Six lessons to the professors of the mathematiques one of geometry the other of astronomy, in the chaires set up by the noble and learned Sir Henry Savile in the University of Oxford.

Of the Faults that Occurre in Demon∣stration. To the same egregious Professors of the Ma∣thematicks in the University of Oxford.

LESSON III.

YOu begin your reprehension of my thirteenth Chapter with a Question. Whereas I divide Proportion into Arithmeticall, and Geometricall; You ask me what proportion it is I so divide. Euclide divides an Angle into Right, Obtuse, and Acute. I may ask you as pertinently what Angle it is he so divides? Or when you divice Animal into Homo, and Br••tum, what Animal that is which you so divide? You see by this how absurd your Q••estion is. But you say the Definition of Proportion which I make at Chap. 11. Art. 3. namely, that Pro∣portion

is the comparison of two Magnitudes, one to another, agreeth not, neither with Arith∣meticall, nor with Geometricall Proportion. I believe you thought so then, but having read what I have said in the end of the last Lesson, if you think so still, your fault will be too great, to be pardoned easily. But why did you think so before? Is it not because there was no Definiti∣on in Euclide of Proportion universall, and because he maketh no mention of Proportion Arith∣meticall, and because you had not in your minds a sufficient notion thereof your selves to supply that Defect? And is not this the cause also, why you put in this Parenthesis (if Arithmeticall Proportion, ought to be called Proportion)? which is a confession that you know not whether there be such a thing as Arithmeticall Proportion or not; notwithstanding, that on all occasions, you speak of Arithmetically Proportionals. Yes, this was it that made you think that Propor∣tion universally, and Proportion Geometricall is the same and yet to say you cannot tell whether they be the same or not. 'Tis no wonder therefore, if in such confusion of the understanding, you apprehend not that the Proportions of two to five, and nine to twelve, are the same; for you are blin∣ded by seeing that they are not the same Proportions Geometricall. Nor doth it help you that I say the Difference is the Proportion, f•••• by Difference you might if you would, have understood the act of Differing.

At the second Article you note for a fault in Method, that after I had used the words Ant••∣cedent and Consequent of a Proportion in some of the precedent Chapters, I define them after∣wards.. I do not believe you say this against your knowledge, but that the eagerness of your malice made you oversee. Therefore go back again to the third A••ticle of Chap. 11. Where ha∣ving defined Correlatives, I add these words, Of which the first is called the Antecedent, the second the Consequent. This is but an oversight, though such as in me, you would not have excused.

At the thirteenth Article you find fault with, that I say that the Proportion of Inequality, whe∣ther it be of Excess, or of Defect, is Quantity, but the Proportion of Equality is not Quan∣ty. 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 his Chamber at Paris, at such time, as the first volume of his Cogitata Physico-Mathematica was almost prin∣ted: In which, because he had not said all he would say of Proportion, he was forced to put the rest into a Generall Preface; which as was his custom, he did read to his Friends, before he sent it to the Pr••ss. In that generall Preface under the Title de Rationibus at{que} Proportionibus, at the Numbers twelve, thirteen, fourteen, he maintaineth against Clavius, that the Composition of Proportions is (as of all other things) a Composition of the Parts to make a Totall; 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 then Nothing; because Equality (he saies) is a term of middle signification, between Ex••ess and Defect. And there also he refuteth the Arguments which Clavius at the end of the nineth Element of Euclide bringeth to the contrary. And though this were approved by divers good Geometricians then present, and never gain-sayed by any since, Yet do not I say it upon the credit of them, but upon sufficient grounds. For it hath been demonstrated by Euto••ius 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 there∣fore we put in order any three numbers, whereof the two last be equall, 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 fro•• this, that one Inequality may be greater then another.

But for the clearing of this Doctrine (which Mersennus cals Intricate) of the composition of Proportions, I observed, first, that any two Quantities, being exposed to sense, their Pro∣portion 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 onely 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 then the second, and the second greater then the third, or the first greater then the second, and the second less then the third, so that to make the first and second equall, if we use addition, we must to make the second and third equall use substraction, 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, no∣thing want Quantity, or as Symbo lists mark them 〈 math 〉〈 math 〉. 0. 0-1 And upon this ground I thought depended the universall truth of this Proposition, that in any rank of Mag∣nitudes of the same kind, the Proportion of the first to the last, was compounded of all the Pro∣portions (in order) of the intermediate Quantities; the want of the proof thereof Sir Henry S••vile ••als (Naevus) 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 Pro∣position, 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 Logick School, that but now begins to learn the Pre∣dicaments. But what do you mean by Aequale, and Inequale? Do you mean Corpus Aequale, and Corpus Inequale? They are both in the Predicament of Substance, neither of them in that of Quantity; Or do you mean Aequalitas, and Inaequalitas? They are both in the Predi∣cament of Relation, neither of them in that of Quantity, and yet both Corpus, and Inaequa∣litas, 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, then if they had said Corpus est tantum, and Inaequalitas tanta, not Tantitas; that is, Bodies and Inequalities are so much, not somuchness; and all intelligent men are contented with that expression, and your selves use it. And the Quantity of Inequality is in the Predicament of Quantity, because the measure of it is in that Line by which one Quantity exceeds the other. But when neither exceedeth other, then there is no Line of Ex∣cess, 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 Aristotles 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 Equali∣ty, not Quantity, my Argument was this; That because one Inequality may be greater or less then another, but one Equality cannot be greater nor less then another, Therefore Ine∣quality 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 Axiome. But whether true or false, you understand not in what sense it is true or false. 'Tis true, that one Inequality is In∣equality, 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, intensions, 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 addi∣cted to the authority of Aristotle, now, then 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 indefinitely taken) is susceptible of greater and less, because one number may be greater then another. And this is a good Argument to prove that Number is Quantity. And do you think the Argu∣ment the worse for this, that one six cannot be greater then another six? After all these childish Arguments which you have hitherto urged, can you perswade any man, or your selves, that you are Logicians?

To the fifth and sixth Article you object, first, that if I had before sufficiently Defined (Ra∣tio). 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 'tis harder to conceive what idem signifies, then what Homo. Besides, idem hath not the same signification alwayes, and with whatsoever it be joyned; it doth not signifie 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 equall 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 my self and Euclide, against this objection. For Euclide also after he had Defined (Ratio Proportion, and that su•…•…ciently as he believed, yet he defines the same Proportion again apart. I know you did not mean in this place to object any thing against Euclide; but you saw not what you were doing. There is within you some spec••all cause of Intenebration, which you •…•…ld do well to look to.

In the nex•• p••••ce you say, when I had defined A•…•…thmeticall Proportions to be the same when the difference is the same; it was to be expected I should define Geometricall 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 exp••cted, that as I defi∣ned one by substraction, 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 Euclide could move you to expect it. Or why should you say it was to be expected? But it seems you have the vanity to place the measure of truth in your own Learning. In Lines incommensurable there may be the same Proportion, when nevertheless there is no Quo∣tient; for setting their Symboles 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 Euclide 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 Proportio∣nal, 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 Proportionall, yet that is not the Definition of the same Proportion, but a Theoreme Demonstrable from it. But this Definition Euclide could not accommodate to Pro∣portion in Generall, 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 mo∣tion, as by an Efficient cause both of the said Lines, and also of their Proportions. Which mo∣tions continuing, the Proportions must needs be all the way the same. And therefore I defined those Magnitudes to have the same Geometricall Proportion, when some cause producing in equall times, Equall Effects, ••id determine both the Proportions. This you say needs an Oedipus to make it understood. You are (I see) no Oedipus; but I do not see any diff••••ulty, neither in the Definition, nor in the Demonstration. That which you call perplexity in the

Explication, is your prejudice, arising from the Symboles in your fancy. For men that pre∣tend no less to naturall Phylosophy, then 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 mens bladders in the Superficies of Geometry, without being able to indure 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 any thing 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 then ever.

For the Corollaries that follow this sixth Article, you say they contain nothing new. Which is not true. For the nineth is new, and the Demonstrations of all the rest are new, being groun∣ded upon a new Definition of Proportion, and the Corollaries themselves for want of a good De∣finition of Proportion, were never before exactly Demonstrated. For the truth of the sixth Definition of the fifth Elem•…•…t of Euclide cannot be known but by this Definition of mine, be∣cause it requires a Triall in all numbers possible, that is to say, an infinite time of Triall, 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 equall; which Triall 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 'tis well, for they are sutable 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 Proposition) are equall 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 wo••ld 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 ta∣king of them together as one totall. This is that absurdity of which Mersennus in the generall Preface to his Cogitata Physico-Mathematica hath convinced Clavius, who at the end of Eu∣clides nineth Element denyeth the composition of Proportion to be a Composition of Parts to make a Totall; Which therefore he denyed, because he did not observe, that the Addition of a Proportion of defect, to a Proportion of Excess, was a Substraction of Magnitude; and because he understood not that to say, Composition is not the making a whole of Parts, was contradicti∣on; which all, but too learned men would as soon as they heard ••bho••re. 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 minde (you say) to contend about the Phrase. Let us therefore see what it is you con∣tend about.

The Proportion (you say) which is compounded of double and triple Proportion, is not (as I would have it) Quintuple, but Sextuple, as in these numbers, six, t••ree, one; where the Pro∣portion of six to three is double, the Proportion of three to one tripl••, and the Proportion of six to one sextuple, not quintuple. Tell me (egregsous Professors) how is six to three double Pro∣portion? 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▪ Euclide at the last Proposition of his nineth Element sayes indeed, that these numbers, one, two, four, eight, are 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 in double Proportion, yet there is no m••n that under∣stands it otherwise, then if he had said in Proportion of the single Quantity, to the double Quan∣tity;

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, nor sextupling of Proportions, but of numbers. You may observe also that Eu∣clide never distinguisheth between double and duplicate, as you do. One word 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 serves him every where for either. Though I confess some curious Grammarians take 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 for duplicate in number, and 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 for double in Quantity; which will not serve your turn. Your Geometry is not your own, but you case your selves with Euclides; in which, as I have shewed 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, ••oaring, have for a time ad∣mired you; yet now that through these holes of your case, I have shewed them your ears, they will be less affrighted. But to exemplifie 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 six∣teen. 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 Quintiple of thirty-two to sixteen. But double and triple added together maketh quintuple. What can be here denyed?

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, AB, AC, AD. The first case I demonstrated thus. 〈 math 〉〈 math 〉 Let it be supposed that the point A were moved uniformly through the whole line AD. The Proportions therefore of AB to AC, and of AC to AD, are determined by the difference of the Times in which they are described. And the Proportion also of AB to AD, is that which is determined by the difference of the Times, in which they are described; but the difference of the Times in which AB and AC are described, together with the difference of the Times, wherein AC and AD are de∣scribed, is the same with the difference of the Times, wherein are described AB and AD. The same cause therefore which determines both the Proportions of AB to AC, and of AC to AD, determines also the Proportion of AB to AD. Wherefore by the Definition of the same Proportion, Article six, the Proportion of AB to AC, together with the Proportion of AC to AD is the same with the Proportion of AB to AD.

Consider now your argumentation against it. Let there be taken (say you) between A and B the Point a, and then in your own words, I argue thus. The difference of the Times wherein are described AB and AC, together with the difference of the Times, wherein are described AC and AD, is the same with the difference of the Times, in which are described aB and aC (namely BD, or BC † CD) wherefore the same cause which determines the two Proportions of AB to AC, and of AC to AD, determines also the Proportion of aB to aD. Let me ask you here whether you suppose the Motion from a to B, or from a to D, to have the same switfness with the motion from A to B, or from A to D. If you do not, then you de∣ny the supposition. If you do, then BC which is the difference of the Times AB and AC, cannot be the difference of the Times in which are described aB and aC, except AB and aB are equall. Let any man judge now whether there be any Paralogism in Orontius that can equall 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 Gamboles, I mean in the Sym∣boles, which have made you call those demonstrations short, which put into words so many as a true demonstration requires, would be longer then any of those of Clavius upon the twelfth Element of Euclide.

To the sixteenth Article you bring no Argument, but fall into a loud Oncethmus (the special Figure wherewith you grace you Oratory) offended with my unexpected crossing of the Do∣ctrine

you teach, that Proportion consisteth it a Quotient. For that being denyed you, your 〈 math 〉〈 math 〉 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 speak∣ing? When did you see any man but your selves publish his Demonstrations by signs not ge∣nerally received, except it were not with intention to demonstrate, but to t••••ch the use of Signes? Had Pappus no Analytiques? Or wanted he the wit to •…•…ten his reckoning by Signes? Or has he not proceeded Analytically in an hundred Problems (especially in his seventh Book) and never used Symboles? Symboles are poor unhandsome (though necessary) sc••ffolds of Demonstration; and ought no more to appear in publique, then the most d••••••rmed neces∣sary business which you do in your Chambers. But why (say you) is this •…•…tion to the Proportion of the greater to the less? Ile tell you; because i••erating 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 then that of one to four, as is Demonstrated by Euclide Elem. 5. Prop. 8. and by con∣sequent the Proportion of one to four, is a Proportion of greater littleness then 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 Corollarie of Art. 28. (which you make absurd enough by re∣hearsing it thus) si quantitas aliqua divisa supponatur in partes aliquot aequales numero infi∣nitas. &c. Do you think that of partes aliquot, or of partes aliquotae, it can be said without absurdity, that they are numero infinitae? And then you say I seem to mean, that if of the Quan∣tity AB, there be supposed a part CB, infinitely little, and that between AC and AB be taken two means, one Arithmeticall AE, the other Geometricall AD, the difference between AD, and AE, will be infinitely little. My meaningis, and is sufficiently expressed, that the said means taken every where (not in one place onely) 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 BE the Arithme∣ticall difference, is greater then BD the Geometricall difference. You bring nothing to prove it, and if you suppose it, you suppose the thing you are to prove. Hitherto you have procee∣ded in such manner with your Elenchus, as that so many objections as you have made, so ma∣ny 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 plain in this manner. A plain Superficies is that which is described by a straight Line so moved, as that every Point thereof describe a severall straight L•…•…. In which you require, first, that instead of describe, I should have said can describe. Why do you not require of Euclide in the Definition of a Cone, instead of (Continetur) is contained, he say (contineri potest) can be contained? It I tell you how one Plain is generated, cannot you apply the same generation to any other Plain? But you object that the Plain 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 Plain already generated. For the motion of a straight Line, whose one Point resting, describeth with the other Points severall perimeters of Circles, may as well describe a Conique Superficies, as a Plain. The Question therefore is, how you will in your Definition take in the Plain which must be generated before you begin to describe your Circle, and before you know what Point to make your Center. This objection therefore is to no purpose; and besides, that

it reflecteth upon the perfect definitions of Euclide before the eleventh Element; it cannot make good his Definition (which is nothing worth) of a Plain Superficies, before his first Ele∣ment.

In the next place you reprehend briefly this Corollarie, that two Plaines cannot inclose a Solid. I should indeed have added, with the base on whose extreams 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, 'tis impossible to include a Solid, with any Number of Plains whatsoever; un∣less you say that Solid is included which nothing at all includes.

At the third Article, where I say of crooked lines, some are every where 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 rectilineall 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 Center 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. 8. 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 cavill from this, that a Circle may touch a Parabola in two Points. Tell me truely, 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 an∣other, the contact is not in a Line, but only in one Point; and then your instance of a Circle and a Parabola, was a wilfull cavill, not befitting a Doctor. If you either read them not, or un∣stood them not, it is your own fault. In the rest that followeth upon this Article, with your Diagram, there is nothing against me, nor any thing of use, novelty, subtilty or learning.

At the seventh Article, where I define both an Angle, simply so called, and an Angle of Con∣tingence, by their severall 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 mo∣tion upon one common Point resting, &c. You ask me to which of these kinds of Angle, I ref••r 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 then Rectilineall? O, how can that Angle which is not comprehended by two straight Lines, be other then Curvilineall? 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 Curvilineall. Which is as well argued, as if a man should say, the House was built by the carriage and motion of Stone and Timber, there∣fore when the carriage and that motion is ended, it is no more a house. Rectilineall and Curvi∣lineall hath nothing to do with the nature of an Angle simply so called, though it be essentiall to an Angle of Contact. The measure of an Angle simply so called is a circumference of a Cir∣cle, and the measure is alwayes the same kind of Quantity with the thing measured. The Re∣ctitude or Curvity of the Lines which drawn from the Center intercept the Arch, is accidenta∣ry to the Angle, which is the same, whether it be drawn by the motion circular of a streight 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 Rectilineall, but computed in the Circumference, not Rectilineall, but mixt; or, if two Circles cut one another, Curvi∣lineall. 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 Rectilineall, and Curvilineall, I respect no more the ge∣neration of it, then 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 nineth Article, he will need no more to make him see that you are utterly ignorant of the na∣ture of an Angle, and that if ignorance be madness, not I, but you are mad; and when an An∣gle 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 conso∣nant to my definition of a Point by a Magnitude not considered. But when you in your trea∣tise de Angulo Contactûs Chap. 3. Pag. 6. Lin. 8. have these words, Though the whole concur∣rent Lines incline to one another, yet they form no Angle any where but in the very point of concourse, You, that deny a Point to be any thing, tell me how two nothings can form an Angle; or if the Angle be not formed neither before the concurrent Lines meet, not 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 continu∣ed 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 14. Chapter.

At the twelfth Article I confess your exception to my universall definition of Parallels to be just, though insolently set down. For it is no fault of ignorance (though it also infect the de∣monstration 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 equall Angles with them both, are equall; 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 equall Angles, are equall, it is both true and universall; and the following Conse∣ctary 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. 14. Art. 12. the Definition of Paral∣lels wanteth somewhat to be supplyed. And of the Consectary, he says, it concludeth not, because it is grounded on the Definition of Parallels. Truely, 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 Proportionall to their Semidiameters; and with Esto, fortasse, recte, omni∣no, noddying to the severall parts thereof, you come at Length to my last inference; There∣fore by (Chap. 13. Art. 6.) the Perimeters and Semidiameters of Circles are Proportionall; which you deny; and therefore deny, because you say it followeth by the same Ratiocination, that Circles also and Spheres are Proportionall to their Semidiameters. For the same distance (you say) of the Perimeter from the Center which determines the magnitude of the Semidia∣meter, determines also the magnitude both of the Circle, and of the Sphere. You acknow∣ledge that Perimeters and Semidiameters have the cause of their determination such as in equal times make equall 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 ac∣cording to the supposition (temporibus aequalibus aequalia facere) to make equall spaces in equall

times? Or when by the turning about of the Semidiameter is described the Plain of a Circle does it (think you) in equall times make the Plains of the interior Circles equall to the plains of the exterior? Or is the Radius that describes the inner Circles equall to the Radius that de∣scribes the exterior? It does not therefore follow from any thing I have said in this demonstra∣tion, that either Spheres, or Plains of Circles, are Proportionall to their Radii. And conse∣quently all that you have said, triumphing in your own Incapacity, is said imprudently by your selves 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 Geo∣metry. 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 meer ignorance and gibberish.

To the fourteenth Article you object, that (in the sixth figure) I assume gratis, that FG, DE, BC, are Proportionall to AF, AD, AB; and you referre it to be judged by the Reader. And to the Reader I referre it also. The not exact drawing of the Figure (which is now amended) is it that deceived you. For AF, FD, DB, are equall by construction. Also AG, GE, EC, are equall by construction. And FG, DK, BH, KE, HI, IC, are equall by Parallelism. And because AF, FG, are as the velocities wherewith they are described; also 2 AF (that is AD) and 2 FG (that is DE) are as the same veloci∣ties. And finally 3 AF (that is AB) and 3 FG (that is BC) are as the same veloci∣ties. It is not therefore assumed gratis, that FG, DE, BC, are Proportionall to AF, AD, AB, 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 DE, first at adventure, and then making AD, two thirds of AB; 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 Cavill.

That which you object to the third Corollarie of Art. 15. was certainly a dream. There is no a•…•…ing of an Angle CDE, for an Angle HDE, or BDE, neither in the Demonstra∣tion, nor in any of the Corollaries. It may be you dream't of somewhat in the twentieth Ar∣ticle of Chap. 16. 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 usefull, it was true.

Article the sixteenth. Here we come to the Controversie concerning the Angle of Contact, which (you say) you have handled, in a speciall Treatise published; and that you have clearly demonstrated in your publick 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 Politicks. If I think that you, or Peletarius, or Clavius, or Euclide have e••red, 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 then a Right-lined Right An∣gle. And that Euclide did not well to leave it so obscure what he meant by Inclination in the Definition of a Plain Angle, seeing else where he attributeth Inclination onely to Acute An∣gles, 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 Euclide, are nevertheless more obscure then he; and also are con∣trary to him. For Euclide 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 lye one on another, to lye 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 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 (Pag. 10. Lin. 22.) and destroyshis opinion. For according to Euclide the greatest Angle is the greatest Inclination, and an Angle equall to two Right An∣gles by this 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 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 believ••) was Euclides meaning; then will the Angle of Contact be no less an Angle then a rectilineall Angle, but onely (as Clavius truly saies it is) Heterogeneous to it; and the doctrine of Clavius more conformable to Euclide then that of Peletarius. Besides, if it be granted you, that there is no inclination of the Circumfe∣rence to the Tangent, yet it does not follow that their concourse doth not form some kind of Angle. For Euclide defineth there but one of the kinds of a Plain Angle. And then you may as much in vain seek for the Proportion of such an Angle to the Angle of Contact, as seek for the Focus, or Parameter of the Parabola of Dives and Lazarus. Your first argument there∣fore is nothing worth, except you make good that which in your second Argument you affirm, namely, That all Plain Angles, not excepting the Angle of Contact, are (Homogeneous) of the same kind. You prove it well enough of other Curvilineall Angles; but when you should prove the same of an Angle of Contact, you have nothing to say but Pag. 17. Lin. 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 be at all shew, 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, name∣ly, that it is the Quantity of that crookedness or flexion, by which a straight Line is bent into an Arch of a Circle equall to it; and that because the crookedness of one Arch may be grea∣ter then the crookedness of another Arch of another Circle equall to it, therefore the Questi∣on Quanta est curvitas, How much is the crookedness, is pertinent, and to be answered by Quantity. And I have also shewn 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 Rectilineall Angle.

But let us see how you answer to that which Clavius has objected already. They are Hetero∣geneous, sayes he, because the Angle of Contact, how oft soever multiplyed, can never ex∣ceed a Rectilineall Angle. To answer which you alleadge, 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 In∣clination before they come to touch, more then a straight Line and an Arch of a Circle? And in the Point of Contact it self, how can it be that there is less Inclination of the two Points of a straight Line and an Arch of a Circle, then 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 AGB and CFB to touch in B, and have a common Tangent through B. Is not the Line CFBGA a crooked Line? And is it not cut by the common Tan∣gent DBE? What is the Quantity of the two Angles FBE and GBD, seeing you say neither DBG nor EBF is an Angle? 'Tis 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 Rectilineall 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 substracted one from ano∣ther, are of the same kind; But an Angle of Contact may be substracted 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 substract, is the Angle of Contact; and that the Plain of the Semi∣circle is the Angle of the Semicircle, which is absurd; though as absurd as it is, you say it di∣rectly in your Elenchus, Pag. 35. Lin. 14. in these words, When Euclide defines a Plain An∣gle to be the Inclination of two Lines, he meaneth not their aggregate, but that which lyes between them. It is true, he meaneth not the aggregate of the two Lines; but that he means that which lyes between them, which is nothing else but an indeterminate Superficies, is false, or Euclide 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 Rectilineall Angle, out of Eucl. 3. 16. Where he saies, it is less then any acute An∣gle. And it follows well, that if it be an Angle, and less then any Rectilineall Angle; it is also of the same kind with it. But to my understanding Euclide meant no more, but that it was nei∣ther greater nor equall; which is as truly said of Heterogeneous, as of Homogeneous Quan∣tities. 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 venter to determine whether two Quantities be Homogeneous or not, without some Definition of Ho∣mogeneous (which is a hard word) that men may understand what it meaneth.

In your eighth Chapter you have nothing but Sir H. Saviles Authority, who had not then resolved what to hold; but esteeming the Angle of contact, first, as others falsely did, by the Superficies that lyes between the Tangent and the Arch, makes the Angle of Contact, and a Rectilineall Angle Homogeneous; and afterwards, because no multiplication of the Angle of Contact can make it equall to the least Rectilineall Angle, with great ingenuity returneth to his former uncertainty.

In your nineth and tenth Chapters you prove with much ado, that the Angles of like Seg∣ments are equall; as if that might not have been taken gratis by Peletarius without Demon∣stration. And yet your Argument contained in the nineth Chapter is not a Demonstration, but a conjecturall 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, ta∣ken from the Parallelism of two concentrique Circles, though the objection be of no moment, yet you have in the same Treatise of yours that which is much more foolish, which is this, Pag. 38. Lin. 12. Non enim magnitu o Anguli, &c. The magnitude of an Angle is not to be estima∣ted by that stradling of the legs, which it hath without the Point of concourse, but by that stradling which it hath in the Point of the concourse it self. I pray you tell me what strad∣ling 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 Plain 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 igno∣••ance, your way of Demonstration by putting N for a great Number of sides of an aequilate∣rall Polygon, is not to be admitted. For though you understand something by it, you de∣monstrate nothing to any Body, but those who understand your Symbolique tongue, which is a very narrow Language. If you had demonstrated it in Irish, or Welsh, though I had not read it, ye•• I should not have blamed you, because you had written to a considerable Number of mankind, which now you do not.

In your l••st Chapters you defend Vitellio without need; for there is no doubt but that whatso∣ever crooked Line be touched by a straight Line, the Angle of Contingence will neither add

any thing to, nor take any thing from a Rectilineall 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 then another; and therefore Heterogeneall, because the measure of an Angle of Con∣tact cannot (congruere) be applyed to the measure of a Rectilineall 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 plain Superficies. And thus you see in how few Lines, and without B••achygraphie, 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 Clavivs 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 Euclide makes a Plain Angle to be that which is contained between the two Lines that form it. 'Tis true, that I do not un∣derstand that Euclide 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 Euclide.

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 faire enough) Touch; or lastly, they are Asymptotes. In which you are first offended with the word It seems. But I allow you that never erre, to be more perempto∣ry then 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 severall kinds of Position, would save me from be∣ing censured for untruth. And yet your instance of two straight Lines in divers Plains, does not make my enumeration insufficient. For those Lines though not Parallels, nor cutting both the Plaines, yet being moved Parallelly from one Plain 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 Plain not parallel, make the same Angle, though not produced till they meet, which they would make if they were so produced. For you have no where proved, nor can prove, that two such Lines do not make an Angle. It is not the actuall concurrence of the Lines, but the Arch of a Circle, drawn upon that point for Center, in which they would meet, if they were produced, and intercepted between them, that constitutes the Angle.

Also your objection con••ernin•• Asymptotes in generall, is absurd. You would have me add, that their distance shall at last be 〈◊〉〈◊〉 then any distance that can be assigned; and so make the de∣finition of the Genus the same with that of the Species. But because you are not Professors of Logick, it is not necessary for me to follow your councell. In like manner, if we under∣stand one Line to be moved towards another alwayes parallely to its self, 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 no where before made such use of the word Figure, as to argue any thing from it; and therefore your objection is just as wise as if you had round 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 hol∣low within, is the same Figure; but you say not why, nor can you ••••••iver 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 any thing without it, because that whole con••ave 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 Paralle••s, 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 supplyed by any student of Geometry, that is not otherwise a fool.

At the fifteenth Chapter Art. 1. Numb. 6. you object as a contradiction, that I make Mo∣tion to be the measure of Time, and yet in other places do usually measure Motion and the affe∣ctions 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, Dyals, Hour-glasses, and the like) have conceived sufficiently, that there cannot be of Time any other measure be∣sides Motion; and that the most universall measure of Motion, is a Line described by some other Motion. Which Line being once exposed to sense, and the motion whereby it was de∣scribed sufficiently explicated, will serve to measure all other Motions and their Time; for Time and Motion (Time being but the mentall 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 design∣ed 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 then its two halfs. How does that follow? Is it impossible when a Line is divided into two halfs that the middle Point should be divided into two halfs 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 alleadge 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 Philoso∣phers or Geometricians, or Logicians, more then are the simplest of rurall people? Or are you not rather less, by as much as he that standeth still in ignorance, is nearer to knowledge, then 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 equall 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?

Notes

• [illustration]