Due Correction FOR MR HOBBES. OR Schoole Discipline, for not saying his Lessons right. In Answer To His Six Lessons, directed to the Professors of Mathematicks. By the Professor of GEOMETRY.

Hobs Leviathan part. 1. chap. 5. pag. 21.
Who is so stupid, as both to mistake in Geometry, and allso to persist in it, when another detects his error to him?

OXFORD, Printed by Leonard Lichfield Printer to the Uni­versity for Tho: Robinson. 1656.

TO THE Right Honourable HENRY Lord Marquesse of Dorchester, Earle of Kingston, Vicount Newark, Lord Pierrepoint, and Manvers, &c.


YOUR Honour may perhaps think it strange that a person so wholly a stranger as I, should tender you such a peece as this: Yet will, I doubt not, acquit me of rudenesse and in­civility in so doing; when you consider, That the adverse party, whom it takes to taske, hath made his appeale hither; and finding himselfe foiled in Latine, hath [Page] here put in his English Bill for some reliefe: And it is but reason that Bill and Answer be filed in the same Court. He had the confidence, to tender his book first to another honorable Person the Earle of Devonshire, with this presump­tion, That though things were not so fully demon­strated as to satisfie every Reader, yet 'twas good enough to satisfie his Lordship, he did not doubt. Which presumption of his was then the more tolerable, because he then thought his demonst [...]a [...]io [...]s good. But when he had been so fully convinced what weake stuffe it was; that now the utmost of his hopes is (for so I understand from his friends) that though he be mistaken in the Mathematicks, yet he hopes to prove himselfe an honest man, (which yet is more I suppose than, by his principles, he need to be:) To make the world believe, that your Lordship doth approve of his Principles, Method, and Manners in those writing; and, that this is the only cause of the favours you have expressed towards him; is so high an affront, as had he not a great confi­dence of your Lorships Magnanimity, to despise it, or Clemency, to pardon it, he would not have offered to a person of so much honour and worth.

Since therefore he hath brought it before you as a controversy, wherein he desires your Lordship to consider and judge, whether he have said his six Lessons aright: I shall not at all demurre to the jurisdi­ction of the court; but as readily admit his Umpar, as allow him the choise of his own Wea­pon; [Page] and so tender your Lordship an English Answer to his English Appeale from my Latine Confutation of his treatise in Latine: That when in the judgement of this own Umpar, he sees himselfe foiled at his own weapons; he may hereafter make choise of French or Dutch, or some other Language, which he may hope to be more favourable to him, than Latine or English hath yet been.

He tells your Lordship, what great feates he hath done in his book; and your Lordship knows as well, by this and my former answer, how they have been defeated.

And then he reckons up certaine positions (some of them absurd enough) and would have you believe them to be our Principles at Oxford: But doth not tell your Lordship where they are to be found in any writings of ours.

Now, (that your Lordship may not seek them there in vaine, where they are not to be found,) I shall briefely shew where the rise of all these ac­cusations lye; in his own writings, not in ours.

First, He had taught us Cap. 13. §. 16. Si ratio detur minoris ad majus, rationesque aliquot addantur ipso aequales non multiplicari proprie, sed submultiplicari dici­tur: ita (que) quando additur primae rationi altera, ratio primae quantitatis ad tertiam, [...]emissis est rationis primae ad secundam. That is, in plaine English, If there be any proportion assigned of a lesse quantity to a greater, and to that proportion be added another proportion equall to it; that proportion that doth result by this addition, [Page] is not the double, but the halfe of that assigned proportion. Now, because this is very absurd, and I had told him so; He would have your Lordship believe, that it was I had said (not he,) that Two equall proporti­ons, are not double to one of the same proportions. Which is his first Charge.

Secondly; He had sayd farther, in the same place, Cap. 13. §. 16. Ratio 2 ad 1 vocatur dupla, & 3 ad 1 tripla, &c. (and he saith true.) But then (forgetting that these were his own words) he would have it thought (Less. 5. p. 42.) absurd to say that the proportion of two to one is double; and asks, is not every double proportion, the double of some proporti­on? And doth here intepret that phrase (of his own) the proportion of two to one is called double, to be all one as to say, That a proportion is double, triple &c. of a number, but not of a proportion. Which is his second charge.

Thirdly he had Cap. 8. §. 13, 14. (without any necessity) layd [...]he whole stresse of Geometry, upon this supposition. That, It is not possible for the same body to possesse at one time a greater, at another time a lesser place. (For, if this be possible, the same body is, by his definition, at the same time equall to a bigger, and to a lesse body than it selfe: as I there shewed by a consequence so cleare that he cannot himselfe deny it.) Which he there first, attempts to prove, (as simply as a man would wish,) but then presently flyes off againe, and say [...] that a thing in it selfe so manifest needs no demon­stration. But sayd I, (without declaring my own [Page] opinion in the case, which what it is he knowe [...] not) An assertion of such huge consequence to his doctrine as this is, and being (as he well knows) generally denyed (whatever he or I think of it) by all those who maintaine Conden­sation & Rarification in a proper sense, (without either vacuum, or the admission and extrusion of a forraigne body;) ought to be well proved, by him that builds so much upon it, and not be assu­med gratis. Now because of this it is, that he tells you in his third charge, That 'tis one of our principles, That the same body without adding to it, or taking from it, is sometimes greater and sometimes lesse. So hainous a matter is it, to require a proofe from him, of what he doth affirme though of never so great consequence.

Fourthly, He tells us Cap. 14. §. 19. (and 'tis true enough) that an Hyperbolick line, and its Asymp­tote, doe still come nearer and nearer till they approach to a distance lesse then any assignable quantity: And conse­quently if infinitely produced, must be supposed to meet, or to have no distance at all; (and so the distance of that hyperbola so produced, from a line parallel to the Asymptote, to be the same with the distance of that Asymptote from the said parallell; that i [...], equall to a given quantity.) And that this is a good inference, we are taught Less. 5. §. 43. as standing on the same ground with the demonstrations of all such Geome­tricians, Ancient and Moderne, as have inferred any thing in the manner following, [viz. If it be not greater [Page] nor lesse, then it is equall. But it is neither greater nor lesse. Therefore &c. If it be greater, say by how much. By so much. 'Tis not greater by so much: Therefore it is not greater. If it be lesse, say by now much &c.] which, being good demonstrations are together with this overthrown, if this inference be not good; that is, if things which differ lesse than any assignable quantity may not be reputed equall, But now, to say thus, That the distance of an Hyperbole, from a streight line drawn beyond its asymptote and the parallell thereunto, doth continually decrease, so as, if it be supposed infinitely produced, it must be sup­posed to be at length the same with that of the Asymptote from the sayd parallell, because nei­ther greater nor lesse by any assignable quantity; (which is but the result of his own assertion) is all one as to say, That a quantity may grow lesse and lesse eternally, so as at last to be equall to another quantity; or which is all one, saith he, that there is a last in Eternity▪ which is his fourth charge: and, what absurdity is in it, falls upon himselfe. Just as, when having told us Cap. 16. §. 20. Punctum inter quantitates nihil est, ut inter numeros Cyphra: And Cap. 14. §. 16. Punctum ad lineam neque rationem habet, neque quanti­tatem ullam: He railes upon me, twenty times over, as if I had somewhere said A point is nothing; only because I say with Euclide, [...].

Fiftly 'tis his usuall language, in designing an angle, to say, it is contained or comprehended by or between the two sides: As for example Cap. 14. [Page] §. 9. (three times in two lines) idem angulus est qui comprehenditur inter AB & AC, cum eo qui comprehendi­tur inter AE & AF, vel inter AB & AF, And §. 15. cor. 1. angul [...]s comprehensos a duabus rectis. And §. 16. angulus qui continetur inter AB & eandem AB &c. (And 'tis well enough so to speake.) But now, forgeting that it was himselfe that sayd so, he delivers it as a principle of ours, That the nature of an angle consists in that which lies between the lines which comprehend it; that is, saith he, An angle is a superficies.

Sixtly; when he had said (absurdly enough,) Cap. 11. § 5. Consistit ratio antecedentis ad consequens, in Differentia, hoc est, in ea Parte majoris qua minus ab eo superatur; sive in majoris (dempto minore) Residuo, &c And again, Ratio binarii ad quinarium est ter­narius, &c. And Cap. 12. § 8. Ratio inaequalium (li­nearum) EF, IG, consistit in differentia GF, (and the like elsewhere.) which is all one as to say, that the nature of proportion consists in a num­ber, a line, an absolute quantity (which how absurd it was I had let him know;) He hath then the impudence to say (as though it had been I, not hee, had so spoken) that, I make Proportion to be a Quotient, a number, an Absolute quantity, &c. or, as he here speaks in his sixth charge, that the Quotient is the proportion of the Division to the Dividend, (as pure non-sense as a body need to read;) Only because I affirm Rationis (Geometricae) aestimatio­nem esse, not penes residuum, but penes quotum: that Geometricall proportion is to be estimated, not according to [Page] the Remainder, but according to the Quotient (which himselfe now knows, though he did not then, to be true enough; for he hath now learned to say so too, Less. 2. p. 16. As the Quotient gives us a measure of the Proportion of the Dividend to the Divisor in Geometricall Proportion; so also the Remainder after Substraction is the measure of Proportion Arithmeticall.) And by these means he goes about to prove him­selfe an honest man: Just like the honest man, who when he had cut a purse, put it slyly into another mans pocket (after he had taken out the mony) that so this other might be hanged for it. And I hope, by that time Your Lordship hath perused the peece which I now tender, you will be able to judge, whether M. Hobs be not as well a good Mathematician, as an Honest Man; much a­like.

Your Lordship hath now the case fair before you; if you shall think it worth the while to take cognizance of it. I shall leave it here, and permit it to your Lordships judgement, whether to per­use and consider it, (which by reason of your good accomplishment in these, as well as in other parts of Learning, you are well able to doe,) or to lay it by for those that will: as being un­willing, by any importune solicitation, to tres­passe upon your Lordships leasure, or divert your thoughts, from matters of more concernment, to consider of such toys as these. Desiring mean while your Lordships favour so far, as to give mee [Page] leave to honour you, and (though I have not hi­therto had the honour to be known to you) to subscribe my selfe,

Your Honours Most Humble Servant, John Wallis.

[Page] [Page 1] DVE CORRECTION for Mr HOBS.

SECT. I. Concerning his Rhetorick and good Languge.

IT seems, M. Hobs, (by the fag end of your Book of Body in English) that you have a mind to say your lesson; and that the Mathematick Professors of Ox­ford should heare you. Truth is, 'tis scarce worth the while ei­ther for you or us. Yet we could be contented, for once, to hear you; (if we thought you would say any thing that were worth hearing) But to make a constant practice of it, or to entertain you as one of our Schollars, I have n [...] mind at all. Because, I fear, you are to old to learne, (though you have as much need as those that be yonger;) and yet will think much to be whipt, when you doe not sa [...] your Lesson right.

But, before we go further, I should ask you; what moved you to say your Lessons in English, when as the Books, a­gainst which you doe chiefely intend them, were written in Latine? But I foresee a faire answer that you might possibly make; (and therefore doe nor much wonder at it.) There [Page 2] be many grave and weighty reasons that might move you thereunto.

As first, because you doe presume, that there may be found divers persons, who may understand rayling in English, that yet doe not understand Mathematicks in Latin: and those being the persons on whom you have greatest hope of do­ing good, you ought to have a speciall regard to them, and apply your selfe to their capacities.

Secondly, because in case you should have attempted an Answer in Latin; you had lost your labour as to the whole design: For then those who should read your answer, would be able also to read that against which you write: and, comparing both together, would presently see to how little purpose all is that you have said. Whereas now your English Readers must be faine to take upon trust what you please to tell them. (Whereby you gain clearly, as to them, the opportunity of misrepresenting at pleasure what you see good.) And for this Reason, if you shall think fit to make any reply to this; I would advise you to doe it in Latin; that so Forrainers, who understand not English, may take upon trust what you shall please to tell them.

But thirdly, and principally (which is the reason of greatest weight) because that when ever you have thought it convenient to repaire to Billingsgate, to leane the art of Well-speaking, for the perfecting of your naturall Rheto­rick; you have not found that any of the Oister-women could teach you to raile in Latin, and therefore it was re­quisite to apply your selfe to such lauguage as they could teach you.

But prithee tell me, in good earnest, (for I cannot think you so simple as you would seem to be,) Whether you doe indeed believe (though you thought good to set a good face upon it, and talk big,) that all that you have said is worth a straw, either as to the defending of your Repu­tation, or the impairing of ours?

As to the Rhetorick and good language of it, (with which I shall first begin) that you can upon all occasions, or without occasion, give the titles of Foole, Beast, Asse, D [...]gge, &c. (which I take to be but barking,) with the rest of your course complements: You may take them, per­haps, to be admirable in their kind; yet are they no better then a man might have at Billingsgate for a box o'th ear. [Page 3] And of no better alloy are those other garnishes; That we un­derstand not what is Quantity, Line, Superficies, Angle, and Pro­portion: (and truly that's a sad case:) That neither of us un­derstand any thing either in Philosophy or in Geometry; (A lack a day!) That you do verily believe (it's pitty you can't per­swade some body else to be of your fai [...]h,) that since the be­ginning of the World there hath not been (and who doubts but you are a good Historian,) nor ever shall be, (and you hope your Prognosticks may be believed, for you would have us think you have been taken for a Conjurer,) so much absurdi­ty written in Geometry, as is to be found in these books of mine, (you should alwaies except your own Learned Works, which doubtlesse are, in this kind, incomparable pieces. But the truth is, you are not altogether out here; for in my Elenchus, which is one of the Books you mention, you may see that there hath been mach absurdity written in Geometry, and, they that read it, may know by whom.) But you have confuted them wholly and clearly (it seems you make cleare work where you come) in two or three leaves, (a quick rid­dance!) That, the negligences of your own you need not be asha­med of, (because you are ashamed of nothing;) That you ve­rily believe there was never seen worse reasoning, then in that Philos [...]phicall Essay, (and that's all the confutation of it:) nor worse Principles then these in our Books of Geometry; (and that's another Article of your Faith) That, by the use of Symbols, and the way of Analysis by squares and cubes, &c. you never saw any thing added to the Science of Geometry; (by which a man may see what a good Geometer you are like to prove;) That the Scab of Symbols, or Gambols, (your tongue is your own, you may call them what you please,) or the Symbolick tongue is harder to understand then Welch or Irish, (no marvaile then, you never saw any thing there­by added to Geometry.) That, to confute your Learned labours, is but to take wing like Beetles, from your egestions; (it seems it was but a shitten piece we had to deale with.) That, what you like not, is worthy to be gilded, but you doe not meane with gold; That Symbols are pior unhandsome scaf, folds of Demonstration; and ought no more to appeare in publike, then the most deformed necessary businesse which you doe in your Chamber; (one would think, by such stuffe as this, together with the ribauldry in your obscene Poem De Mirabilibus Pecci, that you had not learned all your Rhetorick at Bil­lingsgate, [Page 4] but had gone to Turn-ball-street for part of it.) That, your faults are not attended with shame, (It's no com­mendations, to be past shame;) That, you shall without our leaves be bold to say, (who ever doubted but that you be bold enough?) that your selfe are the first that hath made the grounds of Geometry firm and coherent, (as if Geometry were no lesse beholden to you, then Civil Philosophy; which, you say, is not ancienter then your Book de Cive.) That you have rea­son to blash (not for any of your own faults doubtlesse, but) considering the opinion men will have beyond Sea, of the Geo­metry taugh' in Oxford, (no doubt but the University of Ox­ford, if men knew all; are much beholden to you for your tender care of them;) yet withall, that the third definition of the fift of Euclide, is as bad as any thing was ever said in Geo­metry by D. Wallis, (And, if so, then doubtlesse D. Wallis need not be much dismaid; for Euclide hath not been ac­counted hitherto a despicable Author.) But such bumbast as this, and a great deale more of the same kind, I suppose, you doe not take to be Mathematicall demonstrations; nor to prove any thing, but the Forehead and Fury of him that speaks it.

But because the stresse of all this lies only upon what you verily believe, and what you never saw, and what you feare men will think of us beyond Sea: &c. To ease you of this fcar, I think it will not be amisse to let you heare the opinion of others both concerning your selfe and us, and the busi­nesse of Symbols (with which I see no reason why you should be so angry, save that you do not understand them.) that you may see▪ whether others haze the same belief with you. I need not tell you what Morinus and Tacquet think of the businesse. For those you have heard already. I shall only give you an extract of two or three Letters, which I have received from Persons whose face [...] I never saw; nor were they otherwise engaged to deliver an opinion in the case, then that they met with my books abroad: And yet no Clergy men, He assure you.

The first is from a Noble Gentleman of good worth, who hath deserved better of the Mathematicks then ever M. Hobs is like to doe; and whom, I heare, you use to commend. His words are these.

Eodem ibi tempore [Paristis] a Viro Nobili pagella vestra de Circuli Quadratura, Londino mittebatur; simul (que) Hobbii Philoso­losophia [Page 5] Nova. Quam ubi primum examinare concessum est, con­tinuo Paralogismum eum animadverti, quo Parabolicae lineae re­ctam aequare contendit, calculoque refutavi. Deinde alia quoque notavi, quae nihilo saniora erant, authoremque ingenio minime de­faecato praeferebant. Miror te hunç dignum judicasse quem tam prolixe refelleres. Etsi non sine voluptate Elenchum tuum pervolvi, doctum equidem atque acutum.

You see he hath no great opinion of you: He finds you full of Paralogismes: He takes you to be a man of a muddy brain; and wonders only that I thought it worth while to foul my fingers abou [...] [...]uch a piece as yours.

The other is a publick Professor of Mathematicks, of known abilities, and beyond exception; and he speaks yet somewhat fuller to the whole businesse.

Cum aestate praeterita in manus inciderit Thomae Hobbes Ele­mentorum Philosophiae Sectio prima; abs [...]inere non potui quin tra­ctatum istum leviter evolverim. Instigabat me ad hoc, tum Au­thoris hujus celebritas, tum etiam quod plura in eadem tractatu offen debam Geometrica, quae si Philosophiam non excelerent, sal­tem ut quam maxime illustratura forent, opinabar. Sed me illum perlustrante, cum talia ibi invenerim ejus de Algebra sive Ana [...]ys [...] judicia, equibus mihi facile fuit colligere, quod Author hic in ea­dem Arte parum deberet esse versatus; (quandoquidem haec ill [...] Ars existit, ut si liber suus in Geometria egregii ac ardui quid con­tineret, qualia se passim invenisse praetendere mihi videhatur, id ipsum huic Arti, judicio meo, in totum deberet;) Cumque adhuc in perlustrando dum p [...]rgebam, non nulla de rectae ac curvae aequalitate, aliaque complura animadvertebam quorum cognitionem nunquam mihi pollicebar, ac inter seponenda not abam, vel certe si spos ali­qua inveniendi illa mihi superesset, quin Algebram in partes vo­carem non dubitabam: Aliam exinde de ipso [...]pinionem concepi, credens quod illa quae illū ante e [...]proprio penu deprompsisse autuma­bam, non nisi aliorum inventa esse, sed in alium sensum ab eo tra­ducta aut correpta: Ideoque siquid boni in eo comprehenderetur, id quam maxime esse ventilandum ac excutiendum; ac proinde il­lius examen, si vel utile aut necessum judicarem, in commodius tempus mihi esse differendum. Quemadmodum autem haec ita conceperam, ita quoque evenit ut amicus, cui me eo tempore invi­senti dictum tractatum exhibueram, falsitatem plurium illius pro­positionum haud longe post invenerit, illasque uno folio coram om­nibus exponere decreverit. Qui edere ista utiliter rotus, ubi se ad hoc accinxerat, tuum interim, vir▪ Clarissime, Elenchum in lucent [Page 6] proditum vidit, ac postquam te isto munere optime defunctum de­prehendit, a proposito suo destitit. Egregie autem te eum, Vir Clarissime, sed pro merito tamen excepisse ibidem agnovi, ita ut credam eum in posterum a te prudentiorem doctioremque factum, licet ille tibi nullas gratias (judicio meo) pro beneficio isto sit ha­biturus, Inter illa quae in Elencho tuo offendi, nihil expectatio­nem majorem mihi excitavit, quam Arithmetica tua Infinitorum, de qua subinde mentionem facis: Quam novissime in lucem pro­ditam, quamprimum cum caeteris tuis tractatibus vidi, mihi com­paravi, ac multa praeclara & ingeniosa inventa, qualia mihi pro­posueram, continere deprehendi. Perpl [...] et autem quod tum in Arithmetica tua Infinitorum, tum in Sectionibus tuis Conicis per­tractandis, calculum Geometricum ubique adhibueris, tum propter brevitatem, tum quod is (ut ipse mones) demonstrationum om­nium fons existat, atque demonstrationes omnes, solenni modo fa­ctae, certa arte ex illo confi [...]i possint. Id quod prae aliis Clarissimus D. des Cartes in Demonstrationibus suis est molitus, qui neglecta Theorematum ac Lemmatum longa serie, quibus alias in demon­strando difficulter carere liceret, calculo omnia constare voluit; atque in eum finem passim aequationes investigat, quibus rei veritas, ac quomodo illa cognosci possit, absque verborum involueris, brevi­ter atque perspicue ob oculos ponatur. Quae autem de Circuli qua­dratura tradis, utrum scilicet rem acu tetigeris necne nondum examinare mihi contigit: subtilissime autem cum illam prosecu­tus mihi videaris, atque etiam calculo ipsam inquisiveris, non du­bito quin omnium saltem proxime atque accuratissime ad scopum collimaveris.

You see what he thinks of you, and mee, and Symbols. He discerns presently by your judgement of Algebra, what a Geometer you are like to prove; that it must needs be one who understood it not, that rants at that rate; and will yet talke of squaring a Circle, and find a streight line equall to a crooked, and other fine things, without the help of Al­gebra. He sees by a little what the rest is like to prove; ei­ther little worth, or not your own. And therefore, though at first he made hast to get it, yet when he sees what is in it, he thinks your book may well be thrown aside, or at least be examined at leisure. He tells you of another, that, had not my Elenchus prevented him, meant to have been upon the bones of you. He tells you, that my Elen [...]hus, as sharp as it is, is no more then you had deserved. He supposed [Page 7] withall (though therein it seems he was deceived) that you would have learned from thence, more Mathematicks, and more discretion for the future; and yet did believe (as well he might) [...] you would scarce thank me for that favour. He is well enough satisfied also with my other Pieces, (what ever you think of them,) and likes them never the worse for that Scab of Symbols (as you call it) but much the better; (because, though you understand them not, he doth.) And much more to that purpose.

And by this time, I hope, you be pretty well eased of your feare, least the University of Oxford should suffer in the opinion of Learned men beyond Sea, by reason of the Ma­thematicks that we have written. (Nor have you reason to think, that Malmesbury, will be much the more renowned for your skill in that kind.) And, that you may not despise their Testimonies, the persons are very well known to the World, by what Works they have extant in Print, to be no contemptible Mathematicians.

Beside these, I shall, for the satisfaction of your English Readers (who perhaps may not so well understand the words of the Authors above mentioned,) adde an extract of one Letter more; from a noble Gentleman, whom as yet (to my knowledge) I never saw, nor had formerly any the lest intercourse with him by letter or otherwise, though I had before heard of his worth and skill, both in Mathematicks and other learning: And which is more, he is neither of the Clergy; nor any great Admirer of them, beyond other persons of equall worth and Learning. He was pleased, though wholly a stranger to mee, upon view of my Elenchus, to intimate to me by a Letter directed to a third person, That D. Wallis had unhappily guessed, that those propositions which M. Hobs had concerning the measure of Para­b [...]lasters, were not his own, but borrowed from some body else without acknowledging his Author: and signified withall, that they were to be found demonstrated in an exercitation of Caval­lerius, De usu Indivisibilium in Potestatibus Cossicis; (a piece which I then had never read:) And that M. Hobs, endeavouring to demonstrate them anew, had missed in it. For which civility from a person of Quality, to mee a meer stranger, I could doe no lesse then returne him a civill answer of thanks for that favour. In reply to which (having in the mean time [Page 8] seen and perused my Arithmetica Infinitorum) he was pleased to honour me farther with this.

I had not so long deferred &c. but that &c. And I beseech you receive it now from a Person, who much honours your emi­nent Learning and Humanity, and would egerly imbrace an oc­casiō to give you most ample testimony of the esteem he hath for you. I had not, (before &c.) seen your Arithmetica Infinitorum, which alone, although your other labours were not taken in to make up the value, may equall you with the best deservers in the Mathematicks. I was before acquainted with many excellent Propositions therein by you demonstrated (as you partly know,) but admired them, there, as wholly new, not because you had de­monstrated them only another way, but by a generall method, so little touched at by others, so in effect wholly new, and of so rare consequence for entring into the secrets and Soul of Geometry (if my judgement may passe for any thing) as truly I believe the Art may reckon it among the most confiderable advances given it. Sir, I wish all prosperity to your deservings, and humbly thank you for the fair admittance you have given me to your acquaintance and friendship, which I shall preserve with a tendernesse due to a thing so estimable; and believe, Sir, you have Power at your own mea­sure in Yours &c.

This is English, and therefore needs no exposition; your English Reader, whether Mathematician or not, may un­derstand it without help. You see all are not of your opi­nion concerning my scurvy book of Arithmetica Infinitorum.

I will not trouble your patience with reciting more te­stimonies in this kind; (though, the truth is, very many persons of Honour and Worth, and eminent for their skill in these studies, have been forward of their own accord to put more honour upon me in this kind, then were fit in modesty for me to own.) These you have heard already, are more, I presume, then you take any great content in; and the lest of them, were abundantly sufficient to outway your verily believe; upon the strength of which, you have the confidence to utter all those reproaches which in your scur­rilous piece you endeavour to cast upon us; but find them to fall back, and foul your self. And you see withall, both how little reason we have to fear the opinion that men will have beyond Sea, of the Geometry taught at Oxford; and with [Page 9] how much vanity it is that you tel us according to your Rhe­torick, that when you think, how dejected we will be for the fu­ture; and how the grief of so much time irrecoverably lost, and the consideration of how much our friends will be ashamed of us, will accompany us for the rest of our life, you have more compassion for us then we have deserved. No doubt Sir, but you are a very pittifull man! (who have so much compassion for us:) And we are much bound to behold you. But since your cō ­passion of us, is not only more then you think we deserve, but, likewise, more then we think we stand in need of; we are loath your good nature should be injurious to your selfe. And therefore, knowing how much your selfe at present nay need compassion, we desire you to suffer that charity to begin at home, and not to be too lavish of that commodity upon us, of which at present we have so little need and you so much. But, that there may be no love lost between us; know, that we have the like compassion for you, upon the same account. You have but prevented us; and taught us, by your extreme civility, what might have better beseemed us to say. You tell us somewhere, the rea­son, why the Ladies at Billingsgate, amongst all their com­plements, have none readier then that of Whore, because, forsooth, when they remember themselves, they think that like­liest to be true of others. And truly, we have reason to believe, that the anguish of such considerations as those you menti­on, being so frequently present to your own thoughts, makes you so apt to think that others may be tormented in the like manner. (For who are more compassionate to those that feele the toothach, then those that are most tor­mented with it themselves?) For, as your words are else­where, A man of a tender forehead, after so much insolence, and so much contumelious language as yours, grounded upon arrogance and ignorance, would hardly endure to outlive it.

As for our selves; I do not find, that our friends do yet disowne us; or, that we need to feare, in this contest, the fury of our foes. And, whatever diseases you may believe my Conick, Sections, and Arithmetica Infinitorum, to be infe­cted with: I do not see, that wiser Physitians can yet dis­cerne, either the one to be troubled with the Scab, or the other with the Scurvy.

But you tell us, (and that may serve for answer to the Testimonies but now recited) Though the Beasts, that think [Page 10] our railing to he roaring, have for a time admired us; yet, now that you have shewed them your eares, they will be lesse affrighted. Sir, those Persons (as they needed not the sight of your eares, but could tell by the voice what kind of creature brayed in your books: so they) doe not deserve such lan­guage at your hands: And, you would not have said it to their faces. I know your Apology will be, that you say it provoked; and that by Vespasians law, when a man is provoked, it is not uncivill to give ill language. And that we may know you have been provoked, you tell us, how hainous and hazar­dous a thing it is, to speake against some sorts of men, whether that which is said in disgrace be true or false; And by all men of understanding it is taken (not only for a provocation, but for a defiance, and a challenge to open Warre. And truly, so far as that may passe for Law, I cannot deny but that you have been provoked; for sure it is, that much hath been said a­gainst you, and that, as is supposed, to your disgrace, and, I believe, the provocation hath been the greater, because that which hath been said, is true. But is this such a provocati­on as may warrant you, by Vespasians Law, to rave at the next man you meet with? and to revenge your selfe upon him that comes next? Is it such a provocation of M. Hobs, for any man to admire us, that he may thenceforth, with­out incivility, be called a Beast, or what you please? Is it not enough for you to involve the two Professors in the same crime, and consider us every where as one Author, and therefore both responsible, joyntly and severally, for what is said by either, because forsooth, we approve, you say, of one anothers doctrine: but must all that doe but admire us be un­der the same condemnation? It's possible that some of them may admire our folly; (you see, one of them won­ders at my discretion, that I would foule my fingers with you, or think you worth the Answering:) must they be called Beasts also? It seems 'tis a dangerous businesse for a man to admire any who do not admire you.

But I have done with the Rhetorick and good Lan­guage. We have had a tast of it; and that's enough unlesse 'twere better. They that desire to have more of it, may ei­ther read over your book, or goe to Billingsgate, whether they please.

But when men shall heare you rant it after this rate, and talk high; surely they must needs think, that you have very [Page 11] good ground for it, must they not? A shallow foundation would never bear a confidence of such a towring hight. One would hardly believe mee, if I should say, That not­withstanding these Braggadocian words, there is not any one assertion of mine, that you have either overthrown or shaken; nor any one of your own (which I charge to be false,) that you have defended; Yet that's the case. A great cry, and a little wooll! (as the man said when he shore his hoggs.)

Parturiunt Montes.— And that's it we have next to shew.

SECT. II. Concerning his Grammar, and Criticks.

I Shall therefore next after the Rhetorick, consider the Grammar, you'l say, that Grammar should have gone first. It may be so. But it's no great matter for method, when a man deales with you; for you are not so accurate in your own, that you need find fault with anothers.

There be six or seven places (and, I think no more) where you would play the Critick.

First, you tell me pag. 11. that [Punctum est Corpus, quod non consideratur esse Corpus] is not Latin, nor the version of it [a Point is a body, which is not considered to be a body] English. If you had said, it had not been good sense, I would have agreed with you. But why not that, Latin? or this, English? (Nay stay there; you are not to give a reason for what you say. It's enough that you say so.) Quod esse videmus, id vide­tur esse. Quod esse sentimus, id sentitur esse. Quod esse putamus, putatur esse. Quod esse cognoscimus, cognoscitur esse. Quod esse dicimus, dicitur esse. And why not as well, Quod esse conside­ramus, consideratur esse? But what should it have been, if not so? Why thus, Punctum est corpus quod non consideratur ut cor­pus. Very good! Bur Sir, It's one thing, to consider a thing as a body, or as if it were a body, (either of which the words ut corpus may beare;) another thing, to consider that it is a body, which was the notion I had to expresse, and therefore your word would not so well serve my turne, but rather the other. And when we have this to expresse, That though it be a body, and we know it to be a body, yet do not at present actually consider it so to be; (which I take to be [Page 12] neither Irish, nor Welsh, nor, which is worse then either, the Symbolick tongue; but good English;) it is better ren­dred in Latin by esse, then by ut.

Secondly, you tell me pag. 44. I might have left out [Tu vero] to seek an [Ego quidem.] (As though vero might ne­ver be used where there is not a quidem to answer it.) And is not this a worthy objection? But however, to satisfy you, look again and you may see a quidem which answers directly to this vero. My words are these Articulo quarto (cap. 17.) curvilineorum illorum descriptionem aggrederis per puncta. Quae quidem res est non ita magnae difficultatis, ut tanto apparatu, tantisque ambagibus opus sit. Exempli gratia. &c. Tu vero, qua­si per planorum Geometriam id fieri non possit, statim imperas me­diorum quotlibet Geometricorum inventionem. Doe you see the quidem now? Very good!

But before I leave this, (to save my selfe that labour a­non,) I must let your English Reader see, how notoriously you doe here abuse him, (him, I say; for the abusing of me in it, is a matter of nothing) My words were these; In the 4th Article (of your 17 Chap.) you attempt the describing of those curve lines by points, (that is, the finding out as many points as a man pleaseth, by which the said curve lines are to passe, through which, with a steady hand, those lines may be drawn, not Mathematically, but by aim,) which is a matter of no great difficulty, and may be performed without so much adoe as you make, and so much going about the bush. As for example, &c. (and so I go on to shew how those points may be easily found Mathematically, by the Geometry of Plains, that is, by the Rule and Compasse, or by streight lines and circles, without the use of Conick Sections, or o­ther more compounded lines. And, having shewed that, I proceed thus) But you, as though this work (the finding of those points) could not be done by the Geometry of Plains, (as I had shewed it might,) require presently the finding of as ma­ny mean proportionals as you please (viz. more or fewer ac­cording as the nature of those lines shall be;) between two lines assigned: (which by the Geometry of Plains cannot be done:) And so, of a Plain Probleme, you make a Solid and linea­ry Probleme. Which how unbeseeming it is for a Geometrician to doe, you may learne from those words which your selfe cite out of Pappus, pag. 181. (in the English, pag. 233.) Videtur autem non parvum peccatum esse apud Geometras, cum Problema [Page 13] Planum per Conica aut Linearia ab aliquo invenitur. It's judged by Geometers no small fault, for the finding out of a Plain Problem, (as this is,) to have recourse (as you here) to Solid or Lineary Problems. Now these words, one would think, were plain enough for a man of a moderate capacity to un­derstand. And is it not well owl'd of you, to perswade your English Reader that I had here taught, that a man may find as many mean-proportionalls, as one will, by the Geometry of Plaines? (where I said only that the work before spoken of, might be done by the Geometry of Plaines, and there­fore needed not the finding of such Mean-proportionals?) And then (because you doe not know whether or no, as many mean proportionals as one will, may be found by the Geometry of Plains,) you tell us, that you never said it was impossible; (truly if you had said so, I should not have bla­med you for it;) but that the way to doe it was not yet found, (you might have added, nor ever will be,) and therefore it might prove a Solid Problem for any thing I know. Nay truly, Sir, I know very well (though it seems you doe not,) that it is at lest a Solid Probleme, or rather Lineary; and that the way to doe it, Mathematically, by the Geometry of Plains, is neither yet found, nor ever will be. For those Problems which depend upon the resolution of a Cubick or Superior Aequation, not reducible to a Quadratick, (which is the case in hand) can never be resolved by the Geometry of Plains. Which, if, instead of scorning, you had endeavoured to understand, the Analyticks, you might have known too. But this by the way; to save my selfe the labour anon. I returne to your Criticks again.

Thirdly, whereas it is said c. 16. art. 18. Longitudinē percursam cum impetu u [...]ique ipsi BD aequali; I said the word cum were better out, unlesse you would have Impetus to be only a Com­panion and not a cause. For where a causality is imported, though we may use with in English, yet not cum in Latin. To kill with a sword (importing this to have an instrumentall or causall influence, and not only that it hangs by the mans side, while some other weapon is made use of) is not in La­tin Occidere cum gladio, but gladio occidere. So ebrius vino; pallidus ira; incurvus senectute, or, if you will, prae ir [...], ob [...]iram, &c. not cum vino, cum ira, &c. You say, it is better in (though for the most part your selfe leave it out in that constructi­on;) let the Reader judge; for it is not worth contending [Page 14] for. All that you say in defence, is that Impetus is the Ab­lative of the Manner. What then? the question remains, as it was before, whether this Modus do not here import a causall influence? And 'tis evident it doth; for the effect here spoken (that such space be dispatched) doth equally de­pend upon two causes; the one, that the motion be uniforme; the other, that the Impetus be so great. And therefore (since you please to insist upon it, which I did but give a touch at by the way, as in many other places where you take it Pa­tiently,) cum not proper in either place; but either an Ab­lative without a Preposition; or, if you would needs have a preposition, per, prae, pro, propter, ob, or some other which do import a Causality; not cum, which imports only a Con­comitancy.

Fourthly, you say, pag. 61. That you think, I did mistake [praetendit scire] for an Anglicisme. Your words were these at first, (as that Paragraph was first printed, pag. 176.) ta­men quia tu id nescis, nec praetendis scire praeter quam ex auditu, &c. as appears in the torne papers. And then, (after you had new modeld that whole Paragraph, as it now is pag. 174.) tamen quid id nescit, nec praetendit scire &c. This I did and doe still take (not mistake) for an Anglicisme And you cannot deny but that it is so. Where is the mistake then? You say tis a fault in the Impression. Yes that it is; and that twice for failing. But was it not a fault in the Copy first? you say it should have been, praetendit se scire. That, I confesse, helps the matter a little. But why was it not so? The Printer left out se ( [...]es, at both places.) And why? but, because the Author had not put it in? In like manner pag. 222. Tractatus huius partis tertiae, in qua motus & magni­tud [...] per se & abstracte consideravimus, terminum hic statuo. This was the Printers fault too, was it not? or, at least, a fault in the Impression? (Beside much more of the like language up and down) And if you think it worth while to make a catalogue of such phrases; tell me against next time, and I shall be able to furnish you with good store.

There be two places more (to make up the halfe dozen) wherein you would faine play the Critick: of which, I heard from divers persons, you made much boast, long before your book came out; that you had D. Wallis upon the hip; &c. The one was that adducere malleum was no good Latin, because that duco and adduco were words not [Page 15] used but of Animals, and signified only to guide or leade, not to bring or carry. The other was, that I had absurdly deri­ved Empusa from [...] & [...]. It's true, those charges, not­withstanding your first confidences, are not now layd in these words; but the former extenuated, and the latter va­nisht. Yet some nibling there is at both.

The former of these, (which I make the fifth in order) is pag. 51. where you tell me, that Adducis malleum, ut occidas muscam, is not good Latin? But why? when we speak of bodies Animate, Ducere and Adducere, you say, are good. Tis very true. Did any body deny it? But are they not good also, of Bodies inanimate? or other things? (I exspected, that in order to the confutation of my phrase you should have told us, in what cases they had not been good, and that this was one of those; not, in what other cases they are good, as well as this; for that hurts not mee.) May they not be applied as well to a Hammer, as to a Tree? Though this be Animate; not that? You were, I heare, of opinion, when you first made braggs of this notion, (or else your friends belye you) that they were not to be used but of bodies Animate: But, that being notoriously false, some body it seems had rectified that mistake, and informed you better, and therefore you dare not say so now.

But why, now, is not adducis malleum good Latin? Be­cause, forsooth, Ducere and Adducere, when used of bodies Animate, signify to guide or lead; (and sometimes they doe so.) Now though a ninny may lead a ninny, yet not a hammer. Very witty? But I am of opinion, that he who leads M. Hobs, leads both. Or however, if a man may not lead a hammer; yet, I hope, he that hits the naile at head (which M. Hobs seldome doth) may be said to guide his hammer: may he not? The phrase therefore is good, even by your own law. But heark you, man; to lead, you told us, is the signification of the word, when it is used of Animates; why then do you talke of leading a hammer? do you take the hammer to be animate? or would have us take you to be the ninny?

But farther; they singify you say to guide or lead. What then? did I say they do not? Prithee tell me, what they do not signify; not, what they doe; if you meane to over­throw my use of the word. Tis true, sometimes they signi­fy to guide, or lead; viz. with the parties consent, (Fata [Page 16] [...]lentem, du [...]unt, nolentemtrahunt:) yet sometimes the quite contrary; as ducere captivum, Claud. Cic. to take a man Prisoner, or carry him captive, against his will: so ducere in carcerem; du­cere ad supplicium, & deducere, Cic. to bring or carry a man to Prison, to execution, &c. (which for the most part is against his will.) Filia vi abducta, Ter. my daughter was carried away by force. And so, frequently. But suppose they doe, some­times, signify to guide, sometimes to lead; what then? doe they signify nothing else? Is Ducere lineam, Plin. to guide a line, or to lead a line? and not rather to draw a line? Ducere ux [...]rem, Cic. to guide a wife? or to lead a wife? (though perhaps you will cavill at that phrase) and not ra­ther to take a wife? But you say, Of bodies inanimate Addu­cere, is good for Attrahere, which is, to draw to. Very good! But what is it not good for? is it good for nothing else? Du­cere somnos, soporem, somnium, Virg. Hor. Insomnem ducere no­ctem. Virg. Ducere somno diem, noctem ludo; sic horam, horas, tempus, aestatem, aevum, adolescentiam, senectutem, vitam, aeta­tem, coenam, convivium, &c. ducere, producere, traducere, Hor. Virg. Claud. Propert. Ovid. Cic. Sen. Plin. Liv. &c. Do they signify, to leade, to guide, to dran? and not rather, to spend, to continue, to passe over, to passe away, &c.

Well! but however▪ (whatever they may signify else,) Duco, adduco, &c (with the rest of its com [...]ounds,) you would have us believe, (for that's it you drive at, though you dare not speak it out, or be confident to affirme it,) do not signify to take, carry, fetch, or bring, (which you sup­pose to be the sense (aime at) unlesse when used of bodies animate. But that's as false as can be. Adducere fehrem Hor. Adducere Sitim, Virg. Adducere vini tae­dium. Plin &c. doe they signify to lead a fever? or to guide a fever? or to draw a fever, (with cart-ropes, or a team of horses?) and not rather to bring a fever, &c. In my Dictio­nary, duco & adduco, signify to bring, as well as to draw. The truth is, duco, with its compounds, is a word of as great variety and latitude of signification, as almost any the La­tine tongue affords. And, amongst the rest, to bring, fetch, carry, take, (to, from, about, away, before, together, asunder, &c. according as the praeposition wherewith it is compounded doth require) is so exceeding frequent in all Authors (Plautus, Terence, Tully, Caesar, Tacitus, Livy, Pliny, Se­neca, Virgil, Ovid, Horace, Claudian, &c.) that he must [Page 17] needs be either malitiously blind, or a very great stranger to the Latin tongue, that doth not know it, or can have the face to deny it. Rem huc deduxi. Cic. Res eo adducta est, (deducta perducta,) in eum locum, in eum statum, in dabium, in certamen, in controversiam, in periculum, in maximum discrimen, &c. Cic. Liv. Caesar. Plancus ad Cic. &c. Add [...]cta vita in extremum. Tacit. Adducta res in fastidium, Plaut. in judicium, Cic. rem ad mucrones & manus adducere, Tacit. Contracta res est & adducta in augustiam. Cic. Rem co producere. Cic. Ad exitum, ad culmen, ad summum, ad umbilicam, ad extremum ca­sum, &c. Cic. Caesar. Liv. Hor. &c. That is, The matter is brought to that passe, &c. So, Sive enim res ad concerdiam ad­duci potest, sive ad bonorum victoriam &c. Cic. So, ex inordinato in ordinem adduxit, (speaking of Gods bringing the World out of the first Chaos,) and again, Eas primum confusas, postea in ordinem adductas mente divina. Cic. So, aquae ductus, a­quarum deductio, rivorum a fontibus deductio, aquam ad utilita­tem agri deducere, Cic. Aquam ex aliquo loco perducere, Plin. In urbem induxit, idem. To bring water from place to place. (not to draw it, attrahere.) Thus adducere febres, to bring fevers.

Officiosaque sedulitas, & opella forensis
Adducit febres & testamenta resignat.



Ova noctuae, &c. tadium vini adducunt.


Addua [...]ere sitim tempora, Virgi [...]i, [sc. aestiva)


In like manner, febres deducere, to take them away.

Non domus & fundus, non aeris acervus & auri,
Aegroto domini deduxit corpore febres,
Non animo curas.—

So, ‘deducere fastidium.’ Plin And then ‘Febrim (que) reducit,’ Hor. to bring back again. So, ‘Frondosa reducitur aestas.’ Virg.

‘Luctus fortuna reduxit.’ Claud. ‘Reducere exemplum, libertatem, morem,’ &c. Plin. ‘Aurora diem reduxit.’ Virg.

‘Collectasque fugat nubes solem (que) reducit.’ Virg. that is, re­storeth. So, ‘Reducere somnum,’ Hor. ‘Spem mentibus anxiis reducere’ Idem. ‘In memoriam reducere,’ Plin. Cic. Now it would be hard to say, that in all these places Adduco, De­duco, Reduco, &c. are put for Attrabo, Detraho, Retraho, &c. Attrahere febres, attrahere taedium, &c. So ‘Abduxi [...]lavem,’ Plaut. I took or brought away the key (as had every whit, as adducere malleum, to bring a hammer.) So ‘Navis a praedoni­bus abducta,’ Cic. Ter. The ship taken at Sea by Pyrats, and carried away.

[Page 18] ‘Visaque confugiens somnos abduxit imago.’ Ovid.

So (speaking of Hercules loosing the chains whereby Pro­metheus was chained to the rock)

Vincula prensa manu saxis abduxerat imis
Val. Flac.

‘Quidsi de vestro quippiam orem abducere?’ Plaut. What if I should desire to carry away somewhat of yours?

‘Coeperat intendens, abductis montibus, unda Ferre ratem.’ Val. Flac. ‘Abducti montes, id est, semoti. —abducta (que) flumina ponto.’ Idem.

‘Quod ibidem recte custodire poterunt, id ibidem custodiant; quod non poterunt, id auferre atque abducere licebit.’ CIC. Where ab­ducere, all along, is no more then auserre. In like manner, conducere is oft times the same with conferre, congerere. As ‘Veteres quidem scriptores hujus artis, unum in locum conduxit Aristoteles.’ Cic. ‘Partes conducere in unum,’ Lucret. (i. e. in unum corpus componere.) So deducere, to carry forth. ‘Ducere, deducere, producere, funus, exequias.’ Plin. Virg. Stat. Lucan. ‘Deducunt socii naves,’ Virg. And to take away, (the same with tollere, demere, auferre,) as in deducere febrem, deducere fasti­dium, as before. Thus deductio and subtractio for as you use to call it both in English and Latin, Substractio, as if it came from sub and straho) is contrary to Additio, and signifies all kind of Ablation or taking away. ‘Addendo, deducendoque, videre quae reliqui summa fiat.’ Cic. ‘Ʋt, deducta parte tertia, deos reli [...]ua reddatur Africanus de pactis dotalibus. Ʋt centum nummi deducerentur.’ Cic. ‘Sibi deducant drachmam, reddant caetera.’ Cic. ‘ut beneficia integra perveniant, sine ulla deductio­ne.’ Sen. So, ‘deducere cibum.’ Ter. to abate, diminish, or take away; as also, ‘Cibum subducere,’ Cic. ‘Subducere vires,’ Ovid.

[...]t succus [...]ecori & lac subducitur agnis.
Jam mihi subduci facies humana videtur.

‘Ignem subdito; ubi ebullabit vinum, ignem subducito.’ Cato de re rust. ‘Aurum subducitur rerrae.’ Ovid. So, ‘Annulum subduco.’ Plant. ‘Subducere pallium,’ Mart. to take or steale away. ‘De­ducere vela, deducere carbasa,’ Ovid. Luc.

—primaque ab origine mundi,
Ad mea perpetuum deducite tempora carmen.

That is, To bring down from the beginning of the World to his own times.

—a pectore postquam Deduxit vestes.

‘Deducere sibi galerum, vel pileolum,’ Sueton. to putt off, or take off.

Et cum frigida mors animâ subduxerat artus.
—Seductae ex aethere terrae.

Where seducere, is no more but separare. So, ‘diducta Britan­nia mundo.’ Claud. ‘Ante se fossam ducere & jacere vallum,’ Liv. ‘Vallum ducere,’ Idem. ‘fossam, vallum, praeducere,’ Tacit. Sen. ‘perducere,’ Caes. to cast up a wall, a bank, a trench before them. ‘Murum in altitudinem pedum sexdecim perduxit. duas fossas ea altitudine perduxit. munitio de castello in castellum per­ducta.’ Caesar. So, ‘ducere muros,’ Virg. to raise up: and ‘edu­cere turrim. aramque educere coelo certant. sub astra educere. molemque educere coelo.’ idem. to raise up as high as heaven. Thus, ‘educere foetum,’ Cic. Claud. Plin. ‘Educere, producere, faetum, partus, liberos, sobolem, fructus, &c.’ Si [...]ius. Plaut. Hor. &c. To bring forth, So, ‘Educere cirneam vini.’ Plaut, to bring out a flagon of wine, (as bad, I trow, as ‘adducere malleum.) Educere naves ex portu;’ and ‘in terram subducere,’ Caesar.

Ʋn [...]que conspecta livorem ducit ab uva.
—arborea frigus ducebat ab umbra.

‘Animum ducere’ (to take courage) Liv. ‘Ab ipso Ducit opes animumque ferro,’ Hor. ‘Argumenta ducere,’ Quintil. ‘Ducere conjecturam, similitudinem, &c. Cic. ‘Initium, principium, ex­ordium ducere.’ Cic. ‘Ortum, originem ducere,’ Cic. Quint. Hor. (i. e. sumee,) ‘Producere exemplum,’ Juvenal. ‘Ducere cica­tricem.’ Colum. Liv. Ovid. ‘Cicatricem, crustam, rubiginem, callum, obducere.’ Plin. Cic. ‘Obducere velum, torporem, tene­bras,’ Plin. Cic. Quintil. ‘Inducere, introducere, consuetudi­nem, morem, ambitionem, seditionem, discordiam, novos mores,’ Cic. Stat Plin. ‘Qua ratione haec inducis, e [...]dē illa possunt esse quae tollis.’ Cic. ‘Inducere formam membris,’ Ovid. ‘Cuti nitorem,’ Plin. ‘Te­nebras, nubes, noctem,’ Ovid. ‘Senectus inducit rugas,’ Tibul. ‘Tentorium vetus deletum sit, novum inductum,’ Cic. ‘Introdu­cere, quod & in medium afferre, dicitur.’ Bud. Cic. ‘Obliviae poenae ducere.’ Val. Flac. ‘Sollicitae vitae,’ Hor. ‘Nec podagri [...]us, nec articularius est, quem rus ducunt pedes,’ Plaut. (whose feet can carry him, not lead, guide, or draw him.) ‘Transducere arbores,’ (to transplant or remove from place to place,) Co­lum. ‘Quod ex Italia adduxerat.’ Caes. And if these Authori­ties be not enough; it were easy to produce a hundred more, (to justify my use of the word, and bring your new notion [Page 20] to nothing;) wherein Duco (both in it selfe and its com­pounds) signifies to take, bring, fetch, carry, &c. without a­ny regard had at all to your notion of guiding, leading, or [...], that we may see what a deale of impudence and ignorance you discover, when you undertake to play the Critick. And when you have done the best you can, you will not be able to find better words then Adducere malleum, and Reducere, to signify the two contrary motions of the [...]; the one when you strike with it, the other when you take it back to fetch another stroke.

To all these examples I might, if need were, adde your own which though it would be but as anser inter olores; nor would it at all increase the reputation of the phrase, to say [...] you use it: Yet it may serve to shew, that it is not out of i [...]dgement, (because you think so;) but out of malice and a designe of revenge (that you might seem to say some­what, though to little purpose,) that you thus cavill with­out a cause. For duco, adduco, circumduco, and the rest of the compounds, are frequently used by your selfe, in the same [...]nse and construction which you blame in mee. Lineam [...]cere, producere, &c. a puncto, ad punctum, per punctum, &c. are phrases used by your selfe fourty and fourty times. If [...] do not seem to come home to the businesse; that of [...]um-effectum, rem a [...]i [...]uam &c. producere, (to produce, [...]ring forth, bring to passe,) comes somewhat nearer; which [...] at lest twenty times in one page. p. 74. and within three leaves, (cap 9 & 10,) above fifty times: and else­where frequently. So, actus educi poterit, p. 78. partes flui­ [...] educi [...]osse. p. 258. deduci hinc potest. (i. e. inferri) p. 23. [...] inde deducere non possum. p. 248. fluviorum origines [...] possunt. p. 278. ratio quaevis ad rationem linearum reduci [...]. p. 96. linea in se reducta p. 190. quibus & reduci cogi­ [...] nes praeteritae possint. p. 8. copulatio cogitationem inducit. p 20. n [...]men aliquod idoneum inducat. p. 52. phantasma finis [...] thantasmata mediorum. p. 229. in animum inducere non [...] p. 24 [...]. Parallelismus ob eam rem introductus est. p. 246. [...] instantia adduci potest. p. 82. And particularly of [...]dies, in flectione laminae (lege, flexione) capita ejus addu­ [...]ur. p. 2 [...]5. flexio est, manente eadem lineâ, adductio extre­ [...] [...]unctorum, vel diductio, p. 196. terminis diductis, ibid. [...] adductio extremarū linearum. p. 197. cujus puncta ex­t [...]ema diduci non possunt. p. 106. adductio vel diductio termino­rum, [Page 21] ibid. and so again five or six times in that and the next page. So ex cujuspiam corporis circumductione. p. 4. corpus cir­cumductum, ibid. si corpus aliquod circumducatur, ibid. in [...]elli­gi potest planum circumduci, p. 109. si planum circumducat [...], ibid. punctum ambientis quodlibet ab ipso circumducitur, p. 18 [...]. and the like elsewhere. In all which places, by your law, it should have been circumlatio, circumlatus, circumfer [...], circumferri, circumfertur, &c. as it is, p 50. p. 108. and [...] some other places. Now if circumdaci and circumferri, [...] be used promiscuously, and so circumductio and circum [...], &c. why not as well in the same cases adducere and [...] &c.? And if corpus quodpiam, may, without absurdity, be [...] circumduci, why not as well adduci? In like manner, [...] sum est conduci mobile (i. e. simul ferri) ad E ad A, concu [...] duorum motuum &c. p. 193. and moti per certam & design [...] viam conductio facilis, p. 200. with many the like phras [...] which are every whit as bad as adducere malleum. And therefore, you had very little reason to quarrell at that phrase; save that there was nothing else to find fault with, and somewhat you were resolved to say.

And the like is to be said of that other phrase, next be­fore, quod non consideratur esse corpus, which, though it be [...] Latine, when I speak it; yet, with you the same constructi­on comes over and over again, as least a hundred times [...] simulachrum hominis negatur esse verus homo, p. 23. qu [...] [...] gantur esse verae. p. 26. singulae partes singulas lineas conficere [...]telligantur, p 68. si corpus intelligatur moveri,—redigi— [...] escere, ibid. severall times intelligitur quiescere,— [...] 70. agens intelligitur producere effectum, p. 73. du [...] [...] intelliguntur transire, p. 87 ostenderetur ratio esse [...] p. 100. lineae extendi intelligautur, p. 108. intelligatur radius [...]veri, p. 111. si partes fractae intelligantur esse minim [...], p▪ 11 [...]supponatur longitudo esse, p. 131, altitudo ponitur esse in [...] basium triplicata, p. 153. sphaera intelligatur moveri, p. [...] haesio illa supponatur tolli, p. 188. intelligatur radius [...] materia dura, ibid. vis magnetica invenietur esse motus, p▪ [...] ▪ And so punctum, corpus, res aliqua, ponitur, supponitur, inte [...] ­gitur, ostenditur, &c. esse, quiescere, movere, circum [...] &c. p. 62, 64, 68, 75. 85, 106, 112, 115, 110, [...] 141, 142, 147, 155, 171, 182, 183, 184, 188, [...] 239. and many other places: which are every whit [...] as consideratur esse. Yea and consideratur also is by your [...] [Page 32] so used p. 87. Eaedem duae lineae—prout considerantur pro ip­sis magnitudinibus—poni. &c. So that 'twas not judge­ment, but revenge, that put you upon blaming this phrase also. And you care not, all along, how much you bespatter your self, (for, you think, you cannot look much fouler then you doe already,) if you have but hopes to be a little revenged on us. And truly you have that good hap all the way, that there is scarce any thing (right or wrong) that you blame in us, but the same is to be found in your selfe also with much advantage.

But this fault (adducis malleum) you should not, you say, (though it had been one,) have taken notice of in an English man; but that you find me in some places nibling at your Latine. Yes; I thought, that was the matter. You had a mind to be revenged. And ha'nt you done it handsomely? Was there nothing else to fasten upon with more advantage then these poor harmlesse phrases? 'Tis very well. It seems my Latine (though as carelessely written as need to be; for 'twas ne­ver twice written, and scarce once read, before it was prin­ted,) did not much lye open to exception; for if it had, I perceive I should have heard of it with both eares.

But you are offended, it seemes, that I should offer to nibble at your Latine. And truly, if that were a fault, I know not how to help it now. I must needs confesse, I did some times (when I stumbled upon them, but never went out of my way to seek them; for, if so, I might have found enough) correct some phrases, as I went along, (sometime to make sense, where the sentence was lame; sometimes to make it Latine, where the phrase was incongruous or barbarous;) because I did not know, that your being an English man, had given you a peculiar priviledge above others to speak barba­rously without controll. Such as these, nescit, nec pratendit scire praeterquam ex auditu. p. 174. or as it was first printed. p. 176. nescis, ne [...]p praetendis, &c. And accipiat lector tanquam Proble­matice dicta. p. 181. And Placuit quoque ea stare quae merito per­tinent ad vindicem, ibid. So p. 143. (at lest in my book) progressio stabit hoc modo, 0. 1. 2. 3. 4. &c. And diverse other places, which I do not now remember. But you know there be many more, which, had they come in my way, I might have found fault with, as well as these; As that p. 37. falsae sunt,—& multa istiusmodi (propositiones.) And p. 116. definiemus lineam curvam esse eam cujus termini diduci [Page 23] posse intelligimus. And p. 111. quantitas anguli ex quantitate arcus cum perimetri totius quantitate compaeratione aestimatur. (for ex quantitatis—comparatione, or ex quantitate—cōparata [...] p. 115. ducatur a'termino primae, ad terminos caeterarum, rectae lineae. And p. 222. partitertiae, in qua motus & magnitudo consideravi­mus, terminum hic statuo. And p. 224. Ex quo intelligitur esse ea (phantasmata) corporis sentientis mutatio aliqua. So p. 269. Exeuns, for exiens. and p. 3. exemplicatum esse, for exemplo explicatum, aut comprobatum. and p. 51. exemplicativum; and many more of the same stamp (as barbarous every whit, as those of the Schoolemen, which you blame as such, p. 22▪ non sunt itaque eae voces Essentia, Entitas, omnisque illa Bar­baries, ad l'hilosophiam necessarius non est.) I might adde that of p. 20. tanquam diceremus, (as if we should say,) and p. 22. tan­quam possent, and elsewhere, instead of quasi, acsi, (or some such word) or tanquam si, which is Tullies phrase, (tan­quam si tua res agatur. tanquam si Consul esset. tanquam si clausa esset Asia &c.) for tanquam without si▪ signifies but as, not as if: But because I know you are not the first, that have so used it, of modern writers; and that even of the ancients, some of them doe sometimes leave out si, (as in other cases they doe ut;) I shall allow you the same liberty, and passe this by without blame (as passable, though not so accurate.) To these we may adde those elegances p. 32. (syllogismus) stabit sic. p. 49. sed haec dicta sint pro exemplo tantum, and So, p. 269. Ventus aliud non est quam pulsi aeris motus rectus; qui tamen potest esse circularis, vel quomodocunque curvus. And a multitude more of such passages, (which, were it worth while to collect them, might be added as an appendix to Epistolae obscurorum virorum,) of which some are incon­gruous, some barbarous, some bald enough, and some mani­fest contradictions, or otherwise ridiculous But these are but negligences, as you call them, and therefore not attended with shame: for we doubt not but that, if you had particular­ly considered them, you could have mended them. Only, me thinks, he that is so frequent in such language, need not have quarrelled with such harmelesse phrases as adducere malleum, or consideratur esse. But I go on.

The other place (which makes up the halfe dozen) you talked much of it at first, yet before it comes to be printed, 'tis dwindled to nothing. It was, that I had derived your [Page 24] Athenian Empusa, from [...] and [...]; and said it was a kind of Hob goblin that hopped upon one legge, (which you take to be a clinch, forsooth, because your name is Hobs;) and hence it was that the Boys play, now a daies in use, (fox come out of thy hole,) comes to be called Empusa. This derivation you did, at first, cry out upon as very absurd; and you meant to pay me for it: Till you were informed, as I hear, by some of your friends, that the Scholiast of Aristophanes (as good a Critick as M. Hobs) had the same▪ (and so have Eu­stathius, Erasinus, Caelius Rhodiginus, S [...]ephanus, Scapula, Ca­lepine, and others:) and therefore you were advised not to quarrell with it. Whereupon waving your main charge, you only tell mee (pag. ult.) that it doth not become my gravity, to tell you that Empusa, your Daem [...]nium Athenien­se, was a kind of Hob-goblin, that hopped upon one legge; and that thence a boys play, now in use, comes to be called Ludus Empusae. And withall, pray me to tell you, where it was that I read the word Empusa, for the Boys play I spake of? To the Que­stion, I answer, that I read it so used in Junius's Nomencla­tor; Riders, and Thomas's Dictionary; sufficient Authors for such a businesse. And then as for the Clinch you talk of, in Hobs and Hob-goblins, and the jest you suspect in Hobbius, and Hobbi, which you say, is lost to them beyond sea; I hope that losse will never undoe mee: and when you can help me to a better English word for your Daemoniū, thē Hob-goblin; or a better Latin word for Hobbes then Hobbius (whose vo­cative case, in good earnest, is Hobbi,) I shall be content, without any regret, to part with the jest, and the clinch too, to do you a pleasure; Who tell us presently after, that you meant to try your Witt, to do something in that kind. And then shew your selfe as great a Witt, as hitherto a Cri­tick.

There is yet a Seventh passage, p. 14. which may be referred also to this place. The words Mathematicall definition do not please you Those termes or words, which do most properly belong to Mathematicks, we commonly call Mathematicall termes, and the definitions of such termes, in Mathematicks, Mathematicall definitions. And is it not law­full so to do? No, you tell us. But why? Because it doth bewray another kind of Ignorance. What ignorance? An inex­cusable ignorance. How doth it bewray it? It is a marke of ig­norance; of ignorance inexcusable. Ignorance of what? Igno­rance [Page 25] of what are the proper works of the severall parts of Phi­losophy. And, I pray, why so? Because it seems by this, that all this while, I think it is a piece of the Geometry of Euclide, no lesse to make the Definitions he useth, then to inferre from them the Theorems he demonstrates. A great crime, doubtlesse! But how doth it appeare, that I think so? May not a man recommend Hellebor to you, as a good Physicall drug, (be­cause used in Physick, and proper for some diseases,) unlesse he think, it is the Physitians work to make it, as well as to make use of it? But suppose I do; what then? do you be­lieve no body thinks so, but I? or do you believe, that any body thinks otherwise but you? Is it not proper for words of Art, (voces artis,) to be defined and explained in that art to which they belong? is it not proper for a Gram­marian to define Gender, Number, Person, Case, Declension, Coniugation &c. in the sense wherein they are used in Gram­mer? And for a Logician to define Genus, Species, Ʋniver­sale, Individuum, Argumentum, Syllogisinus, &c. in the sense wherein they are used in Logick? And may not those be called Grammaticall, and these Logicall definitions? And for a Mathematitian, to define or tell what is a Triangle, a Cone, a Parabolaster, what is Multiplication, Division, Ex­traction of rootes, what is Binomium, Apotome, Potens duo me­dia, &c. And may not these definitions be called Mathema­ticall? No, by no means, you tell us, to call a Definition Mathematicall, Physicall &c. is a marke of ignorance, of unex­cusable ignorance. (And doe you not think then, that Gor­raeus was a wise man, to write a large Volumne in folio, intituled Definitiones Medicae?) But why a marke of ignorance? Because a Mathematitian, in his definitions teach you but his language (not his art) but teaching language is not Mathema­tick, nor Logick, nor Phisick, nor any other Science, (but some Art perhaps, which men call Grammar.) some men would have thought that to Define, had belonged to Logick; but let it passe for Grammar at present. Do you think, no­thing, is Mathematicall, wherein a man makes use of Gram­mar? Can a man teach Mathematicks, in any language, with­out Grammer? (unlesse, perhaps, in the Symbolick Language, which is worse then Welsh or Irish.) But you say, He that will understand Geometry must understand the termes before he begin: (because a man ought not to go into the water, be­fore he can swim.) Well, But if not his Definitions, what [Page 26] then is it, in Euclide, that is Mathematicall? it is, you tell us, his inferring from them the Theorems he demonstrats. (And why not the solution of Problems also; as well as the inferring of Theorems?) But to infer and to demonstradte, are, I suppose as much the work of Logick; as, to define, is the work of Grā ­mar. And therefore, by the same reason for which you will not allow the Definitions to be Mathematicall, because to teach a language is the work of Grammar, you must also ex­clude the Propositions and Demonstrations, because to inferre and demonstrate, is the work of Logick. And so, nothing in Euclide will be Mathematicall. 'Twill be Grammar and Lo­gick, all of it. And are not these pure Criticismes; think you? Do not these wofull notions of yours, and the language that doth accompany them, shew handsomely together? But enough of this.

SECT. III. Concerning Euclide: and the Principles of Geometry.

WE have seen your Elegances already, in the first Section, and then your Critsicismes in the second. It's time now to look upon your Geometry. And I should here begin with your first Lesson; but that, by what we heard even now, you will not allow me to call it Geometricall, or any peece of Geometry, consisting, as it doth, of Definitions. And yet, what ever the matter is, me thinks you come pretty neer it: for you call them Principles of Geometry. But you'l say, perhaps, they be Principles of Geo­metry, but not Geometricall Principles, (for to call any De­finitions Geometricall, were as bad as to call them Mathema­ticall, which were a marke of ignorance unexcusable.) Acute­ly resolved!

But, whatever else they be, Principles they are without doubt. For, as you define p. 4. A Principle, is, the beginning of something: And no man can deny, but that the first Les­son is a beginning of something: And therefore, a Principle. Now contra principia, we know, non est disputandum. I must take heed therefore, what I say here.

In this Lesson, you take Euclide to task, and give him his [Page 27] Iurry: (And when you have lesson'd him, it is to be hoped, wee will not think much to be lesson'd by you:) And with­all intermingle some Principles of your own, for his and our correction and instruction: such as these,

That [...] can have no place in solid bodies. p. 2. (because you know not how to distinguish between a Me­chanicall and a Mathematicall [...], as knowing no other way of measuring but by the Yard and the Bushell, or at least by the Pound. p. 4. & 13.) And yet you tell us by and by. p. 3. that there may be in bodies, a Coincidence in all points (which coincidence, had it been Greek, would have been as hard a word as [...],) and that this may pro­perly be called [...]: and yet presently p. 4. you tel us again, that [...] hath no place in solids; nay more, nor in circular, or other crooked lines; (as though you did not know, that two equall arches of the same circumference, would [...].)

That the length of T [...]me, is the length of a Body. p. 2. (As though he had not spoken absurdly, that said, Profecto vide, bam fartum, tam Diu, pointing to the length of his arme.)

That an Angle hath quantity, though it he not the Subject of quantity. p. 3. (for there be octo modi habendi.)

That the quantity of an Angle, is the quantity of an Arch. p. 3. (And why not as well of a Sector, since Sectors, as well as Archs, in the same circle, be proportionall to their corre­spondent Angles.)

That 'tis a wonder to you, that Euclide hath not any where defined, what are Equalls, at least, what are equall Bodies. p. 4. (As though every body did not, without a definition, know what the word meanes. Any Clown can tell you, that those bodies are Equall, which are both of the same bignesse.)

That Homogeneous quantities are those which may be compared by [...], or application of their measures to one another. p. 4. (And consequently, two solids cannot be Homogeneous; because, you say, [...] hath no place in solids p. 2. & 4. And also, that incommensurable quantities, cannot be homogene­ous; because by 1 d 10▪ they have no common measure.)

That the quantity of Time, and Line are Homogeneous, p. 4. Because Time is to be measured by the Yard; (or, in your [Page 28] own words, because the quantity of Time, is measured by appli­cation of a line to a line;) But why not, by the Pint? For you know Time may be measured by the Hour-glasse, as well as by the Clock. And though the Hand of a Clock or Diall, determine a Line, yet the sand of an Hour-glasse fills a vessell.

That, Line and Angle have their quantity homogeneous, be­cause their measure is an Arch or Arches of a Circle applicable in every point to one another. p 4. (As though you had forgot, that you told us but now, that [...], or application, hath no place in circular or crooked lines.)

And All hitherto, you say p. 5. is so plain and easy to be un­derstood that we cannot without discovering our ignorance to all men of reason, though no Geometricians, deny it. Nay more, 'Tis new, 'Tis necessary, and 'Tis yours. very good! Now have at Euclide.

Euclid's first definition, [...], &c. A Marke is that of which there is no part; is, you say, to be candidly construed, for his meaning is, that it hath parts, and that a good many. For a marke, or as some put instead of it, [...], which is a marke with a hot iron, is Visible; if visible then it hath quanti­ty; and consequently may be divided into parts innumerable p. 5. (A witty argument! 'Tis visible, therefore 'tis divisible, But could you not as well have said, That A Marke consists of two Nobles? For that is as much to the businesse, as a marke with a hot iron.) Nay more Euclids definition, you say is the same with yours, which is, A point is that Body whose quan­tity is not considered. Lay them both together and look else. A marke is that of which there is no part. A point is that Body, whose quantity is not considered. Just the same to a cow's thumb. They begin both with the letter. As like, as an Apple and a Oyster.

But by the way, how comes a Point on a suddaine to be a Body? you told us just before, in the same page, p. 5. that a Point is neither Substance, nor Quality, and therefore it must be Quantity or else 'tis Nothing. If it be no Substance, how can it be a Body in your language?

But we have not done yet. Prithee tell me, good Tho. (before we leave this point) who twas told thee, that [...] was a marke with a hot iron? for 'tis a notion I never heard till now, (and doe not believe it yet.) Ne­ver [Page 29] believe him againe, that told thee that lye; for, as sure as can be, he did it to abuse thee. [...] signifies a distin­ctive point in writing, made with a pen or quill, not a mark made with a hot Iron, such as they used to brand Rogues and Slaves with; (And accordingly [...], distinguo, interstinguo, inter [...]ung [...], &c. are oft so used;) It is also used of a Mathematicall Point; or somewhat else that is very small: As [...], a moment, or point of time, and the like. What should come in your cap, to make you think, that [...] signifies a mark or brand with a hot iron? I perceive where the businesse lies. 'Twas [...] run in your mind, when you talked of [...], and, because the words are some­what alike, you jumbled them b [...]h together, according to your usuall care and accuratenes [...] [...] as if they had been the same. (Just as when, in Euclide [...] you would have us be­lieve that [...] & [...] [...] is but one word.) Do you not think now, that a boy [...] Westminster Schoole would have been soundly whipt for such a fault? Me thinks I heare his Master ranting it at this rate; How now Sirrah! Is [...] and [...], all one with you? I'le shew you a difference presently. Take him up Boyes. I'le shew you how [...] may be made without a hot Iron, I warrant you. And after a lash or two, thus goes on: [...], is a Point made with a Pen, quoth he (with a lash) will you remember that? 'Tis [...], is a mark with a hot Iron, (lashing again,) think upon that too. Henceforth, quoth he, (setting him down,) Remember the difference between [...] and [...].

The second definition. A line is length which hath no breadth; you would have to be candidly interpreted al­so. If a man, you say, have any ingenuity, he will understand it thus, A line is a body &c. very likely!

The Fourth definition, is this, A streight line is that which lies evenly between its own points. p. 6. Well; how is this to be understood? Nay, this definition is inexcusable. Say you so? let it passe then, and shift for its selfe as well as it can. It hath made a pretty good shift hitherto; perhaps it may outlive this brunt also. But, because you are willing to lend [Page 30] it a helping hand, you say, He meant, perhaps, to call a streight line, that which is all the way from one extreme to another, equal­ly distant from any two or more such lines, as being like and equall have the same extremes. It may be so. Many strange things are possible. But it would have been a great while before I should have thought this to be the meaning of those words.

The seventh definition, you say hath the same faults. Then let that passe too; and answer for it selfe as well as it can.

The eighth, is the Definition of a plain Angle. Against which you object onely this of your own, That by this Defi­nition, two right angles taken together are no Angle. And 'tis granted. Euclide did not intend to call an aggregate of two right angles, by the name of an Angle: And therefore gave such a definition of an [...], as would not take that in. Where's the fault then?

The thirteenth definit [...], A Terme or Bound, is that which is the extreme of any thin [...] [...] you say, is exact, (very good?) But, that it makes against [...] doctrine. What doctrine of mine? viz. that a point is nothing. Who told you, that this is my doctrine? I have said, perhaps, that a Point hath no hignesse; or, that a Point hath no parts, (and so said Euclide in his first definition,) but when or where did I say, it is nothing? But how do you prove hence, that a point hath parts? Because, you say, The extremes of a line are Points. True. What then? A point therefore, you say, is a part. It doth not follow. How prove you this consequence, If an extreme, then a part? But, say you, what in a line is the extreme, but the first or last part? I answer; A Point, which is no part. Have you any more to say?—If you have no more to say, then heare mee. A point is the extreme of a line: Therefore it hath no parts. I prove it thus; because, if that point have parts; then, either all its parts are extreme, and bound the line, or some one, or more: Not all: For they cannot be all utmost; but one must stand beyond another: if onely some, or one; then not the Point, but some part of it, bounds the line, which is con­trary to the supposition. You see, therefore, the Definition doth not make against my doctrine.

The fourteenth Definition of Euclide, you would have abbreviated thus. A figure is quantity every way determined, and then tell us, it is in your opinion as exact a definition of a Figure as can possibly be given. But I am not of your opinion; [Page 31] For by this Definition of yours, a streight line (of a deter­minate length) must as well be a Figure, as a circle. For such a line, having no other dimension but length, if its length be determined, it is every way determined; that is, accor­ding to all the dimensions it hath. (If you object, that it hath no determinate breadth; I answer, the breadth of a streight line is as much determined, as the thicknesse of a Circle, or other plain figure.) And, by the same reason, A Pound, a Pint, a Hundred, an Hour, &c. must be Figures, because they are Quantities every way determined, viz. ac­cording to all the dimensions that those words import. This Definition of Euclide,— (stay a while, the Defini­tion mentioned is not Euclides, nor equivalent to it His [...], imports more then your determined. [...]. should be rendred A figure, is that which is every way encompassed by some bound, or boundes. Which can be only in such a quan­tity as hath locall extension; and that, finite.) But The Defini­tion, you say, (whose soever it be) cannot possibly be im­braced by us who carry double, namely Mathematicks and Theo­logy;) but by you it seems, it may, who carry simple, and care not how destructive your principles are to Theology.) Your Definition, we (whether Theologers or Mathemati­cians) cannot admit; for the reason by us already assigned. But it seems you have a farther reach in it: Lets hear what it is. For this determination, say you, is the same thing with circumscription. A locall determination, intended by Eu­clide, is so. But what then? And whatsoever is any where (ubicunque) Definitivè, is there also Circumscriptivé. How do you prove this? or how doth this follow from the other? —You cannot but know this is generally denyed. Have you any thing to offer by way of proof?—Not a word. Well; but what is it you drive at? You offer no­thing of proofe, for what you affirme (by your own con­fession) against all Divines, or as you call them Theologers. But lets see what you would gather from it. By this means, you say, the distinction is lost, by which Theologers, when they deny God to be in any place, save themselves from being accused of saying he is nowhere; for that which is nowhere is nothing. 'Tis true, that Divines do [...]ay, (and I hope you'l say so too) that God is not bounded, or circumscribed, within the limits of any [Page 32] place; because they say, and do believe, there is no place where he is not. And he that saies the latter, must needs say the former. For to say that God, who is every where, & fills all places; is yet bounded within certain limits; were a con­tradiction. For, to be concluded within certain limits, is to be excluded from all places without those limits; And therefore not to be every where. And if this be not your opinion too, speak out, if you can for shame, that the world may see what you are. Do you believe, that what thing so­ever is at all any where, (not excepting God himselfe) must needs be circumscribed within some certain bounds, so as not to be without or beyond them? And that whatsoever is not, in any place so circumscribed, is no where, and therefore nothing? If so; then whether of the two do you affirme? That God is so circumscribed or concluded within certain limits, and excluded from all others at the same time? Or, That he is not so concluded, and therefore no where, and so nothing? If you say the first, you deny God to be Infinite: If the se­cond you deny him to bee. And, either way, you may with­out injury be affirmed to maintain horrid opinions concerning God. As for that distinction of Definitivè and Circumscrip­tivè, with which you say the Theologers think to save them­selves: You are wholly out in the businesse: Theologers use not that distinction in this case. It's true, that, in the case of Angells, and the Soules of men, there are that affirme them to be in loco definitivè, but not circumscriptivè: because though they be not bodies, and so locally extended per po­sitionem partis extra partem; yet neither are they infinite, or every where, but have a definite, determinate existence, as to be here, and not at the same time elsewhere. But as to God, we neither affirme him to be circumscribed, nor to be confined within any bounds; but to be Infinite and every where. And if any be so absurd as to affirme that God is determined within some place, so as not to be at the same time without or beyond it, whether by Circumscription or Definition, we shall without scruple, (notwithstanding that we carry double,) reject the distinction so applied, and your opinion with it, without fear of being cast out from the so­ciety of all Divines.

But in the mean while, I wonder how this Definition of Euclide comes to have any thing to doe with this businesse. A Figure, saith Euclide, is that which is incompassed within [Page 33] some bound or bounds. Well, what then? Will you assume But God is a figure? and then conclude, That, if God be at all any where, he must be so concluded within bounds? If you do, you argue profanely enough, and deserve as bad Epithites as any have been yet bestowed upon you. We should ra­ther, admitting Euclides definition, argue thus, A figure is concluded within certain bounds; But God is not so concluded, (as being infinite, and so without bounds;) Therefore God is not a Figure: And be neither in danger of being cast out of the Mathematick Schooles, nor yet, from the Society of Schoole-Divines.

The Fifteenth Definition, which is, of a Circle, you grant to be true.

And skip over the rest to the five and twentieth, which is, of Parallell streight lines. This Definition you think to be lesse accurate, and think your own to be better: But of this it will be time enough, if need be, to consider in its proper place.

After this, you let all the Definitions passe untouched, till the third of the Fift Book. Saving that you touch by the way, on the Fourth of the Third Book, which you grant to be true: and the first of the Fift Book, which, you say, may passe for a Definition of an Aliquot part, as was by Euclide intended.

But, the Third Definition of the Fift Book (the Definition of [...], Ratio,) you say, is intollerable. Yea 'tis as bad as any thing was ever said in Geometry by D. Wallis. (Because for­sooth, you can make nothing of it, but this, that Proportion is a what-shall-I call it asnesse or sonesse of two magnitudes &c.) Yet this definition hath hitherto been permitted to passe, and may do still. And when you understand it a little bet­ter, perhaps you may think so too. But of this I have dis­coursed more at large, in a peculiar Treatise against Meibo­mius: and shall therefore forbear to examine it here.

Against the fourth definition, you object nothing, but that the sixt might be spared.

The Fourteenth, you say is good. And tell us farther, that the composition here defined, is not the same composition which he defineth in the fourth def. before the sixth book. And you say true; for this is a composition by Addition, and that is composition by Multiplication. And therefore do not [Page 34] think much if hereafter I shall say, that there be two com­positions of proportion.

To the rest of his definitions you give a generall appro­bation. His Postulata you allow also: and so give over Lessoning of Euclide: But tell us before you part, that A man may easily perceive, that Euclide did not intend, That a point should be (without parts, which you call) nothing; or a line, without latitude; or a Superficies, without thicknesse: though it be evident that he hath defined them so to be. But why must we not think, he meant as he saith? (Because, say you, Lines are not drawn but by Motion, and Motion is of Body only. A pretty argument, and worth Marking! like that above, of [...], a Mark or brand with a hot Iron.

SECT. IV. Concerning the Angle of Contact.

HAving dore Schooling of Euclide; in your second Lesson you fall upon us.

Four peeces of mine, you take to task. p. 10. (My Elenchus of your Geometry; my Treatise concerning the An­gle of Contact; and that of Conick Sections; and my Arithme­tica Infinitorum.) Yet have not been able to find, either one false Proposition, or so much as a false Demonstration; in any one of them. Yet, that you may seem to say some­thing, you'l blunder on, though you break your shinnes for it. And you'd have it thought, that you have wholly and clearly confuted them Ep. Ded. (for you use to make clear work where you goe,) and that I have performed nothing in any of my books. p. 10

This is the charge. Let's see how you can make it good.

Wee'l begin with that of the Angle of Contact; which you undertake in your third Lesson. p. 26.

The subject of that treatise, is, a controversy between Clavius and Peletarius. Clavius is of opinion, that the An­gle of a Semicircle EAC (Fig. 1.) is lesse then the re­ctilineal Right Angle PAC; because that is but a part of this; the other part EAP, the Angle of contact, (which with that of the Semicircle makes the right Angle PAC,) being, as he supposeth, an angle of some bignesse. Peletari­us is of opinion, that the Angle EAC, is equall to PAC; [Page 35] and not a part of it, but the whole; the supposed Angle PAE being, as he thinks, no Angle, or an angle of no bignesse.

This being the state of the controversy: I take Peleta­rius his part. And my first argument is from the nature of a Plain angle, which Euclide defines to be the mutuall inclina­tion of two lines &c. And therefore the lines EA, PA, in the point of concurse A, not being at all inclined each to other; but in the same coincident position without incli­nation; they do not contain an angle. The tendency of the circumference EAN, before it comes at the point A, is to­wards the tangent PT; when it's past that point, the ten­dency is from it; but in the point A, it doth neither tend toward it, nor from it, nor crosse it; and therefore must be either in parallell position, or coincident. And this argu­ment is managed in the 3 and 4 Chapters.

You tell us to this, that Peletarius did not well—Cla­vius did not well—Euclide did not well—That is, You think so. And it's like, You think, I have done worst of all. But I doe not much stand upon your thoughts.

You say particularly, p. 26. That I am more obscure then Euclide. (It may be so.) That I am contrary to him, (That you are to prove.) That I make two lines when they ly upon one another, to lye [...], without inclination: I do so. Shew me if you can, where Euclide saith the contrary. Tell mee, where lines, either in the same or in parallell positions, are by Euclide said to incline or be inclined each to other? to thwart, or crosse each other? According to Euclide, you say, an angle equall to two right angles should be the greatest inclina­tion, and so the greatest angle, where as, by this [...], it should be the least that can be, or rather no angle. Shew me where ever Euclide doth acknowledge any angle to be e­quall to two right angles? or, which is all one, that too contiguous parts of the same right line, are by Euclide said to be inclined to each other, or to contain an angle? Nay he says the quite contrary. For in his definition of a plain angle, he makes it one qualification, that the lines containing it, must be such as are non indirectum positae. And therefore two streight lines in directum positae (such as those must needs be, which are to contain your supposed angle equall to two right angles) cannot, by Euclides definition, contain an [Page 36] Angle. We do not therefore in this disagree.

You adde farther, (as giving this for lost, Though it be gran­ted (as it must needs be) that there be no inclination of the cir­cumference to the tangent (and consequently no Angle; by that definition of Euclide.) yet it doth not follow that they forme no kind of Angle. And why doth it not follow? Be­cause say you, Euclide there defines but one of the kinds of a plain angle. That Euclide doth not there define, an angle in general or all kinds of angles, is very true; for there be many other both superficiall and solid angles, which are not plain angles: But that he doth there define a plain angle in generall, and therefore all kinds of plain angles is evident frō his words. For in the eighth Definition he defines a plain Angle, (as the genus) A plain Angle, saith he, is the mutuall inclination of two lines, &c. and then in the next definition, defines a right lined plain angle, (as one species of it) viz. when both these lines be right lines. It's manifest therefore that he in­tended in the former definition to define a plain angle in generall; whether the lines containing it be streight or crooked. And therefore since the angle of contact falls not within that definition, it is not to be reputed a plain angle. And so my first Argument stands good.

The second, is an Argument of Peletarius, drawn from the first Proposition of the tenth of Euclide; (and enfor­ced likewise by me, from the second proposition of the first of Archimedes de sphaera & cylindro:) To which Clavius rejoyns, that the proposition is to be understood only of Homogeneous quantities; &; of such, grants the argument to pro­ceed. And you; supposing these to be Heterogeneous, say, it is like as to seek for the Focus of the Parabola of Dives and Laza­rus. To your scoff at Scripture, I reply only this, that the Focus of that Parabola is a bad place to be in, & wish you to take heed of it. With Clavius, we joyn issue; granting the propositions cited not to be understood of Heterogeneous quantities; and prove these not to be such; by this argument: If any thing make the angle of Contact PAE, to be heterogene­ous to a rectilineal angle; it must be the crookednesse of the side AE. (for if that side were streight; the angle were recti­lineall;) But that hinders not, (for I prove the angles CAE, and SAE, notwithstanding the same side AE, are homogeneous to right lined angles; as you grant, [Page 37] and Clavius could not deny:) Therefore nothing hinders. And this is done in my fift Chapter.

What Clavius had brought to prove the contrary, is an­swered in the sixth Chapter. And if you had not thought his arguments to be all answered, you should have done well to have undertaken the managing of some one of them. That you mention, doth only, upon supposition that it is a quantity, prove it to be heterogeneall; because not Homogeneall. Which is to beg the question. For we, as well as he, deny it to be a Homogeneall quantity; and therefore conclude it to be no quantity; for heterogeneous it is not. His argument amounts but to this, 'Tis not a quantity Homo­geneous, (by 5 d 5) therefore 'tis a quantity Heterogeneous. I grant his Antecedent, but deny the Consequence (which proceeds only upon supposition that it is a Quantity, which is the thing in question.) He should first have proved it to be a Quantity; which Peletarius and I deny.

In the seventh Chapter I prove, by other arguments, that if the angle of Contact be an angle, it must be homogeneous to rectilineal angles.

1. That which may be added to, or subtracted from, a right lined angle, is homogeneous to it: Because Heteroge­neous quantities are not capable of addition, or subduction. (And this you grant.) But so here; For PAE if an angle, may be added to the angle SAP, making the angle SAE; (which therefore, saies Clavius, is bigger then SAP;) and taken from the angle PAC, leaving the angle EAC, (which therefore, saies Clavius, is lesse then PAC;) There­fore, if an angle, it is homogeneous You grant the major; and deny the minor: that is, you deny the only foundation upon which Clavius builds his opinion; and so yeeld the cause. For he doth upon no other ground maintain the an­gle of the semicircle EAC, to be lesse then the right an­gle PAC, but because the angle of Contact PAE, is a part of it, and therefore the other part EAC, must be lesse then the whole.

2. Those which are to each other as Greater and Lesse, have proportion each to other; and are consequently ho­mogeneous; by the third def. of the fift of Euclide. (and this you grant.) But, the angle of Contact PAE, is lesse then the angle SAP; by the 16 of the third of Euclide; (for his words are, that it is lesse then any right lined angle.) [Page 38] And this Clavius would not deny, but oft affirmes it. Therefore they be homogeneous. All that you have to say is, that though Euclide say it is lesse, yet (to your understan­ding) he doth not mean so. But doth he not, to your understan­ding prove, that the least right lined Angle is bigger th [...]n it? and if so, supposing it to be angle, must it not be Homoge­neous? even by your own concession.

To the third and fourth Arguments in that Chapter, You object nothing; and therefore those, I suppose, you al­low to conclude what is contended for. viz. that the angle of Contact is not Heterogeneous to other plain angles: and therefore, this being the only exception, my first main ar­gument stands good.

The Eight Chapter you say, contains nothing but the Authority of Sir Henry Savile. And you say true; for no more was intended.

The third main Argument is proposed in the ninth Chapter; Because the Angles of semicircles (because like seg­ments) are equall. Whence Peletarius infers, that the Angle of Contact is no quantity. Clavius grants the consequence of the Argument; but denies the Antecedent: affirming DAC (fig. 2,) to be lesse then EAC, though both angles of Se­micircles, this of the bigger, that of the lesse. To this you say, that in my 9 and 10 Chapters I prove with much adoe, that the Angles of like segments are equall: (if I prove it, though with much adoe, then I carry the cause; for that was the on­ly thing denied by Clavius. But you adde) as if that might not have been taken gratis by Peletarius, without demonstration: (Implying thereby, that I need not have proved it.) And this is like your selfe, who care not how you abuse your English Reader. The case is thus. Peletarius had taken it gratis, as a thing that in reason should not have been deni­ed him. Yet 'tis denied by Clavius; and the whole issue of the cause put upon it. Had I not reason then to prove it? Yet I prove it thus; First, that Peletarius had reason to take it gratis, and that it was unreasonable in Clavius to put him upon the proofe; and this is done in the ninth Chapter. But then, because he had denyed it, how unreasonable soe­ver it were so to doe, and withall put the whole issue of the cause upon it; therefore in the tenth Chapter I undertake to prove it by argument. And you grant, I prove It. What should I doe more?

[Page 39] The 11th Chapter clears the same argument from a seem­ing difficulty. And you say nothing to it, but that the ob­jection was of no moment, and needed no answer.

To the Arguments of the 12 and 13 Chapters, (and those are a pretty many, for in one of them are contained six,) your answer is (and that's all) that they 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 (I will not say a lye, though that also be your language, but) manifestly false; and you could not but know it so to bee. For there is not, in those whole Chapters any such thing assumed for proofe; nor doth any one of those arguments depend upon any such notion; but let your notion of Angle be what it can, my arguments will hold their weight. This therefore is nothing but a notorious untruth, wherewith (because you had nothing to say to the Arguments) you meant to abuse your English Reader. But suppose I had said, (as it is like I may sometimes) that an Angle is contained by, or between the two sides; is this any more then to say that the two sides contain the Angle? And doth not every body say so as well as I? Are they not Euclide's own words, 9 d 1. When the lines ( [...]) containing or comprehending the angle be right lines, the angle is called Rectilineall? Nay are they not your own words, cap. 14. § 7. Anguli qui rectis continentur lineis, rectilinei; qui curvis, anguli curvilinei sunt; qui recta & curva continentur, misti? What a doe then doe you make for nothing? Perhaps the word between troubles you. But is not by and between in this case all one? It is to mee; and if you doe not like the one word take the other; 'tis all one to mee, (But, by the way, the phrase, contain be­tween, is not so much as once used in either of those Chap­ters: and therefore that cavill is to no purpose at all, but to abuse your English Reader, who cannot contradict you.) And doth not Euclide's word [...] signify to contain between? and [...], the lines which do comprehend, (or contain between them) the Angle? Nay doe not you your selfe use it again and again, cap. 14. § 9. Ad quantitatem anguli neque longitudo, neque aequalitas aut inaequalitas linearum quae angulum comprehendunt, quicquam faciunt, idem enim angulus est qui comprehenditur inter AB & AC, cum eo qui comprehenditur inter AE & AF, vel inter [Page 40] AB & AF. And again cap. 14. parag. 16. Angulus qui cont [...]netur inter AB & eandem AB &c. And soon af­ter, angulus qui sit inter GB & BK, aequalis est angulo qui sit inter GB & arcum BC. (which is also retained in the Eng­lish.) And so elsewhere. But say you, To say that an angle is contained between the lines that make it, is as much as to say, that it is a plain superficies. And was it so when you wrote those passages last cited? Were you then of opinion that the Angle contained or comprehended between the lines AB and AC, (as you there speak,) was a plain superficies? Or, if those words do not import so much when you speak them, why should you think they doe when I speak them? But, it seems, having nothing else to cavill at, you thought fit to tell your English Reader, who must take it upon trust from you, That I affirme a plain angle, to be a plain superfici­es, because, forsooth, I say (as Euclide and all others doe, and your selfe among the rest,) that it is contained between two lines. You might, with much better Logick, have concluded the contrary For though Euclide, as I doe, said that two streight lines may comprehend an angle, 9 d 1. yet he affirmes, that two streight lines cannot comprehend a superficies, 10 ax 1. And therefore, when I affirme that an angle may be comprehended between two streight lines, you might (at least a sober-man might) have concluded, that I did not take it for a superficies, because that cannot be com­prehended by fewer streight lines then three. But enough of this. And, if this be all you have to say against the Ar­guments of the 12 and 13 Chapters, I hope they may passe for current: and be judged to conclude the cause.

To that of the last chapters (as you speak) where I prove the same from a proposition of Vitellio: (which proposition of his I doe also vindicate from an exception of Cabbaeus:) You object nothing, but that I defend Vitellio without need (and yet I had there told you, that Cabbaeus denies his argu­ment:) for say you there is no doubt but whatsoever c [...]ooked line be touched by a streight line, the angle of contingeuce will nei­ther adde any thing to, nor take any thing from a Rectilineall Right Angle; That is, there is no doubt but that Clavius was in the wrong, and I in the right, all the way: for this was the very thing that was in controversy betwixt us. And so you have brought your confutation to a good Catastro­phe. And thus much for the Angle of Contact.

SECT. V. Arithmetica Infinitorum, Vindicated.

LEt's see now what you have to say against my Arith­metica Infinitorum. Five propositions you there take to taske; the first, the third, the fift, the nineteenth, and the thirty ninth.

The first you, you say, is this Lemma; In a series of quanti­ties arithmetically proportionall, beginning with a point or cyphar, (as for example 0, 1, 2, 3, 4, &c.) to find the proportion of the Aggregate of them all, to the Aggregate of so many times the greatest as there are termes. Very true, this is the first propo­sition; what then? This you say, is to be done by multiplying the greatest into halfe the number of termes. What is to be done thus? finding the proportion? No such matter. That's the way to find the summe, (upon supposition that the propor­tion is already known to be, as 1 to 2,) not to find out what is the Proportion, (supposing it yet unknown,) which the Lemma proposeth to be inquired, and finds it to be as 1 to 2. But 'tis well however that you can at length tell how to gather the summe of such a proportion (after I had taught you in my Elenchus,) for you were, it seems, of an other opinion, when you said Cap. 16. parag. 20. In hujusmodi progressione (0. 1. 2. 3. 4. &c.) summa nume­rorum omnium simul sumptorum, aequalis est semissi ejus nu­meri qui fit a maximo termino ducto in minimum, id est, hoc loco in ciphram. Which you now confesse pag. 41. to be a great error.

You go on, and say, The Demonstration is easie. But how, say you, do I demonstrate it? You should have asked rather, How I find it, (then how I demonstrate it:) for that was it the Lemma proposed. But you are so well acquainted with the Analyticks, that you know not how to distin­guish between the [...], and the [...]. by the first we find out the solution of a Problem; by the second we prove it. Now if you can find a more naturall [...], or way of finding out the solution of this and the other Problems (for I was here shewing a generall method for this and others that follow,) pray let us know it in [Page 42] your next, and I shall thank you for it. But doe not talk of Demonstrating, when I propose the finding out; for, if you doe, I shall say, that's nothing to the purpose.

You tell us next, that an Induction, without a Numeration of all the particulars is not sufficient to inferre a Conclusion. Yes, Sir, if after the Enumeration of some particulars, there comes a generall clause, and the like in other cases, (as here it doth) this may passe for a proofe, till there be a possiblity of giving some instance to the contrary; which, here, you will never be able to doe. And if such an induction may not passe for proofe, there is never a proposition in Eu­clide demonstrated. For all along he takes no other course then such, (or at least grounds his Demonstrations on pro­positions no otherwise demonstrated.) As for instance; he proposeth it in generall 1 e 1. to mak an Equilater triangle on a line given. And then shews you how to doe it upon the line [...] which he there shews you: and leaves you to supply, and the same by the like meanes, may be done upon any other streight line; and then inferres his generall conclusion. Yet I have not heard any man object, that the induction was not sufficient, because he did not actually performe it in all lines possible.

You then aske, whether it be not also true in these numbers, 0, 2, 4, 6, &c. or 0, 7, 14, 21, &c? Yes, and in these also (which perhaps you would little think) 0, √2, √8, √18, √32, √50, &c. But why, say you, doe I then limit it to the numbers 0, 1, 2, 3, 4, &c? I should rather wonder why you think I doe. No wise man would have thought it; when he sees that I speak in generall of any series in continued Arithmeticall progression, that begins with a point or ciphar. And there had been no colour, for you to aske such a question, if, in reciting my proposition, you had not in stead of saying as, for example, said only as. For doubtlesse those are continued. Arithmeticall progressions be­ginning with a Ciphar; and they are also juxta naturalem nu­merorum consecutionem, that is, like progressions to those of the naturall numbers 0, 1, 2, 3, &c.

You ask then (very wisely) whether it will hold in 0, 1, 3, 5, 7. I answer, no. (nor is any such thing affir­med.) because 0, 1, 3, are not Arithmetically proportio­nall.

[Page 43] And when you have done Catechizing us, you then conclude, well, the Lemma is true: (in good time!) that is as much as to say, you were willing to shew your teeth though you cannot bite. What's next?

The first Theorem that I draw from it, is, you say, that a Triangle to a Parallelogram of Equall base and Altitude, is as one to two. Well; and what of this? The conclusion you say is true. Very good, Then two of the five have scaped alrea­dy. But you doe not like of the Demonstration, because of the words as it were (and the like exception you took before, at the word scarce,) which you say, is no phrase of a Geometrician. Yes Sir; a very good phrase, if the Geome­trician doe determine precisely (as I have done) how much by that quasi he intends to limit the accuratè. For I doe not suffer either the scarce, or the as it were, to runne at randome without bounds. I tell you that by quasi linea, or vix aliud quam linea, I doe not meane precisely a Line, but a Parallologramme whose breadth is very small, viz. an ali­quot part of the whole figures altitude, denominated by the number of Parallelogramms. (Which is a determination Geometri­cally precise.) And by triangulum constat quasi, &c. I tell you that I mean, that a Figure, consisting of such Parallelo­gramms, inscribed in a Triangle, whose difference in bignesse from that Triangle is lesse then any assignable quantity, is so con­stituted. As you may see precisely determined in the place to which this Demonstration referres. The words there­fore vix and quasi, being thus determined, are here very good Geometricall words; and your cavills come to no­thing.

My fift Proposition is, you say, The Spirall line is equall to half the circle of the first Revolution. But, in saying so, you say not true. For that is not my proposition; but one of your own, patched together, after your fashion, out of my fift and sixth put together. And, as it stands, I cannot own it. The words of the first revolution, should have been adjoyned to the Spirall line, not to the word circle: to shew how much of the Spirall line is intended. And, instead of halfe the circle, you should have said, halfe the circumference of the first circle; for I did not compare the Spirall line with the Circle (that is, a line, with a Figure,) but that Spirall line with the circumference, (viz. as 1 to 2,) and the Spi­rall Figure, with the Circle, (viz. as 1 to 3) And the cir­cle [Page 44] you intend, is not by mee, or by Archimedes, called the circle of the first revolution, but the first circle; which is con­terminate with the first revolution of that Spirall line. But if you will needs have my fifth and sixth Propositions put together, pray let it be thus, That so much of the Spirall line, in the sense of the proposition, as belongs to the first revolution, is equall to halfe the circumference of the first circle. Now in what sense I take the words Spirall line, in these propositi­ons, is so clearly defined in the Scholium of pag. 10. that it is not possible for any man, unlesse willfully, to mistake mee. viz. That I doe not intend the true spirall of Archime­des, but the aggregate of the arches of infinite like Sectors, con­stituting a figure inscribed within that Spirall of Archimedes. And thus, both those and the other Propositions are true. Nor can you deny them.

But now because you have nothing to say against the Proposition in the true sense of it: you will needs per­swade mee, (because you know what I meant better then my selfe.) that I did not so mean, nor would be understood, so, as I said, I meant and desired to be understood; but that I meant somewhat else. And you have this ground for it; Because in the sense wherein I said I would have it un­derstood, the proposition is true; but you have a desire that it should be false; and therefore it must be understood in some other sense. Let's see therefore what it is, that I may at length know what it was I meant. What Spirall is meant, you say, we shall understand by the construction. Yes, if you take in the whole construction; but not by a peece of it. My construction begins thus, Let a streight line MA, turned about the center M, be supposed, by a uniforme motion, to describe, with its point A, the circumference AOA; whilest a point in the same line, so carried about, is supposed to describe a Spirall line MTA. This is the first part of the constru­ction; (and from hence I inferre, by the way, that the streight lines MT will be proportionall to the angles AMT, and the Archs AO.) This therefore, say you, is the Spirall of Archimedes. Very true: and it was intended so to be. But let's goe on and heare the rest of the construction, (for hi­therto we have had but a part of it.) Which if you may be believed, is this; inscribing in the Circle an infinite multi­tude of Equall angles, and consequently an infinite number of Sectors, whose Archs will therefore be in Arithmeticall propor­tion; [Page 45] (which, you say, is true;) and the aggregate of those archs equall to halfe the circumference AOA. Which, you say, is true also. But, if I had said so, I had lyed; for I know it to be false: (in you it was only an Error, or, as you use to call it, a Negligence; because you thought it had been true.) For this is neither my construction, nor are those things true which you affirme. For, if in a circle, there be a num­ber of Sectors inscribed (whether finite or infinite) both those Sectors, and the Archs of them, are proportionall to their Angles; and therefore, the Angles being equall, the Archs will be equall also, and not Arithmetically proportio­nall: And the Aggregate of those archs, will not be equall to half the circumference AOA, but, to that whole circumference. But my construction was this; Within the Spirall line, de­scribed as above, supposing an infinite multitude of Sectors conti­nually inscribed on equall angles, their Radii AT will be Arith­metically proportionall, viz. as 0, 1, 2, 3, and consequently their archs will be so too. And this, I suppose, is that which you intended to grant as true; Being the result of the second part of my construction. Then followes the third part of the construction (which hath the nature of a Definition; which, till thus much of the construction was past, could not conveniently be expressed,) The Spirall line (intended in the proposition, not that of Archimedes) is supposed therefore to be made up, of the archs of those infinite Sectors, arithmetically proportionall (for so they are already proved to be) beginning with a point or o. (And then goes on the Demonstration.) But the circumference consists of so many archs equall to the biggest of them; as is evident. Therefore (by the se­cond Prop.) that to this, is as one to two. Which is the thing to be proved.

Now this, to some capacities, though not to M. Hobs, would have been easy enough to understand. Yet that it might not lye open to any cavill, or misunderstanding; I thought fit in a particular Scholium, to expresse my mean­ing so fully, as that there might be no possibility of mi­staking what I intended. (And, the truth is, I would have had that Scholium Printed next after the Fifth Proposition. But finding, that, through some neglect, the Printer had there left it out, I gave him order to put it in, at the next convenient place; which was, in the next sheet, at the end of the 12 Proposition: a place proper enough for it.) And [Page 46] you cannot deny, but that my words there, be plain enough to be understood, and not capable of any distortion to any other sense. And that the Proposition in this sense is true, you cannot deny; and so much (I suppose) you intended to grant, when you said, That the aggregate of those archs is equall to halfe the circumference AOA, is true also. Three there­fore of the five are already found to be true.

My 19. Prop. you say, is this Lemma. In a series of Quan­tities, beginning from a point or ciphar, and proceeding accor­ding to the order of square numbers, (as for example 0, 1, 4, 9, 16, &c.) to find what proportion the whole series hath, to so ma­ny times the greatest. 'Tis true; this is my 19 Proposition. What then? I conclude, you say, the proportion is that of 1 to 3. No Sir, I do not conclude it to be so. I conclude it to grea­ter then that of one to three. My words are these, Ratio pro­veniens est ubique major quam subtripla Excessus autem perpetuo decrescit prout numerus terminorum augetur, &c. ut sit rationis provenientis excessus supra subtriplam, ea quam habet unitas ad sextuplum numeri terminorum p [...]st o. That is in plain English thus. The series so increasing, is alwaies more then a third part of so many times the greatest. For it containes evermore, a third part thereof, and moreover, an aliquot part denomina­ted by six times the number of termes following the o. And is not this true? can you have the face to deny it? Wee'l try if you please; take your own instances. Let the series be of three termes 0, 1, 4, the aggregate is 5: the greatest so many times taken, that is 3 times 4, is 12. I say 5 contains of 12, a third part (viz. 4 = ⅓ × 12.) and moreover a part denominated by 6 times 2, (for there are two termes besides o.) that is a twelfth part of the number 12. (viz. 1 = 1/12; × 12.) And is not this true? is not 5 = 4 × 1? Again, let the termes be four, viz, 0, 1, 4, 9. = 14. and the greatest so many times taken 9, 9, 9, 9. = 36. I say that 14 containes ⅓ of 36, (that is 12,) and moreover, (because 3 × 6 = 18) 1/18 of 36, (that is 2.) And is it not true, that 14 is equall to 12 + 2? I think it is. Again, let the termes be five, viz. 0, 1, 4, 9, 16, = 30. and there­fore so many times the greatest is 16, 16, 16, 16, 16, = 80. I say that 30 contains, ⅓ of 80, that is 26 2/1 & moreover 2 ¼ of 80; (because 4 × 6 = 24) that is 3 1/ [...]. And is it [Page 47] not so? is not 26 ⅔ + 3 ½ = 30? you may try it farther if you please. My skill for yours, 'twill hold. (And thats fair odds in a wager.) The Proposition therefore is true thus farre.

Well but I said farther; That though the Proposition be still more then the subtriple; yet the excesse doth still de­crease. Doe you not think that true too? if not, let's try. if the termes be three, you see the proportion is as 5 to 12, that is as ⅓ + 1 ½ to 1. if four, the proportion is as 14 to 36, that is ½ + 1 ½ to 1. if five, then as ⅓ + 1/24 to 1. &c. As we have seen already. But the proportion of ⅓ + 1 1 [...] to 1, is more then of ⅓ + 1 ½ to 1, and yet this more then ⅓ + 1/24 to 1. and so forward. But you forsooth would faine perswade us, that as the number of termes increase, so the proportion increaseth. As if the proportion of ⅓ + 1/24 to 1, were greater then that of ⅓ + 1/18 to one. and yet would pretend to understand proportions, and tell us what M. Oughtreds meaning is &c. as if we did not under­stand M. Oughtred, and his meaning too, better then you. But, by the way, I wonder how you durst touch M. Ough­tred for fear of catching the Scab. For, doubtlesse, his book is as much covered over with the Scah of Symbolls, as any of mine. Which makes me think, you understand his and mine much alike.

I adde farther, (though not in this proportion,) that the proportion doth so decrease, as that (though it be never lesse then a subtriple, yet) the excesse above the subtriple, will by degrees vanish, as the number of termes increaseth, till it grow lesse then any assignable quantity. and it is pro­ved thus: Because the second fraction, which with ⅓ makes up the antecedent of the proportion, whose conse­quent is 1; doth proportionally decrease, as the number of termes doth increase. And therefore, as the number of termes may increase beyond any assignable number: so may the excesse decrease below any assignable quantity. And, if the number of termes be supposed infinite, the proportion will be infinitely near to the subtriple.

But you tell us upon this, (and wittily doubtlesse, as you suppose, by a sly transition from the phrase infinitely near, to that of eternally nearer,) you tell us, I say, that if the [Page 48] proportions come eternally nearer and nearer to the subtriple, (supposing them at first bigger then it, which you should have added, for else the case alters,) they must also come eternally nearer and nearer to the subquadruple, and so to the subquintuple, &c. I grant it. But what then? it doth not follow, that if it come eternally nearer to the subquadruple, then it will come infinitely neare, or nearer then any assignable difference; for it can never, upon that supposition, come nearer to it then the subtriple. Like as the Hyperbole, doth eternally come nearer and nearer to its Asymptote, and consequently, will eternally come nearer also to a parallell that lyes beyond it; but not infinitely near; for, since that it never pas [...]es the Asymptote, though it doe eternally approach, yet it never comes nearer to that Parallell, then the Asymptote doth. And indeed if it should, it could not eternally approach to the Asymptote, but so soon as it is passed it, it would then grow farther and farther from the Asymptote, while it doth approach to the parallell beyond it. And, in the pre­sent case, this proportion which doth eternally approach, and may come infinitely neer to the subtriple, doth indeed eternally approach, but not come infinitely near, to the sub­quadruple. For it never comes nearer to it, then is the sub­triple. And I would not have you think us such weak Ma­thematicians, or such young birds, as to be caught with such chaffe, or not see through so weak a fallacy as that is. And therefore when you inferre, that we may as well con­clude thence, that the proportion, is as one to four, or one to five, &c. (supposing the number of termes infinite) as to con­clude, it is as one to three: We suppose that you would have us think withall, either that you doe not speak in good earnest, or else that you are not well in your wits: For o­therwise, doubtlesse you cannot be so simple as to believe it.

There is but one Proposition more that you undertake to deal with. Which is the 39, viz. this Lemma, In a se­ries of quantities beginning with a point or cipher, and proceeding according to the series of Cubick Numbers, (as for example 0, 1, 8, 27, 64, &c.) to find what proportion the whole series hath to so many times the greatest. And you deal with this, just as you did with the last. First you mis-recite it, and then say 'tis false. I conclude, you say, that they have the pro­portion of 1 to 4. Which is false, I do not so conclude; but [Page 49] that it is more then so; viz. it contains a fourth part, and moreover another aliquot part, denominable by four times the number of termes following the cipher. That is, if the termes be three, the proportion is as ¼ + 1/8 to 1. if four, it is as ¼ + 1/12 to 1. if five, it is as ¼ + 1/16 to 1. And so forward. And if you make triall, you shall find it so to be. (For 0 + 1 + 8 = 9; and 8 + 8 + 8 = 24. Now 9 is equall to ¼ + 1/8 of 24, viz. to 6 + 3. So 0 + 1 + 8 + 27 = 36. and 27 + 27 + 27 + 27 = 108. Now 36 is equall to ¼ + 1/12 of 108, viz. to 27 + 9. So 0 + 1 + 8 + 27 + 64 = 100; and 64 + 64 + 64 + 64 + 64 = 320. Now 100 is equall to 1 [...] + 1/16 of 320, viz. to 80 + 20. And so of the rest.) If you think it to be otherwise; shew, if you can, one in­stance to the contrary. The Proposition therefore is true; but you had not the honesty to report it right. (or else your witts were at wooll-gathering.) And so of all those five propositions which you have taken to taske, there is not any one faulty.

And I should now have done with this businesse, but that I discern, upon these two last Propositions, your rea­son why you are so much out of charity with the Symbo­lick tongue. 'Tis very hard, you have told us diverse times; yet here, it seems, you mean to try what you could doe at it. And 'tis to be hoped, you may, in time, learne the language; for you be come to great A already. (But truly were it not that you must defend your reputation, you tell us, you should not have done so much.) But such pittifull work dost thou make with poor great A, and to so little purpose, that if there were no better use to be made of Symbols, then so, it's pitty they should ever be used at all. And truly, were I great A, before I would be willing to be so abused, I should wish my selfe little a, an hundred times. Yet thus much, I confesse you have done: You have clear­ly convinced me, that you have reason not to be much in love with Symbols. For to what purpose? since you can neither use, nor understand them. And truly, upon this very account, I am apt to think, that much of your 13 chapter, is none of your own.

[Page 50] Well; Arithmetica Infinitorum is come off clear. Wee'l see next what you have to say to Conick Sections.

SECT. VI. My Treatise of Conick Sections vindicated.

AS for my Treatise of Conick Sections, you say, it is so co­vered over with the Scab of Symbols, that you had not the patience to examine whether it be well or ill demon­strated. A very fine way of confutation; and with much case. You have not the patience to examine it, (that is, in plain Eng­lish, you do not understand it,) Ergo I have performed nothing in any of my Books (for that is the inference in the same page, p. 49.) [...]. But, Sir, must I be bound to tell you a tale, and find you ears too? Is it not lawfull for me to write Symbols, till you can understand them? Sir, they were not written for you to read, but for them that can.

However, whether you understand it or not, yet some­what you observe, you say, (though you have not the patience to examine whether it be well or ill.) Pray lets heare your Observations; (for they be like to be wise ones.)

You observe, you say, that I find a Tangent to a point given in a Section, by a Diameter given: (very good; There's no hurt in that, I hope, is there?) and in the next Chapter, I teach the finding of a Diameter. You should have done well to have told us, where to find those Chapters. For I do not remember, that that Treatise is at all divided into Chapters. Well! but suppose I had in one Chapter, by the help of Diam [...]ter given, found a Tangent; in another Chapter, by the help of a Tangent, found a Diameter: Had there been any hurt in all this?

You observe also, you say, that I call the Parameter an Ima­ginary line, as if the place thereof were lesse determined then the Diameter it selfe. (But did you observe, whether I did well or ill, so to call it?) And then, you say, I take a mean pro­poirtionall between the intercepted Diameter, and its contiguous ordinate line, to find it. Pray tell me where you observed that. For, had I observed it, I should have observed it as a great fault; and not said as you doe, And 'tis true, I find it. For, believe mee, that is not the way to find a Parameter. Nor [Page 51] doe I give you any such direction. You may (in a Parabo­la) find the Parameter by taking a third Proportionall, but not by taking a Mean Proportionall, to those two lines. You say, The Parameter hath a determined quantity. Yes doubt­lesse. And, in some Writers, it hath a determined Positi­on too (viz. in the Tangent of the vertex:) But because I make no use of any such position, I give you leave either to draw it where you will, or not to draw it at all. For by a Parameter, I mean only, a line of such a length, where e­ver it be; whether at Rome, or Naples, or in M. Hobs his brain. They that make use of the Parameters position, as inferring any thing from it, must assigne it a certain place. I make use only of its bignesse, and therefore care not where it stand.

Lastly you observe, you say, that I doe not shew how to find the Focus. (nor was) bound to doe.) And that's all.

And is not this a worthy confutation? Yes doubtlesse; worthy of you; For how could you else inferre, That I have performed nothing in any of my Books; if you had not confuted them all.

And thus much of those three Treatises. Which, you see are come off safe and sound, without the losse of leg or limb. And with this advantage, (if M. Hobs his testimo­ny, in point of skill, were worth any thing,) that they have obtained from him as ample a Testimony as he is able to give. viz. That when as he hath imployed the utmost both of his skill and malice, to find what faults he could, he hath not been able to discover any one: (which Testi­mony, from a considerable adversary, would have been worth something; but, from M. Hobs, J confesse, it signifies little:) and all the attempts he hath made to that purpose, have not been so strong, but that a Butter-fly might have broken through them.

SECT. VII. Concerning the Eighth Chapter in M. Hobs his Book of Body.

HItherto we have tryed your skill and valour in point of Assault: And found, that, though you charge as furiously as if you meant to look us dead; yet you come off as poorly as a man could wish. J am apt to think, [Page 52] that your weapons were not well made, and that your Musket was of a bad bore, (for it hath done no executi­on, save only in the recoile;) or else you held it by the wrong end, (like the Jack-an-Ape that peep'd in the gunns mouth to see the bullet come out,) for though it made a great noyse, yet it hath hurt no body but your selfe. My Colleague and I, are both of us alive, and live-like; and Eu­clide sleeps as securely as he did before.

Wee'l try now, how good you are in point of Defense; and see how you can defend your Corpus against my Elen­chus. Perhaps you may have better luck at that.

But, mee thinks, it begins unluckyly. Before you fall to work with Elenchus; you traverse your ground, that you may take it to the best advantage: and distinguish, between faults of Ignorance, and faults of Negligence, (pag. 9.) you tell us that from right Principles to draw false Conclusions (which you are very good at) are but faults of negligence and humane frailty, and such as are not attended with shame, &c. That 'tis only as being lesse awake, &c. (and yet think much to be told, that you discourse as if you were halfe a sleep:) And much more your preface to that purpose. As if the first consideration to be had, in the choice of your ground, were, whence you might with best advantage runne away; (a businesse of ill Omen in the beginning of a Combate;) that when you shall be forced to quit your ground, you may, at least, shew a fair pair of heeles.

My Elenchus, as I then told you, begins at first with some lighter skirmishes, shewing how unhandsome some of your Definitions and Distributions are, giving instance in a few; which though faults had enough, yet are but small ones in comparison of those greater which follow, in false Propositions and Demonstrations.

I begin with that of Chap. 8. § 12. Where you define a line, a length, a point, in this manner. If when a Body is mo­ved, its magnitude (though it alwaies have some) be not all con­sidered, the way it makes is called a Line, or one single dimensi­on; the space through which it passeth is called Length; and the Body it selfe a Point. But what if a Body be not moved? i [...] there then neither Point, nor Line, nor Length? A Point there may be, which is not a Body, much lesse A Body moved: and a Line, or Length, through which no body passeth: And therefore the definitions are not good, because not [Page 53] reciprocall. The Axis of the Earth, is a Line, and that line hath its Length; yet doe I not believe that any Body doth, or ever did, passe directly from the one to the other Pole, to describe that Line. The notion therefore of Motion or Body moved, I then said, was wholly extrinsecall and acci­dentall to the notion of Line, or Length, or of a Point; no waies essentiall or necessary to it, or to the understanding of it: and that therefore it was not convenient, to clog the definitions of these, with the notion of that.

To this you answer, (having waved first, what you at­tempted, as from the example of Euclide,) That, how ever it may be to others, it was fit for you to define a Line by Motion. And I doe acquiesse in that Answer. For, though it would not become any man else so to define it; yet it becomes M. Hobs very well; as well agreeing with his accuratenesse in other things.

I said farther, That the distance of two points though resting, was a Length, as well as the measure of a passage, (and there­fore the notion of a body moved, not necessary to the definition of Length.) To which you answer, that the distance of the two ends of a thread wound up into a Clew, is not the length of the thread. Much to the purpose.

I asked, Whoever defined a Line to be a Body? And you tell mee, you take it for an honour to be the first that doe so. And you may, for ought I know, have also the honour to be the last. And as to that long rant against Euclide; That if a Point have no parts, and so no magnitude; A Line can have no breadth, nor can be drawn (mechanically you mean;) and then there is not in Euclide one Proposition demonstrated, or de­monstrable. We doe not think, that your asseveration a suf­ficient argument, more than we take a word of your mouth to be a slander; but desire some better proofe of that con­sequence before we assent to it. You tell us else where, that A Point is to Magnitude, as a ciphar is to Number (cap. 16 art. 20.) And yet I suppose you will not say that, unlesse a Ciphar have some multitude, as well as a Point some Magni­tude, there is not in Euclide any one Proposition demonstrated. And to the same purpose is that Cap. 14. § 16. An angle of contingence, if compared with an angle simply so called how little so ever, hath such proportion to it, as a point to a Line, that is, (neque rationem, neque quantitatem ullam,) no proportion, nor a­ny quantity at all. Which how well it agrees with your o­ther [Page 54] doctrines, it concerns you to see to, (for if a Point to a Line, have no proportion nor any quantity at all, then is it not a Part thereof;) and how little this comes short of what you so often rant at, as making a Point to be no­thing.

Again, whereas in the place cited (both in Latine and English) you thus define; The Way (of the Body so mo­ved) is called a Line, or one single Dimension; and the Space through which it passeth, is called Length. I argued, that Length, doubtlesse, was one single dimension; and therefore, if one single dimension, as in your definition, be the same with Line; then Length will be a Line, and not therefore need a se­cond definition. Now, to help the matter, in your Lessons; you define thus, The Way is called a Line; and the space gone over by that motion, Length or one single dimension. Whence my argument is yet farther inforced, If one single dimension signify the same with Line, (as in your Book;) and also the same with Length, (as in your Lesson;) then Line and Length signify with you the same thing; & therefore with you, should not have had two distinct and different definitions. Which I take to be ad hominem, a good argument. You an­swer, that to say Line is Length, proceeds from want of under­standing English. It may be so. But what's this to the clear­ing of your Definitions? where those two words are made equivalent. Yet farther, chap. 12. parag. 1. there are, say you, three dimensions, Line (or Length,) Superficies, and Solid. where again Line and Length are made the same. Now whether or no Line be Length, or whether it be for want of understanding English that you affirme it, it concernes you to cleare; for 'tis you, not I, that affirme it so to be.

Your next definition is of Equall Bodies; which you thus define, Equall Bodies, are those which may possesse the same place. Against which definition J objected, That you should rather define a thing, by what it is, then by what it may be: That the notion of Place, was wholly extrinsecall to the notion of Equality; for Time, Tone, Numbers, Pro­portions, and many other quantities are capable of Equali­ty, without any connotation of Place; and the notion of Equality in them, is the same notion with that of Equality in Bodies; (else how can you say, that two Equall Num­bers, and two Equall Bodies, are in the same Proportion;) [Page 55] And therefore, That one good definition of Equality, or E­qualls, in generall; had been much better, then so many particulars, of Equall Bodies, Equall Magnitudes, Equall Motions, Equall Times, Equall Swiftnesse, &c. as you here bring; and yet, when you have all done, there be a great many more Equalls, which you leave undefined: (And your bare assertion, That there is no Subject of Quantity, or of Equality, or of any other Accident, but Body, doth not help the matter at all; for we are not bound to take your word for it:) That, if you would needs mention place, you should rather have defined them by the place they have, then what they may have; & so, defined those bodies to be Equall, which do possesse Equall places, rather then, which may possesse the same place: That a Pyramid, remaining a Py­ramid, may be Equall to a Cube; yet cannot, remaining a Pyramid, possesse the place of that Cube: Or, if you will, That a Pyramidall Atome, though so Adamantine as to be in­capable of any transmutation, (as those who teach the do­ctrine of Atomes doe maintain,) may yet be equall to a Cu­bicall Atome, though not possesse the place thereof: That you might as well have defined a Man, to be one who may be Prince of Transilvania, as to define Equall Bodies, to be those which May possesse the same place. (with much more, of which you take no notice.)

To that last particular, you answer, that 'tis wittily obje­cted, as I count witt, but impertinently. And why imperti­nently? Is not that definition of a Man; as good as yours of Bodies Equall? You think not, Because if so, J must be of opinion, That the possibility of being Prince of Transilvania, is no lesse essentiall to a Man; then the possibility of being in the same Place, is essentiall to Equall Bodies. And truly J am of that opinion. J think it every way as Possible for for any man living, to be Prince of Transilvania; as for the Arctick and Antartick Circles, (or the Segments of the Sphere which they cut off,) be they never so Equall, to possesse the same place. Nor is that possibility lesse essentiall, than this.

You adde, That there is no man (beside such Egregious Geo­metricians as we are) that inquires the Equality of two bodies, but by measure: And, as for liquid bodies, &c. by putting them one after another into the same vessell, that is to say, into the same place; And, as for hard bodies, they inquire their Equality [Page 56] [...]y weight. To which I shall reply nothing at all; because you speak therein so like a Geometrician.

I objected farther, That it is not yet agreed amongst Phi­losophers (and your authority will not decide the contro­versy,) whether or no, the same body may not, by Rarefa­ction and Condensation, (words understood by other men, though you understand them not,) sometimes possesse a bigger, some times a lesser place. We see, that the same Air in the head of a weather-glasse, doth sometimes possesse a bigger, sometimes a lesser part of the glasse, according as the Wea­ther is cold or hot, and you cannot deny, (what ever o­thers may) but that both are filled; for you doe not allow any Vacuum at all. We know, that into a Wind-gun, though it were full (you say) before, yet much more Air may be forced in. And into the Artificiall Fountain, (which you mention Cap. 26. fig 2.) though full of Air, may be for­ced also a great quantity of Water. Now how to salve these Phaenomena, (with many others of the like kind) with­out either allowing Vacuum, which you deny; or Condensa­tion, which you laugh at; (one of which others use to as­signe) because you find it too hard a task for you to under­take, (as well you may,) you leave to a m [...]lius inquirendum p. 144. l. 27. (or in the English, p. 316. l. 34.) Now if it be true, that the same body doth, or possible, that it may, possesse, some time a bigger, sometime a lesse space, (as those who deny Vacuum doe generally affirme,) then, by your definition, the same body (I doe not say may possibly become, but) at present is both bigger, and lesse, and equall to it selfe: Because it hath at present a possibility of posses­sing hereafter both a larger place, by Rarefaction, and a les­ser place, by condensation, than now it doth. And so you, by determining the equality or inequality of Bodies, not by the place they have, but by such place as possibly they may have (upon any supposed metamorphosis or transmu­tation,) doe confound Bigger, and Lesse, and Equall, and so take away the whole foundation of Mathematicks: For if there be no difference between Bigger, Lesse, and Equall, there is no roome either for Mathematicks or Measure. But, whether that opinion of Rarefaction and Condensation be true or not: yet since you cannot deny, but that it is at least a considerable controversy, and, by men as wise, and as good Philosophers as M. Hobs, maintained against you; [Page 57] yea and a Controversy not belonging to Mathematicks but Physicks, or Naturall Philosophy, and there to be determined; it was not wisdome to hang the whole weight of Mathema­ticks, upon so slender a thread, as the decision of that con­troversy in Naturall Philosophy, which whether way it be determined, is wholly impertinent to a Mathematicall De­finition.

To which you reply onely this, (which is easy to say) that Rarefying and Condensing, are but empty words; and that (of which we have spoken already) Mathematicall Defini­tion, is not a good phrase.

To that definition you had annexed this also; Eadem ratione, magnitudo magnitudini, &c. Ʋpon the same account one Magnitude is equall, or greater, or lesser, then another, when the bodies whose they are, are greater, equall, or lesse. These words, I said, must bear one of these two [...]enses, either, that E­quall Bodies, or Bodies equally great, are of equall greatnesse, (which is no very profound notion:) or else, that the mag­nitudes, towlt the lines, superficies, &c. or at least, the length, bredth, &c. of Equall bodies, is Equall, (taking the words for a definition of Equall Lines, Equall Superficies, &c) and this, I said, was manifestly false: for no bodies may be e­quall, whose length, breadth, superficies, &c. are unequall. You say now, that you meant the former, (and I cannot contradict it, for you know your own meaning best, yet you must give me leave to think:) and so leave us without any definition of Equall Lines, Plaines, or Superficies Which yet, considering how oft you are afterwards to. make use of, might have been as worthy of a definition, as some of those equalls that you have defined.

In the next Paragraph, Cap. 8. parag. 14. you undertake to prove, that one and the same Body, is alwaies of one and the same magnitude, and not bigger at one time then another, or at one time fill a bigger place, than it doth at another time. Let's heare how you prove it (for, by what we heard but now, you are much concerned to make good proofe of it, because if there be a possibility of possessing at any time a bigger or lesse place than now it doth, than it is, by your definition, at present bigger or lesse than it selfe.) Your proofe is in these words, For seeing a Body, and the Magni­tude, and the Place thereof, cannot be comprehended in the mind otherwise than as they are coincident, (observe therefore, that [Page 58] this argument doth no more prove, that a Body cannot change its Magnitude, than that it cannot change its Place, for you make Place as much coincident with Body, as you doe Magnitude, and the argument proceeds equally of both:) if any Body be understood to be at rest, that is, to remain in the same place during some time, and the Magnitude thereof be in one part of the time greater, and in another part lesse, that Bodies place, which is one and the same, will be coincident some­time with greater, sometime with lesse magnitude, that is, the same Place will be greater and lesse than it selfe, which is im­possible. This is your whole proof to a word. Now this, I told you, is no sufficient proof, because it proves only that a Body doth not change its quantity so long as it is at rest, and doth precisely keep the same place; (which no body doth af­firme.) And, pray look upon the Argument once again: doth it prove any more than so? But that which you un­dertook to prove was, that it doth never change its magni­tude, but hath alwaies the same, as well when its place is altered, as when it remains in the same place: (for, J sup­pose, you will not deny, but that a Body may change its place.) Those that hold the contrary opinion, doe not say that a body doth change its greatnesse while it doth precisely keep the same place; but that, with change of place, it may change its dimensions too: And to this, if you would have said any thing, you should have applied your argument. And is not this then a just exception to your argument? Will this argument hold, think you, Be­cause a Body doth not change its magnitude so long as it keeps precisely the same place: Therefore, it never changeth its mag­nitude, but hath alwaies the same? This argument hath no appearance of consequence, but only upon this supposi­tion, that a Body doth alwaies keep precisely the same place. And, then, I confesse, the Argument looks like an Argu­ment, in this forme, So long as a body keeps precisely one and the same place, it hath precisely one & the same Magnitude: But a Body doth alwaies keep precisely one and the same Place: There­fore it hath alwaies one and the same Magnitude. And if this be your argument, we allow the form, but deny the matter of it, and say, the Minor ought to be proved. For we are of opinion, that it is possible, for the same Body, not to be alwaies in the same place. If you think otherwise, pray [Page 59] prove it. For 'till that be proved, your present argument is to no purpose.

Sed rem ita per se manifestam, demonstrare opus non esset, &c. But, say you, a thing of it selfe so manifest, would need no De­monstration at all, (a fine facile way of Demonstration, that which you know not how to prove needs no demonstra­tion.) but that you see there are some, whose opinion concerning Bodies and their magnitude, is, that Body may exist separated from its magnitude, (no not so, but that it may change its magnitude, For they doe no more believe that it can ex­sist without Magnitude, than that it can exsist without a Figure: It cannot be but that a finite Body must have al­waies some figure, though not alwaies the same: and so al­waies some Magnitude, but whether alwaies the same or no, you should have proved if you could:) and have grea­ter or lesse magnitude bestowed upon it; (as well as different figures:) Making use of this principle for the explication of the nature of Rarum, and Densum. Since therefore you know there are that do so; why did not you, (at least in your English Editition, after you had notice of the weaknesse of your Latine Argument) bring some good Argument to overthrow that opinion; and not content your selfe to say that it is so manifest of it selfe, as that it needed no demonstrati­on. Especially, (as I then told you) since you doe not allow that Euclide may assume to himselfe gratis without demonstra­tion, That the whole is greater than its part; (those were my words, though you recite them a little otherwise.)

But you say, I know this to be untrue, that is, I lye: My words were these; Non interim Euclidi permittis, ut citra de­monstrationem hoc sibi gratis assumat, Totum esse majus sua par­te: that is, You do not allow it Euclide, that he may without Demonstration assume to himselfe, or challenge, That the whole is greater then its part. Now let your own words be judge, who is the lyar, you or I. Cap. 6. artic. 12, 13. The whole method of Demonstration, you say, is Syntheticall,— beginning with Principles, or primary Propositions. Now such Principles are nothing but Definitions,—And, Besides Definitions, there is no other Proposition that ought to be called Primary or (si paulo severius agere volumus) be received in­to the number of Principles. For those Axioms of Euclide, seeing they may be demonstrated, are no Principles of Demonstration. And accordingly art. 16. you define Demonstration, to [Page 60] be a syllogisme, or series of syllegismes, derived and continued from the Definitions of names, to the last conclusion. And parag. 17. You require to a Demonstration, That, the premi­ses of all Syllogismes be demonstrated from the first Definitions. (And the like cap. 20. parag. 6. diverse times.) So that these Axioms, being no Definitions, nor any Principles of Demonstration, no Demonstration can take rise from them, nor can they be otherwise assumed in demonstration, than as they are themselves deduced or demonstrated from Definitions. And doth not this come home to what I said? And cap. 8. parag. 25. Of which Axioms (omitting the rest) I will only (say you) demonstrate this one, The whole is grea­ter then any part thereof. To the end that the Reader may know, that those Axioms are not indemonstrable, and therefore not Principles of Demonstration. And yet again Less. 1. p. 4. As for the commonly received third sort of Principles, called Common Notions, they are Principles only by permission of him that is a Disciple; who being ingenuous, and coming not to cavill but to learn, is content to receive them (though demonstrable) with­out their demonstration. And again pag. 9. you exclude those common notions called Axioms, from the number of Princi­ples, as being demonstrable from the definitions of their termes, acknowledging no other Principles, but Definitions, and Postulata, (those the only principles of Demonstration; these of Construction.) If therefore they be no Principles of Demonstration; if only principles by permission of the Disciple, and only in curtesy; then, though your selfe possibly may he so gracious or liberall, as to admit of them without their demonstration; Yet the Teacher cannot, without this fa­vour, assume to himselfe, or require them to be granted, as he may doe Principles, without Demonstration. 'Twas not I therefore was the lyar, when I said, You doe not allow that Euclide may assume to himselfe gratis, or require to be granted, without de­monstration, That the whole is greater than its part. For 'tis but in courtesy, if you grant it him, as you may any other true Proposition, and only upon supposition that it may be demonstrated: upon which supposition, you may also al­low all the Propositions in Euclide, for they may be all de­monstrated.

And thus much concerning your eight Chapter.

SECT. VIII. Concerning his 11, and 13 Chapters.

WEE shall next consider what you have to say in defense of your 11 and 13 Chapters, concerning Proportion.

And here after a freak; and then a rant against Euclide; you have a large discourse about Proportion; p. 15, 16. The summe of which, so farre as is to the purpose, is this, That there betwo kinds of Proportion, (as the word is now adaies taken;) the one of which is called Arithmeticall Proportion; the other, Geometricall Proportion: And as the Quotient gives us a measure of the Proportion of the Dividend to the Divisor, in Geo­metricall Proportion; so the Remainder, after subtraction, is the measure of Proportion Arithmeticall. Pag. 16. And thus much is both true and clear, and to the purpose. And had you but thus delivered your doctrine of Proportions, in your Book de Corpore, I should never have found fault with it. But you, not knowing (till you learned it out of my Elenchus,) that the Quotient did as well determine Geo­metricall Proportion, (and give name to it) as the Re­mainder doth Proportion Arithmeticall, were fain to blunder on as well as you could, without it: and put your selfe upon a great many unhandsome shifts, and which will not hold water, to give account, even of Geometricall Proportion, from the Remainder or difference, which was not to be done otherwise then by the Quotient, as you here clearly confesse; For the Measure, you say, of Geometri­call progression, is (not the Remainder, whether absolutely or comparatively considered, but) the Quotient.

But before you come thus farre; you tell us by the way, That I say, that you make proportion to consist in the Remainder, and that I make it consist in the Quotient. As to the former of these, I did not then say, that you make proportion to consist in the Remainder; though if I had said so, I had said true e­nough, for you doe so, more than once. Cap. 11. parag. 7. In ratione inaequalium, say you, ratio minoris ad majus, Defe­ctus; ratio majoris ad minus Excessus dicitur. And again par. 5. Consistit ratio antecedentis ad consequens in differentia, &c. sive in majoris (dempto minore) Refiduo. And. soon after, [Page 62] Ratio binarii ad quinarium est ternarius, &c. You cannot de­ny but that these are your words, and that I blamed you for them, as a piece of non sense; all that you have to say is, that it was too hastily put: & therefore you labour in the Eng­lish a little to disguise it. So cap. 12. art. 8. Cum Ratio inaequalium, per cap. praeced. art. 5. consistit in differe [...]tia ipsa­rum, &c. and again, Ratio inaequalium, EG, EF, consistit in differentia EF, quae est quantitas, (yes, quantitas absoluta, for 'tis a line.) And these, because I did not particularly tell you of them, are yet uncorrected in your English; seeing (by the fifth Article of the precedent Chapter,) the proportion of two unequall magnitudes consists in their difference, &c. And again, the Proportion of unequalls EG, EF, is quantity; for the difference GF, in which it consists is quantity. Now when, you say in expresse words, as in the places cited, The pro­portion of the antecedent to the consequent consists in the Diffe­rence, or the Remainder; it had been no wrong if I had said, as you say I doe, that you make Proportion to consist in the Re­mainder; and that absurdly enough. And then, J pray, to whom belong those reproaches, that are so oft in your mouth, as if somebody did affirme, that Proportion is a Num­ber, an Absolute quantity, &c? is it not your selfe that af­firme it so to be? And doth any body so beside your selfe? And is not then, that (by your own law p. 10,) in your selfe intolerable, which you cannot tolerate in another?

But you adde farther, that I say, that I make it to consist in the Quotient. And is not this abominably false? J neither say so, nor doe so, nor did J give any ground at all for any man (that is in his witts) to believe J did. My words were these, Videmus igitur Rationis aestimationem esse (secundum Te) penes Residuum, non penes Quotum, & Subductione, non Divisione quaerendam esse. (And what reason J had to say so, they that consult the place will see.) Now could any man (who had not a great confidence that his English Reader understands no Latine) be so impudent as to say, that in those words, I say, you make Proportion to consist in the Remainder; and I, in the Quotient? Can any man, that un­derstands, though but a little Latine, (if he be not either out of his witts, or halfe a sleep,) think that these words Rationis aestimatio est penes Quotum, (that is, the Proportion is to be estimated according to the Quotient, or, to use your own words, the quotient gives us the measure of the proportion,) [Page 63] could be thus Englished, proportion consists in the quotient? And that then you should raile at us, quite through your Book, for saying that Proportion is a certain quotient, that it is a number, that it is an absolute quantity, &c. as if we had been so ridiculous as to speak like you. For, that you have so spoken you cannot deny, (and therefore the absurdity what ever it be, lights upon your selfe:) But, to say, that I said so, or any thing to that purpose, till you can shew where I said it, J take to be, (so farre as a word of your mouth can be) a manifest slander. J neither say so, nor think so.

Now some men perhaps may wonder, there should be so great a cry and so little wooll; they would think perhaps, by what you say, that J had somewhere said in expresse termes, that Proportion is a Quotient, or that it consists in the Quotient, or that it is a number, or an ab­solute quantity, or that the quotient is the proportion, or that a Proportion is the double of a Number, but not of a propor­tion, or somewhat that sounds like somewhat of these, when they hear me thus charged, again and again, many a time, and oft; and not that the whole ground of the accusation had been but this, that I said, The proportion is to be estimated by the quotient. And truly 'tis somewhat hard to give a good account of it: yet wee'l try what may be done.

J was told, some years a goe, of a man that had told a lye so often, and with so much confidence, that at length he began to believe it himselfe. And J am almost of opinion, that M. [...]obs having now said it so often over, doth, by this time, be­gin to think, that J had indeed said, somewhere, that the quotient was the proportion. And truly there is some reason why he should: For if he had heard any other man so oft and so confidently affirme it, he would no doubt have be­lieved him: and why should he not as well believe him­selfe.

But moreover; It did perhaps runne in his mind, that he had somewhere read some such words as these, Consistit au­tem Ratio antecedentis ad consequens, in Differentia, hoc est in ea parte majoris qua minus ab eo superatur; sive in majoris (dem­pto minore) Residuo. Or such as these, Ratio binarii ad qui­narium est ternarius. Or else this, Ratio minoris ad majus, Defectus; ratio majoris ad minus, Excessus dicitur. (And well it might: for they are all his own words, Cap 11. parag. 3. & 5. and Cap. 12. parag. 8.) And he might think, that to [Page 64] say thus, was all one, as to affirme Proportion to be a Number, or an Absolute quantity: (And truly I think so too.) And that therefore the expression was very absurd; (For so I had intimated to him in my Elenchus, upon this occasion.) And therefore (forgetting, perhaps, that they were his own words, and not mine.) he doth (like the Woman that called her daughter Bastard, not minding that in so doing shee called her selfe Whore,) exclaim against his own words, as most ridiculous non-sense. And who might doe it better?

Or else, to use his own comparison, like Women of poor and evill education, when they scold; amongst whom the readiest dis­gracefull word is Whore; because, when they remember them­selves, they think that reproach the likeliest to be true; at least, if they be called Whore themselves, though never so truly, they will be sure to call Whore again at all adventures, hit or misse. So M. Hobbs, finding himselfe to have been so absurd, as to make Proportion a Number, or Absolute quan­tity, and that I had blamed him for it; thought, perhaps, it was possible I might, sometime or other, have been as carelesse in my language: and therefore, however, hee'l say so, ('tis easy to say it) and let me disprove it.

If any man, notwithstanding all this, be not satisfied that M. Hobs had reason to say as he doth; truly I cannot help it; he must speak for himselfe: These were the best reasons I could think of▪ And so wee'l goe on.

In your 11 Chap. parag. 3. you gave us in the Latine, (for in the English there be some things altered,) this de­finition of Proportion; Proportion is nothing else but the aequa­lity or in equality of the Antecedent, compared with the consequent, according to magnitude. With this Explication, As for exam­ple, the proportion of Three to Two, is nothing else, but, that Three, is greater then Two, by One: and the proportion of Two to Five, is nothing else, but that Two, is lesse than Five by Three: And therefore in the proportion of Ʋnequalls, the proportion of the Lesse to the Greater is called the Defect; and that of the Greater to the Lesse, the Excesse. And this is your generall definition of Proportion, with the Explication of it; and nor a particular definition of Arithmeticall Proportions, (nor is it at all by you pretended so to be.) And there­fore should have been so ordered, as at least to take in Geo­metricall Proportion; For Geometricall proportion, and simply [Page 65] proportion, are by your selfe made equivalent termes (Less. 2. p. 16. l. 25.) and this, you say, is onely taken notice of by the name of Proportion: And, so the word is constantly used in Euclide, and elsewhere: (And therefore you need not wonder as you doe p. 18. l. 7, that J should say, If Arith­meticall Proportion, ought to be called Proportion; implying that though now that phrase be common, yet that it is a depart­ing from the former use of the word; and that, according to Euclides use of the word Proportion, Arithmeticall Proportion cannot be so called.) Now your Definition and Explication of Proportion, doth wholly leave out Geome­tricall Proportion altogether, (which yet is, if not the only, yet the more principall kind of Proportion.) For it takes no cognizance of the Quotient at all, but only of the Difference, the excesse or defect. And according to your doctrine the Proportion of 3 to 2, is + 1, the excesse of 1; and of 2 to 5, is -3, the defect of three.

From this I inferred, that if the proportion of one quan­tity to another, be nothing else, but the excesse or defect of this to that, (as you teach,) then where ever the excesse or de­fect is the same, there the proportion is the same; and so 3 to 2, must have the same proportion that 5 hath to 4; (You say, p. 17. True, the same Arithmeticall Proportion Very good: But J added farther, of which you did not think fit to take notice,) and on the contrary, where there is not the same defect or the same excesse there is not the same proportion, and conse­quently, there is not the same proportion of 3 to 2 and of 6 to 4. To this you have nothing to say, and therefore say nothing, (but recite halfe my sentence, and leave out the other halfe:) For though, there be not the same Arithmeticall Proportion (as you speak) of 3 to 2, and of 6 to 4; (that is, not the same excesse,) yet there is the same Geometricall Pro­portion; and that you cannot deny to be Proportion, though it doe not come, within your definition.

Now it's true, (but that's another fault, not an excuse) that you do not hold to this sense alwaies, for in the same page art. 5. (in the Latine, I mean) you do clearly con­tradict what you had but now said in art. 3. The proportion, say you, of the Antecedent to the consequent consists in the Dif­ference, or Remainder, not simply (yes simply, if that be true which you said before; for if it be nothing else but the diffe­rence, that is it the difference simply: But if not simply; [Page 66] how then?) but as compared with one of the termes related, &c. For though there be the same difference between 2 and 5, that there is between 9 and 12, yet not the same Proportion. And why not? as well as the same proportion between 3 and 2, and between 4 and 5? as we heard you reply but now. May not we as well say here, as you there, (Les. 2. p. 17.) Is there not the same Arithmeticall Proportion? And is not A­rithmeticall proportion, proportion? But it seems, by this time, you had forgotten your former exposition, whereby in the same page, your definition of Proportion must be so under­stood, as will agree to none but Arithmeticall proportion; now it must bear such a sense as can agree to none but Geo­metricall.

In the English, I confesse, your Translator hath a little mended the matter, and but a little, ('tis but Coblers work at the best;) But however, 'tis good to hear folks mend, though it be but a little: it may come to something in time.

But now of those two senses, which you have given, of the Definition of Proportion, (opposite enough in con­science one to another, though, I suppose, you did not in­tend therein to contradict your selfe,) neither of them will serve your turn. For the Proportion here defined, and so explicated as we have heard, is a Genus, which is, in the beginning of your 13 Chapter, to be distributed into its two Species; Proportion Arithmeticall, and Proportion Geometricall. Now take your definition of Proportion in generall, according to which of your two expositions you please, it cannot be thus distributed. For if Propor [...]on (as you say chap. 11, [...]art. 3.) be nothing else but the excesse or defect, &c. as 3 is lesse then 2 by 1; then it cannot agree to Geometricall proportion, for that is somewhat else. If it be such a comparative difference, as you mention cap. 11. art. 3. it will not agree to Arithmeticall proportion; for according to that sense, you say, 2 to 5, and 9 to 12, are not in the same proportion. I say therefore, that neither of those two expositions, do agree to that generall notion of Pro­portion, which shall be common to both Arithmeticall and Geometricall. And when I aske, which of the two expo­sitions you are willing to stand to. Whether that of Cap. 11. art. 3. or that of Cap, 11. art. 5. (shewing withall that neither of them will serve your turne, for neither of them [Page 67] will take in both Arithmeticall and Geometricall Progres­sion,) you fall a raving in the beginning of your third Lesson, something at Euclide, and something at us, but nothing to the purpose. And then tell us, that when you say the Difference is the Proportion, by Difference, we might if we would, have understood, the act of Differing. That is, wee might understand, as madly as you speak. Your words were these, Cap. 11. art. 5. Consistit autem Ratio in Dif­ferentia, sive Residuo, &c. ita ratio binarii ad quinarium est ter­narius, &c. Would you have us understand Residuum, and Ternarius, to be the Act of Differing? And C. 12. art. 8. Ratio inaequaliū (EG, EF) consistit in differentia GF. Would you have us understand that line GF, to be the act of differing? You say, we might if we would. But you'ld think us very simple if we should. To as good purpose is it, that you tell your English Reader (for you think you may tell him any thing,) that [...] say, that (thus much of) your Definition, Ch. 11. Art. 1. [Pro­portion is the Comparison of two Magnitudes one to another,] a­grees neither with Arithmeticall nor Geometricall proportion. For I said nothing of any such words, good or bad. And 'twere much if I should: for I can find no such words there.

At the second Article (chap. 13.) I note, you say, for a fault in method, that after you had used the words, Antecedent, and Consequent of a Proportion, in the precedent Chapters, you now define them. 'Tis true, I did take notice of it, but I said withall, that this was but a small fault in comparison of many others. But what if I did? You do not believe, you say that I spake this against my knowledge. No; why should you for you know 'tis true. Have you not used the words ma­ny times before in the precedent chapters? And doe you not define them here? And is not this a fault in Method? Do Mathematicians use, when they have taken a Terme for two or three chapters together, to be of a known significa­tion, and sufficiently understood, come at length to define it? you say, you had before defined it chap. 11. art. 3. 'Tis true you had there defined the Antecedent and Consequent of Cor­relatives; (which definitions might have served well enough for the Antecedent and consequent in Proportions too, for those are Correlatives, and you need not have brought any new ones.) But where was my oversight? Did I deny this? I did not blame you for using the words before you had defined them, [Page 68] (nor would I have blamed you, if they had not been de­fined at all;) But for defining them after you had thus long used them. For, if they had now, ever since the beginning of the 11 Chapter, been taken for words of a known significati­on, and as such frequently used, (which you do not deny, and your definitions at that place do but aggravate, not ex­tenuate, this charge,) then, I say, it was immethodicall and superfluous to define them in the 13 chapter. Nor was it my oversight to say so. And the like impertinent answer you give p. 51. where I blamed you (not for omitting in the 19 chapter, but) for defining in the 24 chapter, those termes which were of frequent use in the 19 chapter. But wee go on.

You tell us, Chap. 13. art. 3. That the proportion of Ine­quality is Quantity, but that of Equality is not. Which I said was very absurd; and that the one did no more belong to the Praedicament of Quantity than the other; and that it is to bee, of both equally, either denied or affirmed: And that your argument for it, (That One equality is not greater or lesse then another; but of proportions of inequality, one may be more or lesse unequall:) might as well conclude that Oblique angles, be quantities, but not Right angles, for these be all equall, and equally Right; but not those. For answer to this, you fall a ranting at Aristotle, at Praedicaments, and the L [...]gick Schooles, &c. And then you tell us the Greater and Lesser cannot be attributed to Right Angles, because a Right Angle is a Quantity determined, (as though the quantity of the Proportion of Equality were not so too.) What you alledge out of Mersennus, was but his mistake. Composition of Pro­portion is a work of Multiplication, not of Addition, as appears by the definition of it 5 d 6. and to argue, that Proportion of equality is as Nothing, because in composition of Proportions it doth not increase or diminish another proportion; is but as to conclude that, 1, a Ʋnity, is Nothing, because in Multiplication it doth neither increase nor diminish the quantity multiplyed thereby. But of this mistake of Mersennus, I have spoken already in the end of another Treatise, already Printed, against Meibomius; and vindicated Clavius suffici­ently from what both Mersennus and Meibomius allege a­gainst him.

To the fourth Article, where you define Greater and Lesser Proportion; I said nothing (because it were endlesse [Page 69] to note all the faults I see) though those definitions are liable enough to censure. Greater Proportion, you say, is the proportion of a greater Antecedent to the same Consequent, or of the same Antecedent to a lesse Consequent. And Lesse Propor­tion, is the proportion of a lesse Antecedent to the same Consequent, or of the same Antecedent to a greater consequent. Yet we know, that the proportion of an Ell to a Yard, is lesse then that of a Pottle to a Pint, (and this therefore greater then that,) though neither the Antecedents nor the Consequents, be ei­ther the same, or Equall, or Homogeneous.

To the 5 and 6 Articles, where you define the same Pro­portion. I said First, that, had Proportion been well defi­ned before, you might have spared these definitions of the same proportion. For having before defined (as well as you could) what is Proportion (both Arithmeticall, and Geome­tricall;) and withall told us, art. 4. that by the same pro­portion was meant Equall proportions; and having also defined before (after your fashion) what are Equalls chap. 8. and what is the Same chap. 11. Why should you think (if those definitions were such as they should have been) that wee needed another definition of the Same, or Equall Proportions? But, since you were resolved to doe works of Supereroga­tion; I ask why, having defined the same Arithmeticall pro­portion, art: 5. by the Equality of the Differences; you did not also define the same Geometricall Proportion, art. 6, by the Equality of the Quotients? For by the Same, you say, you mean Equall, art. 4. Now universally all quantities are Equall, that are measured by the same number of the same Measures (Less: 1 p: 4.) and therefore those are the same or equall Proportions, which have the same or equall Measures: And you know now (though perhaps you did not then) that as the Quotient gives us a measure of the Proportion in Geometricall Proportion, so the Remainder is the Measure of Proportion Arithmeticall. (Les: 2. p. 16.) And therefore, as, in the one, you define the same or equall proportion, by the Equality of the Remainder; so you should in the other, by the equality of the Quotient, (that is, in both places by the equality of its measure:) And not have brought us such an imbrangled definition as this. viz: One Geometricall progressi­on is the same with another, when a cause in equall times tro­ducing equall effects, determining the proportion, may be assigned the same in both, or as your English hath it, when the same [Page 70] cause producing equall effects in equall times, determines both the proportions. So that, to prove, that 4 to 2, and 6 to 3, are in the same Geometricall proportion, we must call in the help of Time, and Motion, and Velocity, and Ʋniformity, &c. which are wholly extrinsecall to it; and why, but because, forsooth, there is no effect in Nature which is not produced in Time by Motion, (as though some Motion, in some Time or other, had made this to be a true Proposition, that 4 is the double of 2: and therefore if we cannot find what motion did make it so, we must imagine some that might have made it.) I need not tell you, that, if this be a good reason, you should upon the same account, have found out as bad a de­finition for the same Arithmeticall proportion: (for that 8 to 6, and 12 to 10, are in the same Arithmeticall proportion, is, doubtlesse, as much as that other of Geometricall pro­portion, an effect which nature hath at some Time or other pro­duced by Motion.) But, since you have waved this considera­tion of nature in the definition of the same Arithmeticall pro­portion, which you define by the equality of the Remainders; I said, it might have been expected, that you might have done so in the definition of the same Geometricall proportion [...], and accordingly defined it, by the Equality of the Quotients. But you are very angry with me, for saying, It might have been expected. And truly I could almost find in my heart to confesse that this was a fault. For though it might have been expected from another man; yet it was not to be expe­cted from M. Hobs; for his witt is not like the witt of o­ther men, He is the First (he tells us) that hath made the grounds of Geometry firm and coherent. But why was it not to be exspected? Because, you say, It is impossible to define (Geo­metricall) proportion universally by comparing Quotients. (Im­possible, I confesse, is a hard word; but yet, I hope, it may be.) But why is it impossible? more than it is impossible to define Arithmeticall proportion universally by comparing of Remainders? Because, forsooth, In quantities incommensura­ble there may be the same proportion, where neverthelesse there is no Quotient: (Very good! But why no quotient?) for quo­tient there is none but in Aliquot parts. (Gooder, and gooder!) But, I pray, is not A / B as good a Quotient, as A-B is a Remainder? whether the quantities be commensurable, or Incommensurable? No, you say; For setting their Symbols [Page 71] one above another with a line between, doth not make a Quotient. But why not? as well, as setting their Symbols one after another, with a line between, makes a Remainder? For, if the quantities be incōmensurable, the Remainder is no more explicable in Rationall numbers, then is the quotient. If from 3 you subduct √2, the Remainder is but 3 − [...]2. If you divide 3 by √2, the quotient is 3/√2;. And is not his as much a Quotient, as that a Remainder? and as well designed? Yet this is all you have to say to the businesse: The rest is but Ranting, or vapouring. But, however, we are much deceived, you tell us, if we think, with pricking of Bladders to let out their vapour; for we see, you say, we make them swell more then ever. What? till they bu [...]st? I hope not so. (Crepent licet, modo non Rumpantur.) I have heard, I con­fesse, that a Toad would swell the more for being pricked; but I never knew that a Bladder would, till now.

The next thing that troubles you, is, that I said, that the Corollaries of these two Articles taught us nothing new. (There be as I recon five and nine; fourteen in all.) Yes, you say, the ninth Corollary of the sixth Article is new: (No; it is not. We are taught the same by the second of the fifth of Euclid; and by the converse of the eleventh prop. of the sixth chapter of M. Oughtred's Clavis;) and the rest were never before exactly demonstrated. What? none of them? That's much. You mean, I suppose not all. And that I am content to believe: For they are not all true. As for exam­ple; The second Corollary of the fifth Article, is thus de­livered Universally, If there be never so many magnitudes A­rithmetically proportional, (whether in continuall or interrup­ted proportion; for you doe not limit it to either, more then you had done that next before it, which you cannot deny to be understood of both) the summe of them all will be equall to the product of halfe number of Terms, multiplied by the summe of the extremes. And then that we may be sure it is not intended only of cōtinual proportion, you give instance in proportion discontinued, For (say you) if A. B∷C. D∷E. F. be Arithmetically proportionall (though but discontinued, for so your Symbols import, both in the Latine and the English, least we might think it had been the Printers fault, and not the Authors;) the couples A + F, B + E, C + D, (you say) will be equall to one an other. This, [Page 72] though it be true of continued Arithmeticall proporti­on, yet of discont [...]nued proportion, as you here affirme it, it is notoriously false. For how doth it appeare, that C+D, is equall to A + F. For instance, let the termes be these 2. 1 ∷ 20. 19 ∷ 3. 2. in arithmeticall proportion. is 20+19, equall to 2 + 2? or to 1 + 3? It's no marvell then that this was never before exactly demonstrated. But we are taught nothing new by this. For though this be new and be years, yet we cannot learn it. Wee'l go on therefore: and see what you say next of the thirteenth Article.

Wee began, as I said, with slighter skirmishings; about Definitions &c. The skirmish now growes hotter; when I charge you with false propositions and demonstrations; and that you be touched to the quick, we may guesse by the loud out-cry; In objecting against the thirteenth, and sixteenth Articles, we doe at once bewray both the greatest Igno­rance, & the greatest Malice, &c (and so on, for a whole leafe or more;) Now this Ignorance h [...]wrayd, was your own, viz. that you had given us false demonstrations &c. and then is it not spightfully done of us to discover them? Well; let's see what 'tis that makes you cry out so fiercely.

The proposition is this, Of three quantities that have pro­portion to one another, (suppose AB, AC, AD; or 6, 3, 1;) the proportion of


the first to the second, and of the second to the third taken together, are equall to the proportion of the first to the third. That is, said I, The proper­tion compounded of that of the first to the second, (suppose 6 to 3. which is double,) and that of the second to the third (viz. 3 to 1, which is treble,) is equall to that of the first to the third, (viz. 6 to 1, which is sextuple.) And was not this your meaning? (I am su [...]e 'tis either thus or worse) This composition, I said, was such as Euclide defines 5 d 6; which is done by multiplying the quantities of the propor­tions: viz. 6/3 × 3/1 = 6/1, (not by adding them; for so 6/3 + 3/1 = 2/1 + 3/1 = 5/1.) Did I not explaine your meaning right? I [...]meant no hurt in saying this was your meaning; for the meaning was a good meaning; and the proporsion so meant, is a good proporsion; (but, if you mean otherwise, the proposition is false:) and, doubtlesse, 'twas a good meaning too, when you meant [Page 73] to demonstrate it; (all the mischiefe was, you could not do what, you meant to doe.) If this be your meaning (as J am sure it is or should be,) what is it that troubles you? You doe not like the word Composition: that's one thing. Well then let it be called Addition for once, J told you then, J would not content for the name; (but you know 'tis such an Additon of Proportions, as is made by multiply­ing of the quantities; as appeares by the very words of the definition 5 d 6) Then you doe not like that J should say the proportion of 6 to 3. is double; and that of 3 to 1, treble. Tell me (say you) egregious Professors, How is 6 to 3 double proportion? The answer is easy, (though perhaps you will not like it;) The proportion of 6 to 3, or 2 to 1, is that which is commonly called Double; and that of 3 to 1, is is commonly called Treble; And if you will not believe me, pray believe your own words, Corp. pag. 110. l. 5, 6. Ratio 2 ad 1. vocatur Dupla; et 3 ad 1 Tripla. You tell us then, We may observe that Euclide never distinguisheth be­tween Double and Duplicate (no more then other Greek writers do between [...] and [...].) one word (you say, serves him every where for either. You might as well bid us put out our eyes; or else believe that [...] and [...] ▪ are the same words. Perhaps you thought so when you wrote your booke in Latine; but, since that time you have been better instructed, and have learned at length to distinguish between Double and Duplicate, as we shall heare anon. But let's goe on. All this hitherto hath been but scuffling, and little to the purpose, though there you make the greatest out cry, (like a lapwhing, when shee's furthest off her nest.) we are now comming to a close grapple. (and 'tis like to prove as had as a Cornish hugge.) Your de­monstration, I said, was false (and that greeves you.) The strength of it, as I told you, lyes in this, The difference of AB, AC, (be they Lines or Times, chuse you whether, for by construction the times and lines are made proportio­nall,) together with the difference of AC, AD, taken together, are equall to


the difference of AB, AD; therefore the proportion of AB, to AC, and of AC, to AD, taken together is equall to that of AB to AD. That [Page 74] this is the strength of your demonstration you doe not de­ny. Now that consequence I denyed; affirming that from that equality of the difference, you could not inferre the equality of Geometricall proportion; (and, of Arithme­ticall, the question is not; nor is pretended to be.) And J gave this instance to the contrary, to shew the weaknesse of your Argument; Taking between A and B, any point at pleasure suppose a; you may as well conclude the pro­portion of aB to aD, aS of AB to AD. to be compounded of that of AB to AC, and of AC to AD. For, (in your own words.) the difference of AB, AC, with that of AC, AD, are equall to the difference (not only of AB, AD, but even of) aB, aD; and therefore the proportions of those, to that of these. Now all that you have to say against it, (for I doe suppose, as you would have me, the motion to be equally swift all the way,) is this, The difference of AB, AC, [...]annot


be the same with the difference of aB, aC, except AB and aB are equall. And here we joyne issue. The difference of AB, AC, say I, is BC; and the difference of aB, aC, is the same BC; though AB, aB, are not equall. The case is ripe for a verdict. Let the Jury judge. And now you may, if you will, go on to rant at Ignorance and Malice, at Sym­bols and Gambols, at double and duplicate, at asses and eares, at Cla [...]ius, Orontius, and too learned men, or whom you will; haeret lateri lethalis arundo. But thus 'tis, when men will needs have Geometricall proportion, to be estimated by Differences, and not by Quotients.

(I told you moreover that your demonstration was but Petitio principii, and shewed wherein, with some other faults which you take no notice of, because you had no­thing to say to them. And shewed you how your 13, 14, and 15, articles with all their Corollaries, (which fill up a matter of 4 pages.) might have been to better purpose de­livered in so many lines. But this is no great fault with you, who think the farthest way about, the nearest way home.)

At the 16 Article the case is as bad or worse. The cry goes on still. This is all Ignorance and Malice too. And a huge out cry against Quotients, and Symbols, and a loud On [...]ethmus as you call it. But not a word to the purpose of what was objected; (except only one clause wherein you [Page 75] tell us how absurd you mean to be by and by.) The busi­nesse is this, Euclide (10 d 5) defines Duplicate, and Tri­plicate proportion, &c. in this manner, If three magnitudes be in continuall proportion, the first to the last hath duplicate pro­portion of what it hath to the second; if four, triplicate; &c. (and that indifferently whether the first or last be the bigger.) Now you (that you might shew your selfe wiser then Eu­clide, and be the first that ever made the grounds of Geometry firm and coherent,) thought it was to be limited to this case only, when the first quantity is the greatest. And therefore thus define, The proportion of a greater quantity to a lesse (very warily) is said to be multiplied by a number, when other propor­tions equall to it, be added. And therefore if the quantities (con­tinued in the same proportion) be three; the proportion of the first to the last is Double, of what it hath to the second; if four, Treble, &c. (which most men, you say, call duplicate, tripli­cate, &c.) But if the proportion be of the lesse to the greater (of which Euclide, it seems, was not aware) and there be an ad­dition of more proportions equall to it, it is not properly said to be multiplied, but submultiplied (that is, divided; which yet you tell us, by and by, is to be done by taking mean proportio­nalls.) So that of three quantities (so continued) the propor­tion of the first to the last, is halfe of what it hath to the second; if four, a third part, &c. which are commonly called subdupli­cate, subtriplicate, &c. Now this, I told you, was foul great mistake, and such a one as should not have procee­ded from a Reformer of the Mathematicks. And, to use your own distinction (Less. 2. p. 9.) 'tis a fault not of Neg­ligence, but of Ignorance, or want of understanding principles: and therefore an ill favoured fault, and, by your own rule, to be attended with shame. I shewd you there (and you be­lieve me now) that in the numbers 1, 3, 9, 27, &c. the proportion of 1 to 9, though lesse, was not subduplicate to that of 1 to 3, but duplicate, as truly as the proportion of 9 to 1 is duplicate to that of 3 to 1; and that of 1 to 27 was triplicate, not subtriplicate, of that of 1 to 3; Of which I gave you this demonstration, (though it seems, you did not understand it, and therefore say, I bring no Argument.) Because 1/9 = ⅓ × ⅓, and 1/27 = ⅓ × ⅓ × ⅓, as well as 9 [...] = 2/1 × 3/1, add 27/1 = 3/1 × 3/1 × 3/1. And the subduplicate of 1 to 3, is not, as you suppose, that of 1 to 9, but of 1 to [Page 76] √3. Now this was so unlucky a mistake, or Ignorance, in a thing so fundamentall, that (as I then told you, and you have since found to be true) an hundred to one, but it would doe you a deal of mischief all along. And it was the touching upon this fore place, that gawled you so much but now, and put you beside your patience.

But let's see now how you behave your selfe. A loud rant we have, as if it were grievous doctrine I had taught, and your own had been much better. But not a word to the purpose save only this 'Tis absurd to say, that taking the same quantity twice, should make it lesse. But though you say so, you doe not think so. For when you have done your rant, you goe slyly, (without saying a word of it, or ac­knowledging any error,) and put out that whole sixteenth Article, which we had in the Latine, giving us in the English another instead of it, quite of another tenour, and quite contrary to what you had before. And now a proportion of the lesse to the greater, (as well as of the greater to the lesse,) being twice taken, shall be duplicate, (not subduplicate as before;) and thrice taken, (not subtri­plicate, but) triplicate. Now (because you say it,) it is not ab­surd to say, that taking the same quantity twice, should make it lesse; (though when I said it, it was absurd.) Now A pro­portion is said to be multiplyed by number, not submultiplyed, when it is so often taken as there be unities in that number. (Whether it be of the greater to the lesse, or of the lesse to the greater;) And if the proportion be the greater to the lesse, then shall also the quantity of the proportion be increased by the multiplication; but when the proportion is of the lesse to the grea­ter, then as the number increaseth, the quantity of the proportion diminisheth; For it is no absurdity now, to say that taking the same quantity twice makes it lesse. And truly now, methinks, thou sayst thy lesson pretty well; I could find in my heart to spit in thy mouth and make much of thee, hadst thou not railed at him that taught thee; which is but a trick of an ungratefull schollar: But let's goe on, and see whether this good fit will hold? As in these numbers, 4, 2, 1. the proportion of 4 to 1, is not only the duplicate of 4 to 2, but also twice as great. (Nay that is good againe; he hath learned that there is a difference between Duplicate and twice as great. Surely this is not he, (or else the world's well [Page 77] amended with him,) that laughed at the distinction of Duplicate and Double. Well, let's heare some more of it.) But, inverting the order of those numbers thus, 1, 2, 4, the pro­portion of 1 to 2, is greater than that of 1 to 4; and therefore though the proportion of 1 to 4, be the duplicate of 1 to 2, yet it is not twice so great as that of 1 to 2, but contrarily the halfe of it. In good truth; a prety apt Schollar: for one of his inches; He says just as I bid him. Well, well! the world's well amended with T. H. The [...]'s hopes he may come to good. Yee see he learnes apace. He may be a Mathematician in time; though I say't that should not say't. I confesse he hath his faults still, as well as other men, (you must not think he can mend all at once,) The whole article is not so good throughout, at this bit at the beginning. He hath got a naughty trick of saying The proportion of equality is no quan­tity, (but he hath been whipt for already;) He makes it stand for a Cyphar, (but that's a thing of nothing: It should have been but 1, and that's not much more.) And he tells us that the proportion of 9 to 4 is not onely duplicate, of 9 to 6, but also the Double, or twice as greate. And again, that the propor­tion of—4 to—6, is double to the proportion of—4 to—9, &c. which would have deserved whipping at another time; but because he said the rest so well, I'le spare him for this once. He doth, it seems, believe there is a difference between double and duplicate, though he doe not yet know what it is; he will learn against next time. And to the like purpose is that which follows; If there be more quantities then three (it's no matter how many) as A, B, C, D, in conti­nued proportion, what ever the proportion be, so that A be the least; it may be made appeare that the proportion of A to B, is triple magnitude, though subtriple in multitude, to the propor­tion of A to D. But however he shall be spared for this bout; because I said so; and I will be as good as my word.

We have but one touch more, and I have done with this Chapter. 'Tis at the Corollary of the 28 Article. Here you find fault first with the word aliquot; and ask whe­ther I think that partes aliquot can be numero infinitae? And I think they may. Where there are more then one, there be at least aliquot, whether few or many. What I objected a­gainst that Corollary, was not against the truth of it, for it is obvious and facile; but that it needed not so much a doe, as to be ushered in with three teadious Propositions, art: [Page 78] 26. 27. 28. Which, I said, (and you do not deny it,) seems to be put in only in order to this Corollary. In Art 28. you were come thus farre, If from a line (AB) be cut off a part (suppose AC) and between that and the whole, be taken two meanes, the one Geometricall, the other Arithmeticall (AD, AE;) the greater the part


is that is cut off AC (and con­sequently the remainder CB, the lesse,) the lesse will be the difference between those two meanes (AD, AE.) Hence at length you come to this Corollary, That if the line (AB) be divided into equall parts infinite in number, (and so the part remaining CB, one of them, infinitely small,) the difference between the Arithmeticall and Geometricall means, will be infi­nitely little. This I said might be proved universally, not only of these two, the Arithmeticall and Geometricall Meanes, but of any two Meanes whatsoever. For supposing any two lines AD,


AE, which are each of them greater then AC, but lesse then AB, (as all means must be) your points DE, (whether D, or E, stand first, it matters not) must needs fall between C and B: And there­fore the difference DE, cannot be bigger than CB, but rather a part thereof, and so lesse. Now the whole CB is supposed infinitely little, and therefore its part DE can­not be bigger. And is not this as well proved as if I had premised in order to it, three whole Articles, and spent three pages about it? You say, (and that's all you object against it) that I doe not prove, that BE the Arithmeticall difference is lesse than BD the Geometricall difference. No; Nor ought I so to doe. For the thing to be proved was not which of the two is greatest, but that (whether soever were the greater) the difference between the two is infinitely small: which is done sufficiently without that other. And are not you then a wise Mathematician to make such an ob­jection?

SECT. IX. Concerning his 14. and 15 Chapters.

IN Your 14 Chapter, Art: 2. I found fault with your definition of a Plain, to be that which is described by a streight line so moved as that every point of it describe a streight line. I told you, it is not necessary, much lesse es­sentiall, to be so described, (and you confesse it;) and ma­ny plains there are which are not so described. The defi­nition therefore is not good.

Again. You had said in the first Article: Two streight lines cannot include a superficies. (Right,) And then Art: 2. Two plain superficies cannot include a solid. No, said I, nor yet Three. 'Twas simply done then to name but two. And you confesse it to be a fault; but not a fault to be ashamed of.

Again, you had said Art: 1. That a streight line and a croo­ked, cannot be coincident, no not in the least part. And then Art: 3. You tell us of some crooked lines which have parts that are not crooked. This I noted for a contradiction; because with those parts not crooked, a streight line may be coin­cident. And you cannot deny it. Therefore in the Eng­lish, instead of crooked, in the former place, you put perpe­tually crooked; which though it be but a botch, helps the matter a little.

In the fourth Art. In the description of a circle, by car­rying round a Radius; you define the Center to be that point which is not moved. Now a Point you had before defined cap. 8. art. 12. to be a Body moved &c. So that to say, the Point which is not moved, is as much as to say, the Body moved &c. which is not moved. Which seems to me a contradicti­ction. To this objection, you say only that which I must say to your answer, viz: It is foolish.

Art: 6. you say, If two streight lines touch one another in any one point, they will be contiguous through their whole length. No, nor alwaies;


The streight lines AB, BD, touch in one point B, and in that only. And the streight lines AC, BD, are contiguous only in their common part BC, not through their whole length. [Page 80] Yet are they such contiguous lines as your proposition meanes, viz. such as meeting in some one point, will not cut one other though never so much produced.

You said farther, Crooked incongruous lines cannot touch each other, save only in one point. Yes, said I, a Circle may touch a Parabola in two points. And you confesse it. But say, you meant that each contact is not in a line, but only in one point. Perhaps you meant so, (though yet I question whether you did then think of more con­tacts then one:) but why then did you not say so? (I mean, in the Latine? for in the English, upon this notice it is a little mended) But I reply, Yes, if those incongru­ous curve lines, have but some parts which are not crooked, (as even now you told us,) they may touch in a line. Yea & incongruous lines continually crooked, may in some pasts of them agree, though not congruous all the way, and there­fore touch in a line. And therefore even yet, it is not ac­curate.

But you'l say (as pag. 10.) Such faults as these, are not attended with shame, unlesse they be very frequent. What you mean by very frequent, I cannot tell; but, mee thinks, 'tis very ugly to have them come thus thick.

Art 7. you divide a superficiall Angle, into an Angle simply so called, and an Angle of contingence. Which you define in this manner; Two streight lines applied to each other, and contiguous in their whole length, being separated or pulled open in such manner, that their concurrence in one point remains; If it be by way of circular motion, whose center is the point of concurrence, and the lines retain their streightnesse; the quan­tity of this divergence is an angle simply so called: If by conti­nuall flexion in every imaginable point; an angle of contingence▪ I asked; to which of these two you referre the angle made by a right line cutting a circle? or whether you doe [...] take that to be a superficiall angle. You say, to an angle [...] so called, that is, as we heard but now, to an angle made by two lines which retain their streightnesse, (though one [...] them be crooked.) And then, you tell us that Rectilin [...] and Curvilincall hath nothing to doe with the nature of an angle simply so called: When yet your definition requires, that the lines retain their streightnesse. I will ask, you say, (yes I do ask; and do you give a wise answer if you can;) How can that angle which is generated by the divergence of two streigh [Page 81] lines, [whose streightnesse remains,] be other then Rectili­neall? You say, A house may remain a house, though the carriage of the timber cease. Much to the purpose! How do you ap­ply the similitude? Even so, the lines retain their streightnesse, though they be crooked, is that it? Or is it thus, Even so, the Angle remains an angle made by lines retaining their streight­nesse, when they be crooked? Perhaps you mean thus, The Angle being once made by the divergence of streight lines, re­mains an Angle though one or both of those lines be afterwards made crooked. Very good! but doth it remain the same An­gle? the same quantity of divergence? (for so you define an angle,) doth not (in your account,) the bowing of one of the lines (the other remaining as it was) alter the quanti­ty of divergence, (measurable by the Arch of a circle, as you determine) from what it was before such bowing? though yet that very bowing alone, by your doctrine, be enough to make an Angle of it selfe? Well, let it be so for once, (though it should not be so, by your principles.) But how­ever, though this should be allowed, yet at least, so long as the Angle is in making, the lines must be streight. Tell me then, J prithee, how a Sphericall Angle comes to be an An­gle simply so called. Is a sphericall Angle made by the diver­gence of streight lines or of cooked? Can it be made a sphericall Angle so long as the lines retain their streightnesse? It seemes so: for an Angle properly so called, that is, an Angle made by the divergence of streight lines, whose streightnesse re­mains, is distributed into Plain and others, (as though all Right lined angles, were not Plain Angles;) and then again into Rectilineall, Curvilineall, and mixt; as though these were, species of Rightlined Angles. Do you think it possible to make an Angle Sphericall, Curvilineall, or mixed, so long as the lines retain their streightnesse? do you think these things will ever hold together? or is this to make the prin­ciples of Geometry firm and coherent? You were better say, as the truth is, that when you formed that definition of an Angle simply so called, you had your eye only upon a Right­lined Angle, and fitted your definition thereunto; but when afterward, under the same name, you took in curvi­lineall and mixt angles, you should have altered the defi­nition, but neglected it: And then apply your ordinary apology▪ That it was indeed a fault, but not such an one [Page 82] as you need be ashamed of. But, to goe about to defend it, is more ridiculous then the thing it selfe.

At the ninth Article, I had shewed how simply you de­fined the quantity of an Angle, your definition as you call it, is this: The quantity of an Angle, is an Arch of a circle de­termined by its proportion to the whole perimeter. An Angle was before defined to be the Quantity of Divergence; That which you define now is the quantity of an angle, that is, the quantity of the quantity of divergence. Very handsomely! Then in stead of, the quantity of an Angle is measured by an Arch; you say, the quantity of an Angle is an Arch. Again, it is, you say the Arch of a circle: But what Arch? and of what circle? for you determine neither. You mean, I sup­pose, that Circle whose center is the Angular point; but you doe not say so: and, you mean also, the Arch of that circle intercepted between the two streight lines contain­ing the angle; But then you should have said so, as well as meant so. For, as the definition now runs, neither Arch, nor circle, is determined. Next you say, that this quanti­ty is to be determined (for so the words must be constru­ed to make sense of them) by the proportion of that Arch to the whole Perimeter: That is, what proportion that inter­cepted Arch hath to the whole perimeter; such proportion hath that Angle to—what? you do not tell us, to what. As for instance, suppose the Arch be a quadrant or quarter of the whole perimeter; the Angle is then a quar­ter of—somewhat no doubt; but you doe not tell us of what, Is it a quarter of an Angle? or a quarter of an Arch? or a quarter of a Circle? No; 'tis a quarter of four right Angles. 'Tis that, you should have said. Now are not these faults enough for one poor definition? They are but Negligences, you'l say: but they be scurvy ones; and there be enough of them, for lesse then two lines. But whether to commit so many negligences, in lesse then two lines, be so very frequent, as that they be attended with shame, I leave for others to judge. You should have said thus, as I then told you, (but I see you are not alwaies willing to learne;) The quantity of a Rectilineall angle, in proportion to four Right angles, is determined by the proportion of an Arch of a Circle (whose center is the Angular point) intercepted between the two streight lines containing that angle, to the whole circumference. But, it seems, you had ra­ther [Page 83] keep your own definition, with all its faults, then seem to be taught by mee: Though yet you have nothing to say in defence of any one of them; and therefore (as you use to doe in such cases) take no notice of them in your answer at all; as if no such exceptions had been made.

The like exceptions, I said, ly against the 18 Article. And you take the like care neither to mend them, nor to take notice of them.

At the 12 Art. I shewed, what a pittifull definition you had brought of Parallells; and that the Consectary from it was false, and the Demonstration thereof a sad one. You confesse all: But are not pleased that I should triumph.

Your emendation which you intimate, by inserting the same way; will do some good in the consectary, but will not make good the definition. Your new definition in the English, is little better then that of the Latine. The con­sectary, as it is now mended in the English, is true; but the demōstration of it hath many of the same faults, though not all, that I noted in the Latine: and doth not at all conclude the truth of the consectary, from that definition. As ap­pears by what I objected formerly. What you attempt to prove of two lines, you should have proved universally of any two; for so much your definition requires.

At the 13 Art. you bring a sorry argument to prove The Perimeters of Circles to be proportionable to their Semidiame­ters. The strength of the argument lies in this, The bignesse of the Perimeter is determined by its distance from the center; and the length of the Semidiameter is determined likewise by the same distance; therefore, since the same cause determines both effects, the Perimeters are proportionall to their Semidiame­ters. This consequence I deny; because, not only the big­nesse of the Perimeter, but of the circle also is determined by the same cause; as also the superficies and the solid con­tent of a spheare. For that distance of the circumference and Center, determines the greatnesse of all these. And therefore, by your argument, circles, and spheares, &c. must be proportionall to their semidiameters: which is absurd. To which retort, because you can answer nothing; you d [...]e, according to your usuall Rhetorick, fall to ranting.

At the 14 Article, I said, that your argument was but [Page 84] petitio principii. You say, There was a fault in the figure, (that it was not exactly drawn) which is now amended. True; but there is a worse fault in the demonstration, which is not amended yet. For though you have altered your Figure, and your demonstration too; yet the fault remaines. And 'twas this, not the figure, which I found fault with. For you do not prove that BH, BI, BC, (fig. 6.) are proportio­nall to AF, AD, AB, but upon supposition that FG, DE, BC, were so: which was the thing at first to be pro­ved. You say, that AF, FD, DE, are equall by construction. (True.) And, that FG, DK, BH, KE, HI, IC, are equall by Parallelism. But this is not true. The Parallelism proves that FG, DK, BH, are equall; and that KE, HI, are also equall; but not that either of these two, are equall to either of those three, (or to IC:) unlesse you first suppose that DE, is the double of DK, or FG, as AD is the double of AF, which is the very thing to be proved.

You tell me; There was another fault (yes, three or four for failing) which I might have excepted against. But the weight of the demonstration did not ly there; and I did not intend to trouble the Reader with every petty fault; (for then I should never have done:) especially in this and the next Article; where I did not then repeat your Figure at all; and therefore did briefly intimate where the fault lay: which had been direction enough for an intelligent man to have [...]ound it out: But because J did not point with a festcue to every letter, you had not the wit to under­stand it.

In like manner Art. 15. when I told you the third Corol­lary was false, and shewed you briefly the ground of your mistake; because J did not, with a festcue point from letter to letter, you were not able to spell out the meaning; but, as being lesse awake, thought it had been a dream. You had told us, that (in your 7 figure) the angles KBC, GCD, HDE, &c. were as 1, 2, 3, &c. And 'tis true. Thence you undertake in your third Corollary to give account of the bending of a streight line into the circumference of a circle; namely, by its fraction continually increasing according to the sayd numbers 1, 2, 3, &c. But how so? For, say you, the streight line KB being broken at B according to any angle, as that of KBC, and again at C according to the double of that Angle, and at D according to the treble &c. 'twill containe [Page 85] a rectilineall figure; But if the parts so broken be considered as the least that can be, that is, as so many points, 'twill be a cir­cumference. This, I said was false, and that the ground of your mistake was, that for the Angle BDE and its Remain­der HDE, you took CDE and its remainder.

And J need not say more; verbum sapienti a word for a wise man, had been enough; but, for you it seemes, it was not. You, like a man halfe a sleep, took it to be a dreame. Therefore, if you please to rub up your eyes a little, and take a festcue I will, for your better noddification, point to the letters as we goe along, and teach you to spell it out. The tangent line BK, continued indefinitely both ways, being broken at B, according to the Angle KBC, will lye in BCG: Now this line BCG being broken at C, ac­cording to the Angle GCD which is the double of KBC, its part CG, will lye in CD continued, CDδ And hi­therto you be right. But this continuation of CD, is not DH, as you seem to suppose, but Dδ which will fall be­tween DH, DE. When therefore this line CDδ comes to be broken againe at D, that its continuation may lye in DE, the faction will not be according to the Angle HDE (which indeed is the triple of KBC) but accor­ding to the angle δDE: which will be lesse then HDE, because it is evident that CD cuts BH, And indeed the very same Angle of fraction with that at C; For seeing the angle CDE, is equal to BCD, by construction, the subtenses being taken equall; the adjacent angles (anguli [...]) must be equall also, that is δDE = GCD And therefore the angle of fraction at D, precisely equall with that at C; not as 3 to 2, as you suppose. And by the same reason the angle of fraction at E must be equall to that at D; not as 4 to 3, as you suppose. And so the Angles of fraction at C, D, E &c are not as 2, 3, 4, &c. but are all equall. You see therefore, if you be yet awake, that it was not a dreame of mine, but a reall mistake of yours, to take HDE for the angle of fraction of CD. And conse­quently that your proposition was false. And this fas [...]ho [...]d was the occasion of another falsehood in the 20. Article of the 16. Chapter. (which since you have blotted out.) for there you cite this proposition as the foundation of that: And whereas you say, You cannot guesse what that propo­sition was, (and yet are very sure that it was true,) for that [Page 86] you have no coppy of that article either printed or written. If you have not, J am sure you may have, for there be enough that have. For your book sold in sheets unbound, had commonly that article amongst the rest, and by that meanes it came to me. And, rather then you should be farre to seek for it, I have recited that whole article verbatim, yea to a letter, in its due place in my Elenchus; and proved it to be false.

Against your opinion concerning the Angle of Contact, (in the 16. Article,) J said little; because J think it needs no refutation. Your opinion is this, That the angle PAD, (Fig. 2. Sect. 3) is bigger then the angle PAE, as being divi­ded by the line AE. But the angle EAC, is not bigger then the angle DAC, nor is divided by the line DA, but both of them equall as well to each other, as to the angle PAC, and also to the angle GAC. That this is your opinion, is evident. They that like it may imbrace it, for all me: And I hope, they that like it not may leave it.

The rest of what concernes this businesse, is considered before in its proper place.

At the 18. Art beside what is common to this and the seventh, J noted for a fault, and you doe not deny it so to be, that you deliver it as Euclide's opinion, that a Solid angle is but an Aggregate of plain Angles. Jt may be your opinion; but surely 'twas none of Euclide's. If you had thought it had; you should have here if you could, pro­duced somewhat out of Euclide where he declares such an opinion.

At the 19. Article All the ways by which two lines respect one an other, or all the variety of their position, seem, you sayd, to be comprised under four kinds; For they are etiher Parallells; or (if produced at least) make an angle; or (if bigge enough) be Contingents; or lastly are asymptotes. By Asymptots you mean (not all such as never meet, for then Prallells would fall under this kind; but) such as will come always nearer and nearer together, but never touch one another (you might have added this other character; that they doe so approach each other, as that at length their distance will be lesse than any assignable quantity. But it seems you allow your Asymptotes a greater latitude: And doe in your English, determine your meaning so to be: And that, I suppose, because you had neglected to put in, that limitation, in the latine; [Page 87] and therefore were not willing upon my intimation to mend it in the English. For none else that I knew, speak of any other lines under the name of Asymptotes, but such as doe not only eternally approach, but do approach also infi­nitely neare, And, I have reason to believe, from your sim­ple objection Less. 5. p. 48. l. 23. that you thought those two must needs go together, viz. that whatsoever quanti­ties doe eternally approach, must needs at last come infinitely neare. But however wee'l be content, if you would have it so, to take Asymptotes at what latitude you will give it them.) You say now, that I am offended at the word it seems. No, Sir, no offence at all. I am not at all angry, that, to you, it should seem so. I said but, that to mee, it seemed otherwise; (And, I hope you are not offended that all things did not seem to me, as they did to you: For I perceive, that by this time, it seems otherwise to you also. Which hath made you in the English, to give us this Article new moulded.) I shewed you then, ma­ny other positions of lines, which doe not agree to any of your four kinds. And you confesse it. And some of them such, as will not be salved with your new botch. As they that please to compare them will soon find.

J touched at some other faults; as, That the definition of points alike situate, (art. 31.) seemed very uncouth.

That the word Figure, which is defined art. 22. had been oft used long before it was defined; (which though it be, with you a small fault, yet a fault it is.) And you con­fesse it.

That by your definition a solid spheare, and a spheare made hollow within, is the same figure. (For your defini­tion takes notice of no superficies, but that within which they are included: your words are, intra quam solidum includitur. You say, It is my shall [...]wnesse, to think, those points which are in the concave superficies of a hallowed sphear not to be contiguous to any thing without it, because that whole concave superficies is within the whole spheare. It may be my shallownesse perhaps; but it is I confesse, my opinion, that this concave superficies being, as you say all within a spheare, (and therefore may be contiguous to somewhat within the spheare,) is not con­tiguous to any thing without it, (if it be, tell me to what? and how it can be contiguous when the whole thicknesse [Page 88] of the spheare is between? unlesse you think it can touch at a distance:) Nor, is that superficies intra quam sphaera in­cluditur: For if, as you say, that whole superficies be within the whole sphear, how can the spheare be within that superficies? You should rather have confest, as the truth is, that you did not think of a solides being contained by two or more superficies, not contiguous to one an another: and [...], had not provided for that case.

I excepted likewise against your definition of Like t [...]ngs, cited here out of Cap. 9. art. 2. Those things, you de [...]ine to be Like, which differ only in magitude. They do not, I say, al­waies differ in this; for it is possible like things may be equall (And therefore if they differ in nothing else, they differ not ut all.) And sometimes again they may differ in somewhat else; at least in position. Else what needs your next definion, of similia similiter posita? if it were not possible for similia to be dissimiliter posita? To which exception (because you had nothing to say) you say nothing

So your definition of like figures alike placed, I said was false: you confesse it is so, (and therefore amend it in the English.) You confesse you say, there wants something which should have been added; but call we Foole for taking notice of it: Or else, you call your selfe Foole, for not supplying it; For you say, that it might easily be supplyed by any student in Geometry, that is not otherwise a Foole. But, rather then fall out for it, wee'l divide the Foole between us; and cry Ambo. 'Twas I, like a foole, took notice of that to be wanting, which you like a Foole, omitted, when you should have supplyed it.

The 15. Chapter, because it contained but little Mathe­maticall, I did but touch at; leaving that for my worthy Collegue to take to taske, with the rest of your Philosophy. Which he hath done to purpose. Yet some few things J noted as a rast of the rest.

J noted that (contrary to others who define Time to be the measure of motion) you determine Motion to be the measure of time; And yet (contrary to your own deter­mination) you do frequently make Time the measure of motion; measuring both motion, and its affections (swift­nesse, slownesse, uniformity, &c.) by Time. You confesse it to be so: But raile at us for minding Books, more than Clocks and hour-glasses. And then (contrary to both) you [Page 89] tell us, that time and motion have but one dimension which is a line. And at last would perswade your English Reader, that I would have you measure swiftnesse and slownesse, by longer and shorter motion: But they that understand Latine, can find nothing to that purpose: I only told you what you did, (and how absurd that was,) not, what I would have you do▪

Then, because it still runnes into you mind, that I had some where said, That a point is nothing (though no body can tell where;) you fall againe upon that. For my part, though I oft affirm that a Mathematicall point, hath no parts▪ yet J never denyed it to stand for as much at least, as a cy­phar doth in numbers; and you allow it noe more, (c. 16; art. 20.) your words are these Punctum inter quantitates nihil est, ut inter numeros cyphra. Is it then J, or you? that say a point is nothing?

You told us soon after, that All endeavour (for even that is motion) whether strong or weak, is propogated to infinite distance. As if (said J) the sk [...]pping of a Flea did propagate a motion as farre as the Indies. You ask, how we know it? If you meane, How we know that it is so; Truely, J doe not know that at all. If you meane, how we know that it follows from what you affirme; It is so evident a conse­quence from the words alleadged, that you need not aske; Or, if those words be not enough those that follow be yet fuller, Procedit ergo omnis conatus, sive in Va [...]uo, sive in plano, non modo ad distantiam quantamvis, sed etiam in tempore quantulocunque, id [...]est, in instanti. That [...]s, All endeavour of mo­tion whether the space be Full or Emty, is continued, not only to as great a distance as is imaginable, but in as little a time, that is, in an instant. But if your meaning be, what do I say to the contrary? Truely I say nothing to the contrary. They that have a minde to believe it, may.

Then you goe on to catechise us; What is your name? Are you Philosophers? or Geometricians? or Logicians? &c. (Nay, never aske that question, we know you are good at giving names, without asking) I hope, the next question will be, Who gave you that name? And truely as to many of the names you give us, a man might easily believe, yourself were the Godfather, you call us so often by your own names.

Lastly, Of two things moving with equall swiftnesse, that, say you, strikes hardest which is bignesse. No, say I, but that [Page 90] which is heaviest. A bullet of Lead, though but with equall speed, strikes harder then a blown Bladder. If any man think otherwise let him try.

SECT. X. Concerning his 16 Chapter.

IN the 16 Chapter, I said, there were 20 Articles; you say, but 19. 'Tis easily reconciled. There be twenty in my book; and there were 20 in yours too, before the last was cut or torne out: now, it seems, in yours there are but nineteen.

Well; but, be they twenty, or be they nineteen; twenty to one but the greatest number of them be naught. I do confidently affirme, you say, that all but three are false. Nay, that's false, to begin with. I said, that, all but three were unsound. Some of them be non-sense, or absurd; some be false; some undemonstrated; all unsound; at least, within three: And I have already proved them so to be. But you (you say) do affirme, that they are all true, and truly demon­strated. And that's answer enough to all my arguments. What need you say any more? If that be true, doubtlesse you have the better on't. But let's trie a little, if we cannot find one unsound amongst them.

Your first Proposition as it stands yet in the Latine, you say, is this, The velocity of any Body moved, during any Time, is so much, as is the product of the Impetus in one point of Time, mul­tiplied into the whole Time. Well, I hope at least the first is sound, is it not? In one Point, you say; but which one? Is it any one? or some one? Nay 'tis but some one, not any one; but, which one, you tell us not. What say you to this? Is it sound? This, you confesse, without supplying what is want­ing, is not intelligible. Very good! Habemus confitentem re­rum. To the first [...] Article as it is uncorrected in the Latine, [...] ob­ject, you say, That meaning by Impetus, some middle impetus, and assigning none, you determine nothing▪ Well what say you to that? you say, 'tis true. And then you rant at us for not mending it, (as though we were bound to mend your faults) yet look again, and you'l find J did. J told you what you should have said; as well as what you said amisse. But e­nough of this. Here's one fault confessed.

[Page 91] In the same Article; you would have the Impetus applied ordinately to any streight line, making an angle with it. J asked, How an impetus can be ordinately applied to a Line? or make an Angle with it? Absurdly, you say; and that's the answer. And J told you how this should have been mended too.

You tell me that Archimedes and others say, Let such a line be the Time, and again p. 36. l. 16. Let the line AB be the Time. Very likely! just as when we say, Let the Time be A. That is, Let it be so designed; or, Let the Line AB, or the letter A, be the Symbole of the Time. What then? Doth it therefore follow, that either Lines or Letters be homoge­neous to Time? No such matter. Their Symbols may be Homogeneous though the Things be not. You say farther, in the same Article: If the Impetus increase uniformely, the whole velocity of the motion shall be represented by a Triangle, one side whereof is the whole Time, and the other the greatest Impe­tus, (Well! & what shall be the third side? or what angle shall these contain? Do you think that the assigning of two sides, without an Angle, will sufficiently determine the bignesse of a triangle? But lets go on.)

Or else &c. Or lastly by a Parallelogramme having for one side a mean proportionall between the greatest Impetus and the halfe thereof. Well, but what for the other side? And, what Angle? Is a Parallelogramme, said J, sufficiently deter­mined, be the assignement of but one side, and never an angle? what think you? is this sound? It was indeed a very great o­versight, you confesse, to designe a Parallelogram by one only side. And is not all this sufficient to prove the first Article unsound? if it be not, wee'l go on, for there be more faults yet.

For, say you, these two parallelograms are equall both each to other, and to the (fore mentioned) Triangle (without ha­ving any consideration of Angles at all) as is demonstrated in the Elements of Geometry. This, I say, is notoriously false: For a Triangle of which nothing is determined but two sides: and a Parallelogramme, of which the sides only are determined, but nothing concerning the Angles: can never by any Geometry, be demonstrated to be equall. This therfore is not only unsound, but false. And all this J told you before. What an impudence then is it, when you knew all this, to affirm, that they be, all true and all truly demonstra­ted, [Page 92] when the very first of them is thus notoriously faulty! But we have not done yet.

It might be hoped, that this confessed oversight is, at lest mended in the English: (especially since you tell us that one from beyond sea hath taught you how to mend it) No such matter. For the Amendment is as bad or worse then what we had before. For now it runs thus. The whole velocity shall be represented by a Triangle &c. (as be­fore) or else by a Parallelogram, one of whose sides is the whole Time of motion; and the other, half the greatest Impetus: Or lastly, by a Parallelogram, having for one si [...]e a mean pro­portionall between the whole Time and the halfe of that Time; and for the other side the halfe of the greatest impetus. For both these Parallelograms are equall to one another, and seve­rally equall to the Triangle which is made of the whole line of Time, and the greatest acquired impetus. As is demonstrated in the Elements of Geometry. Now this, you shall see, is pit­tifully faise. Let the time be T; and the greatest impetus, I: and let the Angles be supposed all Right Angles (for such your Figures represent, though your text says nothing of them.) The Altitude therefore of the triangle, is T, (the whole time:) the Basis I, (the greatest impet [...]s:) and consequently the Area thereof is one halfe of T × I: that is ½ IT. Again the Altitude of the former Paral­lelogram, T, (the whole time,) its Basis, ½ [...], (half the greatest impetus,) and therefore the area T × ½ I, or ½ IT; equall to that of the Triangle Lets see now whe­ther the last Parallelogram be equall to either of these, as you affirm. The Altitude you will have to be a mean pro­portionall between the whole Time and its halfe: that is, between T&½ T; It is therefore the root of T × ½ T, that is the root of ½ Tq, that is √½ Tq, or T√½: The Basis you will have to be one half of the greatest Impetus, that is ½ I: And con­sequently, the Area must be ½ I × √½ Tq, or ½ I × T √½, or ½ IT√½. But ½ IT√½ is not equall to ½ IT: Therefore this Parallelogram is not equall either to the former, or to the Triangle. 'Tis false therefore which you affirmed. Quod erat demonstrandum.

[Page 93] Now what do you think of the businesse? is not the matter well amended? 'Twas bad before, now 'tis worse. When you told us but of one side, and left us to guesse the other, 'twas at our perill if we did not guesse right, and 'twas to be hoped, you meant well, though you forgot to set it down. But, now you tell us, what you meant, we find that you neither said well, nor meant well: For what you now say is clearly false. The two Parallelograms which you affirm to be equall, are no more equall then the Side and the Diagonall of a Square; but just in the same propor­tion; viz. as √½ to 1. Nay was it not a pure piece of wisdome in you, that, when you had been taught from be­yond Sea, as you tell us, how it should have been mended, you had not yet the wisdome to take good counsell; but, trusting to your own little wit, have made it worse than it was? it falls out very unluckily, you see, that when you af­firmed so confidently, that they are all true, and all truly de­monstrated, the very first of them should be so wretchedly faulty. But enough of this. Wee'l try whether the next will prove better.

In the second Article you give us this Proposition. In every uniform motion, the lengths passed over are to one another, as the product of the ones Impetus multiplied into its time, to the product of the others Impetus multiplied into its time. And why not, said J, (without any more adoe) as the time to the time? Which needed no other demonstration than to cite the definition of Ʋniform motion, (viz. which doth in e­quall or proportionall times, dispatch equall or proportionall lengths.) What need had you to cumber the Proposition with Impetus and Multiplication, and Products, when they might as well be spared? and then put your selfe to the trouble of a long and needlesse demonstration, when the bare citing of a definition would have served the turne? You answer, That the product of the Time and [...]mpetus, to the product of the Time and Impetus, is also as the Time to the Time. and therefore the Proposition is true. Yes doubtlesse; and therefore I did not find fault with it, as false; but as foo­lish, to make such a busle to no purpose. For, by your own confession, the proportion of the lengths dispatched, is as well designed by the termes alone, as by those multiplica­tions and products.

But there is another fault which J f [...] with your pro­position; [Page 94] [...] told you that, instead of, in every uniforme motion, you should have said, (and, that you might have said it safely, as the rest of the wordsly,) in all uniform motions; for you make use of this proposition afterwards, not only in comparing divers parts of any the same uniforme moti­on, but in comparing divers motions one with another. But at this you are highly offended, that J should under­stand to what purpose this Proposition is brought, better than your selfe; and that J should presume to tell you, what you ought to have said. (And, on the other hand, when J do not do so, you blame mee, that J do not to my reprehension adde a correction: So that, it seems, you are nei­ther well, full nor fasting: J must neither do it, nor let it alone.) And then you go on to rant, after your fashion, at Wit and Mystery, and times and wayes, and steddy brains, at rea­ding thoughts, and noise of words, at step and stumble, &c. And yet, for all the anger, (when the heats over) you think best to take my counsell; and therefore say in the English, just as J said it should have been in the Latine.

The proposition then being thus to be understood, (though at first, ill worded,) the demonstration, I said, would not hold. For though it will doe well enough (yea more then enough; for you might have spared halfe of it;) in comparing severall parts of the same motion, and in comparing severall motions of the same swiftnesse; yet for the comparing of uniforme motions in generall, it will not serve by no meanes; for you do assume at the first dash, that the motions compared have the same Impetus. Now this must not be allowed. For it's very possible (as you now know, since, J told you, though before you see­med to be ignorant of it, as J then convinced you;) that two motions may be both uniforme; and yet not have both the same Impetus. Your proposition therefore (as it was to be understood) was not truly demonstrated.

Now, because this was very evident, and not to be de­nied; therefore you thought it best to make no words of it, but mend it as well as you could. And so, in the Eng­lish, you have mended the proposition, as J bid you; and given us a new demonstration, which is pretty good; But not yet without fault. For in stead of the length AF (fig. 1.) you should have said, the length DG: for the length should have been taken in the line DE, which, according to [Page 95] your construction, is the line of Lengths; not in the line AB, which is, by construction, the line of Times. So im­possible a thing is it, for you to mend one fault and not to make another.

But if all these faults be not enough to make this Article unsound, there is yet another, before we leave. Since there­fore you say, in uniforme motion, the Lengths dispatched are to one another, as the Times in which they are dispatched; it will also be, by permutation, as time to length, so time to length. This consequence I denied; because Permutation of pro­portion hath place only in Homogenealls, no [...] in Hetero­genealls; (and referred you for farther instruction concer­ning it, to what Clavious hath on the 16. Prop. of the 15. of Euclide.) You tell me, that I think, line and time are He­terogeneous. Yes, and you think so to if you be not a foole. If not, pray tell me how many yards long is an hour? Or, How much line will make a day?

Well, lets try a third Article. (For the two first you see be nought, that's a bad begining.)

Art. 3. In motion uniformely accelerated from rest, (that is, when the Impetus increaseth in proportion to the times) the length run over in one time, is to the length run over in an other time. (In the English for Impetus, you have put mean Impetus, and so in some other propositions; but that neither mends nor mars the businesse.)

To this, first you dream of an objection, and then think of an answer to it. I object you say, that the Lengths run over, are in that proportion which the Impetus hath to the Impetus. Prithee tell me, where I made that objection to this ar­ticle; and i'le confesse 'twas simply done. But 'till then, i'le say 'tis done like your selfe, to say so however. (For 'tis lawfull with you to say any thing, true or false) Your English Reader, perhaps, may think 'tis true.

Next, You aske, you say, where it is that you say or dreame, that the lengths run over are in proportion of the Impetus to the Times? But prithee, why dost thou aske me such a questi­on? Am I bound to give an account of all thy dreames? Perhaps you dreamed that I had charged you with such a saying; But, look again, and you'l find that's but a dreame as well as the rest.

That which I said was this, The parallell line FH, BI, (fig. 1.) do shew what proportion the Impetus at F hath to [Page 96] the Impetus at B; to wit, the same with the time AF, to the time AB: (And is not this your meaning, when you say the Impetus increaseth in proportion to the times?) But, though those (and other parallell lines) do define what proportion the severall Impetus have to each other; yet they do not designe (by permutation of proportion, as you fancied in the Corollary of the precedent article) what proportion the severall Impetus have to the Times; be­cause they be Heterogeneous, and do not admit of that permutation. And these are the words, which gave occa­sion to those your two dreames. And then (as if between sleeping and waking) you ask, if it be you or [...] that dream? Had you been well awake, you needed not have asked the question.

The objections that I made to it, were these.

First, that in stead of motu accelerato, (accelerate moti­on,) you should have said, motibus acceleratis (accelerate motions,) because you speake of more than one. You say, there is no such matter: and bid mee give an instance. J will so, and that without going farther then your present [...]. Let AB (say you) represent a Time &c. Againe, let AF represent another time &c. And in each of these times you suppose a Motion, which motions this proposi­tion compares. Therefore, say I, there must be at least two Motions, because two Times; unlesse you will say, that one and the same motion may be now, and anon too.

I objected farther, that the demonstration doth nor prove the proportion; except only in one case, to which you do not restraine it. For the whose stresse of your demonstration, (in the Latine) lyes upon this, that the triangles ABI, AFK, be like triangles (where you inferre, that the space dispatched in the latter time AK, is to that of the former time AB, as the triangle ABI, to the trian­gle AFK, that is in the duplicate proportion of the times AB, AF.) Which supposeth that the second motion in the time AF, doth acquire the same Impetus which the first motion had acquired in equall time. Whereas it is possi­ble, that, of two motions, each of them uniformly accele­rated, the one of them may in half the time acquire as great a swiftnesse, as the other doth in the whole time; If there­fore the latter motion in the same AF, do acquire a swift­nesse equall to that of the former in the time AB, (which [Page 97] may very well be, for the words uniformly accelerated, doe imply only the manner of acceleration, not the degree of cele­rity; as your selfe now discern, though then you did not,) the triangles will be, not ABI, AFK, but ABI, AFH; which are not like triangles, but unlike; and so the de­monstration falls. You should have provided in your pro­position, not only that the two motions, (the one in the time AB, the other in the time AF,) be each of them u­niformly accelerated, but that they be both equally swift. Which when you have neglected to take care of, you af­firm that universally, which will hold only in one case.

But the truth is, 'tis evident enough, by this and divers other Articles, that you took the manner of acceleration, (viz. if in the same, in the duplicate, or triplicate, &c. proportion to the times,) had sufficiently determined the speed also. And therefore took it for granted, that the motion in the time AF, if uniformly accelerated, must needs attain precisely the same degree of the celerity, that the other motion in the time AB, uniformly also accele­rated, had attained in equall time. (Which to be a very great mistake, you now doe apprehend.) Otherwise you would not have let these Articles ly so naked without such provision; nor would you, (as in the 13 Article, and those that follow,) undertake, by the manner of acce­leration, and the last acquired Impetus, to determine the time of motion. Whereas, in the same manner of accele­ration (whether uniformly, or in the duplicate, or tri­plicate, or quadruplicate proportion;) any assignable im­petus or degree of celerity, may be attained in any assigna­ble time whatever.

I objected farther, that because, as hath been shewed, the Triangle AFK, or AFH, is not necessarily like to the triangle ABI, therefore it doth not follow that the length passed over, will be in duplicate proportion to the time. For unlesse the triangles be alike, the proportion of them will not be duplicate to that of their homologous sides.

Now these two Objections were clear and full, (and did destroy your whole demonstration;) and this you dis­cerned well enough, though you did not think fit to make any reply or confession; (but invent some other objecti­ons, which I never made, that you might seem to answer to somewhat.) And therefore in the English, without [Page 98] making any words of it, you mend it. And instead of those words in the Latine, As the triangle ABI, to the triangle AFK, that is, in duplicate proportion of the time AB to AF: you say in the English. As the triangle ABI, to the tri­angle AFK, that is, if the triangles be like in the duplicate proportion of the time AB, to the time AF; but, if unlike in the proportion compounded &c. (which is a clear confession of all those objections. But let's go on. Compounded of what?) of AB, to Bi, and of AK, to AF. No such mat­ter; of AF to FK, (that's it you would have said:) not, of AK to AF. There's one fault therefore; but thats not all. Of AB to AF, and of BI to FK; thats it you should have said: for AB to BI, the Time to the Impetus, hath no proportion at all; but are Heterogeneous, as I have of­ten told you. There's a second fault therefore in your e­mendation. And is not this Tinker-like, to mend one hole and make two? Nay there is a third yet, which is the worst of all.

In the mending of this fault, (though you had not mis­sed in it,) you have discovered another, which you did your endeavour, but now, to hide. I said in the proposi­tion for motion, you should have said motions; because it was intended of more than one compared. You tell me, there's no such matter; meaning, I suppose, the latter motion in the time AF, was but part of that former motion in the time AB: But if, as you now confesse, the triangle AFK, be not necessarily alike triangle to ABI, (but that the point K may fall either within or without the line AI,) then must this be not only another, but an unlike motion to the former: viz. either faster or slower, though uniformly accelerated as that was. Do not you know that old rule; Oportet esse memorem. But this 'tis, when men will commit faults, and then deny them. And yet presently after, by going about to mend them, betray themselves.

Much such luck you have in mending the Corollary. You had said in the Latine, In motion uniformly accelerated, the lengths transmitted are in the duplicate proportion of their times. This, I said, was true in one case, (viz. in equall celeri­ties,) but not universally. Therefore you, to mend the matter, in the English make it worse; In motion uniformly accelerated, say you, the proportion of the lengths transmitted, to that of their Times, (No, but the proportion of the length [Page 99] transmitted, one to the other,) is compounded of the propor­tions of the Times to the Times, and Impetus to Impetus.

There be more faults in this Article; but I am weary of the businesse; let's go to the next.

The fourth Article hath all the faults that the third hath, (which are enough as wee have seen already,) and some more.

First, for motu accelerato, you should have said motibus acceleratis; because you compare more motions then one.

Secondly, the Motion performed in the time AF, (Fig. 2.) though accelerate according to the duplicate propor­tion of the times, as well as that in the time AB; yet may that be either swifter or slower than this; (because as we have often said, the manner of acceleration doth not de­termine the degree of celerity;) And therefore the point K which determines its greatest Impetus, doth not necessa­rily fall in the Parabolicall line, but may fall either within or without it: according as the celerity is lesse or more.

Thirdly, And therefore it doth not follow, that the Lengths dispatched by such motion, are in triplicate pro­portion to their Times. For this only depends upon sup­position that the point K in the second motion, must needs fall in the Parabola AI, designed by the first motion.

Now these two latter faults, in the former Article, you did endeavour to amend in the English: But because, it seems, here it was harder to doe, you have left them as they were before. That these were faults, you were clear­ly convinced of; and do as good as confesse, by your at­tempt to mend them in the third Article. But because you saw it was impossible for one of your capacity to think of mending all; you resolve to give over mending, and (which is the easier of the two) resolve to try the strength of your brow.

But, as if there were a necessity of growing worse and worse; beside those, common to this and the third article, here is an addition of more faults, as foul as any of them.

In your demonstration; your stresse lyes upon this argu­ment, Seeing the proportion of FK to BI, is supposed dupli­cate to that of AE to AB, (which yet is a false supposi­tion; for the ordinate lines in a Parabola are not in du­plicate, but in subduplicate proportion to the diameters: [Page 100] But, suppose it true, what then?) that of AB to AF, will be duplicate to that of BI to FK. That is, Because the Or­dinate lines in a Parabola, are in duplicate proportion to the Dia­meters; therefore those Diameters are in duplicate proportion to those Ordinate lines. Which if it be not absurd enough, I would it were. First, the proportion of the Ordinates, must be duplicate to that of the Diameters, (because M. Hobs will have it so;) and then (by the virtue of Hocus Pocus) this must be duplicate to that.

To this you make no reply: but inslead thereof, disguise the matter in your Lesson, by putting double for duplicate, as if they were all one; (though yet Chap. 13. art. 16. wee have, in the English, a long harangue of your own to shew the difference between them;) and then raile at those that first brought up the distinction; and tell us, (which is notoriously false) that Euclide never used but one word for Double and Duplicate; (that is [...], and [...], are with M. Hobs but one and the same word.) But what is all this to the ra­king off that absurdity with which you are here char­ged?

Next I shewed you, that your whole argument was grounded upon a false supposition; viz. that the veloci­ty of the motion in hand, was to be designed by the Semi­parabola AKB; and that the ordinate lines in that Semi­parabola, (by which you would have the increasing Impe­tus to be designed) did increase in duplicate proportion to their Diameters (by which you designe the Times▪) both which are false. For, these ordinate lines, are well known (to all but M. Hobs) to increase in the Subdupli­cate (not the Duplicate) proportion of the Diameters: And consequently that Semiparabola can never expresse the Aggregate of the Impetus thus increasing.

I did farther demonstrate, that the point K, ought to fall within the Triangle ABI, not without it; and there­fore not in the Parabolicall line by you designed. The de­monstration was easy. For if the time AF be one halfe of AB, that is, as 1 to 2: the Impetus increasing in dupli­cate proportion to the times, must be as 1 to 4; and there­fore FK will be but a quarter of BI. But because AF is halfe of AB, therefore FN will be halfe of BI. And [Page 101] consequently FK (a quarter) will be lesse then FN, which is the halfe of BI. Which because you saw too evident to be contradicted▪ you thought it best (as your usuall cu­stome is in such cases,) to raise at it in stead of answe­ring it.

I shewed you farther, that the Aggregate of all the Impe­tus in a motion thus accelerated, or the whole Velocity, was not ⅔ of the Parallelogram AI, but only ⅓ of it. For this aggregate is not to be designed by a Semiparabola, but by the complement of a Semiparabola. And many o­ther mistakes▪ consequent thereunto. And indeed so many, as that dispairing of mending them all, you resolved to let them stand as they were.

Yet I shewed you withall the chiefe ground of all these mistakes, and how they might have been mended. But it appeares you had not the wit to understand it, and there­fore durst not venture upon it. But have left this whole article such an Hodge podge of errors, as would turne a quea [...]ie stomach, but to examine it.

And your Corollarys are false also.

In the first Corollary, 'Tis false which you affirme, that the proportion of the parabola ABI to the parabola AFK, is triplicate to the proportion of the times, AB to AF, (as it is in the English.) or of the Impetus BI to FK, (as it is in the Latine.) This exception you confesse to be just, yet leave it uncorrected in the English; because you know not how to mend it; without giving your selfe the ly in the rest. For as badde as it is, it follows, with the rest of your doctrine. It must all stand or fall together.

The second Corollary, (at least, if understood of the Parabola,) is also false; for the segments of a parabola (of equall height) successively from the Vertex, are not as the numbers 7, 19, 37, &c. the difference of the Cubes 1. 8, 27, 64, &c. but us the differences of these surd nūbers 1, √8, √27, √24. &c. That which you alledge to justify your selfe; that the parts of the Parabola cut off are as the cubes of their bases; is but a repetition of the same error. They are not as the Cubes of their Bases, but as the square roots of such Cubes.

The third Corrollary is wholly false, A motion so acce­lerated doth not dispach two thirds; but one third, of [Page 102] what a uniform motion would have done, with an Impetus equall to the greatest of those so increasing. You say, I give no demonstration of it. (It may be so; and it's all one to me, whether you believe it to be true or no. You may think, if you please, that the Corollary is true still; it will not hurt me.) Yet if you considered what had been said before, you should have seen the reason: viz. because the aggre­gate of the Impetus did not constitute a semiparabole, but the complement of a semiparabola, which is not ⅔ but ⅓ of the Parallelogram.

The fift article hath the same faults with the fourth; and runnes all upon the same mistakes.

The main foundation of all these continued errors, was, I told you, the ignorance of what is proportion duplicate, triplicate, subduplicate, subtriplicate, &c. Of three numbers in continuall proportion, if the first be the lest, the proportion of the first to the second is duplicate, of what it hath to the third, not subduplicate: That was your opinion Cap. 13. § 16. of the Latine. In the English, you have retracted that error in part; yet retaine all the ill consequences that followed from it.

Next, you suppose the Aggregates of the Impetus increa­sing in the duplicate, triplicate &c. proportion of the times, to be designed by the Parabola, and Parabolasters, (as if their ordinates did increase in the duplicate, tripli­cate, &c. proportion of their Diameters; cujus contrarium verum est;) whereas you should have designed them by the complements of those figures, But you aske me what line that (complement) is? No Line, good Sir, but a Figure, which with the figure of the Semiparabola &c. doth com­pleare the Parallelogram. You ought therefore (as I then told you, but you understood it not,) to have described your Parabola the other way; that the convex (not the concave) of the parabolicall line should haue been to­wards the line of times AB. so should the point K have fallen between N and F; and the convex of the Parabola with AT (the tangent) and BI (a parallel of the Diame­ter,) have contained the complement of that parabola, whose diameter therefore must have been AC, and its Ordinate CI.

Next, in pursuance of this error, you make the whole velocity, in these accelerations (in duplicate, tripli­cate [Page 103] &c. proportion of the times) to be ⅔, ¾, &c. of the velocity of an uniforme motion with the greatest acquired Impetus, (because the Parabola and Parabolasters, have such proportion to their Para [...]lelograms) whereas they are indeed but ⅓, ¼, &c. thereof; for such is the proportion of the complements of those figures, to their Parallelograms. Now upon these false principles, with many more conso­nant hereunto, you ground not only the doctrine of the fourth and fifth Articles, but also most of those that follow; especially the thirteenth and thenceforth to the end of the Chapter: which are all therefore of as little worth as these.

But enough of this. The first five Articles therefore are found to be unsound; and many ways faulty.

The sixth, seventh and eighth Articles, I did let passe for sound: And you quarrell with me for so doing. But I said withall, you might have delivered as much to better purpose in three lines, as there you did in five pages. (Be­side such petty errors all along as it were endlesse every where to take notice of) which gives you a new occasi­on to raile at Symbols.

After these three, there is not one sound Article to the end of the Chapter, and what those were before, we have heard already.

The ninth article is this, If a thing be moved by two Mo­vents at once, concurring in what angle soever, of which the one is moved uniformely, the other with motion uniformely accele­eated from rest, till it acqu [...]e an Impetus equall to that of the Ʋniforme motion; the line in which the thing moved is carried, will be the crooked line of a semiparabola. Very good! but of what semiparabola? (for hitherto, we have nothing but a proporsion of Galilaeo's transcribed.) You tell us, [...]t shall be that Semiparabola, whose Busis is the Impetus last ac­quired; And this is the whole designation of your Para­bola.

To this designation I objected many things.

First, that the Basis of a Semiparabala is not an Impetus but a Line: and therefore 'tis absurd to talke of a Semi­parabola whose Basis is an Impetus.

Secondly, if it be said that an Impetus may be designed by a line; I grant it; (a line may be the Symbol of an Im­petus, [Page 104] as well as a Letter▪) but this line, is what line you please; (for any Impetus may be designed by any line at pleasure:) & so, to say that It is a Semiparabola, whose basis is that line which designes the Impetus: is all one as to say, it is a Semipa­rabola, whose basis is what line you please. So that we have not so much as the Basis of this Semiparabola deter­mined.

Thirdly, suppose that the Base had been determined, (as it is not) yet it is a simple thing to think that determi­ning the basis, doth determine the Parabola. For there may be infinite Parabola's described upon the same Base. You doe not tell us what Altitude, what Diameter, nor what Inclination this Parabola is to have.

Now to this you keep a bawling; but say nothing to the businesse. You tell us, that you had said, what angle so­ever. That is, you supposed your Mevents to concurre in what angle [...]soever; but you sayd nothing of what was to be the angle of inclination in the Parabola. You might have said indeed, it was to be the same with that of the Movents: But you did not; and therefore I blam'd you for omit­ting it.

Then, as to the Diameter, you might have said (but you did not) that the line of the acccelerate motion, would be the diameter. 'Twas another fault therefore not to say so; for that had been requisite, to the determining of the Parabola.

But when you had so said; this had but determined the Position of the diameter, not its magnitude: it may be long or short, at pleasure notwithstanding this.

Then as to the altitude of it; this remaines as much unde­termined as the rest. You tell us neither where the Vertex is, nor how farre it is supposed to be distant from the Base. you might have said, (but you did not,) that the point of Rest, where the two motions begunne, was the vertex. (And twas your fault you did not say so in the latine, as you have now done in the English.) But had you so said, you had not thereby determined either the Altitude, or the Diameters length.

You say, The vertex and Base being given, I had not the wit to see that the altitude of the Parabola is determined. No true­ly; nor have I yet. But it seems you had so little wit, as to think it was. Had the vertex and Base been, positione data: [Page 105] I confesse, it had been determined: (For then I had been told how farre off from the Base, the Vertex had been.) But when the Base is only magnitudine data, there, is no such thing determined. For a base of such a bignesse, may be within an Inch, and it may be above an E [...]l from the Vertex, according as the Parameter is greater or lesse. Now you doe not pretend any other designation of the Base, then that it be equall to such an Impetus; which de­termines only the bignesse of it, not the distance from the vertex. So that the altit [...]de, notwithstanding this flamme, remaines undetermined. (And must do so, whatever you think, till you do determine the degree of celerity, which answers to the Parameter of the Parabola; as well as the manner of acceleration, which only determines that it is a Parabola, but not what Parabola. The proposition there­fore is extreamly imperfect; nor doth determine that which it did undertake to determine.

The figure is yet worse. You suppose the line AB, (fig. 6.) by uniforme motion, to have dispatched the length AC, or BD, and so ly in CD; in the same time that the line AC, by motion uniformely accelerated, dispatcheth, the length AB, or CD, to come and lye on CD. That is, (because AB, according to your figure, it about twise the length of AC,) the motion accelerated doth, in the same time, dispatch about twise the length of what is dispatched by the uniforme motion. But it is evident, the accelerate motion is all the way, to the very last point, slower than the uniforme, (for by supposition, it doth not till the last point, attain to that Impetus or swiftnesse, with which the uniform motion was carryed all the way.) Therefore according to you, a slower motion doth, in the same time, dispatch a a greater length then the swifter, Which is absurd enough: And to which you make no reply.

The demonstration also (saving what you have from Galileo) I then shewed you to be faulty; and you reply nothing in its vindication and therefore I need not repeat it. You have in the English a little disguised the proposition, but to little purpose. The Parabola which you undertake to determine, remaines as undetermined as it was before. And the figure the same with all its faults: And the de­monstration no whit mended. So much of this Article as [Page 106] yo [...] tooke out of Galileo was good, before you spoild it; but the next is all naught.

Your tenth Article doth but repeat all the faults of the ninth, and you have nothing more to say in the vindicati­on of this then of that. The Parabolaster here, remaines as undetermined as the Porabola there; your Figure (fig. 6.) makes the flower motion in the same time to dispatch the greater length; your demonstration is faulty as that was. Nay you have not here, so much as disguised it in your English, as you did the former; but left it as it was in the Latine. So that this falls under the same condemnation with the former.

I hinted also, that we have here a great talke of Para­bolasters which are not to be defined till the next Chapter. But that's a small fault. Your English helps it, by sending us thither for the definiton.

Your eleventh Article undertakes to give us a generall rule, to find what kind of line shall be made by the motion of a body carryed by the concurse of any two Movents, the one of them Ʋniformely, the other with acccleration, but in such proportion of Spaces and Times as are explicable by Numbers, as Duplicate, Triplicate &c. or such as may be designed by any broken num­ber whatsoever. Your rule for this, sends us to the Table of Chap, 17▪ art. 3. to seek there a Fraction whose Denominator is to be the summe of the Exponents of Length and Time; and its Numerator, the exponent of the Length. Upon this I proposed you a case which falls within your proposition, but not within your Rule: (to shew that your Rule did not performe what you undertook to performe by it.) Let the motions, sayd I, be, the one, uniforme; the other acce­lerate, so as that the spaces be in subduplicate proportion to the Times; or, in your language, as 1 to 2. We are therefore, by your Rule, to seeek in the Table the fraction ⅓. But there's no such fraction to be found (nor any lesse then ½.) Your rule therefore doth not serve the turne. Well▪ let's heare what you have to say for your selfe. Did I not see (you aske) that the Table is only of those figures which are de­scribed by the concourse of a motion Ʋniforme, with a motion accelerated. Yes I did, see that the table is only of such: Nay more, I saw (which is more to your purpose) that the pro­position is only of such; (though yet if need be, I could [Page 107] shew you how the same figures might be described by mo­tion retarded as well as motion accelerated,) & therefore I proposed such a case; viz. an acceleration in the subduplicate proportion of the times, that is after the rate √1▪ √2. √3. √4. &c. which is the subduplicate rate of 1, 2, 3, 4, &c. I had no reason therefore, say you, to look forin thae Takle. That is, I had no reason to expect, that your Rule should performe what you undertake. But why no reason to expect it? For my case is of motion uniforme concurring with motion retarded. No▪ such matter, (nor be you so simple to think so, whatever you here pretend;) for √1. √2. √3. √4. &c. is no decreasing progression, but increasing: for √2, is more then √1 & √3, & more then √2. & so on. But why should you think it is not so? Because forsooth. I do not make the pro­portion of the spaces to that of the times duplicate, but subduplicate. Very good [...] But if times be proposed in a series increasing as 1, 2, 3, 4, &c. will not the subduplicate rate be increasing also, as well as the duplicate? that is, doe not the Rootes of these numbers continually increase, as well as their Squares? Think againe and you'l see they doe. Well, but however, though this table will not serve the turne, yet the [...]ase may be solved, you tell us, another way. No doubt of it. I could have told you so before. (For though you knew not how to resolve it; I did; and therefore directed▪ you to the 64. Prop of my Arithmetica Infinitorum; where you have the case resolved more universally then it is by you proposed; viz. where the exponent of the rate of accele­ration is not explicable by numbers; but even by surd ro [...]tes, or other irrationall quantities.) But what be­comes of your rule in the mean while, which sent us to that Table for solution? where, you now tell us, (for I had told you so before) it is not to be hard? This eleventh Article therefore, is like the rest. Nor is it at all amended in the English.

Your twelfth proposition, I said, was wretchedly false; And I say so, still. But, you say, you have left it standing un­altered; (& yet that's false too; for your English hath a con­siderable alteration from what was in the Latine, though not much for the better) Your words were these▪ If motion be made by the concourse of two movents, whereof one is moved [Page 108] uniformely, the other with any acceleration whatever (for which you say in the English, the other beginning with Rest in the Angle of concurse, with any acceleration whatsoever) the movent which is moved uniformely shall put forward the thing so moved, in the severall parallel spaces, lesse, than if both motions had been Ʋniforme. I gave instance to the contrary, (in fig. 5.) The streight line AND, may be described by a compound of two uniforme motions; and the parabolick line AGD, by a motion compounded of two, the one uniforme and the other accelerated, (neither of which you can deny, for you affirme both, at art. 8, and 9.) But within the Paralells AC, EF, the thing moved (contrary to your assertion) is more put forward by this, than by that motion (for EG, is greater then EN,) The p [...]oposition therefore, in this case is false. Yonr answer is, that other Geometricians find no fault with it. It may be so. But is there any Geometrician (who hath well examined it) will say 'tis true? and that, in all cases? In some cases I told you, it may happen to be true; and in in other cases it will be certainely false: (And I told you also, when, and where.) And I did in the case proposed prove it so to be; and you can say nothing to the demonstration. You would indeed tell me of another case wherein, you think it is true. But what's that to the purpose? When I give instance to the contrary of a uni­versall proposition, you must allow me to lay the case as I think good (so as it be within the limites of that univer­sall) and not as you would have me. The proposition therefore is demonstrated to be false. And you have no­thing to say in vindication of it.

The thirteenth Article doth propose a Problem as ridi­culous as a man would desire to read. 'Tis this Let AB (fig. 8.) be a Length transmitted with uniforme motion in the Time AC: And let it be required to find another length which shall be transmitted in the same time with motion uniformly accelerated, so as the Impetus (or, as in the English, the line of the impetus) last acquired be equall to the streight line AC. The Answer say J to this Probleme, is what length you please. (And you might as well have propounded, A quantity being assigned which is equall to its foure quarters; let it be required to find another quantity which is equall to its two halves. Or thus A parallelogram being proposed of a known Base and Altitude; let it be required to find what may be the [Page 109] altitude of a triangle on the same base. Where, what quantity you will, doth serve for answer to the former: And, what alti­tude you will for the latter. And, what length you will, is the answer to your Problem.) For there is no length assignable, which may not, in any assignable Time, be dispatched by a motion uniformly accelerated, whose last Impetus shall be what you please. And 'tis but as if you should have asked; What may be the height of that Parabola, or Triangle, whose Basis is equall to AC?

The Problem being thus ridiculous, it cannot be expe­cted that the construction or demonstration should be bet­ter. And truly 'tis pittifull stuffe all of it: as J then shewed. And you do not so much as attempt any thing by way of answer, to justify either your construction or demonstra­tion.

You ask here, (for you have no more witt then to pro­pose such a question,) granting that a Parabola may be de­scribed upon a Base given; and yet have any Altitude, or any Di­ameter one will: (which you say who doubts?) How it will hence follow, that when a Parabolicall line is described (is to be described, you should have said; for the Problem is of somewhat to be done, not, of somewhat done already,) by two motions, the one uniform, the other uniformly accelerated from rest; That the determining the Base, doth not also deter­mine the whole Parabola? J answer. Because every Parabola may be so described; (which if you did not know before, you may now learn of me:) And therefore, since that, up­on a Base given, a Parabola may be described of any altitude (as you grant;) and that every Parabola may be so descri­bed: the determining of the Base, doth not determine the Altitude of a Parabola so to be described; more then the Altitude of a Parabola simpliciter.

But if you would have done any thing to acquit your selfe of the charge in this Article, (of proposing a Ridicu­lous, Nugatorious Problem:) You should have assigned some Length, which by a motion so accelerated, and acquiring such an Impetus, could not have been dispatched in a Time assigned. Till then; I say, it may dispatch what length you please: And therefore your Problem is as ridiculous as a man could wish.

There be divers other petty faults, that J took notice of by the way; as that those words, so as the Impetus acquired [Page 110] be equall to a Time (as if heterogeneous things could be e­quall.) And, those words, as duplicate proportion is to single proportion, so let the line AH be to the line AI. (which is as pure nonsense as need to be:) As if there were one certain Proportion of the Duplicate proportion, to the single Pro­portion. You tell us, upon second thoughts, in your Eng­lish, cap. 13. art. 16. that Duplicate proportion is sometime greater then the single; and that it is sometimes lesse: And yet you would here have us think that it is alwaies as 2 to 1. The proportion of 9 to 1, is duplicate of that of 3 to 1: And the proportion of 4 to 1, is duplicate of that of 2 to 1. But there is not the same proportion of the proportion 2/1 to the propor [...]ion [...]/1, that there is of the pro­portion 4/1 to the proportion 2/1▪ but that is triple this double: (for nine times as many, is the triple of three times as many; and four times as many, is but the double of twice as many.) But this you cannot understand, and there­fore call for help from somebody that is more ready in Sym­bols. It seems a man must speak to you in words at length, and not in figures. And truly, all's little enough to make you understand it.

The 14, 15, and 16 Articles are just like the 13: and as ridiculous as it. What was there objected, you confesse, may as well be objected to these. But that hath been proved to be ridiculous: and therefore so are these. Any length being gi­ven, which, in a Time given, is dispatched with uniform motion; To find out what length will be dispatched in the same time with motion so accelerated, as that the Lengths dispatched be conti­nually in triplicate proportion to that of their times. (so Art. 14.) or quadruplicate, quintuplicate, &c. (ibidem.) or as any number to any number. (so Art. 15. 16.) and the Impetus last acquired equall to the Time given. That's the Problem. The Solution should have been; What length you please. Take where you will you cannot take amisse. If you say, 'tis an Inch, you say true: If you say, 'tis an Ell, you say true: And if you say 'tis a thousand miles, no body can contradict you. For it may be what you please.

And is it not a wise thing of you then, for the design­ing of an Arbitrary Quantity, a What-you-will, to bring a parcell of Constructions, and Demonstrations, with finding of Mean Proportionalls, as many as one please; for a mat­ter [Page 111] of two leaves together? And, when you have done all, 'tis but, (as you were,) What you will.

J noted farther that in all these Articles 13, 14, 15, 16, as in those before Art. 9, 10, 11. & those following 17, 18, 19. You doe every where make the slower motion, in the same time, dispatch the greater length. Which I did clearly demonstrate. To this you reply nothing to the purpose: But cavill, that you might seem to say something. You say, I corrupt your Article by putting Movens for Mobile. But there's no such matter; for in the place alleadged (Art. 1.) Movens is your own word, not mine. You say, 'tis no mat­ter whether AB or AC (in the fifth figure) be the greater. Yes it is; it's impossible that AB, according to your sup­position, should be so bigge as AC; and yet, you have made it almost twice as big. You say, you speak of the concurse of two movents; very true. But each of those movents have their se­verall pace assigned them; & therefore you should not have made the slower movent to rid more ground. And then you would tell mee, what I think; and then talk of hard speculations, of edge and wit and malice &c. But nothing to the purpose. For when you have all done, its evident, and you cannot deny, that in your 5 and 6 and 11 Figures, AB is made welnigh twice as long as AC; and so again in your 8, 9, and 10, Figures AH much longer then AB; and yet these longer lines designe the length, dispatched by the slower motions in the same time. For the motion accele­rate, which doth not till its last moment attain the swiftnesse, with which the uniform motion proceeds all the way, must needs be slower then that uniform motion. But this was a fault which I might safely have let passe; for these Articles were ridiculous enough before.

In the 17 Article, I shewed first, that the Proposition, as it was proposed, was not perfect sense. Then, that, the sense being supplied, the Proposition was false. And lastly, that your Demonstration had at lest fourteen faults, and most of them such, as that any one was sufficient to over­throw the Demonstration.

The Proposition was this, If in a time given, a Body run over two lengths, one with Ʋniform, the other with accelerated motion, in any proportion of the length to the time, And again in a part of that time, it run over parts of those lengths with the same motions; the excesse of the whole longitude above the whole [Page 112] (to what?) is the same proportion with the excesse of the part a­bove the part, to what? Is this good sense? No; you con­fesse there was somewhat left out in that Proposition, but say, it was absurdly done to reprehend it. Very good! It seems you must have the liberty to speak non-sense without controll.

Well; but how is the sense to be supplied? we made two or three essays the last time, and found never a one would hold water, but which way soever we turned it, the Proposition was false. We have two proportions de­signed only by their Antecedents, and we are to guesse at the consequents. The best conjecture I could make was this; As the excesse of the whole above the whole, is to one of those wholes; so is the ex [...]esse [...] of the part above the part, to one of those parts, (respectively.) That is (calling the greatest whole G, and its part g: and the lesser whole L, and its part l.) as G − L, to G; so g − l, to g. Or se­condly thus; as G − L, to L; so g − l, to l. But both these are found false. My next conjecture was from the [...], (but there I was fain to leave my proposition quite, and take up new Antecedents, as well as seek new conse­quents,) and that directs me to such an Analogisme (p. 140. l. 39.) I say that as AH to AB, so AB, to AI; but this is ambiguous, because, AB comming twice, once as a whole, and another time as a part, sit doth not appear which is which; therefore here be two conjectures more; viz. a third thus, as the whole to the whole, so the part to the part; (that is G. L∷ g. l.) Or fourthly thus, as the whole to its part, so the whole to its part. (that is G. g∷ L. l.) But these two are both false also. My next attempt was from the [...], (but here also I must desert the proposition too, and seek new antecedents as well as consequents,) where I find it thus (p. 141. l. 7, 9.) as AH to AB, so is the excesse of AH above AB, to the excesse of AB above AI: which was to be demonstrated. that sends me to a fifth, sixth, seventh and eighth analogisme (because it doth not appear which AB is the whole, and which the part;) the fifth thus, as the whole to the whole, so the excesse of the whole above the whole, to the excesse of the part above the part, (taking AB in the two first places for the whole) that is G. L∷ G − L. g − l. The sixth thus, as the whole to the whole, so the excesse of the whole above its part, to the excesse of the [Page 113] whole above its part (taking AB in the first and last place, for the whole,) that is G. L∷ G − g. L − l. The se­venth thus, as the whole to its part, so the excesse of the whole above the whole, to the excesse of the part above the part, (ta­king AB in the first and last place for a part,) that is G. g∷ G − L. g − l. The eighth thus, as the whole is to its part, so the excesse of the whole above its part, to the excesse of the other whole above its part, (taking AB in the two first places, for the part,) that is G. g∷ G − g. L − l. But these four be all false likewise, as well as those before. Now all these eight conjectures are of equall probability (though all false) it cannot be said which of them is more like to be the sense intended than the other. And yet, forsooth, when, by talking non-sense, you leave us at this uncertainty of conjecture, it is (you say) absurdly done to reprehend it. I confesse, if any one of these Analogismes had been true, we might have guessed that to be your meaning: but when they be all equally probable, and equally false, which should we take? Well, but 'tis to be hoped, that now you will tell us. You tell us therefore (Less. p. 38.) it should be thus, as the excesse of the whole above its part, to the excesse of the other wh [...]le above its part, so that whole, to this whole: which affords us a ninth analogisme, G − g. L − l∷G. L. which is coincident with my sixth conjecture. And yet again (Less. p. 39.) you tell us, that the proposition is now made (in the English) according to the demonstration (that is; both false,) and there we find it thus, the whole to the whole, as the part to the part; that is G. L∷g. l. which allso is coincident with our third conjecture. But which soever of all these analogismes you take, the Propo­sition is false, and therefore the demonstration must needs be so too.

Now to prove that this Proposition is false, which way soever you turne it, (either as it was before, or as it is now,) I made use of the figure of your first article, and proceeded to this purpose. Let the whole time (fig. 1.) be AB, an hour, (that is, because I would not have you mi­stake mee, as you doe Archimedes, let the line AB repre­sent an hour, or, be the symboll of an hour; for I would not have you think that I take a line to be an hour; but to represent an houre; and the letters AB to represent that line, not to be that line; like as at another time we take [Page 114] a letter, without a line, to represent an houre:) and part of that time AF, halfe an houre. Let also the continued Impetus of the Uniform motion (I mean the Symboll of it) be AC, or BI: which BI also is to be (the Symbol of) the last acquired Impetus of the motion accelerated. And this acceleration we will suppose at present (as your selfe do in your [...]) to be uniform acceleration. The velocity therefore of the whole uniform motion, will be represented by the Parallelogram ACIB; (by the first article;) and of it's part, ACHF; (by the same ar­ticle;) The velocity of the whole uniformly accelerated motion, will be the Triangle AIB; and of its part, AKF; (by the same article.) Since therefore the lengths dispatched be proportionall to those velocities; the whole length of uniform motion, to the whole of the accelerate, will be as the Parallelogram ACIB, to the Triangle AIB, that is, as 2 to 1. (viz. the length of the uniform motion, bigger than that of the accelerate; whereas your figure and demonstration, do all the way suppose the con­trary;) so that if the uniform motion do in an houre di­spatch 16 yards, the accelerate will in the same time di­spatch 8 yards, (that is G = 16, & L = 8.) Again, the length dispatched by the uniform motion in the whole time; to that in half the time, is as the Parallelogram ACIB, to the Parallelogram ACHF; that is, as 2 to 1; so that if G (as before) be 16, then is g = 8. Lastly; the length dispatched by the accelerate motion in the whole time, to that in halfe the time, is as the Triangle AIB, to the Triangle AKF; that is as 4 to 1, (for the sides AB, to AF, being as 2 to 1, and the triangles in duplicate proportion to their sides, the triangles will be as 4 to 1:) So that if L (as before) be 8, then is l = 2. Now ha­ving thus found the measures of these four lengths; (viz. G = 16. L = 8. g = 8. l = 2.) You shall see that those Analogismes are all false; not one true amongst them. The first is this, G − L. G ∷ g − l. g. that is 16 − 8 = 8. 16 ∷ 8 − 2 = 6. 8. or 8. 16 ∷ 6. 8. But this is false. The second this, G − L. Lg − l. l. that is 16 − 8. 8 ∷ 8 − 2. 2. or 8. 8 ∷ 6. 2. But this is false also The third this, G. L ∷ g. l. that is 16. 8 ∷ 8. 2. and this also is false. The fourth this, G. g ∷ L. l. that is 16. 8 ∷ 8. 2. and this is also as false as the other. The [Page 115] fifth is this, G. L∷G − L. g − l. that is 16. 8∷16 − 8. 8 − 2. or 16. 8∷8. 6. which is also false. The sixth this, G. L∷G − g. L − l. that is 16. 8∷16 − 8. 8 − 2, or 16. 8∷8. 6. which is like the rest. The se­venth is this G. g∷G − L. g − l. that is, 16. 8∷16 − 8. 8 − 2 or 16. 8∷8. 6. false also. The eighth is this. G. g∷ G − g. L − l. that is 16. 8∷16 − 8. 8 − 2. or 16. 8∷8. 6. which is also false. The ninth is this, G − g. L − l∷G. L. that is 16 − 8. 8 − 2∷16. 8. or 8. 6∷16. 8. The tenth is like the third, G. L∷g. l, that is 16. 8∷8. 2. all false. The proposition therefore, turne it which way you will, is a false Proposition. And yet you have the Impudence to tell us (though you knew this before, for I told it you last time, and brought the same demonstration, to which you have not replied one word) that 'tis all true, and truly demonstrated.

Do you think 'tis worth while after all this, to examine your demonstration? 'Tis a sad one, I confesse; but tis yours, and therefore it may perhaps be beautifull in your eye. The last time we looked upon it, we found it had at least fourteen grosse faults: (and most of them such, as were singly enough to destroy it:) enough in conscience for one poor demonstration. (And had you not been good at it, a man would have wondred how you could have made so many ex tempore.) Since that time, 'tis quite defunct. And there is a young one start up in stead of it. But 'tis of the same breed, and tis not two pence to choose, whether this or that. Your new demonstration runs it self out of breath at the first dash. You had told us (Art. 3. Co­roll. 3) In motion Ʋniformly accelerated from rest, (such as is one of these) the length transmitted (as here AH, fig 8.) is to another length (viz. AB,) transmitted uniformly in the same time, but with such Impetus as was acquired by the acccle­rated motion in the last point of that time (just the case in hand) as a Triangle to a Parallelogram which have their alti­tude and base common, that is, as 1 to 2, for the Parallelogram is double of the Triangle. So that AH, in your figure, should be but just half as bigge as AB; and you have made it all­most twise as big. And upon this foundation depends the whole demonstration. For if that fault were mended, your whole construction comes to nothing. And is not [Page 116] this demonstration then well amended? especially when you had faire warning of it the last time.

And then you send us to the demonstration of the 13 Article for confirmation of this, whereas that Article hath been cashiered long agoe, and the demonstration with it. But thus 'tis when men will not take warning.

At length you fall to raating, (as you use to do when you be vexed;) about skill, and diligence, and too much trusting; about discretion, Hyperbole's, and Sir H. Savile, &c. And tell us that when a beast (Joseph Scaliger) is slain by a Lion (Clavius) 'tis easy for any of the fowles of the aire (Sir H. S.) to settle upon, and peck him. And Vespasian's law, no doubt, will bear you out in all this. Only this I must tell you, that Sir H. Savile, had confuted Joseph Scaliger's Cyclometry, as well as Clavius; and, I suppose, before him. Which if you have not seen, I have.

In the 18 Article, we have this Proposition. If, in any Parallelogram, (suppose ACDB, fig. 11.) two sides contain­ing an angle be moved to the sides opposite to them, (as AB to CD, and AC to BD,) one of them (AB) with uniform motion, the other (AC) with motion uniformly accelerated: that side which is moved uniformly (AB) will effect as much, with its concurse through the whole length, as it would do if the other motion were also uniform, (or were not at all. For what e­ver the other motion be, the motion of AB to CD, car­rieth the thing moved with it from side to side, and that's all. What point of the opposite side it shall come to, de­pends upon the other transverse motion, not upon this at all. And this is so easy that no body would deny it. If you mean any thing more then this, that it shall carry it just to the opposite side & no farther, your demonstration doth not at all reach it▪ But you go on) and the length transmitted by it in the same time, a mean proportionall between the whole and the halfe, of what? Till you tell us of what? I say, as I said before, that these words have no sense.

The construction and demonstration of this propositi­on, I remember, we made sad work with, the last time we had to doe with them, as well as with those of the for­mer Article; which will be now too long to repeat. The whole weight of the Demonstration lies, severally, upon at lest these three Pillars, of which if any one do but fail, the whole demonstration falls. First, upon the strength [Page 117] of the 13 Article, which we have destroyed long agoe. Secondly upon the 12 Article, which we have also long since proved to be false. Thirdly, upon this learned as­sertion, the streight line FB will be the excesse by which the (lesser) length transmitted by AC with motion uniform­ly accelerated, till it acquire the impetus BD, will exceed the (greater) length transmitted by the same AC in the same time with uniform motion, and with the Impetus every where equall to BD. Which destroys it selfe. For if the accele­rated motion, as is supposed, do not till its last moment acquire that speed with which the uniforme motion is moved all the way; then that must needs be slower than this; and consequently dispatch a lesser length in the same time: whereas you according to your discretion, make the length dispatched by that slower motion, to be more then that of the swifter in the same time, and tell us the excesse is FB. And then to helpe the matter, when I presse you with this absurdity, you tell us you speak of motions in concurse: as though in concurse, the slower motion did in the same time, caeteris paribus, dispatch a greater length than the swifter, though out of concurse the swifter motion did dispatch a greater length than the slower▪ Now either of these three, much more all of them, doth wholly destroy the strength of your demon­stration. Yet they that desire to see more may consult what I sayd before.

The ninteenth Article doth not pretend to any other strength than that of the eighteenth. And therefore falls with it.

The twentieth Article I did before prove to be false and frivolous. (it depended upon Chap. 14. Art. 15. Corol. 3. which Corrollary I have there consuted.) You say nothing by way of vindication of what I excepted against; only passe your word for it, that it is true. Yet withall confesse, there is a great error; and that error say I, though there were nothing else, would make that article unsound. But this article you say, was never published (yet 'tis as good as most of those that were in this Chapter; for i'le undertake for it, there he above a dozen worse;) and therefore it was inhuman­ly done, you say, to take notice of it. Truly, if the proposi­tion were a good proposition, as you say it was. J think J [Page 118] did you a courtesy to publish it for you, that you may have the credit of it; yet J should not have done it, had it not been publike before. If you would not have it ta­ken notice of, you should have taken care not to send it abroad. For it hath been commonly sold with the rest of your book (to many more persons beside my selfe;) they that would, might teare it out (as some did) and they that would, might keep it in, as J did.

Well, (be the number of articles 20, or be they 19,) before the sixth there was none sound, (but either in whole or in part unsound,) and from the eighth there hath been none sound; therefore there have not been above three sound at the most. Quod erat demonstrandum.

SECT. XI. Concerning his 17. Chapter.

THE Reader by this time may perhaps be weary, as well as J; and think it but dull work to busy him­selfe upon such an inquiry, where the result is but this, That M. Hobbs his Geometry is nothing worth; which (if he had any himselfe) he knew before. To save him therefore, and myselfe the labour, wee'l make quicker work in what's behind.

In the 17. Chapter, some of the Propositions are true and good; (and truely I wondred at first where you had them, but since I know:) But the demonstrations are foolish and ridiculous. The Propositions therefore are your own (you know where you stole them;) and the Demonstrations are of your own making; (for there be scarce such to be found any where else.)

What you say to the first Article comes to this result; that I should say, It is well known, that, in Proportion, Double is one thing, and Duplicate another. And you aske, To whom it is known? (it seems it was not known to you:) And tell us, that they are words that signify the same thing; and, that they differ (in what subject soever) you never heard till now. It's very possible that this may be true; that you did never know the difference between those two words till I taught you. (But this was your ignorance not my fault.) But now, you know there is a difference. And [Page 119] therefore (contrary to what you had affirmed in the Latin) you tell us in your English, Chap. 13. art. 16. p. 121. l. 7. &c. and p. 122. l. 26. &c. that the proportion of 4 to 1, to that 4 to 2 &c. is not only Duplicate, but also double or twise as great. But on the contrary, the proportion of 1 to 4, to that of 1 to 2, &c. though it be duplicate, it is not the double, or twise as great, but contrarily the halfe of it; and that of 1 to 2, to that of 1 to 4, &c. is Double you say, and yet not duplicate but subduplicate. Now if you never heard of such a difference till you heard it from me, then you are indebted to me for that peece of knowledge: and have no reason to quarrell with me, as you use to doe, for saying you did not understand what was duplicate and subdupli­cate proportion; for you confesse you did not, but tooke it to be the same with double and subduple, and never heard that they did differ till now.

In the second Article, because it is fundamentall to those that follow, I took the paines first to shew how un­handsomely the proposition and [...] were contrived; and then to shatter your demonstration all to pieces; and shewed it to be as simple a thing as ever was put together, (unlesse by you, or some such like your selfe.)

As to the first, you tell us, that, to proceed which way you pleased was in your own choice. And I take that for a suffici­ent answer. You did it, as well as you could; and they that can do better may.

As to the Demonstration, you keep a vapouring (no­thing to the purpose,) as if it were a good demonstration. and not confuted. Yet, when you have done, (because you knew it to be naught) you leave it quite out in the Eng­lish, and give us another (as bad) in stead of it. That is, you confesse the charge. Your fundamentall Proposition was not demonstrated; and so this whole chapter comes to nothing.

But however, 'tis to be hoped, that your new demon­stration is a good one; is it not? No, 'Tis as bad as the other. Only 'tis not so long: And of a bad thing, (you know,) the lesse the better. It begins thus, The proportion of the complement BEFCD, (fig. 1.) to the deficient figure ABEFC, is all the proportions of DB to OE, and DB to QF, and of all the lines parallell to DB, terminated in the line BEFC, to all the parallells to AB terminated in [Page 120] the same points of the line BEFC. Now for this (besides that it is a piece of non-sense) you send us for proof to the second Article of the 15. Chapter, where there is nothing at all to that purpose. Then you go on. And seeing the propor­tion of DB to OE, and of DB to QF, &c, are every where triplicate to the proportion of AB to GE, and of AB to HF &c. the proportions of HF to AB, and of GE to AB, &c. are triplicate (no, but subtriplicate) of the proportions of QF to DB, and of OE to DB &c. Now this is but the same Bull that hath been baited fo often. viz. because the diame­ters (DB, OE, QF, &c. that is CA, CG, CH,) are in the triplicate proportion of the Ordinates (AB, GE, HF,) therefore these Ordinates are in the triplicate proportion of those diameters. You might as well have sayd, seeing that 6 is the triple of two therefore 2 is the triple of 6. But let's hear the rest, for there is not much behind.) And therefore the deficient fig. ABEFC, which is the aggregate of all the lines HF, GE, AB, &c. is triple to the complement BEFCD, made of all the lines QF, OE, DB, &c. A very good consequence! Because the Ordinates are in triplicate proportion to the diameters (yet that is false too, for they are in subtriplicate) there­fore the figure is triple to its complement? But how doe you prove this consequence? Nay, not a word of proof. We must take your word for it. Well then, of this last Enthy­mem, (which was directly to have concluded the question,) the Antecedent is false▪ and the consequence at lest not proved (I might have said false also, for so it is.) And this is your new demonstration.

The third article, I sayd, falls with the second; for having no other foundation but that, (nor do you pretend to other) that being undemonstrated (for your former demonstration your selfe have thrown away, and your new one we have now shewen to be nothing worth,) this must be undemonstrated too.

In the fourth Article, you attempt the drawing of these Curve lines, by point; and to that purpose require the finding of as many mean proportionalls as one will, (like as you had before done Cap. 16. 6. 16. for the finding out an arbitrary line to be taken at pleasure: Which I told you was simply done, because that without such mean pro­portionalls, (that is, without the effection of solid & Lineary problems,) it might have been done by the Geometry of [Page 121] Plains, that is with Rule and Compasse. And I shewed you how. To which you have nothing to reply, but, that I made use of one of your figures (to save my selfe the labour of cutting a new one,) that is, I made better use of your figure then you could doe.

The fifth proposition (beside that it is built upon the se­cond, and therefore falls with it,) is inferred only from the Corrollary of the 28. article of the 13. Chapter, (nor doth your English produce any other proof,) where, sayd I, there is not a word to that purpose. And you confesse it.

The 6, 7, 8, & 9. Art. do not pretend to other foundation than the second; & therefore till that be proved, fall with it.

The 10. Article is a sad one, as may be seen by what I did object against it, as you say, for almost three leaves toge­ther. One fault amongst the rest you take notice of, and you would have your Reader think that's all; though there be above twenty more. 'Tis this, Because (in fig. 6.) B C is to BF for so your words are, though your Lesson mis-recite them, in triplicate proportion of CD, to FE; therefore, inver­ting, FE, to CD is in triplicate proportion of BF to CB. And doe you not take this to be a fault? No, you say, this I did object then (Yes and doe so still, as absurd enough:) But now, you say, you have taught me; (what a hard hap have I, that I cannot learn;) That of three quantities, (you should rather have taken foure; but however three shall serve for this turne,) beginning at the lest, (suppose 1, 2, 8,) if the third to the first (8 to 1) be in triplicate proportion of the second to the first (of 2 to 1) also, by conversion, the first to the second (1 to 2) shall be in triplicate proportion of the first to the third, of 1 to 8. This is that you would have had me learne. But, good Sir, you have forgotten that, since that time, you have unlearned it your selfe. For your 16. artic. of Chap. 13. as it now stands corrected in the English, teacheth us another doctrine; viz. that if 1, 2, 4, 8, bee continually proportionall, 1 to 8 shall be as well triplicate (though not bigger) of 1 to 2, (not this triplicate of that,) as 8 to 1 is of 2 to 1. The case is now altered from what it was in the Latine. And therefore you are quite in a wrong box, when, in your English, you cite Chapt. 13. Art. 16, to patronize this absurdity. For in so doing you doe but cut your own throat. You must now learne to sing another song; called Palinodia. Well, this is one of [Page 122] the faults of this article. They that have a minde to see the rest of them, may consult what I said before; where I have noted a parcell of two dozen.

In the 11. Article, you doe but undertake to demonstrate a proposition of Archimedes. Your demonstration (besides that it depends upon the second Article which is yet un­demonstrated) is otherwise also faulty, as I then told you. And therefore to say, that I allow this to be demonstrated, if your second bad been demonstrated; is an untruth. For I told you then, that your manner of inferring this from that, is very absurd.

The 12 Article (like all the rest, since the second, beside their other faults,) depends upon the second; and there­fore, till that be demonstrated, this must fall with it.

In the 13. Art. you undertake to demonstrate this Proposi­tion of Archimedes; that the Superficies of any portion of a Sphere, is equall to that circle, whose Radius is a streight line drawn from the pole of the portion to the circumference of its base. Your de­monstration, I said, was of no force; but might as well be ap­plyed to a portion of any Conoeid, Parabolicall, Hyperboli­call, Ellipticall, or any other, as to the portion of a sphere. By the truth of this, say you, let any man judg of your and my Geo­metry. Content, 'Tis but transcribing your demonstration; & inserting the words Conoeid, Vertex, section by the Axis, &c. where you have Sphere, Pole, great Circle &c. which termes: in the Conoeid, answer to those in the Sphere, and the worke is done.

Let BAC, (in the seventh figure,) be a portion of a spheare, or Conoeid, Parabolicall, Hyperbolicall, Ellipticall, &c. whose Axis is AE, and whose basis is BC; and let AB be the streight line drawn from the Pole, or vertex, A, to the base in B: and let AD, equall to AB, touch the Great circle, (or Section made by a plain passing through the Axis of the Conoeid,) BAC, in the Pole, or vertex, A. It is to be pro­ved that a Circle made by the Radius AD, is equall to the su­perficies of the portion BAC.

Let the plain AEBD be understood to make a revolution about the Axis AE. And it is manifest, that, by the streight line AD, a circle will be described; and, by the Arch, or Section, AB, the superficies of a Sphere, or Conoeid mentioned; and lastly, by the subtense AB, the superficies of a right Cone.

Now, seeing both the streight line AB, and the Arch or Se­ction [Page 123] AB, make one and the same revolution; and both of them have the same extreme points A & B: The cause why the Sphe­ricall or Conoeidicall Superficies which is made by the Arch or Section, is greater then the Conicall superficies which is made by the subtense, is, that AB the Arch or Section, is greater then AB the subtense: And the cause why it is greater, consists in this, that although they be both drawn from A to B, yet the subtense is drawen streight, but the arch or Section an­gularly; namely, according to that angle which the arch or Se­ction makes with the Subtense; which angle is equall to the an­gle DAB. For the Angle of Contact, whether of Circles or other crooked lines, addes nothing to the angle at the segment: as hath been shewn, as to Circles, in the 14 Chapter of the 16 article: and as to all other crooked lines, Lesson 3. pag. 28. lin. ult. Wherefore the magnitude of the angle DAB, is the cause why the superficies of the portion described by the Arch or Section AB, is greater than the superficies of the right Cone described by the Subtense AB.

Again, the cause why the Circle described by the tangent AD, is greater then the superficies of the right Cone described by the subtense AB, (notwithstanding that the Tangent and Subtense are equall, and both moved round in the same time,) is this, that AD stands at right angles to the axis, but AB obliquely; which obliquity consists in the same angle: DAB.

Seeing therefore that the quantity of the angle DAB, is that which makes the excesse both of the Superficies of the Por­tion, and of the Circle made by the Radius AD, above the su­perficies of the Right Cone described by the Subtense AB: It followes, that both the Superficies of the Portion, and that of the Circle, do equally exceed the Superficies of the Cone. Wherefore the Circle made by AD or AB, and the Sphericall or Cono­eidicall Superficies made by the arch or Section AB, are e­quall to one another. Which was to be proved.

Shew me now if you can, (for you have pawned all your Geometry, upon this one issue,) where the Demon­stration halts more on my part then it doth on yours? Or, where is it, that it doth not as strongly proceed in the case of any Conoeid, as of a Sphere? All that you can think of by way of exception (and you have had time to think on't ever since I wrote last,) amounts to no more but this (which yet is nothing to the purpose) you ask, In case the crooked line AB, were not the arch of a Circle, whether do I [Page 124] think, that the angles which it makes with the Subtense AB, at the points A & B, must needs be equall? I say, that (its possible, that in some cases, it may be so; and J could for a need, shew you where; and therefore, at least as to those cases, you are clearely gone; for you had nothing else to say for your selfe; but) this is nothing at all to the purpose whether they be or no; For the angle at B, what ever it be, comes not into consideration at all; nor is so much as once named in all the demonstration; So that its equa­lity or inequallity, with that at A, makes nothing at all to the businesse. And therefore your exception is not worth a straw. Think of a better against the next time; or else all your Geometry is forfeited. And they are like to have a great purchase that get it, are they not?

At the 14. Article; (having before, Art. 4. under­tooke to teach the way of drawing and continuing those curve lines, by points: and directed us (for the word require doth not please you) for that end to take mean proportionalls;) you now tell us how that may be done; viz. by these curve lines first drawn. I asked, whether this were not to commit a circle? You tell me, No. But mean while take no notice of that which was the main objecti­on; viz. That this constructiō of yours was but going about the bush; for, upon supposition that we had those lines al­ready drawn, the finding of mean proportionalls by them might be performed with much more ease than the way you take. And I shewed you, How.

But that which sticks most in your stomach, is a clause in the close of this Chap. I told you that some considerable Propositions of this Chapter (and I could have told you which) were true, (though you had missed in your de­monstration,) however you came by them. But that I was confident they were none of your own. (and you know, I guessed right.) And least you should think I dealt un­worthily to intimate that you had them elsewhere: un­lesse I could shew you where: I told you, that I did no worse than those that a while before, had hanged a man for stealing a horse from an unknown person. There was evi­dence enough that the horse was stolen; though they did not know from whom. So, though I knew not whence you had taken them, yet I have ground enough to judge they were not your own. And since that time, (and be­fore [Page 125] that book was fully printed,) I found whence you had them; namely out of Mersennus, (as I told you then pag. 132, 133, 134.) And to take them out of Mersennus, was all one as to rob a Carrier; for there were at lest three men had right to the goods, (and some of them if they had been asked, would scarce have given way that you should publish their inventions in your own name,) Des Chartes, Fermat, and Robervall: And perhaps a fourth had as much right as any one of these; and that is Caval­lerio, who (though, I then did not know it) hath (contrary to what you affirme, that they were never demonstrated by any but you selfe; and that as wisely as one could wish:) demonstrated those propositions in a Tractate of his De usu, Indivisibilium in potestatibus Cossicis. But though the thing be true enough and you cannot deny it, yet you doe not like the Comparison. And would have me consider, who it was, was hanged upon Hamans Gallows? And truly J could tell you that too, for a need. The first letter of his name was H. But enough of this.

SECT. XII. Concerning his 18, 19, 20. Chapters.

WELL! We have made pretty quick work with the 17 Chapter. With the 18 we shall be yet quicker.

The charge against this Chapter, was, that it was all false. And, you confesse it. Not one true Article in the whole.

But, you tell us, in the English 'tis all well. It is now so corrected in the English as that I shall not be able (if I can sufficiently imagine motion, that is, if I can be giddy enough,) to reprehend. Very well! ('Tis a good hearing when men grow better.) They that have a mind to believe it, may: I am not bound to undeceive them. We have had experi­ence all along, that you have a speciall knack at mending. (as sowr Ale doth in summer.) You grant that I have truly demonstrated, what was before, to be all false. You would have me do so again, would you? Very good! When I have nothing else to doe I'le consider of it. They that think it worth the while, may take the pains, to examine it a second time. For my part, I think I have bestowed as much pains upon it already, as it deserves, (and some­what [Page 126] more:) And all the amendment that I find, is this▪ that whereas before wee had three false articles, now we have but two; and the number of true ones, just as many as we had before, viz. never a one.

In the 19 Chapter there were faults enough in consci­ence (for a matter of no greater difficulty than that was;) I noted some of them (and left the Reader to pick up the rest:) Two or three of the lighter touches, (about me­thod,) you take notice of, and make a businesse to justify or excuse them; and the main exceptions (as you use to do) you passe over with a light touch, and a way.

I told you, in the beginning of it, that your Chapters hang together like a rope of sand. And 'tis true enough, for they have no connexion at all. There are so few hooked atomes, that a man cannot tell how to tacke them together.

Next, that having in your 24 Chapter undertaken to shew us, what is the Angle of Incidence; and, what, the Angle of Reflexion; and, that the Angles of Incidence and of Reflexion are equall: you do, in pursuance of that assertion, in this 19 chapter, shew us the consequences thereof. Upon this I asked; why not, either this after that; or that before this? You tell me, that (think I what I will,) you think that method still the best; (to set the Cart before the Horse.) Then you tell us, that I say, you define not here. (Nay that's false, I did not say so; and 'tis not the first time that I have taken you tripping in this kind;) but many Chapters after, (that I said, I do confesse; and you know 'tis true;) what an Angle of Incidence, and what an Angle of Reflexion is. And then, talk against hast, and oversight. But if your selfe had not been over hasty, (or rather willfully perverted my words,) you might have seen (and you know it well e­nough) that I blamed you here, and two or three times be­fore, not so much for using words, before you had defined them (for this fault, as J remember, J mentioned but once; and there you took it patiently:) but for defining words so long after you had used them. For when words, for two or three chapters together, have been supposed, and frequently so used, as of known signification, (whether they had been before defined, or not,) it is ridiculous for a Mathen atici­an to come dropping in with definitions of them at latter end, (as your fashion is,) like mustard after meat. For these definitions should either have come in due time, or else not at all.

[Page 127] The two first Articles are very triviall. And yet (as if it were impossible for you, be the way never so plaine, not to stumble) there wants, at least in the English, a determi­nation in the second Corollary; and yet (as if that were to make amends for t'other) there's one too much. If up­on any point (say you) between B and D, fig. 2. (yes, or any where else upon the same streight line, produced either way, though not between those points,) there fall (from the point A, you should have said,) a streight line, as AC, whose reflected line is CH, this also produced beyond C, will fall upon F. Here, I say, that limitation between B and D, is redundant; and that from the point A, is wanting. For though C. be taken at pleasure, yet A is not, And if it come not from A, its reflex will not come at F.

The third, fourth, and fifth Articles, I told you were false. (viz. The Propositions affirme that universally, which holds true but in some particular case. And the demon­strations, proceed ex falsis suppositis, supposing that to be, which is not; or is, in many cases, impossible.) And this you confesse to be true; but take it unkindly to be told of it. You have endeavoured a little to patch up the businesse in the English, but not so as to hold water. For they are yet lyable to divers exceptions if it were worth the while to unravell them.

The eight Article was ridiculous enough. It makes a huge businesse to no purpose. (You spend the best part of two pages to resolve a Problem which might as well have been dispatched in two lines.) And you doe as good as confesse it All you say against it, is but this, that Adduco is not Latine for to Bring.

The Twent [...]eth Chapter will be soon dispatched. This Chap. all but the two last Art. is wholly new, as it is now in the English: that which you had before in the Latine, being wholly routed & beat out the field▪ (& your Problema­tice dictum into the bargain.) We had in your Latine three attempts for the squaring of a circle; but they all came to naught, and are now vanished. In your Lesson, you give us a fourth; endeavouring to new mould and rally one of the former, which I had before routed; And pre­tende to vindicate it from the exceptions I had made to it: But not an answer to any one of them; nor is this new attempt better than the former, but retaines [Page 128] most of the fundamentall errors therein; And when you have all done, you cashier it your selfe and dare not insist upon it. Beside this, you have in your English, yet three attempts more; and much a doe there is with long and perplexed figures to no purpose. They are by your own confession but Aggressions; and you doe not your selfe be­lieve them to be exact. You doe not, I suppose, think it worth the while for me to confute them, (or if you doe I doe not;) for to what purpose? That you have attemp­ted it, (seven times over,) no man can deny; That your attempts come to nothing, your selfe confesse; Only, you think it convenient to let the Reader know what paines you have taken to no purpose. For my part, J doe not in­tend to follow you in all your new freakes: nor think my selfe ingaged to confute false quadratures as oft as you shall make them. I have done enough already, to let the world see, how little 'tis that you understand in Geometry, and how much they deceive them selves who expect any great matter from you.

Your two last Articles stand as they were, and so doth my answer to them. Your attempt of finding a streight line equall to a Spirall; is but an attempt, as well as that of squaring a Circle. Your rant at Analyticks, with which you con­clude it, (like doggs barking at the Moon,) hurts no bo­dy but your selfe. That Art will live when you be dead; and those that know it, will not think it ever a whit the worse for your not understanding it, or rayling at it.

The following Chapters I did let alone before, and so shall doe now: not because J like them better than those that went before, or think the matter therein to be true; but because I have done enough already. I told you then, that much of them was taken out of others, (and I told you from whence, and you cannot deny it;) And that much was false. Now this, (because I did not think it worth while to insist upon a particular confutation,) you say, is a Lye; yet, you know that Tacquet hath confuted much of it; and, to so good purpose, as that beside other alterations, you have been fain, you tell us, to make the twenty-fourth Chapter almost all new. But whether you have made it better or no, than it was before, they that have a mind thereto, may, if they please, take the pains to exa­mine. [Page]


  • fig. 1.
  • fig. 2.



  • fig. 6.
  • fig. 7

Cap. XVI.

  • fig. [...].
  • fig. 2.
  • fig [...] 6.
  • fig. 8.
  • fig. 11.


  • fig. [...].
  • fig. 6.
  • fig. 7.

Place this at the end

SECT. XIII. Concerning his last Lesson.

YOur last Lesson, little concernes mee; but is directed mainly against my Reverend and Learned Collegue; Who hath allready answered to it as much as he thinks it doth deserve, yet a touch or two there is where­in I am concerned.

You had, in your Latine, a railing rant against Vindex, (and though you thought fit to [...] of that, 20 Chapter, yet placuit ea stare quae pertinent ad Vindicem. But in the English that is expunged also; And now he is left to learn [...], out of your Lessons.) And in order to this, J [...] in my Elenchus, (p. [...] 117. 122.) recited verbatim out of his Vindiciae, those [...], which, it seems, stuck▪ so much in your stomack; concerning M. Warners papers; that the Reader might see how small a matter would put you into a rage. (Which you knew well enough, and can upon no pretence plead ignorance of it. For it is the very same, which both the [...] [...] in your Lessons, you re­ferre to, and rant at.) But [...] forsooth, upon this, (ac­cording to your usuall honesty) you would have your Rea­der believe, that J had there related some personall dis­course, which Vindex, creeping into your company un­known, had sometime had with you: and then rant at the incivility of such a carriage, and (with a fling at Moranus into the bargain) raile a [...] it for allmost two whole pages together, p. 57, 58, 59. Wherein, whether your Civility or Honesty, be more com [...]cuous, let the Reader judge.

In like manner, because J cited a passage concerning Rohervall, out of Mersen [...], you suspect, p. 59. that some­body, you know not who, hath most magnanimously interpreted to me in [...] d [...]sgrace, what passed between you and him in the Cloister of the Convent.

Which is a suspicion like to that of p. 57. that some of our Philosophers that were at Paris at the same time with you, may perhaps have accused you to us of bragging or ostentation. As though there were not ground enough in your wri­tings, to evidence that, to any man, without any such re­lation. But, mean while, J wonder how you behaved your [Page 130] selfe at Paris, that you should be so Jealous least somebody there should tell tales.

And all this is but a little to disguise the businesse, as if I had not by what is extant in Print, in those places cited out of Mersennus (Hydraulic. prop. 25 Cor. 2. Ballistic. prop. 32, Mechanic. praef. punct. 3. & 4. Reflex. Physico-Math. cap. 1: art. 5.) made it evident, that all or most of what was worth any thing in your Mathematicks, was manifestly stollen from Gasilaeo, Robervall, Cartesius, Fermat, &c And [...] them as I perceive by somewhat but now come to [...] him, doth not stick to call you [...], for so doing: and, if some of [...] were [...] doubt▪ not but they would be ready enough to do the like [...]

Now this is all, ( [...] what was sufficiently [...] at before) that in this [...] concerns mee. And, for what concernes my [...], you have already from himselfe received sufficient [...].

I know now no exception remaining, unlesse like his, who putting a Bond in suit when the Defendant made proof of Payment▪ replyed, [...] the Condition of the Ob­ligation was that he should [...], Satisfy, and Pay; and therefore, though the [...] all pay'd, yet forasmuch the Plaintife was not [...] the Bond was forfeit. Now J hope the Reader can bear witnesse, that you have been, by this time, sufficiently Pay'd; and, J hope, Satisfyed; But, if we must never have done till you be Contented, I am a­fraid we shall dye in your debt.



PAge 1. line 5. language, p. 2. l. 24. learn. p. 5. l. 32. dele quod. p. 6. l. 32. finding. p. 9. l. 13. suffer your. p. 17. l. 2 [...]. Plin. p. 18. l 24. dos. p. 19. l. 24. sumere. p. 38. l 12. second. p. 45. l. ult. 13. p. 46. l. 31. 4 † 1. p. 52. l, 35. not at all, p. 57. l. 22. for two. p. 61. l. 35. art. 3. p. 64. l. 39. proportion. p. 66. l. 34. art. 5. p. 67. l. 1. proportion. ibid. l. 17. art. 3. p. 68. l. 25. that Greater. p. 71, l. 33, half the. p. 72. l. 39. proposition. p. 75, l. pen. and. p. 78, l. 23, the points. p. 80, l. 3. one another. p. 92, l. 36. √½, or. p. 95, l. 13. of the 5 [...] ibid. l. 22. adde, as the product of one Impetus into its Time, to the product of the other Impetus into its Time. p. 97, l. 13. thought. ibid. l. 18. of celerity. p. 99, l. 13, it be. p. 103. l. 32, proposition. p. 106, l. 2, the rest. p. 107, l. 6, that Table. ibid. l. 13, and √3 is more. ibid. l. 32, not to be had▪

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