Chapter X (3)

At the thirteenth article you find fault with, that I say _that the proportion of inequality, whether it be of excess or of defect, is quantity, but the proportion of equality is not quantity_. Whether that which you say, or that which I say, be the truth, is a question worthy of a very strict examination. The first time I heard it argued, was in Mersennus’ chamber at Paris, at such time as the first volume of his _Cogitata Physico-Mathematica_ was almost printed; in which, because he had not said all he would say of proportion, he was forced to put the rest into a general preface, which, as was his custom, he did read to his friends before he sent it to the press. In that general preface, under the title _De Rationibus atque Proportionibus_, at the numbers twelve, thirteen, fourteen, he maintaineth against Clavius, _that the composition of proportion is_ (as of all other things) _a composition of the parts to make a total_, and _that the proportion of equality answereth in quantity to_ non-ens, _or nothing; the proportion of excess, to_ ens, _or quantity; and the proportion of defect, to less than nothing; because equality_ (he says) _is a term of middle signification between excess and defect_. And there also he refuteth the arguments which Clavius, at the end of the ninth Element of Euclid, bringeth to the contrary. And though this were approved by divers good geometricians then present, and never gainsaid by any since, yet do not I say it upon the credit of them, but upon sufficient grounds. For it hath been demonstrated by Eutocius, that _if there be three magnitudes, the proportion of the first to the third is compounded of the proportions of the first to the second, and of the second to the third_; which also I demonstrate in this article. And if there were never so many magnitudes ranked, it might be likewise demonstrated, that the proportion of the first to the last is compounded of the proportions of the first to the second, and of the second to the third, and of the third to the fourth, and so on to the last. If, therefore, we put in order any three numbers, whereof the two last be equal, as four, seven, seven, the proportion of four the first to seven the last, will be compounded of the proportions of four the first to seven the second, and of seven the second to seven the third. Wherefore the proportion of seven to seven (which is of equality) addeth nothing to the proportion of four the first, to seven the second; and consequently the proportion of seven to seven hath no quantity; but that the proportion of inequality hath quantity, I prove it from this, that one inequality may be greater than another.

But for the clearing of this doctrine (which Mersennus calls intricate) of the composition of proportions, I observed, first, that any two quantities, being exposed to sense, their proportion was also exposed; which is not intricate. Again, I observed that if besides the two exposed quantities, there were exposed a third, so as the first were the least, and the third the greatest, or the first the greatest, and the third the least, that not only the proportions of the first to the second, but also (because the differences and the quantities proceed the same way) the proportion of the first to the last is exposed by composition, or addition of the differences; nor is there any intricacy in this. But when the first is less than the second, and the second greater than the third, or the first greater than the second, and the second less than the third, so that to make the first and second equal, if we use addition, we must, to make the second and third equal, use subtraction; then comes in the intricacy, which cannot be extricated, but by such as know the truth of this doctrine which I now delivered out of Mersennus, namely, that the proportions of excess, equality, and defect, are as _quantity_, _not-quantity_, _nothing want quantity_; or as symbolists mark them 0+1 . 0 . 0-1. And upon this ground I thought depended the universal truth of this proposition, that in any rank of magnitudes of the same kind, the proportion of the first to the last, was compounded of all the proportions (in order) of the intermediate quantities; the want of the proof thereof, Sir Henry Savile calls (_nævus_) a mole in the body of geometry. This proposition is demonstrated at the thirteenth article of this chapter.

But before we come thither, I must examine the arguments you bring to confute this proposition, that the _proportion of inequality is quantity, of equality, not quantity_.

And first, you object that equality and inequality are in the same predicament: a pretty argument to flesh a young scholar in the logic school, that but now begins to learn the predicaments. But what do you mean by _æquale_ and _inequale_? Do you mean _corpus æquale_, and _corpus inequale_? They are both in the predicament of substance, neither of them in that of quantity. Or do you mean _æqualitas_ and _inæqualitas_? They are both in the predicament of relation, neither of them in that of quantity; and yet both _corpus_ and _inæqualitas_, though neither of them be quantity, may be _quanta_, that is, both of them have quantity. And when men say body is quantity, or inequality is quantity, they are no otherwise understood, than if they had said _corpus est tantum_, and _inæqualitas tanta_, not _tantitas_; that is, bodies and inequalities are _so much_, not _somuchness_; and all intelligent men are contented with that expression, and yourselves use it. And the quantity of inequality is in the predicament of quantity, because the measure of it is in that line by which one quantity exceeds the other. But when neither exceedeth the other, then there is no line of excess, or defect by which the equality can be measured, or said to be _so much_, or be called quantity. Philosophy teacheth us how to range our words; but Aristotle’s ranging them in his predicaments doth not teach philosophy; and therefore no argument taken from thence, can become a doctor and a professor of geometry.

To prove that the proportion of inequality was quantity, but the proportion of equality not quantity, my argument was this: that _because one inequality may be greater or less than another, but one equality cannot be greater nor less than another: therefore inequality hath quantity, or is tanta, and equality not_. Here you come in again with your predicaments, and object, that to be susceptible of _magis_ and _minus_, belongs not to quantity, but to quality; but without any proof, as if you took it for an axiom. But whether true or false, you understand not in what sense it is true or false. It is true that one inequality is inequality, _as well_ as another; as one heat is heat _as well_ as another, but not _as great_. _Tam_, but not _tantus_. But so it is also in the predicament of quantity; one line is as well a line as another, but not so great. All degrees, intentions, and remissions of quality, are greater or less quantity of force, and measured by lines, superficies, or solid quantity, which are properly in the predicament of quantity. You see how wise a thing it is to argue from the predicaments of Aristotle, which you understand not; and yet you pretend to be less addicted to the authority of Aristotle now than heretofore.

In the next place you say, I may as well conclude from the not susception of _greater_ and _less_, that a right angle is not quantity, but an oblique one is. Very learnedly. As if to be _greater_ or _less_, could be attributed to a quantity once determined. Number (that is, number indefinitively taken) is susceptible of _greater_ and _less_, because one number may be greater than another; and this is a good argument to prove that number is quantity. And do you think the argument the worse for this, that one six cannot be greater than another six? After all these childish arguments which you have hitherto urged, can you persuade any man, or yourselves, that you are logicians?

To the fifth and sixth article you object, first, _that if I had before sufficiently defined_ (ratio) _proportion, I needed not again define what is_ (eadem ratio) _the same proportion_; and ask me _whether when I have defined_ man, _I use to define anew what is the_ same man? You think when you have the definition of _homo_, you have also the definition of _idem homo_, when it is harder to conceive what _idem_ signifies, than what _homo_. Besides, _idem_ hath not the same signification always, and with whatsoever it be joined; it doth not signify the same with _homo_, that it doth with _ratio_. For with _homo_ it signifies the same _individual man_, but with _ratio_ it signifies a like, or an equal proportion: and both (_ratio_) _proportion_ and (_idem_) _the same_, being defined, there will still be need of another definition for (_eadem ratio_) _the same proportion_; and this is enough to defend both myself and Euclid, against this objection: for Euclid also, after he had defined (_ratio_) _proportion_, and that sufficiently, as he believed, yet he defines _the same proportion_ again apart. I know you did not mean in this place to object anything against Euclid, but you saw not what you were doing. There is within you some special cause of intenebration, which you should do well to look to.

In the next place you say, when I had defined arithmetical proportions to be the same when the difference is the same; it was to be expected I should define geometrical proportions to be then the same, when the antecedents are of their consequents _totuple_ or _tantuple_, that is, equimultiple (for _tantuplum_ signifies nothing). In plain words, you expected, that as I defined one by subtraction, I should define the other by the quotient in division. But why should you expect a definition of the same proportion by the quotient? Neither reason nor the authority of Euclid could move you to expect it. Or why should you say _it was to be expected_? But it seems you have the vanity to place the measure of truth in your own learning. In lines incommensurable there may be the same proportion, when, nevertheless, there is no quotient; for setting their symbols one above another doth not make a quotient: for quotient there is none, but in _aliquot parts_. It is therefore impossible to define proportion universally, by comparing quotients. This incommensurability of magnitudes was it that confounded Euclid in the framing of his definition of proportion at the fifth Element. For when he came to numbers, he defined the _same proportion_ irreprehensibly thus: _numbers are then proportional, when the first of the second and the third of the fourth are equimultiple, or the same part, or the same parts_; and yet there is in this definition no mention at all of a quotient. For though it be true, that if in dividing two numbers you make the same quotient, the dividends and the divisors are proportional, yet that is not the definition of the same proportion, but a theorem demonstrable from it. But this definition Euclid could not accommodate to proportion in general, because of incommensurability.

To supply this want, I thought it necessary to seek out some way, whereby the proportion of two lines, commensurable or incommensurable, might be continued perpetually the same. And this I found might be done by the proportion of two lines described by some uniform motion, as by an efficient cause both of the said lines, and also of their proportions; which motions continuing, the proportions must needs be all the way the same. And therefore I defined those magnitudes to have the same geometrical proportion, _when some cause producing in equal times equal effects, did determine both the proportions_. This, you say, needs an Œdipus to make it understood. You are, I see, no Œdipus; but I do not see any difficulty, neither in the definition nor in the demonstration. That which you call perplexity in the explication, is your prejudice, arising from the symbols in your fancy. For men that pretend no less to natural philosophy than to geometry, to find fault with bringing motion and time into a definition, when there is no effect in nature which is not produced in time by motion, is a shame. But you swim upon other men’s bladders in the superficies of geometry, without being able to endure diving, which is no fault of mine; and therefore I shall, without your leave, be bold to say, I am the first that hath made the grounds of geometry firm and coherent. Whether I have added anything to the edifice or not, I leave to be judged by the readers. You see, you that profess with the pricking of bladders the letting out of their vapour, how much you are deceived. You make them swell more than ever.

For the corollaries that follow this sixth article, you say they contain nothing new. Which is not true. For the ninth is new, and the demonstrations of all the rest are new, being grounded upon a new definition of proportion; and the corollaries themselves, for want of a good definition of proportion, were never before exactly demonstrated. For the truth of the sixth definition of the fifth Element of Euclid cannot be known but by this definition of mine; because it requires a trial in all numbers possible, that is to say, an infinite time of trial, whether the quimultiples of the first and third, and of the second and fourth, in all multiplications, do together exceed, together come short, and are together equal; which trial is impossible.

In objecting against the thirteenth and sixteenth article, I observe that you bewray together, both the greatest ignorance and the greatest malice; and it is well, for they are suitable to one another, and fit for one and the same man. In the thirteenth article my proposition is this: _If there be three magnitudes that have proportion one to another, the proportions of the first to the second, and of the second to the third, taken together_ (as one proportion), _are equal to the proportion of the first to the third_. This demonstrated, there is taken away one of those moles which Sir Henry Savile complaineth of in the body of geometry. Let us see now what you say, both against the enunciation and against the demonstration.

Against the enunciation you object, _that other men would say_ (not the proportions of the first to the second, and of the second to the third, taken together, &c. but) _the proportion which is compounded of the proportion of the first to the second, and of the second to the third_, &c. Is not the compounding of any two things whatsoever the finding of the sum of them both, or the taking of them together as one total? This is that absurdity of which Mersennus, in the general preface to his _Cogitata Physico-Mathematica_, hath convinced Clavius, who, at the end of Euclid’s ninth Element, denieth the composition of proportion to be a composition of parts to make a total; which, therefore, he denied, because he did not observe, that the addition of a proportion of defect to a proportion of excess, was a subtraction of magnitude; and because he understood not that to say, composition is not the making a whole of parts, was contradiction; which all but too learned men would as soon as they heard abhor. Therefore, in saying that other men would not speak in that manner, you say in effect they would speak absurdly. You do well to mark what other geometricians say; but you would do better if you could by your own meditation upon the things themselves, examine the truth of what they say. But you have no mind, you say, to contend about the phrase. Let us see, therefore, what it is you contend about.

_The proportion_, you say, _which is compounded of double and triple proportion, is not_, as I would have it, _quintuple, but sextuple_, as in these numbers, six, three, one; where the proportion of six to three is double, the proportion of three to one triple, and the proportion of six to one sextuple, not quintuple. Tell me, egregious professors, how is six to three double proportion? Is six to three the double of a number, or the double of some proportion? All men know the number six is double to the number three, and the number three triple to an unity. But is the question here of compounding numbers, or of compounding proportions? Euclid, at the last proposition of his ninth Element, says indeed, that these numbers, one, two, four, eight, are ἐν διπλασίονι ἀναλογία, in double proportion; yet there is no man that understands it otherwise, than if he had said in proportion of the single quantity to the double quantity; and after the same rate, if he had said three, nine, twenty-seven, &c. had been in triple proportion, all men would have understood it, of the proportion of any quantity to its triple. Your instance, therefore, of six, three, one, is here impertinent, there being in them no doubling, no tripling, no sextupling of proportions, but of numbers. You may observe also, that Euclid never distinguished between double and duplicate, as you do. One word διπλάσιον serves him every where for either. Though, I confess, some curious grammarians take διπλάσιον for duplicate in number, and διπλοῦν for double in quantity; which will not serve your turn. Your geometry is not your own, but you case yourselves with Euclid’s; in which, as I have showed you, there be some few great holes; and you by misunderstanding him, as in this place, have made them greater. Though the beasts that think your railing roaring, have for a time admired you; yet now that through these holes of your case I have showed them your ears, they will be less affrighted. But to exemplify the composition of proportions, take these numbers, thirty-two, eight, one, and then you shall see that the proportion of thirty-two to one is the sum of the proportions of thirty-two to eight, and of eight to one. For the proportion of thirty-two to eight is double the proportion of thirty-two to sixteen; and the proportion of eight to one, is triple the proportion of thirty-two to sixteen; and the proportion of thirty-two to one is quintuple of thirty-two to sixteen; but double and triple added together maketh quintuple. What can be here denied?

My demonstration consisteth of three cases: the first is when both the proportions are of defect, which is then when the first quantity is the least; as in these three quantities, A B, A C, A D. The first case I demonstrated thus: (A B C D)/(a) Let it be supposed that the point A were moved uniformly through the whole line A D. The proportions, therefore, of A B to A C, and of A C to A D, are determined by the difference of the times in which they are described. And the proportion also of A B to A D, is that which is determined by the difference of the times in which they are described; but the difference of the times in which A B and A C are described, together with the difference of the times wherein A C and A D are described, is the same with the difference of the times wherein are described A B and A D. The same cause, therefore, which determines both the proportions of A B to A C, and of A C to A D, determines also the proportion of A B to A D. Wherefore, by the definition of _the same proportion_, article six, the proportion of A B to A C, together with the proportion of A C to A D, is the same with the proportion of A B to A D.

Consider now your argumentation against it. “_Let there be taken_,” say you, “_between A and B the point_ a; and then in your own words, I argue thus: _The difference of the times wherein are described A B and A C, together with the difference of the times wherein are described A C and A D, is the same with the difference of the times in which are described_ a _B and_ a _C (namely, B D, or B C + C D_); wherefore, the same cause which determines the two proportions of A B to A C, and of A C to A D, determines also the proportion of a _B to_ a _D_.” Let me ask you here whether you suppose the motion from _a_ to B, or from _a_ to D, to have the same swiftness with the motion from A to B, or from A to D? If you do not, then you deny the supposition. If you do, then B C, which is the difference of the times A B and A C, cannot be the difference of the times in which are described _a_ B and _a_ C, except A B and _a_ B are equal. Let any man judge now whether there be any paralogism in Orontius that can equal this. And whether all that follows in the rest of this, and the next two whole pages, be not all a kind of raving upon the ignorance of what is the meaning of proportion, which you also make more ill-favoured by writing it; not in language, but in _gambols_; I mean in the symbols, which have made you call those demonstrations short, which put into words so many as a true demonstration requires, would be longer than any of those of Clavius upon the twelfth Element of Euclid.

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.

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.