reasoning in very many questions of geometry.

Euclid never used but one word both for _double_ and _duplicate_. It is

the same fault when men call half a quantity _subduplicate_, and a third

part _subtriplicate_ of the whole, with intention (as in this case) to

make them pass for words of signification different from the _half_ and

the _third part_. Besides, from my definition of proportion (which is

clear, and easy to be understood by all men, but such as have read the

geometry of others unluckily) I can demonstrate the same evidently and

briefly thus. My definition is this, _proportion is the quantity of one

magnitude taken comparatively to another_. Let there be therefore three

quantities, 1, 2, 4, in continual proportion. Seeing therefore the

quantity of four in respect of one, is twice as great as the quantity of

the same four in respect of 2, it followeth manifestly that the quantity

of 1 in respect of 4, is twice as little as the quantity of the same 1

in respect of 2; and consequently the quantity of 1 in respect of 2, is

twice as great as the quantity of the same 1 in respect of 4; which is

the thing I maintain in this question. Would not a man that employs his

time at bowls, choose rather to have the advantage given him of three in

nine, than of one in nine? And why, but that three is a greater quantity

in respect of nine, than is one? Which is as much as to say, three to

nine hath a greater proportion than one to nine; as is demonstrated by

Euclid, El. v. prop. 8. Is it not therefore (you that profess

mathematics, and theology, and practise the depression of the truth in

both) well owled of you, to teach the contrary? But where you say “_that

the point K_ (in the second figure of the table belonging to this

sixteenth chapter) _is not in the parabolical line whose diameter is A

B, and base B I, but in the parabolical line of the complement of my

semiparabola_ (_as I may learn from the twenty-third proposition of

your_ Arithmetica Infinitorum) _whose diameter is A C, and base I C_.”

What line is that? Is it the same line with that of my semiparabola, or

not the same? If the same, why find you fault? If not the same, you

ought to have made a semiparabola on the diameter A C, and base I C, and

following my construction made it appear that K is not in the line

wherein I say it is; which you have not done, nor could do.

Then again, running on in the same blindness of passion, you pretend I

make the proportion of B I to F K double to that of A B to A F, and then

confute it; when you knew I made the proportion of F K to B I, double to

that of F N, to B I, that is, of A F to A B; and this was it you should

have confuted. That which followeth is but a triumphing in your own

ignorance, wherein you also say, “_that all that I afterwards build upon

this doctrine is false_.” You see whether it be like to prove so or not.

As for your _Arithmetica Infinitorum_, I shall then read to you a piece

of a lesson on it when I come to your objections against the next

Chapter. In the mean time let me tell you, it is not likely you should

be great geometricians, that know not what is quantity, nor measure, nor

straight, nor angle, nor homogeneous, nor heterogeneous, nor proportion,

as I have already made appear in this and the former lessons.

To the first corollary of this fourth article your exception I confess

is just, and (which I wonder at) without any incivility. But this argues

not ignorance, but security. For who is there that ever read any thing

in the Conics, that knows not that the parts of a parabola cut off by

lines parallel to the base, are in triplicate proportion to their bases?

But having hitherto designed the time by the diameter, and the impetus

by the base; and in the next chapter (where I was to calculate the

proportion of the parabola, to the parallelogram) intending to design

the time by the base, I mistook and put the diameter again for the time;

which any man but you might as easily have corrected as reprehended.

To the second corollary, which is this, _that the lengths run over in

equal times by motion so accelerated, as that the impetus increase in

double proportion to their times, are as the differences of the cubic

numbers beginning at unity, that is, as seven, nineteen, thirty-seven,

&c._ you say it is false. But why? “_Because_” say you “_portions of the

parabola of equal altitude, taken from the beginning, are not as those

numbers seven, nineteen, thirty-seven, &c._” Does this, think you,

contradict any thing in this proposition of mine? Yes, because, you

think, the lengths gone over in equal times, are the same with the parts

of the diameter cut off from the vertex, and proportional to the numbers

one, two, three, &c. Whereas the lengths run over, are as the aggregates

of their velocities, that is, as the parts of the parabola itself, that

is, as the cubes of their bases, that is, as the numbers one, eight,

twenty-seven, sixty-four, &c., and consequently the lengths run over in

equal times, are as the differences of those cubic numbers, one, eight,

twenty-seven, sixty-four, whose differences are seven, nineteen,

thirty-seven, &c. The cause of your mistake was, that you cannot yet,

nor perhaps ever will, contemplate time and motion (which requireth a

steady brain) without confusion.

The third corollary you also say is false, “_whether it be meant of

motion uniformly accelerated_ (as the words are) _or_ (_as perhaps_, you

say, _I meant it_) _of such motion as is accelerated in double

proportion to the time_.” You need not say perhaps I meant it. The words

of the proposition are enough to make the meaning of the corollary

understood. But so also you say it is false. Methinks you should have

offered some little proof to make it seem so. You think your authority

will carry it. But on the contrary I believe rather that they that shall

see how your other objections hitherto have sped, will the rather think

it true, because you think it false. The demonstration as it is, is

evident enough; and therefore I saw no cause to change a word of it.

To the fifth article you object nothing, but that it dependeth on this

proposition (Chap. XIII. art. 16): “_That when three quantities are in

continual proportion, and the first is the least, as in these numbers,

four, six, nine, the proportion of the first to the second, is double to

the proportion of the same first to the last_;” which is there

demonstrated, and in the former lessons so amply explicated, as no man

can make any further doubt of the truth of it. And you will, I doubt

not, assent unto it. But in what estate of mind will you be then? A man

of a tender forehead after so much insolence, and so much contumelious

language grounded upon arrogance and ignorance, would hardly endure to

outlive it. In this vanity of yours, you ask me whether I be angry, or

blush, or can endure to hear you. I have some reason to be angry; for

what man can be so patient as not to be moved with so many injuries? And

I have some reason to blush, considering the opinion men will have

beyond sea, (when they shall see this in Latin) of the geometry taught

in Oxford. But to read the worst you can say against me, I can endure,

as easily at least, as to read any thing you have written in your

treatises of the _Angle of Contact_, of the _Conic Sections_, or your

_Arithmetica Infinitorum_.

The sixth, seventh, eighth articles, you say are sound. True. But never

the more to be thought so for your approbation, but the less; because

you are not fit, neither to reprehend, nor praise; and because all that

you have hitherto condemned as false, hath been proved true. Then you

show me how you could demonstrate the sixth and seventh articles a

shorter way. But though there be your symbols, yet no man is obliged to

take them for demonstration. And though they be granted to be dumb

demonstrations, yet when they are taught to speak as they ought to do,

they will be longer demonstrations than these of mine.

To the ninth article, which is this, “_If a body be moved by two movents

at once, concurring in what angle soever, of which, one is moved

uniformly, the other, with motion uniformly accelerated from rest, till

it acquire an impetus equal to that of the uniform motion, the line in

which the body is carried, shall be the crooked line of a

semiparabola_,” you lift up your voice again, and ask, _what latitude?

what diameter? what inclination_ of the diameter to the ordinate lines?

If your founder should see this, or the like objections of yours, he

would think his money ill bestowed. When I say, _in what angle soever_,

you ask, _in what angle?_ When I say _two movents, one uniform, the

other uniformly accelerated, make the body describe a semiparabolical

line_; you ask, _which is the diameter?_ as not knowing that the

accelerated motion describes the diameter, and the other a parallel to

the base. And when I say _the two movents meet in a point, from which

point both the motions begin, and one of them from rest_, you ask me

_what is the altitude?_ As if that point where the motion begins from

rest were not the vertex; or that the vertex and base being given, you

had not wit enough to see that the altitude of the parabola is

determined? When Galileo’s proposition, which is the same with this of

mine, supposed no more but a body moved by these two motions, to prove

the line described to be the crooked line of a semiparabola, I never

thought of asking him what altitude, nor what diameter, nor what angle,

nor what base, had his parabola. And when Archimedes said, let the line

A B be the time, I should never have said to him, _do you think time to

be a line_, as you ask me whether I think impetus can be the base of a

parabola. And why, but because I am not so egregious a mathematician, as

you are. In this giddiness of yours, caused by looking upon this

intricate business of motion, and of time, and the concourse of motion

uniform, and uniformly accelerated, you rave upon the numbers 1, 4, 9,

16, &c. without reference to any thing that I had said; insomuch as any

one that had seen how much you have been deceived in them before, in

your scurvy book of _Arithmetica Infinitorum_, would presently conclude,

that this objection was nothing else but a fit of the same madness which

possessed you there.

My tenth article is like my ninth; and your objections to it are the

same which are to the former. Therefore you must take for answer just

the same which I have given to your objection there.

To the eleventh, you say first, you have done it better at the

sixty-fourth article of your _Arithmetica Infinitorum_. But what you

have done there, shall be examined when I come to the defence of my next

chapter. And whereas I direct the reader for the finding of the

proportions of the complements of those figures to the figures

themselves, to the table of art. 3, Chap, XVII., you say that if the

increase of the _spaces_, were to the increase of the times, as one to

two, then the complement should be to the parallelogram as one to three,

and say you find not (1)/(3) in the table. Did you not see that the

table is only of those figures which are described by the concourse of a

motion uniform with a motion accelerated? You had no reason therefore to

look for (1)/(3) in that table; for your case is of motion uniform

concurring with motion retarded, because you make not the proportions of

the spaces to the proportions of the times as two to one, but the

contrary; so that your objection ariseth from want of observing what you

read. But I “_may learn_” you say, “_these, and greater matters than

these, in your twenty-third and sixty-fourth propositions of your_

Arithmetica Infinitorum.” This, which you say here is a great absurdity;

but if you mean I shall find greater there, I will not say against you.

This (1)/(3) you looked for, belongs to the complements of the figures

calculated in that table; which because you are not able to find out of

yourselves, I will direct you to them. Your case is of (1)/(3) for the

complement of a parabola. Take the denominator of the fraction which

belongs to the parabola, namely three, and for numerator take the

numerator of the fraction which belongs to the triangle, namely one, and

you have the fraction sought. And in like manner for the complement of

any other figure. As, for example, of the second parabolaster, whose

fraction hath for denominator five, take the numerator of the fraction

of the same triangle which is one, and you have (1)/(3) for the fraction

sought for; and so of the rest, taking always one for the numerator.

The twelfth article, which you say is miserably false, I have left

standing unaltered. For not comprehending the sense of the proposition,

you make a figure of your own, and fight against your own fancied

motions, different from mine. Other geometricians that understand the

construction better, find no fault. And if you had in your own fifth

figure drawn a line through N parallel to A E, and upon that line

supposed your accelerated motion, you would quickly have seen that in

the time A E, the body moved from rest in A, would have fallen short of

the diagonal A D; and that all your extravagant pursuing of your own

mistake had been absurd.

My thirteenth article you say is ridiculous. But why? “_The impetus last

acquired cannot_” you say, “_be equal to a time_.” But the quantity of

the impetus may be equal to the quantity of a time, seeing they are both

measured by line. And when they are measured by the same described line,

each of their quantities is equal to that same line, and consequently to

one another. But when I meet with this kind of objection again, since I

have so often already shown it to be frivolous, and no less to be

objected against all the ancients that ever demonstrated any thing by

motion, than against me, I purpose to neglect it.

Secondly, you object “_that motion uniformly accelerated does no more

determine swiftness, than motion uniform_.” True; you needed not have

used sixteen lines to set down that. But suppose I add, as I do, so as

the last acquired impetus be equal to the time. _But that_, you say, _is

not sense_; which is the objection I am to neglect. But, you say again,

supposing it sense, this limitation helps me nothing. Why? _Because_,

you say, _a parabola may be described upon a base given, and yet have

any altitude, or any diameter one will_. Who doubts it? But how follows

it from thence, that when a parabolical line is described by two

motions, one uniform, the other uniformly accelerated from rest, that

the determining of the base does not also determine the whole parabola?

But fifthly, you say, _that this equality of the impetus to the time

does not determine the base_. Why not? _Because_, you say, _it is an

error proceeding from this, that I understand not what is_ ratio

subduplicata. I looked for this. I have shown and inculcated

sufficiently before, but the error is on your side; and therefore must

tell you, that this objection, and also a great part of the rest of your

errors in geometry, proceedeth from this, that you know not what

proportion is. But see how wisely you argue about this duplication of

proportion. For thus you say _verbatim_. “_Stay a little. What

proportion has duplicate proportion to single proportion? Is it always

the same? I think not for example, duplicate proportion_ (4)/(1) = (2 in

2)/(1 in 1) _is double to the single_ (2)/(1). _Duplicate proportion_

(9)/(1) = (3 in 3)/(1 in 1) _is triple to its single_ (3)/(1).” Let any

man, even of them that are most ready in your symbols, say in your

behalf (if he be not ashamed) that the proportion of nine to one is

triple to the proportion of three to one, as you do.

In the fourteenth, fifteenth, and sixteenth articles, you bid me repeat

your objections to the thirteenth. I have done it; and find that what

you have objected to the thirteenth, may as well be objected to these;

and consequently, that my answer there will also serve me here.

Therefore, if you can endure it, read the same answer over again.

But you have not yet done, you say, with these articles. Therefore

(after you had for a while spoken perplextly, conjecturing, not without

just cause, that I could not understand you) you say that to the end I

may the better perceive your meaning, I should take the example

following. “_Let a movent (in the first figure of this chapter) be moved

uniformly in the time A B, with the continual impetus A C, or B I, whose

whole velocity shall therefore be the parallelogram A C I B. And another

movent be uniformly accelerated, so as in the time A B it acquire the

same impetus B I. Now as the whole velocity, is to the whole velocity,

so is the length run over, to the length run over._” All this I

acknowledge to be according to my sense, saving that your putting your

word _movens_ instead of my word _mobile_ hath corrupted this article.

For in the first article, I meddle not with motion by concourse, wherein

only I have to do with two movents to make one motion; but in this I do,

wherein my word is not _movens_ but _mobile_; by which it is easy to

perceive you understand not this proposition. Then you proceed: “_But

the length run over by that accelerated motion is greater than the

length run over by that uniform motion._” Where do I say that? You

answer, “_in the ninth and thirteenth article, in making A B (in the

fifth figure) greater than A C; and A H (in the eighth figure) greater

than A B; and consequently, the triangle A B I, greater than the

parallelogram A C I B_.” That consequently is without consequence; for

it importeth nothing at all in this demonstration, whether A B, or A C

in the fifth figure be the greater. Besides I speak there of the

concourse of two movents, that describe the parabolical line A G D;

where the increasing impetus (because it increaseth as the times) will

be designed by the ordinate lines in the parabola A G D B. And if both

the motions in A B and A C were uniform, the aggregate of the impetus

would be designed by the triangle A B D, which is less than the

parallelogram A C D B. But you thought that the motion made by A C

uniformly, is the same with the motion made uniformly in the same time

by the motions in A B and A C concurring; so likewise, in the eighth

figure, there is nothing hinders A H from being greater than A B, unless

I had said that A B had been described in the time A C with the whole

impetus A C maintained entire; of which there is nothing in the

proposition, nor would at all have been pertinent to it. Therefore all

this new undertaking of the thirteenth, fourteenth, fifteenth, and

sixteenth articles, is to as little purpose as your former objections.

But I perceive that these new and hard speculations, though they turn

the edge of your wit, turn not the edge of your malice.

At the seventeenth article, you show again the same confusion. Return to

the eighth figure: “_if in a time given a body run over two lengths, one

with uniform, the other with accelerated motion_”; as for example, if in

the same time A C, a body, run over the line A B with uniform motion,

and the line A H with motion accelerated; “_and again in a part of that

time it run over a part of the length A H, with uniform motion, and

another part of the same with motion accelerated_;” as for example, in

the time A M it run over with uniform motion the line A I, and with

motion accelerated the line A B. _I say the excess of the whole A H

above the part A B, is to the excess of the whole A B above the part A

I, as the whole A H to the whole A B._ But first you will say, that

these words _as the whole A H to the whole A B_, are left out in the

proposition. But you acknowledge that it was my meaning; and you see it

is expressed before I come to the demonstration. And therefore it was

absurdly done to reprehend it. Let us therefore pass to the

demonstration. Draw I K parallel to A C, and make up the parallelogram A

I K M. And supposing first the acceleration to be uniform, divide I K in

the midst at N; and between I N, and I K, take a mean proportional I L.

_And the straight line A L, drawn and produced, shall cut the line B D

in F, and the line C G in G_ (which lines C G, and B D, as also H G and

B F, are determined, though you could not carry it so long in memory, by

the demonstration of the thirteenth article). _For seeing A B is

described by motion uniformly accelerated, and A I by motion uniform in

the same time A M; and I L is a mean proportional between I N (the half

of I K) and I K; therefore by the demonstration of the thirteenth

article, A I is a mean proportional between A B and the half of A B,

namely A O. Again, because A B is described by uniform motion, and A H

by motion uniformly accelerated, both of them in the same time A C, B F

is a mean proportional between B D and half B D, namely B E; therefore

by the demonstration of the same thirteenth article, the straight line A

L F produced will fall on G; and the line A H will be to the line A B,

as the line A B to the line A I. And consequently as A H to A B, so H B

to B I; which was to be demonstrated._ And by the like demonstration the

same may be proved, where the acceleration is in any other proportion

that can be assigned in numbers, saving that whereas this demonstration

dependeth on the construction of the thirteenth article, if the motion

had been accelerated in double proportion to the times, it would have

depended on the fourteenth, where the lines are determined. Which

determinations being not repeated, but declared before, in the

thirteenth article, to which this diagram belongeth, you take no notice

of, but go back to a figure belonging to another article, where there

was no use of these determinations. But because I see that the words of

the proposition, are as of four motions, and not of two motions made by

twice two movents, I must pardon them that have not rightly understood

my meaning; and I have now made the proposition according to the

demonstration. Which being done, all that you have said in very near two

leaves of your _Elenchus_ comes to nothing; and the fault you find comes

to no more than a too much trusting to the skill and diligence of the

reader. And whereas after you had sufficiently troubled yourself upon

this occasion, you add, “_that if Sir H. Savile had read my Geometry, he

had never given that censure of Joseph Scaliger, in his lecture upon

Euclid, that he was the worst geometrician of all mortal men, not

exceptioning so much as Orontius, but that praise should have been kept

for me_.” You see by this time, at least others do, how little I ought

to value that opinion; and that though I be the least of geometricians,

yet my geometry is to yours as 1 to 0. I recite these words of yours, to

let the world see your indiscretion in mentioning so needlessly that

passage of your founder. It is well known that Joseph Scaliger deserved

as well of the state of learning, as any man before or since him; and

that though he failed in his ratiocination concerning the quadrature of

the circle, yet there appears in that very failing so much knowledge of

geometry, that Sir H. Savile could not but see that there were mortal

men very many that had less; and consequently he knew that that censure

of his in a rigid sense (without the license of an hyperbole) was

unjust. But who is there that will approve of such hyperboles to the

dishonour of any but of unworthy persons, or think Joseph Scaliger

unworthy of honour from learned men? Besides, it was not Sir H. Savile

that confuted that false quadrature, but Clavius. What honour was it

then for him to triumph in the victory of another? When a beast is slain

by a lion, is it not easy for any of the fowls of the air to settle

upon, and peck him? Lastly, though it were a great error in Scaliger,

yet it was not so great a fault as the least sin; and I believe that a

public contumely done to any worthy person after his death, is not the

least of sins. Judge therefore whether you have not done indiscreetly,

in reviving the only fault, perhaps that any man living can lay to your

founder’s charge; and yet this error of Scaliger’s was no greater than

one of your own of the like nature, in making the true spiral of

Archimedes equal to half the circumference of the circle of the first

revolution; and then thinking to cover your fault by calling it

afterwards an aggregate of arches of circles (which is no spiral at all

of any kind) you do not repair but double the absurdity. What would Sir

Henry Savile have said to this?

The eighteenth article is this, “_in any parallelogram, if the two sides

that contain the angle be moved to their opposite sides, the one

uniformly, the other uniformly accelerated; the side that is moved

uniformly, by its concourse through all its longitude, hath the same

effect which it would have if the other motion were also uniform, and

the line described were a mean proportional between the whole length,

and the half of the same_.”

To the proposition you object first, “_that it is all one whether the

other motion be uniform or not, because the effect of each of their

motions, is but to carry the body to the opposite side_.” But do you

think that whatsoever be the motions, the body shall be carried by their

concourse always to the same point of the opposite side? If not, then

the effect is not all one when a motion is made by the concourse of two

motions uniform and accelerated, and when it is made by the concourse of

two uniform or of two accelerated motions.

Secondly, you say that these words, _and the line described were a mean

proportional between the whole length, and the half of the same_, have

no sense, or that you are deceived. True. For you are deceived; or

rather you have not understanding enough distinctly to conceive variety

of motions though distinctly expressed. For when a line is gone over

with motion uniformly accelerated, you cannot understand how a mean

proportional can be taken between it and its half; or if you can, you

cannot conceive that that mean can be gone over with uniform motion in

the same time that the whole line was run over by motion uniformly

accelerated. Yet these are things conceivable, and your want of

understanding must be made my fault.

My demonstration is this, _in the parallelogram A B C D, (Fig. 11). Let

the side A B be conceived to be moved uniformly till it lie in C D; and

let the time of that motion be A C, or B D. And in the same time let it

be conceived that A C is moved with uniform acceleration, till it lie in

B D._ To which you object, _that then the acceleration last acquired

must be far greater than that wherewith A B is moved uniformly: else it

shall never come to the place you would have it in the same time_. What

proof bring you for this? None here. Where then? Nowhere that I

remember. On the contrary I have proved (Art. 9 of the chapter) that the

line described by the concourse of those two motions, namely, uniform

from A B to C D, and uniformly accelerated from A C to B D, is the

crooked line of the semiparabola A H D. And though I had not, yet it is

well known that the same is demonstrated by Galileo. And seeing it is

manifest that in what proportion the motion is accelerated in the line A

B, in the same proportion the impetus beginning from rest in A is

increased in the same times (which impetus is designed all the way by

the ordinate lines of the semiparabola), the greatest impetus acquired

must needs be the base of the semiparabola, namely B D, equal to A C,

which designs the whole time. I cannot therefore imagine what should

make you say without proof, that the greatest acquired impetus is

greater than that which is designed by the base B D. Next you say, “you

see not to what end I divide A B in the middle at E.” No wonder; for you

have seen nothing all the way. Others would see it is necessary for the

demonstration; as also that the point F is not to be taken arbitrarily;

and likewise that the thirteenth article, which you admit not for proof,

is sufficiently demonstrated, and your objections to it answered. By the

way you advise me, where I say _percursam eodem motu uniformi, cum

impetu ubique_, &c. to blot out _cum_; because the _impetus_ is not a

_companion_ in the way, but the _cause_. Pardon me in that I cannot take

your learned counsel; for the word _motu uniformi_ is the ablative of

the _cause_, and _impetu_ the ablative of the _manner_. But to come

again to your objections, you say, I make “_a greater space run over in

the same time by the slower motion than by the swifter_.” How does that

appear? _because there is no doubt, but the swiftness is greater where

the greatest impetus is always maintained, than where it is attained to

in the same time from rest_. True, but that is, when they are considered

asunder without concourse, but not then when by the concourse they

debilitate one another, and describe a third line different from both

the lines, which they would describe singly. In this place I compare

their effects as contributing to the description of the parabolical line

A H D. What the effects of their several motions are, when they are

considered asunder, is sufficiently shown before in the first article.

You should first have gotten into your minds the perfect and distinct

ideas of all the motions mentioned in this chapter, and then have

ventured upon the censure of them, but not before. And then you would

have seen that the body moved from A, describeth not the line A C, nor

the line A B, but a third, namely the semiparabolical line A H D.

Again, where I say, _Wherefore, if the whole A B be uniformly moved to C

D, in the same time wherein A C is moved uniformly to F G_; you ask me

“_whether with the same impetus or not?_” How is it possible that in the

same time two unequal lengths should be passed over the same impetus?

“_But why_,” say you, “_do you not tell us with what impetus A C comes

to F G?_” What need is there of that, when all men know that in uniform

motion and the same time, impetus is to impetus, as length to length?

Which to have expressed had not been pertinent to the demonstration.

That which follows in the demonstration, _rursus suppono quod latus A

C_, &c. to these words, _ut ostensum est_, _Art. 12_, you confute with

saying you have proved that article to be false. But you may see now, if

you please, at the same place that I have proved your objection to be

frivolous.

After this you run on without any argument against the rest of the

demonstration, showing nothing all the way, but that the variety and

concourse of motions, the speculations whereof you have not been used

to, have made you giddy.

To the nineteenth article you apply the same objection which you made to

the eighteenth. Which having been answered, it appears that from the

very beginning of your Elenchus to this place all your objections

(except such as are made to three or four mistakes of small importance

in setting down my mind), are mere paralogisms, and such are less

pardonable than any paralogism in Orontius, both because the subject as

less difficult is more easily mastered, and because the same faults are

most shamefully committed by a reprehender than by any other man.

I had once added to these nineteen articles a twentieth, which was this:

“_If from a point in the circumference there be drawn a cord, and a

tangent equal to it, the angle which they make shall be double to the

aggregate of all the angles made by the cords of all the equal arches

into which the arch given can possibly be divided_.” Which proposition

is true, and I did when I writ it think I might have use of it. But be

it, or the demonstration of it true or false, seeing it was not

published by me, it is somewhat barbarous to charge me with the faults

thereof. No doctor of humanity but would have thought it a poor and

wretched malice, publicly to examine and censure papers of geometry

never published, by what means soever they came into his hands. I must

confess that in these words, _in such kind of progression arithmetical_

(that is, which begins with 0) _the sum of all the numbers taken

together, is equal to half the number that is made by multiplying the

greatest into the least_, there is a great error; for by this account

these numbers, 0, 1, 2, 3, 4, taken together, should be equal to

nothing. I should have said they are equal to that number which is made

by multiplying half the greatest into the number of the terms. There was

therefore, if those words were mine (for truly I have no copy of them,

nor have had since the book was printed, and I have no great reason, as

any man may see, to trust your faith) a great error in the writing, but

not an erroneous opinion in the writer. The demonstration so corrected

is true. And the angles that have the proportions of the numbers 1, 2,

3, 4, are in the table of your _Elenchus_ , fig. 12, the angles G A D, H

D E, I E F, K F B. And if the divisions were infinite, so that the first

were not to be reckoned but as a cypher, the angle C A B would be double

to them altogether. This mistake of mine, and the finding that I had

made no use of it in the whole book, was the cause why I thought fit to

leave it quite out. But your professorships, could not forbear to take

occasion thereby, to commend your zeal against _Leviathan_ to your

doctorships of divinity, by censuring it.

==========

OF THE FAULTS THAT OCCUR IN

DEMONSTRATION.

TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN

THE UNIVERSITY OF OXFORD.

LESSON V.

At the seventeenth chapter, your first exception is to the definition of

proportional proportions, which is this: “_Four proportions are then

proportional, when the first is to the second, as the third to the

fourth_.” The reader will hardly believe that your exception is in

earnest. You say, I mean not by proportionality the “_quantity of the

proportions_.” Yes I do. Therefore I say again, that _four proportions

are then proportional, when the quantity of the first proportion, is to

the quantity of the second proportion, as the quantity of the third

proportion, to the quantity of the fourth proportion_. Is not my meaning

now plainly enough expressed? Or is it not the same definition with the

former. But what do I mean, you will say, by the quantity of a

proportion? I mean the determined greatness of it, that is, for example,

in these numbers, the quantity of the proportion of two to three, is the

same with the quantity of the proportion of four to six, or six to nine;

and again, the quantity of the proportion of six to four, is the same

with the quantity of the proportion of nine to six, or of three to two.

But now what do you mean by the quantity of a proportion? You mean that

two and three, are the quantities of the proportion of two to three (for

so Euclid calls them) and that six and four are the quantities of the

proportion of six to four, which is the same with the proportion of

three to two. And by this rule, one and the same proportion shall have

an infinite number of quantities; and consequently the quantity of a

proportion can never be determined. I call one proportion double to

another, when one is equal to twice the other; as the proportion of four

to one, is double to the proportion of two to one. You call that

proportion double where one number, line, or quantity absolute, is

double to the other; so that with you the proportion of two to one is a

double proportion. It is easy to understand how the number two is double

to one, but to what, I pray you, is double the proportion of two to one,

or of one to two? Is not every double proportion double to some

proportion? See whether this geometry of yours can be taken by any man

of sound mind for sense. “_But it is known_,” you say, “_that in

proportions, double is one thing, and duplicate another_;” so that it

seems to you, that in talking of proportion men are allowed to speak

senselessly. “_It is known_,” you say. To whom? It is indeed in use at

this day to call _double duplicate_, and _triple triplicate_. And it is

well enough; for they are words that signify the same thing, but that

they differ (in what subject soever) I never heard till now. I am sure

that Euclid, whom you have undertaken to expound, maketh no such

difference. And even there where he putteth these numbers, one, two,

four, eight, &c. for numbers in _double_ proportion (which is the last

proposition of the ninth element) he meaneth not that one to two, or two

to one, is a _double_ proportion, but that every number in that

progression is _double_ to the number next before it; and yet he does

not call it _analogia dupla_, but _duplicate_. This distinction in

proportions between _double_ and _duplicate_, proceeded long after from

want of knowledge that the proportion of one to two is _double_ to the

proportion of one to four; and this from ignorance of the different

nature of proportions of _excess_, and proportions of _defect_. And you

that have nothing but by tradition saw not the absurdities that did hang

thereon.

In the second article I make E K, (fig. 1) the third part of L K, which

you say is false; and consequently the proposition undemonstrated. And

thus you prove it false: “_Let A C be to G C, or G K to G L, as eight to

one_ (_for seeing the point G is taken arbitrarily, we may place it

where we will, &c._)” and upon this placing of G arbitrarily, you prove

well enough that E K is not a third part of L K. But you did not then

observe, that I make _the altitude A G, less than any quantity given_,

and by consequence E K to differ from a third part by a less difference

than any quantity that can be given. Therefore as yet the demonstration

proceedeth well enough. But perceiving your oversight, you thought fit

(though before, you thought this confutation sufficient) to endeavour to

confute it another way; but with much more evidence of ignorance. For

when I come to say, _the proportion therefore between A C and G C is

triple, in arithmetical proportion, to the proportion between G K and G

E, &c._ you say, “_the proportion of A C to G C is the proportion of

identity, as also that of G K to G E.”_ But why? Does my construction

make it so? Do not I make G C less than A C, though with less difference

than any quantity that can be assigned? And then where I say, _therefore

E K is the third part of L K_, you come in, by parenthesis, with (_or a

fourth, or a fifth, &c._). Upon what ground? Because you think it will

pass for current, without proof, that a point is nothing. Which if it

do, geometry also shall pass for nothing, as having no ground nor

beginning but in nothing. But I have already in a former lesson

sufficiently showed you the consequence of that opinion. To which I may

add, that it destroys the method of _indivisibles_, invented by

Bonaventura; and upon which, not well understood, you have grounded all

your scurvy book of _Arithmetica Infinitorum_; where your indivisibles

have nothing to do, but as they are supposed to have quantity, that is

to say, to be _divisibles_. You allow, it seems, your own nothings to be

somethings, and yet will not allow my somethings to be considered as

nothing. The rest of your objections having no other ground than this,

“_that a point is nothing_,” my whole demonstration standeth firm; and

so do the demonstrations of all such geometricians, ancient and modern,

as have inferred any thing in the manner following, viz. _If it be not

greater nor less, then it is equal. But it is neither greater nor less.

Therefore, &c. If it be greater, say by how much. By so much. It is not

greater by so much. Therefore it is not greater. If it be less, say how

much, &c._ Which being good demonstrations are together with mine

overthrown by the nothingness of your _point_, or rather of your

understanding; upon which you nevertheless have the vanity of advising

me what to do, if I demonstrate the same again; meaning I should come to

your false, impossible, and absurd method of _Arithmetica Infinitorum_,

worthy to be gilded, I do not mean with gold.

And for your question, why I set the base of my figure upwards, you may

be sure it was not because I was afraid to say, that the proportions of

the ordinate lines beginning at the vertex were triplicate, or otherwise

multiplicate of the proportions of the intercepted parts of the

diameter. For I never doubted to call double duplicate, nor triple

triplicate, &c., or if I had, I should have avoided it afterwards at the

tenth article of the same chapter. But because when I went about to

compare the proportions of the ordinate lines with those of their

contiguous diameters, the first thing I considered in them was in what

manner the base grew less and less till it vanished into a point. And

though the base had been placed below, it had not therefore required any

change in the demonstration. But I was the more apt to place the base

uppermost, because the motion began at the base, and ended at the

vertex. To proceed which way I pleased was in my own choice; and it is

of grace that I give you any account of it at all.

To the third article, together with its table, you say, “_it falls in

the ruin of the second; and that the same is to be understood of the

sixth, seventh, eighth, and ninth_.” For confutation whereof I need to

say no more, but that they all stand good by the confutation of your

objections to the second.

To the fourth article you say, “_the description of those curvilineal

figures is easy_.” True, to some men; and now that I have showed you the

way, it is easy enough for you also. For the way you propound is wholly

transcribed out of the figure of the second article, which article you

had before rejected. For seeing the lines H F, G E, A B, &c. are equal

to the lines C Q, C O, C D; and the lines Q F, O E, B D, equal to the

lines C H, C G, C A; the proportion of D B to O E, will be triple (that

is, triplicate) to the proportion of C O to G E; and the proportion of D

B to Q F, triple to the proportion of C D to C Q; and consequently,

because the complement B D C F E B is made by the decrease of A C in

triple proportion to that of the decrease of C D, it will be (by the

second article) a third part of the figure A B E F C A. So that it comes

all to one pass, whether we take triple proportion in decreasing to make

the complement, or triple proportion in increasing to make the figure;

for the proportion of H F to B A, is triple to the proportion of C H to

C A. Wherefore you have done no more but what you have seen first done,

saving that from your construction you prove not the figure to be triple

to the complement; perhaps because you have proved the contrary in your

_Arithmetica Infinitorum_. But your way differs from mine, in that you

call the proportion subtriplicate, which I call triplicate; as if the

divers naming of the same thing made it differ from itself. You might as

well have said briefly, the proposition is true, but ill proved, because

I call the proportion of one or two triple, or triplicate of that of one

to eight; which you say is false, and hath infected the fourth, fifth,

ninth, tenth, eleventh, thirteenth, fourteenth, fifteenth, sixteenth,

seventeenth, and nineteenth articles of the sixteenth chapter. But I

say, and you know now, that it is true; and that all those articles are

demonstrated.

Lastly you add, “_Tu vero, in presente articulo, &c. id est, you bid

find as many mean proportionals as one will, between two given lines; as

if that could not be done by the geometry of planes, &c._” You might

have left out _Tu vero_ to seek an _Ego quidem_. But tell me, do you

think that you can find two mean proportionals (which is less than as

many as one will) by the geometry of planes? We shall see anon how you

go about it. I never said it was impossible, and if you look upon the

places cited by you more attentively, you will find yourself mistaken.

But I say, the way to do it has not been yet found out, and therefore it

may prove a solid problem for anything you know.

The fifth article you reject, because it citeth the corollary of the

twenty-eighth article of the thirteenth chapter, where there is never a

word to that purpose. But there is in the twenty-sixth article; which

was my own fault, though you knew not but it might have been the

printer’s.

To the tenth you object for almost three leaves together, against these

words of mine, _because_, in the sixth figure, _B C is to B F in

triplicate proportion of C D to F E, therefore inverting, F E is to C D

in triplicate proportion of B F to B C_. This you objected then. But now

that I have taught you so much geometry, as to know _that of three

quantities, beginning at the least, if the third be to the first in

triplicate proportion of the second to the first, also by conversion the

first to the second shall be in triplicate proportion of the first to

the third_; if it were to do again, you would not object it.

My eleventh article you would allow for demonstrated, if my second had

been demonstrated, upon which it dependeth. Therefore seeing your

objections to that article are sufficiently answered, this article also

is to be allowed.

The twelfth also is allowed upon the same reason. What falsities you

shall find in such following propositions as depend upon the same second

article, we shall then see when I come to the places where you object

against them.

To the thirteenth article you object, “_that the same demonstration may

be as well applied to a portion of any conoeides, parabolical,

hyperbolical, elliptical, or any other, as to the portion of a sphere_.”

By the truth of this let any man judge of your and my geometry. Your

comparison of the sphere and conoeides, so far holds good, as to prove

that the superficies of the conoeides is greater than the superficies of

the cone described by the subtense of the parabolical, hyperbolical, or

elliptical line. But when I come to say, that _the cause of the excess

of the superficies of the portion of the sphere above the superficies of

the cone, consists in the angle D A B, and the cause of the excess of

the circle made upon the tangent A D, above the superficies of the same

cone, consists in the magnitude of the same angle D A B_, how will you

apply this to your conoeides? For suppose that the crooked line A B (in

the seventh figure) were not an arch of a circle, do you think that the

angles which it maketh with the subtense A B, at the points A and B,

must needs be equal? Or if they be not, does the excess of the

superficies of the circle upon A D above the superficies of the cone, or

the excess of the superficies of the portion of the conoeides above the

superficies of the same cone, consist in the angle D A B, or rather in

the magnitude of the two unequal angles D A B, and A B A? You should

have drawn some other crooked line, and made tangents to it through A

and B, and you would presently have seen your error. See how you can

answer this; for if this demonstration of mine stand firm, I may be bold

to say, though the same be well demonstrated by Archimedes, that this

way of mine is more natural, as proceeding immediately from the natural

efficient causes of the effect contained in the conclusion; and besides,

more brief and more easy to be followed by the fancy of the reader.

To the fourteenth article you say that I “_commit a circle in that I

require in the fourth article the finding of two mean proportionals, and

come not till now to show how it is to be done_.” Nor now neither. But

in the mean time you commit two mistakes in saying so. The place cited

by you in the fourth article is, in the Latin, p. 215, line 26, in the

English, p. 255, line 24. Let any reader judge whether that be a

requiring it, or a supposing it to be done; this is your first mistake.

The second is, that in this place the proportion itself, which is, “_If

these deficient figures could be described in a parallelogram

exquisitely, there might be found thereby between any two lines given,

as many mean proportionals as one would_,” is a theorem, upon

supposition of these crooked lines exquisitely drawn; but you take it

for a problem.

And proceeding in that error, you undertake the invention of two mean

proportionals, using therein my first figure, which is of the same

construction with the eighth that belongeth to this fourteenth article.

Your construction is, “_Let there be taken in the diameter C A, (fig. 1)

the two given lines, or two others proportional to them, as C H, C G,

and their ordinate lines H F, G E (which by construction are in

subtriplicate proportion of the intercepted diameters). These lines will

show the proportions which those four proportionals are to have._” But

how will you find the length of H F or G E, the ordinate lines? Will you

not do it by so drawing the crooked line C F E, as it may pass through

both the points F and E? You may make it pass through one of them, but

to make it pass through the other, you must find two mean proportionals

between G K and G L, or between H I and H P; which you cannot do, unless

the crooked line be exactly drawn; which it cannot be by the geometry of

planes. Go shew this demonstration of yours to Orontius, and see what he

will say to it.

I am now come to an end of your objections to the seventeenth chapter,

where you have an epiphonema not to be passed over in silence. But

because you pretend to the demonstration of some of these propositions

by another method in your _Arithmetica Infinitorum_, I shall first try

whether you be able to defend those demonstrations as well as I have

done these of mine by the method of motion.

The first proposition of your _Arithmetica Infinitorum_ is this lemma:

“_In a series, or row of quantities, arithmetically proportional,

beginning at a point or cypher, as 0, 1, 2, 3, 4, &c. to find the

proportion of the aggregate of them all, to the aggregate of so many

times the greatest, as there are terms_.” This is to be done by

multiplying the greatest into half the number of the terms.

The demonstration is easy. But how do you demonstrate the same? “_The

most simple way_,” say you, “_of finding this and some other problems,

is to do the thing itself a little way, and to observe and compare the

appearing proportions, and then by induction to conclude it

universally_.” Egregious logicians and geometricians, that think an

induction, without a numeration of all the particulars sufficient, to

infer a conclusion universal, and fit to be received for a geometrical

demonstration! But why do you limit it to the natural consecution of the

numbers, 0, 1, 2, 3, 4, &c? Is it not also true in these numbers, 0, 2,

4, 6, &c. or in these, 0, 7, 14, 21, &c? Or in any numbers where the

difference of nothing and the first number is equal to the difference

between the first and second, and between the second and third, &c.?

Again, are not these quantities, 1, 3, 5, 7, &c. in continual proportion

arithmetical? And if you put before them a cypher thus, 0, 1, 3, 5, 7,

do you think that the sum of them is equal to the half of five times

seven? Therefore though your lemma be true, and by me (Chap. XIII. art.

5) demonstrated; yet you did not know why it is true; which also appears

most evidently in the first proposition of your _Conic Sections_ , where

first you have this, “_that a parallelogram whose altitude is infinitely

little, that is to say, none, is scarce anything else but a line_.” Is

this the language of geometry? How do you determine this word _scarce_?

The least altitude, is somewhat or nothing. If somewhat, then the first

character of your arithmetical progression must not be a cypher; and

consequently the first eighteen propositions of this your _Arithmetica

Infinitorum_ are all nought. If nothing, then your whole figure is

without altitude, and consequently your understanding nought. Again, in

the same proposition, you say thus: “_We will sometimes call those

parallelograms rather by the name of lines than of parallelograms, at

least when there is no consideration of a determinate altitude; but

where there is a consideration of a determinate altitude (which will

happen sometimes) there that little altitude shall be so far considered,

as that being infinitely multiplied it may be equal to the altitude of

the whole figure._” See here in what a confusion you are when you resist

the truth. When you consider no determinate altitude, that is no

quantity of altitude, then you say your parallelogram shall be called a

line. But when the altitude is determined, that is, when it is quantity,

then you will call it a parallelogram. Is not this the very same

doctrine which you so much wonder at and reprehend in me, in your

objections to my eighth chapter, and your word _considered_ used as I

used it? It is very ugly in one that so bitterly reprehendeth a doctrine

in another, to be driven upon the same himself by the force of truth

when he thinks not on it. Again, seeing you admit in any case those

infinitely little altitudes to be quantity, what need you this

limitation of yours, “_so far forth as that by multiplication they may

be made equal to the altitude of the whole figure_?” May not the half,

the third, the fourth, or the fifth part, &c. be made equal to the whole

by multiplication? Why could you not have said plainly, _so far forth as

that every one of those infinitely little altitudes be not only

something but an aliquot part of the whole_? So you will have an

_infinitely little_ altitude, that is to say, _a point to be both

nothing and something and an aliquot part_. And all this proceeds from

not understanding the ground of your profession. Well, the lemma is

true. Let us see the theorems you draw from it. The first is (p. 3)

“_that a triangle to a parallelogram of equal base and altitude is as

one to two_.” The conclusion is true, but how know you that?

“_Because_,” say you, “_the triangle consists as it were_ [_as it were_,

is no phrase of a geometrician] _of an infinite number of straight

parallel lines_.” Does it so? Then by your own doctrine, which is, that

“_lines have no breadth_,” the altitude of your triangle consisteth of

an infinite number of no altitudes, that is of an infinite number of

nothings, and consequently the area of your triangle has no quantity. If

you say that by the parallels you mean infinitely little parallelograms,

you are never the better; for if infinitely little, either they are

nothing, or if somewhat, yet seeing that no two sides of a triangle are

parallel, those parallels cannot be parallelograms. I see they may be

counted for parallelograms by not considering the quantity of their

altitudes in the demonstration. But you are barred of that plea, by your

spiteful arguing against it in your _Elenchus_ . Therefore this third

proposition, and with it the fourth, is undemonstrated.

Your fifth proposition is, “_the spiral line is equal to half the circle

of the first revolution_.” But what spiral line? We shall understand

that by your construction, which is this: “_The straight line M A_ [in

your figure which I have placed at the end of the fifth lesson] _turned

round (the point M remaining unmoved) is supposed to describe with its

point A the circle A O A, whilst some point, in the same M A, whilst it

goes about, is supposed to be moved uniformly from M to A, describing

the spiral line_.” This therefore, is the spiral line of Archimedes; and

your proposition affirms it to be equal to the half of the circle A O A;

which you perceived not long after to be false. But thinking it had been

true, you go about to prove it, “_by inscribing in the circle an

infinite multitude of equal angles, and consequently an infinite number

of sectors, whose arches will therefore be in arithmetical proportion_;”

which is true. “_And the aggregate of those arches equal to half the

circumference A O A_;” which is true also. And thence you conclude

“_that the spiral line is equal to half the circumference of the circle

A O A_;” which is false. For the aggregate of that infinite number of

infinitely little arches, is not the spiral line made by your

construction, seeing by your construction the line you make is

manifestly the spiral of Archimedes; whereas no number, though infinite,

of arches of circles, how little soever, is any kind of spiral at all;

and though you call it a spiral, that is but a patch to cover your

fault, and deceiveth no man but yourself. Besides, you saw not how

absurd it was, for you that hold a point to be absolutely nothing, to

make an infinite number of equal angles (the radius increasing as the

number of angles increaseth) and then to say, “_that the arches of the

sectors whose angles they are, are as_ 0, 1, 2, 3, 4, &c.” For you make

the first angle 0, and all the rest equal to it; and so make 0, 0, 0, 0,

0, &c. to be the same progression with 0, 1, 2, 3, 4, &c. The influence

of this absurdity reacheth to the end of the eighteenth proposition. So

many are therefore false, or nothing worth. And you needed not to wonder

that the doctrine contained in them was omitted by Archimedes, who never

was so senseless as to think a spiral line was compounded of arches of

circles.

Your nineteenth proposition is this other lemma: “_In a series, or a

row, of quantities, beginning from a point, or cypher, and proceeding

according to the order of the square numbers, as_ 0, 1, 4, 9, 16, _&c.

to find what proportion the whole series hath to so many times the

greatest_.” And you conclude “_the proportions to be that of 1 to 3_.”

Which is false, as you shall presently see. First, let the series of

squares with the prefixed cypher, and under every one of them the

greatest 4 be (0 . 1 . 4)/(4 . 4 . 4). And you have for the sum of the

squares 5, and for thrice the greatest 12, the third part whereof is 4.

But 5 is greater than 4, by 1, that is, by one twelfth of 12; which

quantity is somewhat, let it be called A. Again, let the row of squares

be lengthened one term further, and the greatepm divst set under every

one of them as (0 . 1 . 4 . 9)/(9 . 9 . 9 . 9). The sum of the squares

is 14, and the sum of four times the greatest is 36, whereof the third

part is 12. But 14 is greater than 12 by two unities, that is, by two

twelfths of 12, that is, by 2 A. The difference therefore between the

sum of the squares, and the sum of so many times the greatest square, is

greater, when the cypher is followed by three squares, than when by but

two. Again, let the row have five terms, as in these numbers (0 . 1

. 4 . 9 . 16)/(16 . 16 . 16 . 16 . 16) with the greatest five times

described, and the sum of the squares will be 30, the sum of all the

greatest will be 80. The third part whereof is 26(2)/(3). But 30 is

greater than 26(2)/(3) by 3(1)/(3), that is, by three twelfths of

twelve, and (1)/(3) of a twelfth, that is, by 3(1)/(3) A. Likewise in

the series continued to six places with the greatest six times

subscribed, as ( 0 . 1 . 4 . 9 . 16 . 25)/(25 . 25 . 25 . 25 . 25

. 25) the sum of the squares is 55, and the sum of the greatest six

times taken is 150, the third part whereof is 50. But 55 is greater than

50 by 5, that is, by five-twelfths of 12, that is by 5 A. And so

continually as the row groweth longer, the excess also of the aggregate

of the squares above the third part of the aggregate of so many times

the greatest square, growing greater. And consequently if the number of

the squares were infinite, their sum would be so far from being equal to

the third part of the aggregate of the greatest as often taken, as that

it would be greater than it by a quantity greater than any that can be

given or named.

That which deceived you was partly this, that you think, as you do in

your _Elenchus_ , that these fractions (1)/(12) (1)/(18) (1)/(24)

(1)/(30) (1)/(36) &c. are proportions, as if (1)/(12) were the

proportion of one to twelve, and consequently (2)/(12) double the

proportion of one to twelve; which is as unintelligible as

school-divinity; and I assure you, far from the meaning of Mr. Ougthred

in the sixth chapter of his _Clavis Mathematica_, where he says that

4(3)/(7) is the proportion of 31 to 7; for his meaning is, that the

proportion of 4(3)/(7) to one, is the proportion of 31 to 7; whereas if

he meant as you do, then 8(6)/(7) should be double the proportion of 31

to 7. Partly also because you think (as in the end of the twentieth

proposition) that if the proportion of the numerators of these fractions

(1)/(12) (1)/(18) (1)/(24) (1)/(30) (1)/(36) to their denominators

decrease eternally, they shall so vanish at last as to leave the

proportion of the sum of all the squares to the sum of the greatest so

often taken, (that is, an infinite number of times), as one to three, or

the sum of the greatest to the sum of the increasing squares, as three

to one; for which there is no more reason than for four to one, or five

to one, or any other such proportion. For if the proportions come

eternally nearer and nearer to the subtriple, they must needs also come

nearer and nearer to subquadruple; and you may as well conclude thence

that the upper quantities shall be to the lower quantities as one to

four, or as one to five, &c. as conclude they are as one to three. You

can see without admonition, what effect this false ground of yours will

produce in the whole structure of your _Arithmetica Infinitorum_; and

how it makes all that you have said unto the end of your thirty-eighth

proposition, undemonstrated, and much of it false.

The thirty-ninth is this other lemma: “_In a series of quantities

beginning with a point or cypher, and proceeding according to the series

of the cubic numbers, as O. 1. 8. 27. 64, &c. to find the proportion of

the sum of the cubes to the sum of the greatest cube, so many times

taken as there be terms_.” And you conclude that “_they have a

proportion of 1 to 4_;” which is false.

Let the first series be of three terms subscribed with the greatest (0.

1. 8.)/(8. 8. 8.); the sum of the cubes is nine; the sum of all the

greatest is 24; a quarter whereof is 6. But 9 is greater than 6 by three

unities. An unity is something. Let it be therefore A. Therefore the row

of cubes is greater than a quarter of three times eight, by three A.

Again, let the series have four terms, as (0. 1. 8. 27)/(27. 27. 27.

27); the sum of the cubes is 36; a quarter of the sum of all the

greatest is twenty-seven. But thirty-six is greater than twenty-seven by

nine, that is, by 9 A. The excess therefore of the sum of the cubes

above the fourth part of the sum of all the greatest, is increased by

the increase of the number of terms. Again, let the terms be five, as

(0. 1. 8. 27. 64)/(64. 64. 64. 64. 64), the sum of the cubes is one

hundred; the sum of all the greatest three hundred and twenty; a quarter

whereof is eighty. But one hundred is greater than eighty by twenty,

that is, by 20 A. So you see that this lemma also is false. And yet

there is grounded upon it all that which you have of comparing parabolas

and paraboloeides with the parallelograms wherein they are accommodated.

And therefore though it be true, that the parabola is (2)/(3) and the

cubical paraboloeides (3)/(4) of their parallelograms respectively, yet

it is more than you were certain of when you referred me, for the

learning of geometry, to this book of yours. Besides, any man may

perceive that without these two lemmas (which are mingled with all your

compounded series with their excesses) there is nothing demonstrated to

the end of your book: which to prosecute particularly, were but a vain

expense of time. Truly, were it not that I must defend my reputation, I

should not have showed the world how little there is of sound doctrine

in any of your books. For when I think how dejected you will be for the

future, and how the grief of so much time irrecoverably lost, together

with the conscience of taking so great a stipend, for mis-teaching the

young men of the University, and the consideration of how much your

friends will be ashamed of you, will accompany you for the rest of your

life, I have more compassion for you than you have deserved. Your

treatise of the _Angle of Contact_ , I have before confuted in a very

few leaves. And for that of your _Conic Sections_ , it is so covered

over with the scab of symbols, that I had not the patience to examine

whether it be well or ill demonstrated.

Yet I observed thus much, that you find a tangent to a point given in

the section by a diameter given; and in the next chapter after, you

teach the finding of a diameter, which is not artificially done.

I observe also, that you call the _parameter_ an imaginary line, as if

the place thereof were less determined than the diameter itself; and

then you take a mean proportional between the intercepted diameter, and

its contiguous ordinate line, to find it. And it is true, you find it:

but the parameter has a determined quantity, to be found without taking

a mean proportional. For the diameter and half the section being given,

draw a tangent through the vertex, and dividing the angle in the midst

which is made by the diameter and tangent, the line that so divideth the

angle, will cut the crooked line. From the intersection draw a line (if

it be a parabola) parallel to the diameter, and that line shall cut off

in the tangent from the vertex the parameter sought. But if the section

be an ellipsis, or an hyperbole, you may use the same method, saving

that the line drawn from the intersection must not be parallel, but must

pass through the end of the transverse diameter, and then also it shall

cut off a part of the tangent, which measured from the vertex is the

parameter. So that there is no more reason to call the parameter an

imaginary line than the diameter.

Lastly, I observe that in all this your new method of conics, you show

not how to find the _burning points_, which writers call the _foci_ and

_umbilici_ of the section, which are of all other things belonging to

the conics most useful in philosophy. Why therefore were they not as

worthy of your pains as the rest, for the rest also have already been

demonstrated by others? You know the focus of the parabola is in the

axis distant from the vertex a quarter of the parameter. Know also that

the focus of an hyperbole, is in the axis, distant from the vertex, as

much as the hypotenusal of a rectangled triangle, whose one side is half

the transverse axis, the other side half the mean proportional between

the whole transverse axis and the parameter, is greater than half the

transverse axis.

The cause why you have performed nothing in any of your books (saving

that in your _Elenchus_ you have spied a few negligences of mine, which

I need not be ashamed of) is this, that you understood not what is

_quantity_, _line_, _superficies_, _angle_, and _proportion_; without

which you cannot have the science of any one proposition in geometry.

From this one and first definition of Euclid, “_a point is that whereof

there is no part_,” understood by Sextus Empiricus, as you understand

it, that is to say misunderstood, Sextus Empiricus had utterly destroyed

most of the rest, and demonstrated, that in geometry there is no

science, and by that means you have betrayed the most evident of the

sciences to the sceptics. But as I understand it for _that whereof no

part is reckoned_, his arguments have no force at all, and geometry is

redeemed. If a line have no latitude, how shall a cylinder rolling on a

plane, which it toucheth not but in a line, describe a superficies? How

can you affirm that any of those things can be without quantity, whereof

the one may be greater or less than the other? But in the common contact

of divers circles the external circle maketh with the common tangent a

less angle of contact than the internal. Why then is it not quantity? An

angle is made by the concourse of two lines from several regions,

concurring, by their generation, in one and the same point. How then can

you say the angle of contact is no angle? One measure cannot be

applicable at once to the angle of contact, and angle of conversion. How

then can you infer, if they be both angles, that they must be

homogeneous? Proportion is the relation of two quantities. How then can

a quotient or fraction, which is quantity absolute, be a proportion? But

to come at last to your _Epiphonema_ , wherein, though I have perfectly

demonstrated all those propositions concerning the proportion of

parabolasters to their parallelograms, and you have demonstrated none of

them (as you cannot now but plainly see), but committed most gross

paralogisms, how could you be so transported with pride, as insolently

to compare the setting of them forth as mine, to the act of him that

steals a horse, and comes to the gallows for it. You have read, I think,

of the gallows set up by Haman. Remember therefore also who was hanged

upon it.

After your dejection I shall comfort you a little, a very little, with

this, that whereas this eighteenth chapter containeth two problems, one,

“_the finding of a straight line equal to the crooked line of a

semi-parabola_;” the other, “_the finding of straight lines equal to the

crooked lines of the parabolasters, in the table of the third article of

the seventeenth chapter_;” you have truly demonstrated that they are

both false; and another hath also demonstrated the same another way.

Nevertheless, the fault was not in my method, but in a mistake of one

line for another and such as was not hard to correct; and is now so

corrected in the English as you shall not be able (if you can

sufficiently imagine motions) to reprehend. The fault was this, that in

the triangles which have the same base and altitude with the parabola

and parabolaster, I take for designation of the mean uniform impetus, a

mean proportional, in the first figure, between the whole diameter and

its half, and, in the second figure, a mean proportional between the

whole diameter and its third part; which was manifestly false, and

contrary to what I had shown in the sixteenth chapter. Whereas I ought

to have taken the half of the base, as now I have done, and thereby

exhibited the straight lines equal to those crooked lines, as I

undertook to do. Which error therefore proceeded not from want of skill,

but from want of care; and what I promised (as bold as you say the

promise was), I have now performed.

The rest of your exceptions to this chapter, are to these words in the

end: “_There be some that say, that though there be equality between a

straight and crooked line, yet now, they say, after the fall of Adam, it

cannot be found without the especial help of divine grace_.” And you say

you think there be none that say so. I am not bound to tell you who they

are. Nevertheless, that other men may see the spirit of an ambitious

part of the clergy, I will tell you where I read it. It is in the

_Prolegomena_ of Lalovera, a Jesuit, to his Quadrature of the Circle, p.

13 and 14, in these words: “_Quamvis circuli tetragonismus sit_ φύσει

_possibilis, an tamen etiam_ πρός ἡμᾶς, _hoc est, post Adæ lapsum homo

ejus scientiam absque speciali divinæ gratiæ auxilio, possit comparare,

jure merito inquirunt theologi, pronunciantque; hanc veritatem tanta

esse caligine involutam ut illam videre nemo possit, nisi ignorantiæ ex

primi parentis prævaricatione propagatas tenebras indebitus divinæ lucis

radius dissipet; quod verissimum esse sentio_.” Wherein I observed that

he, supposing he had found that quadrature, would have us believe it was

not by the ordinary and natural help of God (whereby one man reasoneth,

judgeth and remembereth better than another), but by a special (which

must be a supernatural) help of God, that he hath given to him of the

order of Jesus above others that have attempted the same in vain.

Insinuating thereby, as handsomely as he could, a special love of God

towards the Jesuits. But you taking no notice of the word _special_,

would have men think I held, that human sciences might be acquired

without any help of God. And thereupon proceed in a great deal of ill

language to the end of your objections to this chapter. But I shall take

notice of your manners for altogether in my next lesson.

At the nineteenth chapter you see not, you say, the method. Like enough.

In this chapter I consider not the cause of reflection, which consisteth

in the resistance of bodies natural; but I consider the consequences,

arising from the supposition of the equality of the angle of reflection,

to that of incidence; leaving the causes both of reflection, and of

refraction, to be handled together in the twenty-fourth chapter. Which

method, think what you will, I still think best.

Secondly, you say I define not, here, but many chapters after, what an

angle of incidence, and what an angle of reflection is. Had you not been

more hasty than diligent readers, you had found that those definitions

of the angle of incidence, and of reflection, were here set down in the

first article, and not deferred to the twenty-fourth. Let not therefore

your own oversight be any more brought in for an objection.

Thirdly, you say there is no great difficulty in the business of this

chapter. It may be so, now it is down; but before it was done, I doubt

not but you that are a professor would have done the same, as well as

you have done that of the _Angle of Contact_ , or the business of your

_Arithmetica Infinitorum_ . But what a novice in geometry would have

done I cannot tell.

To the third, fourth, and fifth article, you object a want of

determination; and show it by instance, as to the third article. But

what those determinations should be, you determine not, because you

could not. The words in the third article, are first these, _if there

fall two straight lines parallel, &c._ which is too general. It should

be, _if there fall the same way two straight lines parallel, &c._ Next

these, _their reflected lines produced inwards shall make an angle, &c._

This also is too general. I should have said, _their reflected lines

produced inwards, if they meet within, shall make an angle, &c._ Which

done, both this article and the fourth and fifth are fully demonstrated.

And without it, an intelligent reader had been satisfied, supplying the

want himself by the construction.

To the eighth, you object only the too great length and labour of it,

because you can do it a shorter way. Perhaps so now, as being easy to

shorten many of the demonstrations both of Euclid, and other the best

geometricians that are or have been. And this is all you had to say to

my nineteenth chapter. Before I proceed, I must put you in mind that

these words of yours, “_adducis malleum, ut occidas muscam_,” are not

good Latin, _malleum affers_, _malleum adhibes_, _malleo uteris_, are

good. When you speak of bringing bodies animate, _ducere_ and _adducere_

are good, for there _to bring_, is _to guide or lead_. And of bodies

inanimate, _adducere_ is good for _attrahere_, which is to draw to. But

when you bring a hammer, will you say _adduco malleum_, _I lead a

hammer_? A man may lead another man, and a ninny may be said to lead

another ninny, but not a hammer. Nevertheless, I should not have thought

fit to reprehend this fault upon this occasion in an Englishman, nor to

take notice of it, but that I find you in some places nibbling, but

causelessly, at my Latin.

Concerning the twentieth chapter, before I answer to the objections

against the propositions themselves, I must answer to the exception you

first take to these words of mine, “_Quæ de dimensione circuli et

angulorum pronuntiata sunt tanquam exactè inventa, accipiat lector

tanquam dicta_ _problematicè._” To which you say thus: “_We are wont in

geometry to call some propositions theorems, others problems, &c. of

which a theorem is that wherein some assertion is propounded to be

proved; a problem that wherein something is commanded to be done_.” Do

you mean _to be done_, and not proved? By your favour, a problem in all

ancient writers signifies no more but a proposition uttered, to the end

to have it, by them to whom it is uttered, examined whether it be true

or not true, faisable or not faisable; and differs not amongst

geometricians from a theorem but in the manner of propounding. For this

proposition, _to make an equilateral triangle_, so propounded they call

a problem. But if propounded thus: _If upon the ends of a straight line

given be described two circles, whose radius is the same straight line,

and there be drawn from the intersection of the circles to their two

centres, two straight lines, there will be made an equilateral

triangle_, then they call it a theorem; and yet the proposition is the

same. Therefore these words, _accipiat lector tanquam dicta

problematicè_ signify plainly this, that I would have the reader, take

for propounded to him to examine, whether from my construction the

quadrature of the circle can be truly inferred or not; and this is not

to bid him, as you interpret it, to square the circle. And if you

believe that _problematicè_ signifies probably, you have been very

negligent in observing the sense of the ancient Greek philosophers in

the word problem. Therefore your _solemus in geometria_, &c. is nothing

to the purpose; nor had it been though you had spoken more properly, and

said _solent_, leaving out yourselves.

[Illustration:

_Six Lessons._

_Vol. VII. Eng. p.310_II. 325_

]