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. Wallis, John, 1616-1703. 1656 Approx. 363 KB of XML-encoded text transcribed from 73 1-bit group-IV TIFF page images. Text Creation Partnership, Ann Arbor, MI ; Oxford (UK) : 2008-09 (EEBO-TCP Phase 1). A97051 Wing W576 Thomason E1577_1 ESTC R204165 99863848 99863848 116063 This keyboarded and encoded edition of the work described above is co-owned by the institutions providing financial support to the Early English Books Online Text Creation Partnership. This Phase I text is available for reuse, according to the terms of Creative Commons 0 1.0 Universal . The text can be copied, modified, distributed and performed, even for commercial purposes, all without asking permission. Early English books online. (EEBO-TCP ; phase 1, no. A97051) Transcribed from: (Early English Books Online ; image set 116063) Images scanned from microfilm: (Thomason Tracts ; 199:E1577[1]) 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. Wallis, John, 1616-1703. [12], 130, [2] p., folded plate Printed by Leonard Lichfield printer to the University for Tho: Robinson., Oxford, : 1656. Dedication signed: John Wallis. A reply to: Hobbes, Thomas. Six lessons to the professors of the mathematiques. With a final errata leaf. Annotation on Thomason copy: "7ber [i.e. September] 26". Reproduction of the original in the British Library. Created by converting TCP files to TEI P5 using tcp2tei.xsl, TEI @ Oxford. Re-processed by University of Nebraska-Lincoln and Northwestern, with changes to facilitate morpho-syntactic tagging. 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Users should bear in mind that in all likelihood such instances will never have been looked at by a TCP editor. The texts were encoded and linked to page images in accordance with level 4 of the TEI in Libraries guidelines. Copies of the texts have been issued variously as SGML (TCP schema; ASCII text with mnemonic sdata character entities); displayable XML (TCP schema; characters represented either as UTF-8 Unicode or text strings within braces); or lossless XML (TEI P5, characters represented either as UTF-8 Unicode or TEI g elements). Keying and markup guidelines are available at the Text Creation Partnership web site . eng Hobbes, Thomas, 1588-1679. -- Six lessons to the professors of the mathematiques. Geometry -- Early works to 1800. Mathematics -- Early works to 1800. 2007-04 TCP Assigned for keying and markup 2007-04 Aptara Keyed and coded from ProQuest page images 2007-05 Emma (Leeson) Huber Sampled and proofread 2007-05 Emma (Leeson) Huber Text and markup reviewed and edited 2008-02 pfs Batch review (QC) and XML conversion Due Correction FOR M R 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. MY LORD , 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 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 't was 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 ; 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 : itaque 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 , 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 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 't is 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 't is 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 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 . 'T is 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 't is 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. § . 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 't is 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 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 leave to honour you , and ( though I have not hitherto had the honour to be known to you ) to subscribe my selfe , MY LORD , Your Honours Most Humble Servant , John Wallis . Oxford . Oct. 15. 1656. 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 , 't is 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 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 . 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 , 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 ; simulque Hobbii Philosolosophia 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 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 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 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 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 cōpassion 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 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 't is 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 't were 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 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 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 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 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 't is 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 t is 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 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 . T is 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 . T is true , sometimes they signify to guide , or lead ; viz. with the parties consent , ( Fata ●●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 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 . Hor. So , Ova noctuae , &c. tadium vini adducunt . Hor. Addua●ere sitim tempora , Virgi●i , [ sc . aestiva ) Hor. In like manner , febres deducere , to take them away . Non domus & fundus , non aeris acervus & auri , Aegroto domini deduxit corpore febres , Non animo curas . — Hor. So , deducere fastidium . Plin And then Febrimque 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 solemque 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 . 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 . — abductaque 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. Vt , deducta parte tertia , deos reli●ua reddatur Africanus de pactis dotalibus . Vt 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 . Virg. Jam mihi subduci facies humana videtur . Ovid. 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 . Ovid. That is , To bring down from the beginning of the World to his own times . — a pectore postquam Deduxit vestes . Ovid. Deducere sibi galerum , vel pileolum , Sueton. to putt off , or take off . Et cum frigida mors animâ subduxerat artus . Virg. — Seductae ex aethere terrae . Ovid. 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 . Vn●que conspecta livorem ducit ab uva . Hor. — arborea frigus ducebat ab umbra . Ovid. 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 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 , 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 〈◊〉 so used p. 87. Eaedem duae lineae — prout considerantur pro ipsis magnitudinibus — poni . &c. So that 't was 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 ? 'T is very well . It seems my Latine ( though as carelessely written as need to be ; for 't was 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 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 , 't is dwindled to nothing . It was , that I had derived your 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 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 , Vniversale , 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 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 Grāmar . 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 . 'T will 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 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 't is 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 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 , 'T is new , 'T is necessary , and 'T is 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 ! 'T is visible , therefore 't is 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 't is 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 t was told thee , that 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 was a marke with a hot iron ? for 't is a notion I never heard till now , ( and doe not believe it yet . ) Never 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 . 'T was 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 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 ? 'T is 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 , 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 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 't is 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 ; 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 : Le ts 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 le ts 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 . 'T is 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 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 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 't is 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 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 ; 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 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 , 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 , 'T is not a quantity Homogeneous , ( by 5 d 5 ) therefore 't is 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 . ) 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 't is 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 ? 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 ; 't is 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 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 't is 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 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 . 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 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 ; ( 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 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 . 'T is 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 not so ? is not 26 ⅔ + 3 ½ = 30 ? you may try it farther if you please . My skill for yours , 't will hold . ( And that 's 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 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 't is false . I conclude , you say , that they have the proportion of 1 to 4. Which is false , I do not so conclude ; but 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 . 'T is very hard , you have told us diverse times ; yet here , it seems , you mean to try what you could doe at it . And 't is 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 . 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 le ts 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 't is true , I find it . For , believe mee , that is not the way to find a Parameter . Nor 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 , 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 't is 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 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 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 't is 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 ; ) 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 't is 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 ●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 ; 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. Vpon 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 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 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 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 . 'T was 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 't is 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 , 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 't is 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 , ) 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 't is 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 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 , ( 't is 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 Vnequalls , 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 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 ; 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 , ( 't is but Coblers work at the best ; ) But however , 't is 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 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 't were 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 . 'T is 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 't is 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. 'T is 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 , ( 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 Vnity , 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 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 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 Vniformity , &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 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 , 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 't is 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 't is 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 , 't was a good meaning too , when you meant 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 't is 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 't is 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 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 't is , 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 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. ) 't is 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 √ 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 'T is 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 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 . 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 . 'T was 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 . 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 , 't is 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 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 Right-lined 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 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 ; 't is a quarter of four right Angles . 'T is 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 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 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 't was 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 't is 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. 't will containe a rectilineall figure ; But if the parts so broken be considered as the least that can be , that is , as so many points , 't will 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 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 't was 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 ; 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 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. 'T was 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 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 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. 'T is 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 't is 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 , 't is 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 . 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 le ts 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 , 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 Le ts 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 . 'T is false therefore which you affirmed . Quod erat demonstrandum . Now what do you think of the businesse ? is not the matter well amended ? 'T was bad before , now 't is worse . When you told us but of one side , and left us to guesse the other , 't was at our perill if we did not guesse right , and 't was 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 Vniform 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 ; ● 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 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 , le ts 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 't was simply done . But 'till then , I 'le say 't is done like your selfe , to say so however . ( For 't is lawfull with you to say any thing , true or false ) Your English Reader , perhaps , may think 't is 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 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 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 , 't is 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 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 that 's not all . Of AB to AF , and of BI to FK ; that 's 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 't is , 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 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 : 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 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 , 'T is 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 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 &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 Vniforme 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 't is 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 , 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 . 'T was 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 t was 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 : 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 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 Vniformely , 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 Vniforme , 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 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 for ⅓ in 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 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 Vniforme . 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 't is 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 . 'T is 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 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 't is 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 't is 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 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 , 't is an Inch , you say true : If you say , 't is an Ell , you say true : And if you say 't is 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 of two leaves together ? And , when you have done all , 't is 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 , 't is 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 Vniform , 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 ( 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 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 't is 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 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. L ∷ g − 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 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 't is all true , and truly demonstrated . Do you think 't is worth while after all this , to examine your demonstration ? 'T is a sad one , I confesse ; but t is 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 , 't is quite defunct . And there is a young one start up in stead of it . But 't is of the same breed , and t is 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 Vniformly 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 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 't is 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 ) 't is 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 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 't is 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 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 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 , 't is to be hoped , that your new demonstration is a good one ; is it not ? No , 'T is as bad as the other . Only 't is 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 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 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 . 'T is 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 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 , 'T is 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 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 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 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 't is 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 ! ( 'T is 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 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 't is 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 't is 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 't is 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 . 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 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 't is 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 . SECT . IV. fig. 1. fig. 2. SECT . V. CAP : XIV fig. 6. fig. 7 Cap. XVI . fig. ● . fig. 2. fig● 6. fig. 8. fig. 11. CAP. XVII . 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 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 . FINIS . ERRATA . 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 ▪