to the Ideas which they signifie Besides if you but consider how none of the Antients ever used any of them in their published demonstrations of Geometry nor in their Books of Arithmetique more then for the Rootes and Potestates themselves and how bad success you have had your self in the unskilfull using of them you will not I think for the future be so much in love with them as to demonstrate by them that first part you promise of your Opera Matheâ—atica In which if you make not amends for that which you have already published you will much disgrace those Mathematicians you address your Epistles to or otherwise have commended as also the Universities as to this kinde of Learning in the sight of learned men beyond Sea And thus having examined your panier of Mathematiques and finding in it no knowledge neither of Quantity nor of measure nor of Proportion nor of Time nor of Motion nor of any thing but only of certain Characters as if a Hen had been scraping there I take out my hand again to put it in to your other panier of Theology and good Manners In the mean time I will trust the objections made by you the Astronomer wherein there is neither close reasoning nor good stile nor sharpness of wit to impose upon any man to the discretion of all sorts of Readers LESS V. Of MANNERS To the same egregious Professors of the Mathematicks in the University of Oxford LESSON VI. HAving in the precedent Lessons maintained the Truth of my Geometry and sufficiently made appear that your objections against it are but so many errors of your own proceeding from misunderstanding of the Porpositions you have read in Euclide and other Masters of Geometry I leave it to your consideration to whom belong according to your own sentence the unhandsome attributes you so often give me upon supposition that you your selves are in the right and I mistaken and come now to purge my self of those greater accusations which concern my Manners It cannot be expected there should be much Science of any kinde in a man that wanteth Judgement nor Judgement in a man that knoweth not the Manners due to a publique disputation in writing wherein the scope of either party ought to be no other then the examination and manifestation of the truth For whatsoever is added of contumely eiâ—…er directly or scommatically is want of Charity and uncivil unless it be done by way of Reddition from him that is first provoked to it I say unless it be by way of Reddition for so was the Judgement given by the Emperor Vespasian in a quarrell between a Senato and a Knight of Rome which had given him ill language For when the Knight had proved that the first ill language proceeded from the Senator the Emperor acquitted him in these words Maledici senctor ibus non oportere remalâ—dicere fas civtle esse Nevertheless now a dayes uncivill words are commonly and bitterly used by all that write in matter of Controversie especially in Divinity excepting now and then such writers as have been more then ordinarily well bred and have observed how hainous and haâ—ardous a thing such câ—ntumely is amongst some sorts of men whether that which is said in disgrace be true or false For evill words by all men of understanding are taken for a defiance and a challenge to open war But that you should have bserved so much who are yet in your mothers belly was not a thing to be much expected The faults in Manners you lay to my charge are these 1. Self conceit 2. That I will be very angry with all men that do not presently submit to my Dictates 3. That I had my Doctrine concerning Vision out of papers which I had in my hands of Mr. Warners 4. That I have injured the Universities 5. That I am an Enemy to Religion These are great faults but such as I cannot yet confess And therefore I must as well as I can seek out the grounds upon which you build your Accusation Which grounds seeing you are not acquainted with my conveâ—sation must be either in my published writings or reported to you by honest men and without suspition of interest in reporting it As for my self-conceit and ostentation you shall finde no such matter in my writings That which you alleadge from thence is first that in the Epistle Dedicatory I say of my Book de Corpore Though it be little yet it is full and if good may go for great great enough When a man presenting a gift great or small to his betters adorneth it the best he can to make it the more acceptable he that thinks this to be Ostentation and self-conceit is little versed in the common actions of humane life And in the same Epistle where I say of Civill Philosophy it is no antienter then my Book de Cive these words are added I say it provoked and that my detractors may see they lose their labour But that which is truly said and upon provocation is not boasting but defence A short sum of that Book of mine now publiquely in French done by a Gentleman I never saw carrieth the Title of Ethiques demonstrated The Book it self translated into French hath not onely a great testimony from the Translator Serberius but also from Gassendus and Mersennus who being both of the Roman Religion had no cause to praise it or the Divines of England have no cause to finde fault with it Besides you know that the Doctrine therein contained is generally received by all but those of the Clergy who think their interest concerned in being made subordinate to the Civil Poweâ— whose testimonies therefore are invalide Why therefore if I commend it also against them that dispraise it publiquely do you call it boasting You have heard you say that I had promised the Quadrature of the Circle c. You heard then that which was not true I have been asked sometimes by such as saw the Figure before me what I was doing and I was not aâ—…aid to say I was seeking for the solution of that Probleme but not that I had done it And afterwards being asked of the success I have said I thought it done This is not boasting and yet it was enough when told again to make a fool believe 't was boasting But you the Astronomer in the Epistle before your Philosophicall Essay say you had a great expectation of my Philosophicall and Mathematicall works before they were published It may be so Is that my fault can a man raise a great expectation of himself by boasting If he could neither of you would be long before you raised it of your selves saving that what you have already published has made it now too late For I verily believe there was never seen worse reasoning then in that Philoâ—ophicall Essay which any judicious Reader would believe proceeded from a Praevaricator rather then from a man that believed himself nor worse Principles then those in
Body Of this kinde of Magnitudes and Quantities the Subject is Body And because for the computing of the Magnitudes of Bodies it is not necessary that the Bodies themselves should be present the Ideas and memory of them supplying their presence we reckon upon those Imaginary Bodies which are the Quantities themselves and say the length is so great the breadth so great c. which in truth is no better then to say the length is so long or the breadth so broad c. But in the mind of an inteligent man it breedeth no mistake Besides the Quantity of Bodies there is a Quantity of Time For seeing men without absurdity do ask how much it is by answering Tantum so much they make manifest there is a a quantity that belongeth unto Time namely a Length And because Length cannot be an accident of Time which is it self an accident it is the accident of a Body and consequently the length of the Time is the Length of the Body by which Length or Line we determine how much the Time is supposing some Body to be moved over it Also we not improperly ask with how much Swiftness a Body is moved and consequently there is a Quantity of Motion or Swiftness and the measure of that Quantity is also a line But then again we must suppose another motion which determineth the time of the former Also of Force there is a Question of How much which is to be answered by So much that is by Quantity If the Force consist in Swiftness the Determination is the same with that of Swiftness namely by a Line if in Swiftness and Quantity of Body joyntly then by a Line and a Solid or if in quantity of Body onely as Weight by a Solid onely So also is Number Quantity but in no other sense then as a line is Quantity divided into equall parts Of an Angle which is of two Lines whatsoever they be meeting in one point the digression or openess in other points it may be asked how great is that digression This Question is answered also by Quantity An Angle therefore hath Quantity though it be not the subject of Quantity for the body onely can be the subiect in which Body those â—…ing line are marked And because two lines may be made to divaricate by two causes one when having one end common and immoveable they depart one from another at the other ends circularly and this is called surply an Angle and the Quantity thereof is the Quantity of the Arch which the two lines intercept The other cause is the bending of a straight line into a circular or other crooked line till it touch the place of the same line whilst it was straight in one onely point And this is called an Angle of contingence And because the more it is bent the more it digresseth from the Tangent it may be asked how much more and therefore the answer must be made by Quantity and consequently an Angle of Contingence hath its Quantity as well as that which is called simply an Angle And in case the digression of two such crooked lines from the Tangent be uniform as in Circles the Quantity of their digression may be determined For if a straight line be drawn from the point of Contact the digression of the lesser Circle will be to the digression of the greater Circle as the part of the line drawn from the point of Contact and intercepted by the Circumference of the greater Circle is to the part of the same line intercepted by the Circumference of the Lesser Circle or which is all one as the greater Radius is to the lesser Radius You may guess by this what will become of that part of your last Book where you handle the Question of the Quantity of an Angle of Contingence Also there lyeth a Question of how much Comparatively one magnitude is to another magnitude as how much water is in a Tun in respect of the Ocean how much in respect of a Piâ— little in the first respect much in the Latter Therefore the Answer must be made by some respective Quantity This respective Quantity is called Ratio and Proportion and is determined by the Quantity of their differences and if their differences be compared also with the Quantities themselves that differ it is called simply Proportion or Proportion Geometricall But if the differences be not so compared then it is called Proportion Arithmeticall And where the difference is none there is no Quantity of the Proportion which in this case is but a bare comparison Also concerning Heat Light and divers other Qualities which have degrees there lyeth a question of how much to be answered by a so much and consequently they have their Quantities though the same with the Quantity of Swiftness because the intensions and remissions of such Qualities are but the intensions and remissions of the Swiftness of that motion by which the Agent produceth such a quality And as Quantity may be considered in all the operations of Nature so also doth Geometry run quite thorow the whole body of Naturall Philosophy To the Principles of Geometry the definition appertaineth also of a Mâ—asure which is this One Quantity is the Measure of another Quantity when it or the Multiple of it is Coiâ—cident in all points with the other Quantity In which Definition in stead of that 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 of Euclid I put Coincidence For that superposition of Quantities by which they render the word 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 cannot be understood of Bodies but only of Lines and Superficies Nevertheless many Bodies may be Coincident successively with one and the same place and that place will be their Measure as we see practised continually in the measuring of Liquid Bodies which Art of Mâ—asuâ—ing may properly be called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 but not Superposition Also the definitions of Greater Less and Equall are necessary Principles of Geometry For it cannot be imagined that any Geometrician should demonstrate to us the Equality and Inâ—quality of magnitudes except he tell us first what those words do signifie And it is a wonder to me ' that Euclide hath not any where defined what are Equals or at least what are Equall Bodies but serveth his turn throughout with that forementioned 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 which hath no place in Solids nor in Time nor in Swiftness nor in Circular or other crooked lines and therefore no wonder to me why Geometry hath not proceeded to the calculation neither of crooked lines nor sufficiently of Motion nor of many other things that have proportion to one another Equall Bodies Superficies and Lines are those of which every one is capable of being co-incident with the place of every one of the rest And Equall Times wherein with one and the same Motion Equall lines are described And Equally Swift are those Motions by which we run over equall spaces in any time determined by any other motion And universally all Quantities are
they are to be used is that which we call Philosophiaprima It is not the work of a Geometrician as a Geometrician to Define what is Equality or Proportion or any other word he useth though it be the work of the same man as a man His Geometricall part is To draw from them as many true and usefull Theoremes as he can You object secondly That a Pyramis may be equall to a Cube whilst it is a Pyramis True And so also whilst it is a Pyramis it hath a possibility by flexion and transposition of parts to become a Cube and to be put into the place where another Cube equall to it was before This is to argue like a child that hath not yet the perfect understanding of any Language In the third and fourth objection you teach me to define Equall Bodies it I will needs define them by place by the Equality of place and to say that Bodies are Equall that have Equall places Teach others if you can to measure their grain not by the same but Equall Bushels In the fifth objection you except against the word can in that I say that Bodies are Equall which can fill the same place For the greater Body can you say fill the place of the less though not reciprocally the less of the greater It is true that though the place of the less can never be the place of the greate r yet it may be filled by a part of the greater But 't is not then the greater Body that filleth the place of the less but a part of it that is to say a less Body Howsoever to take away from simple men this straw they stumble at I have now put the Definition of Equal Bodies into these words Equall Bodies are those whereof every one can fill the place of every other And if my Definition displease you propound your own either of Equall Bodies or of Equals simply But you have none Take therefore this of mine The sixth is a very admirable exception What say you if the same Body can sometimes take up a greater sometimes a lesser place as by Rarefaction and Condensation I understand very well that Bodies may be somtimes thin and sometimes thick as they chance to stand closer together or further from one another So in the Mathematick-Schools when you read your Learned Lectures you have a thick or thronging Audience of Disciples which in a great Church would be but a very thin company I understand how thick and thin may be attributed to bodies in the Plurall as to a company but I understand not how any one of them is thicker in the School then in the Church or how any one of them taketh up a greater room in the School when he can get in then in the street For I conceive the Dimensions of the Body and of the Place whether the place be filled with Gold or with Air to be coincident and the same and consequently both the Quantity of the Air and the Quantity of the Gold to be severally equall to the Quantity of the place and therefore also by the first Axiome of Euclide equall to one another insomuch as if the same Air should be by Condensation contained in a part of the place it had the dimensions of it would be the same with the dimensions of part of the place that is should be less then they were and by consequence the Quantity less And then either the same body must be less also or we must make a difference between greater Bodies and Bodies of greater Quantity which no man doth that hath not lost his wits by trusting them with absurd teachers When you receive Salary if the Steward give you for every shilling a piece of six pence and then say every shilling is condensed into the room of a six pence I believe you would like this Doctrine of yours much the worse You see how by your ignorance you confound the affairs of mankind as far forth as they give credit to your opinions though it be but little For nature abhorres even empty words such as are in the meaning you assign them Rarefying and Condensing And you would be as well understood if you should say coining words by your own power that the same Body might take up sometimes a greater sometimes a lesser place by Wallifaction and Wardensation as by Rarefaction and Condensation You see how admirable this your objection is In the seventh objection you bewray another kind of Ignorance which is the Ignorance of what are the proper works of the severall parts of Philosophy Though it were out of doubt say you that the same Body cannot have several Magnitudes yet seeing it is matter of Natural Philosophy nor hath any thing to do with the present business to what purpose is it to mention it in a Mathematicall Definition It seems by this that all this while you think it is a piece of the Geometry of Euclide no less to make the Definitions he useth then to infer from them the Theorems he demonstrateth Which is not true For he that telleth you in what sense you are to take the Appelations of those things which he nameth in his discourse teacheth you but his Language that afterwards he may teach you his Art But teaching of Language is not Mathematick nor Logick nor Physick nor any other Science and therefore to call a Definition as you do Mathematicall or Physicall is a mark of Ignorance in a Professor unexcusable All Doctrine begins at the understanding of words and proceeds by Reasoning till it conclude in Science He that will learn Geometry must understand the Termes before he begin which that he may do the Master demonstrateth nothing but useth his Naturall prudence onely as all men do when they endeavour to make their meaning clearly known For words understood are but the seed and no part of the harvest of Philosophy And this seed was it which Aristotle went about to sow in his twelve Books of Metaphysicks and in his eight Books concerning the Hearing of Naturall Philosophy And in these Books he defineth Time Place Substance or Essence Quantity Relation c. that from thence might be taken the Definitions of the most generall words for Principles in the severall parts of Science So that all Definitions proceed from common understanding of which it any man rightly write he may properly call his writing Philosophia prima that is the Seeds or the Grounds of Philosophy And this is the Method I have used defining Place Magnitude and the other the most generall Appellations in that part which I intitle Philosophia prima But you now not understanding this talk of Mathematicall Definitions You will say perhaps that others do the same as well as you It may be so But the appeaching of others does not make your ignorance the less In the eighth place you object not but ask me why I define equall Bodies apart I will tell you Because all other things which are said
propagated to the Indies If I ask you how you know it you may wonder perhaps but answer you cannot Are you Philosophers or Geometricians or Logicians more then are the simplest of rurall people Or are you not rather less by as much as he that standeth still in ignorance is nearer to knowledge then he that runneth from it by erroneous learning And lastly what an absurd objection is it which you make to the eighth Article where I say that when two Bodies of equall magnitude fall upon a third Body that which falls with greater velocity imprints the greater motion You object that not so much the magnitude is to be considered as the weight as if the weight made no difference in the velocity when notwithstanding weight is nothing else but motion downward Tell me when a weighty body thrown upwards worketh on the Body it meeteth with do you not then think it worketh the more for the greatness and the less for the weight Of the Faults that Occurre in Demonstration To the same egregious Professors of the Mathematicks in the University of Oxford LESSON IIII. OF twenty Articles which you say of nineteen which I say make the six teenth Chapter you except but three and confidently affirm the rest are false On the contrary except three or four faults such as any Geometrician may see proceed not from ignonorance of the Subject or from want of the Art of Demonstration and such as any man might have mended of himself but from security I affirm that they are all true and truly Demonstrated and that all your objections proceed from meer ignorance of the Mathematiques The first fault you find is this that I express not Art 1. what Impetus it is which I would have to be multiplyed into the Time The last Article of my thirteenth Chapter was this If there be a Number of Quantities propounded howsoever equall or unequall to one another and there be another Quantity which so often taken as there be Quantities propounded is equall to their whole sum that Quantity I call the mean Arithmeticall of them all Which Definition I did there insert to serve me in the explication of those Propositions of which the sixteenth Chapter consisteth but did not use it here as I intended My first Proposition therefore as it standeth yet in the Latine being this The velocity of any Body moved during any Time is so much as is the product of the Impetus in one Point of Time multiplyed into the whole Time to a man that hath not skill enough to supply what is wanting is not intelligible Therefore I have caused it in the English to go thus The velocity of any Body in whatsoever Time moved hath its Quantity determined by the sum of all the severall Impetus Quicknesses which it hath in the severall Points of the Time of the Bodies motion Aâ—d adâ—ed that all the Impetus together taken through the whole Time is the same thing with the Mean Impetus which Mean is defined Chapter 13. Art 29. multiplyed into the whole Time To this first Article as it is uncorrected in the Latine you object That meaning by Impetus some middle Impetus and assigning none I determine nothing And 't is true But if you had been Geometricians â—…ficient to be Professors you would have shewed your skill much better by making it appear that this middle Impetus could be none but that which being taken so often as there bâ— Points in the Line of Time would be equall to the sum of all the severall Impetus taken in the Points of Time respectively which you could not do To the Corollary you ask first how Impetus can be ordinately applyed to a Line Absurdly For does not Arâ—himedes sometimes say and with him many other excellent Geometricians let such a Line be the Time And do they not mean that that Line or the motion over it is the measure of the Time And may not also a Line serve to measure the swiftness of a Motion You thought you say onely Lines ought to be said to be ordinately applyed to Lines Which I easily believe for I see you understand not that a Line though it be not the Time it self may be the quantity of a Time You thought also all you have said in your Elenchus in your Doctrine of the Angle of Contact in your Arithmetica Infinitorum and in your Coniques is true and yet it is almost all proved false and the rest nothing worth Secondly you object that I design a Parallelogram by one onely side It was indeed a great oversight and argueth somewhat against the man but nothing against his Art For he is not worthy to be thought a Geometrician that cannot supply such a fault as that and correct his Book himself Though you could not do it yet another from beyond Sea took notice of the same fault in this manuer He maketh a Parallelogram of but one side it should be thus Veâ— denique per Parallelogrâ—… 〈◊〉 〈◊〉 latus est medium proportionale intâ—r Impetum maximum five ultimò acquisitum impetûs ejusdem maximi semissem alterum vero latus medium proportionale inter totum tempus ejusâ—'em totius temporis semissem Which I therefore repeat that you may learn good manners and know that they who reprehend ought also when they can to add to their reprehension the correction At the second Article you are pleased to advise me instead of In omni motu uniformi to put in In omnibus motibus uniformibus You have a strange opinion of your own Judgement to think you know to what end another man useth any word better then himself My intention was onely to consider motions uniform and motions from rest uniformly or regularly accelerated that I might thereby compute the lengths of crooked Lines such as are described by any of those motions And therefore it was enough to prove this Theoreme to be true in all uniform or uniformly accelerated Motion not Motions though it be true also in the Plurall It seems you think a man must write all he knows whether it conduce or not to his intended purpose But that you may know that I was not as you think ignorant how far it might be extended you may read it Demonstrated at the same Article in the English universally Against the demonstration it self you run to another Article namely the thirteenth which is this Probleme The length being given which is passed over in a given Time by uniform Motion to findet e length which shall be passed over by Motion uniformely accelerated in the same Time so as that the Impetus last acquired be equall to the Time Which you recite imperfectly thereby to make it seem that such a Length is not determined Whether you did this out of ignorance or on purpose thinking it a piece of wit as your pretended mysterie which goes immediately before I cannot tell foâ— in neither place can any wit be espied by any but your selves
understanding must be made my fault My Demonstration is this In the Parallelogram A B C D Fig. 11. Let the side A B be conceived to be moved uniformly till it lye in C D and let the Time of that Motion be A C or B D. And in the same Time let it be conceived that A C is moved with uniform acceleration till it lye in B D. To which you object that then the acceleration last acquired must be far greater then that wherewith A B is moved uniformly else it shall never come to the place you would have it in the same Time What proof bring you for this None here Where then No where that I remember On the contrary I have proved Art 9. of this Chapter that the Line described by the concourse of those two Motions namely uniform from A B to C D and uniformly accelerated from A C to B D is the crooked Line of the Sâ—miparabola A H D. And though I had not yet it is well known that the same is demonstrated by Galileo And seeing it is manifest that in what Proportion the Motion is accelerated in the Line A B in the same Proportion the Impetus beginning from Rest in A is encreased in the same Times which Impetus is designed all the way by the ordinate Lines of the Semi-Parabola the greatest Impetus acquired must needs be the Base of the Semiparabola namely B D equal to A C which designs the whole Time I cannot therefore imagine what should make you say without proof that the greatest acquired Impetus is greater then that which is designed by the Base B D. Next you say you see not to what end I divide A B in the middle at E. No wonder for you have seen nothing all the way Others would sâ—â— it is necessary for the Demonstration as also that the Point F is not to be taken arbitrarily and likewise that the thirteenth Article which you admit not for proof is sufficiently demonstrated and your objections to it answered By the way you advise me where I say percursam codem motu uniformi cum Impetu ubique c. to blot out cum because the Impetus is not a companion in the way but the cause Pardon me in that I cannot take your Learned Counsell for the word motu uniformi is the Ablative of the Cause and Impetu the Ablative of the Manner But to come again to your objections you say I make a greater space run over in the same Time by the slower Motion then by the swifter How does that appear because there is no doubt but the swiftness is greater where the greatest Impetus is alwaies maintained then where it is attained to in the same Time from Rest. True But that is when they are considered asunder without concourse but not then when by the concourse they debilitate one another and describe a third Line different from both the Lines which they would describe singly In this place I compare their effects as contributing to the the description of the Parabolicall Line A H D. What the effects of their severall Motions are when they are considered asunder is sufficiently shewn bâ—fore in the first Article You should first have gotten into your mindes the perfect and distinct Ideás of all the Motions mentioned in this Chapter and then have ventured upon the censure of them but not before And then you would have seen that the Body moved from A describeth not the Line A C nor the Line A B but a third namely the Semiparabolicall Line A H D. Again where I say Wherefore if the whole A B be uniformly moved to C D in the same Time wherein A C is moved uniformly to F G you ask me whether with the same Impetus or not How is it possible that in the same time two unequall Lengths should be passed over with the same Impetus But why say you do you not tell us with what Impetus A C comes to F G What need is there of that when all men know that in unifoâ—m Motion and the same Time Impetus is to Impetus as Length to Length Which to have expressed had not â—een pertinent to the Demonstration That which follows in the Demonstration rursus suppono quod latus A C c. to these words ut ostensum est Art 12 You confute with saying you have proved that Article to be false But you may see now if you please at the same place that I have proved your objections to be frivolous After this you run on without any Argument against the rest of the Demonstration shewing nothing all the way but that the variety and concourse of Motions the Speculations whereof you have not been used to have made you giddy To the nineteenth Article you apply the same objection which you made to the eighteenth Which having been answered it appeares that from the very beginning of your Elenchus to this place all your objections except such as are made to three or four mistakes of small importance in setting down my mind are meer Paralogisms and such as are less pardonable then any Paralogism in Orontius both because the Subject as less difficult is more easily mastered and because the same faults are more shamefully committed by a Reprehender then by any other man I had once added to these nineteen Articles a twentieth which was this If from a Point in the Circumference there be drawn a Chord and a Tangent equall to it the Angle which they make shall be double to the aggregate of all the Angles made by the chords of all the equal Arches into which the Arch given canpossibly be divided Which Proposition is true and I did when I writ it think I might have use of it But be it or the Demonstration of it true or false seeing it was not published by me it is somewhat barbarous to charge me with the faults thereof No Doctor of Humanity but would have thought it a poor and wretched malice publiquely to examine and censure papers of Geometry never published by what means soever they came into his hands I must confess that in these words in such kind of Progression Arithmeticall that is which begins with o the sum of all the Numbers taken together is equall to half the Number that is made by multiplying the greatest into the least there is a great error for by this account these Numbers 0 1 2 3 4 taken together should be equall to nothing I should have said they are equall to that Number which is made by multiplying half the greatest into the Number of the Terms There was therefore if those words were mine for truly I have no Copy of them nor have had since the Book was Printed and I have no great reason as any man may see to trust your Faith a great error in the writing but not an erroneous opinion in the writer The Demonstration so corrected is true And the Angles that have the Proportions of the Numbers
Chapter of his Clavis Mathematicae where he sayes that 43 7 is the Proportion of 31 to 7 for his meaning is that the Proportion of 43 7 to one is the Proportion of 31 to 7 whereas if he meant as you do then 86 7 should be double the Proportion of 31 to 7. Partly also because you think as in the end of the twentieth Proposition that if the Proportion of the Numerators of these Fractions to their Denominators decrease eternally they shall so vanish at last as to leave the Proportion of the sum of all the Squares to the sum of the greatest so often taken that is an infinite Number of times as one to three or the sum of the greatest to the sum of the increasing Squares as three to one for which there is no more reason then for four to one or five to one or any other such Proportion For if the Proportions come eternally nearer and nearer to the subtriple they must needs also come nearer and nearer to subquadruple and you may as well conclude thence that the upper Quantities shall be to the Lower Quantities as one to four or as one to five c. as conclude they are as one to three You can see without admonition what effect this false ground of yours will produce in the whole structure of your Arithmetica Infinitorum and how it makes all that you have said unto the end of your thirty-eighth Proposition undemonstrated and much of it false The thirty-nineth is this other Lemma In a Series of Quantities beginning with a Point or Cypher and proceeding according to the Series of the Cubique Numbers as 0. 1. 8. 27. 64 c. to finde the Proportion of the sum of the Cubes to the sum of the greatest Cube so many times taken as there be Terms And you conclude that they have the Proportion of 1 to 4 which is false Let the first Series be of three terms subscribed with the greatest the sum of the Cubes is nine the sum of all the greatest is 24 a quarter whereof is 6. But 9 is greater then 6 by three unities An unity is something Let it be therefore A. Therefore the Row of Cubes is greater then a quarter of three times eight by three A. Again let the Series have four terms as the sum of the Cubes is 36 a quarter of the sum of all the greatest is twenty-seven But thirty-six is greater then twenty-seven by nineâ— that is by 9 A. The excess therefore of the sum of the Cubes above the fourth part of the sum of all the greatest is increased by the increase of the Number of terms Again let the terms be five as the sum of the Cubes is one hundred the sum of all the greatest three hundred and twenty a quarter whereof is eighty But one hundred is greater then eighty by twenty that is by 20 A. So you see that this Lemma also is false And yet there is grounded upon it all that which you have of comparing Parabolas and Paraboloeides with the Parallelograms wherein they are accommodated And therefore though it be true that the Parabola is â…” and the Cubicall Paraboloeides ¾ of their Parallelograms respectively ' yet it is more then you were certain of when you referred me for the learning of Geometry to this Book of yours Besides any man may perceive that without these two Lemmas which are mingled with all your compounded Series with their excesses there is nothing demonstrated to the end of your Book Which to prosecute particularly were but a vain expence of time Truly were it not that I must defend my reputation I should not have shewed the world how little there is of sound Doctrine in any of your Books For when I think how dejected you will be for the future and how the grief of so much time irrecoverably lost together with the conscience of taking so great a stipend for mis-teaching the young men of the University the consideration of how much your friends wil be ashamed of you will accompany you for the rest of your life I have more compassion for you then you have deserved Your Treatise of the Angle of Contact I have before confuted in a very few leaves And for that of your Conique Sections it is so covered over with the scab of Symboles that I had not the patience to examine whether it be well or ill demonstrated Yet I observed thus much that you find a Tangent to a Point given in the Section by a Diameter given and in the next Chapter after you teach the finding of a Diameter which is not artificially done I observe also that you call the Parameter an Imaginary Line as if the place thereof were less determined then the Diameter it self and then you take a mean Proportionall between the intercepted Diameter and its contiguous ordinate Line to find it And t is true you find itâ— But the Parameter has a determined Quantity to be found without taking a mean Proportional For the Diameter and half the Section being given draw a Tangent through the Vertex and dividing the Angle in the midst which is made by the Diameter and Tangent the Line that so divideth the Angle will cut the crooked Line From 〈◊〉 intersection draw a Line if it be a Parabola Parallel to the Diameter and that Line shall cut off in the Tangent from the Vertex the Parameter sought But if the Section be an Eâ—lipsis or an Hyperbole you may use the same Method saving that the Line drawn from the intersection must not be Parallel but must pass through the end of the transverse Diameter and then also it shall cut off a part of the Tangent which measured from the Vertex is the Parameter So that there is no more reason to call the Parameter an Imaginary Line then the Diameter Lastly I observe that in all this your new Method of Coniques you shew not how to find the Burning Points which writers call the Foci and Umbilici of the Section which are of all other things belonging to the Coniques most usefull in Philosophy Why therefore were they not as worthy of your pains as the rest for the rest also have already been demonstrated by others You know the Focus of the Parabola is in the Axis distant from the Vertex a quarter of the Parameter Know also that the Focus of an Hyperbole is in the Axis distant from the Vertex as much as the Hypotenusall of a rectangled Triangle whose one side is half the transverse Axis the other side half the mean Proportionall between the whole transverse Axis and the Parameter is greater then half the transverse Axis The cause why you have performed nothing in any of your Books saving that in your Elenâ—…hus you have spied a few negligences of mine which I need not be ashamed of is this that you understood not what is Quantity Line Superâ—…ies Angle and Proportion without which you cannot have the Science
a Judge of such matters But what did he That pretious time which was bat little because he was to depart again presently for Flanders he bestowed wholly in venting his own childish Opinions not suffering the Doctor scarce to speak losing thereby the benefit he came for and discovering that he came not to hear what others could say but to show to others how learned he was himself already Why else did he take so little time and so mispend it Or why returned he not again But when he had talked away his time and found though patiently and civilly heard he was not much admired he took occasion writing against me to be revenged of D. Harvey by sleighting his learning publiquely and tels me that his learning was onely Experiments which he sayes I say have no more certainty then Civil Histories Which is false My words are Ante hos nihil certi in Physicâ erat praeter Experimenta â—uique sua Historias Naturales si tamen â—ae dicendae â—ertae sint quae Civilibus Historiis ceâ—tiores nâ—â— sunâ— Where I except expressly nom uncertainty the Experiments that every man maketh to himself But you see the â—ere-cut by which vain Glory joyned with Ignorance passeth quickly over to Eâ—vy and Contumely Thus it seems by your own confession I was used by Uinâ—â—x He comes with some of my acquaintance in a Visit. What he said I know not but if he â—iscouâ—sed then as in his Philosophicall Essay he writeth I will be bold to say of my self I was so far from morosity oâ— to use his Phrase from being tetricall as I may very well have a good opinion of my own patience And if there passed between us the discourse you mention in your Elenchus Pag. 116. it was an incivility in him so great that without great civility I could not have abstained from bidding him be gone That which passed between us you say was this I complained that whereas I made Sense nothing but a perception of Motion in the Organ nevertheless the Philosophy Schools through all Europelâ—d by the Text of A istotle teach another Doctrin namely that Sensation is performed by Species This is a little mistaken For I do glory not complain that whereas all the Universities of Europe hold Sensation to proceede from Species I 〈◊〉 it to be a perception of Motion in the Organ The answer of Vindex you say was That the other Hypothesis whereby Sense was explicated by the Principles of Motion was commonly admitted here before my Book came out as having been sufficiently delivered by Des Carâ—â—s Gassendus and Sir Kenelme Digby before I had published any thing in this kinde This then it seems was it that made me angry Truly I remember not any angry word that ever I uttered in all my life to any man that came to see me though some of them have troubled me with very impertinent discourse and with those that argued with me how â—pertinently sâ—ever I alwayes thought it more civility to be somewhat earnest in the defence of my opinion then by obstinate and affected silence to let them see I contemned them or heaâ—kned not to what they said Iâ— I were earnest in making good that the manner of Sensation by such Motion as I had explicated in my Leviathan is in none oâ— the Authors by him named it was not Anger but a caâ—e of not offending him with any signe of the contempt â—hich his discourse deserved But it was Incivility in him to make use of a Visiâ— which all men take for a pâ—ofessiâ—n of Friendship to tell me that that which I had already published for my own was found before by Des Cartes Gassendus and Sir Kenelme Digby But let any man read Des Cartes he shall finde that he attributeth no Motion at all to the object of sense but an inclination to action which inclination no man can imagine what it meaneth And for Gassendus and S. Kenelme Digby it is manifest by their writings that their opinions are not different from that or Epicurus which is very different from mine Oâ— if these two or any of those I conversed with at Paris had prevented me in publiâ—â—ing my own Dâ—â—trine yet since it was there known and declared for mine by Mersennus in the Pâ—eface to his Ballistica of which the three fiâ—st leaves are imployed wholly in the setting 〈◊〉 of my opinion concerning sense and the rest of the faculties of the Sâ—ul they ought not therefoâ—e to be said to have found it out before me And consequently this answer which you say was given me by Vindex was nothing else but Untruth and Envy and because it was done by way of Visit Incivility But you have not alleadged nor can alleadge any words of mine from which can be drawn that I am so angry as you say I am with those that submit not to my Dictates Though the discipline of the University be never so good yet certainly this behavour of yours and his are no good Arguments to make it thought so But you the Professor of Geometry that out of my words spoken against Vindex in my 20 Chapter argue my angry humour do just as well as when in your Arithmetica Infinitâ—rum from the continuall increase of the excess of the row of Squares above the third part of the aggregate of the greatest you conclude they shall at last be equall to it For though you knew that Vindex had given me first the woâ—st words that possibly can be given yet you would have that return of mine to be a Demonstration of an angâ—y humouâ— not then knowing what I told you even now in the beginning of this Lesson of the sentence given by Vespasian But to this Point I shail speak again hereafter Your third Accusation is That I had my doctrine of Vision which I pretend to be my own out of papers which I had a long time in my hands of Mr. Warners I never had sight of any of Mr. Warn papers in all my life but that of Vision by Refraction which by his approbation I carryed with me to Paris and caused it to be printed under his own name at the end of Mersennus his Cogitata Physico-Mathematica which you may have there seen and another Treatise of the Proportions of Alloy in Gold and Silver coine which is nothing to the present purpose In all my conversation wiâ—h him I never heard him speak of any thing he had written or was writing de Penicillo optico And it was from me that he first heard it mentioned that Light and Colour were but Fancy Which he imbraced presently as a truth and told me it would remove a rub he was then come to in the discovery of the place of the Image If after my going hence he made any use of it though he had it from me and not I from him it was well dâ—ne But wheresoever you finde my Principles make use of them if you can to demonstrate all the Symptomes