Equall that are measured by the same number of the same Measures It is necessary also to the Science of Geometry to define what Quantities are of one and the same kind which they call Homogeneous the want of which definitions hath produced those wranglings which your Book De Angulo Contactus will not make to cease about the Angle of Contingence Homogeneous quantities are those which may be compared by 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 application of their Measures to one another So that Solids and Superficies are Heterogeneous quantities because there is no coincidence or application of those two dimensions No more is there of Line and Superficies nor of Line and Solid which are therefore Heterogeneous But Lines and Lines Superficies and Superficies Solids and Solids are Homogeneous Homogeneous also are Line and the Quantity of Time because the Quantity of Time is measured by application of a Line to a Line for though Time be no Line yet the Quantity of Time is a Line and the length of two Times is compared by the length of two Lines Weight and Solid have their Quantity Homogeneous because they measure one another by application to the beam of a Ballaâ—… Line and Angle simply so called have their Quantity Homogeneous because their measure is an Arch or Arches of a Circle applicable in every point to one another The Quantity of an Angle sinply so called and the Quantity of an Angle of Contingence are Heterogeneous For the measures by which two Angles simply so called are compared are in two coincident Atches of the same Circle but the measure by which an Angle of Coatingence is measured is a straight line intercepted between the point of Contact and the Circumference of the Circle and therefore one of them is not applicable to the other and consequently of these two sorts of Angles the Quantities are Heterogeneous The Quantities of two Angles of Contingence are Homogeneous for they may be measured by the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 of two Lines whereof one extream is common namely the point of Contact the other Extreams are in the Arches of the two Circles Besides this knowledge of what is Quantity and Measure and their severall sorts it behoveth a Geometrician to know why and of what they are called Principles For not every Proposition that is evident is therefore a Principle A Principle is the beginning of something And because Definitions are the beginnings or first Propositions of Demonstration they are therefore called Principles Principles I say of Demonstration But there be also necessary to the teaching of Geometry other Principles which are not the beginnings of Demonstration but of Construction commonly called Petitions as that it may be granted that a man can draw a straight Line and produce it and with any Radius on any Center describe a Circle and the like For that a man may be able to describe a square he must first be able to draw a straight line and before he can describe an Aequilaterall Triangle he must be able first to describe a Circle And these Petitions are therefore properly called Principles not of Demonstration but of Operation As for the commonly received third sort of Principles called Common Notions they are Principles onely by permission of him that is the Discipleâ— who being â—…nuous and comming not to cavill but to learn is content to receive them though Demonstrable without their Demonstrations And though Definitions be the onely Principles of Demonstration yet it is not true that every Definition is a Principle For a man may so precisely determine the signification of a word as not to be mistaken yet may his Definition be such as shall never serve for proof of any Theoreme nor ever enter into any demonstration such as are some of the Definitions of Euclide and consequently can be no beginnings of Demonstration that is to say no Principles All that hitherto hath been said is so plain and easie to be understood that you cannot most Egregious Professors without discovering your ignorance to all men of reason though no Geometricians deny it And the same saving that the words are all to be found in Dictionaties new also to him that means to learn not onely the Practice but also the Science of Geometry necessary and though it grieve you mine And now I come to the Definitions of Euclide The first is of a Point 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 c. Signum est eujus est pars nulla that is to say a. Marke is that of which there is no part Which definition not onely to a candid but also to a rigid construer is sound and usefull But to one that neither will interpret candidly nor can interpret accurately is neither usefull nor true Theologers say the Soul hath no part and that an Angel hath no part yet do not think that Soul or Angel is a point A mark or as some put instead of it 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 which is a mark with a hot Iron is visible if visible then it hath Quantity and consequently may be divided into parts innumerable That which is indivisible is no Quantity and if a point be not Quantity seeing it is neither substance nor Quality it is nothing And if Euclide had meant it so in his definition as you pretend he did he might have defined it more briefly but ridiculously thus a Point is nothing Sir Henry Savile was better pleased with the Candid interpretation of Proclus that would have it understood respectively to the matter of Geometry But what meaneth this respectively to the matter of Geometry It meaneth this that no Argument in any Geometricall demonstration should be taken from the Division Quantity or any part of a Point which is as much as to say a Point is that whose Quantity is not drawn into the demonstration of any Geometricall conclusion or which is all one whose Quantity is not considered An accurate interpreter might make good the definition thus a Point is that which is undivided and this is properly the same with cujus non est pars for there is a great difference between undivided and in â—ivisible that is between cujus non est pars and cujus non potest esse pars Division is an Act of the understanding the understanding therefore is that which maketh parts and there is no part where there is no consideration bat of one And consequently Euclides definition of a Point is accurately true and the same with mine which is that a Point is that Body whose Quantity is not considered And considered is that as I have defined it Chap. 1. at the end of the third Article which is not put to account in demonstration Euclide therefore seemeth not to be of your opinion that say a Point is nothing But why then doth he never use this definition in the Demonstâ—ation of any Proposition Whether he useth it expressly or no I remember not but the 16th Proposition of the third Book without the force of this
to be equall are said to be so from the equality of Bodies as two lines are said to be equall when they be coincident with the Length of one and the same Body and equall Times which are measured by equall Lengths of Body by the same Motion And the reason is because there is no Subject of Quantity or of Equality or of any other accident but Body all which I thought certainly was evident enough to any uncorrupted Judgement and therefore that I needed first to define Equality in the Subject thereof which is Body and then to declare in what sense it was attributed to Time Motion and other things that are not Body The ninth objection is an egregious cavill Having set down the Definition of Equall Bodies I considered that some men might not allow the attribute of Equality to any things but those which are the Subjects of Quantity because there is no Equality but in respect of Quantity And to speak rigidly Magnum Magnitudo are not the same thing for that which is great is properly a Body whereof greatness is an Accident In what sense therefore might you object can an Accident have Quantity For their sakes therefore that have not Judgement enough to perceive in what sense men say the Length is so Long or the Superficies so broad c. I added these words Eâdem ratione quâ scilicet corpora dicuntur aequalia Magnitudo magnitudini aequalis dicitur that is in the same manner as Bodies are said to be equal their magnitudes also are said to be equall Which is no more then to say when Bodies are Equall their Magnitudes also are called Equall When Bodies are Equall in Length their Lengths are also called Equall And when Bodies are Equall in Superficies their Superficies are also called Equall All which is common speech as well amongst Mathematicians as amongst common people and though improper cannot be altered nor needeth to be altered to intelligent men Nevertheless I did think fit to put in that clause that men might know what it is we call Equality as well in Magnitudes as in Magnis that is in Bodies Which you so interpret as if it bore this sense that when Bodies are Equall their Superficies also must be Equall contrary to your own knowledge onely to take hold of a new occasion of reviling How unhandsome and unmanly this is I leave to be judged by any Reader that hath had the fortune to see the world and converse with honest men Against the fourteenth Article where I prove that the same Body hath alwayes the same magnitude you object nothing but this that though it be granted that the same Boâ—y hath the same magnitude while it resteth yet I bring nothing to prove that when it changeth place it may not also change its Magnitude by being enlarged or coâ—tracted There is no doubt but to a Body whether at rest or in motion â—oâ—e Body may be added or part of it taken away But then it is not the same Body unless the Whole and the Part be all one It the Schools had not set your wit awry you could never have been so stupid as not to see the weakness of such objections That which you add in the end of your objections to this eighth Chapter that I allow not Euclide this Axiom gratis that the Whole is greater then a Part you know to be untrue At my eleventh Chapter you enter into dispute with me about the nature of Proportion Upon the truth of your Doctrine therein and partly upon the truth of your opinions concerning the Definitions of a Point and of a Line dependeth the Question whether you have any Geometry or none and the truth of all the Demonstrations you have in your other Books namely of the Angle of Contact and Arithmetica Infinitorum Here I say you eâ—ter how you wil get out your reputation saved we shall seâ— hereafter When a man asketh what Proportion one Quantity hath to another he asketh how great or how little the one is comparatively to or in respect of the other When a Gâ—ometrician prefixeth before his Demonstrations a Dâ—â—inition he doth it not as a part of his Geometry but of naturall evidence not to be demonstrated by Argument but to be understood in understanding the Language wherein it is set down though the matter may nevertheless if besides Geometry he have wit be of some help to his Disciple to make him understand it the sooner But when there is no â—ignificant Definition prefixed as in this case where Euclides Definition of Proportion That it is a whatsâ—iâ—aâ—t habituâ—e of two Quantities c. is in significant and you alledge no other every one that will learn Geometry must gather the Definition from observing how the word to be defined is most constantly used in common speech But in common speech if a man â—hall ask how much for example is six in respect of four and one man answer that it is greater by two and another that it is greater by half of four or by a third of six he that asked the question will be satisfied by one of them though perhaps by one of them now and by the other another time as being the onely man that knoweth why he himself did ask the Question But if a man should answer as you would do that the Proportion of six to two is of thâ—se numbers a certain Quotient he would receive but little satisfaction Between the said answers to this Question How much is six in respect of four there is this difference He that answereth that it is more by two compareth not two with four nor with six for two is the name of a Quantity absolute But he that answereth it is more by half of four or by a third of six compareth the difference with one of the differing Quantities For halfs and thirds c are names of Quantity compared From hence there ariseth two Species or kinds of Ratio Proportion into which the generall word Proportion may be divided The one whereof namely that wherein the Difference is not compared with either of the differing Quantities is called Ratio Aritbmetica Arithmeticall Proportion the other Ratio Geometrica Geometricall Proportion and because this latter is onely taken notice of by the name of Proportion simply Proportion Having considered this I defined Proportion Chap. 11. Arti. 3. in this manner Ratio est Relatio Antecedentis ad Consequens secundum magnitudinem Proportion is the Relation of the Antecedent to the Consequent in Magnitude having immediately before defined Relatives Antecedent and Consequenâ— in the same Article and by way of explication added that such Relation was nothing else but that one of the Quantities was equall to the other or exceeded it by some Quantity or was by some Quantity exceeded by it And for exemplification of the same I added further that the Proportion of three to two was that three exceeded two by a unity but said not that the unity or
Argument but fall into a loud Oncethmus the special Figure wherewith you grace you Oratory offended with my unexpected crossing of the Doctrine you teach that Proportion consisteth it a Quotient For that being denyed you your comes to nothing that is to just as much as it is worth But are not you very simple men to say that all Mathematicians speak so when it is not speaking When did you see any man but your selves publish his Demonstrations by signs not generally received except it were not with intention to demonstrate but to tâ—â—ch the use of Signes Had Pappus no Analytiques Or wanted he the wit to â—…ten his reckoning by Signes Or has he not proceeded Analytically in an hundred Problems especially in his seventh Book and never used Symboles Symboles are poor unhandsome though necessary scâ—ffolds of Demonstration and ought no more to appear in publique then the most dâ—â—â—rmed necessary business which you do in your Chambers But why say you is this â—…tion to the Proportion of the greater to the less I le tell you because iâ—erating of the Proportion of the less to the greater is a making of the Proportion less and the defect greater And it is absurd to say that the taking of the same Quantity twice should make it less And thence it is that in Quantities which begin with the less as one two four the Proportion of one to two is greater then that of one to four as is Demonstrated by Euclide Elem. 5. Prop. 8. and by consequent the Proportion of one to four is a Proportion of greater littleness then that of one to two And who is there that when he knoweth that the respective greatness of four to one is double to that of the respective greatness of four to two or of two to one will not presently acknowledge that the respective greatness of one to two or two to four is double to the respective greatness of one to four But this was too deep for such men as take their opinions not from weighing but from reading Lastly you object against the Corollarie of Art 28. which you make absurd enough by rehearsing it thus si quantitas aliqua divisa supponatur in partes aliquot aequales numero infinitas c. Do you think that of partes aliquot or of partes aliquotae it can be said without absurdity that they are numero infinitae And then you say I seem to mean that if of the Quantity AB there be supposed a part CB infinitely little and that between AC and AB be taken two means one Arithmeticall AE the other Geometricall AD the difference between AD and AE will be infinitely little My meaningis and is sufficiently expressed that the said means taken every where not in one place onely will be the same throughout And you that say there needed not so much pains to prove it and think you do it shorter prove it not at all For why may not I pretend against your demonstration that BE the Arithmeticall difference is greater then BD the Geometricall difference You bring nothing to prove it and if you suppose it you suppose the thing you are to prove Hitherto you have proceeded in such manner with your Elenchus as that so many objections as you have made so many false Propositions you have advanced Which is a peculiar excellence of yours that for so great a stipend as you receive you will give place to no man living for the number and grossness of errors you teach your Scholars At the fourteenth Chapter your first exception is to the second Article where I define a plain in this manner A plain Superficies is that which is described by a straight Line so moved as that every Point thereof describe a severall straight Lâ—… In which you require first that instead of describe I should have said can describe Why do you not require of Euclide in the Definition of a Cone instead of Continetur is contained he say contineri potest can be contained It I tell you how one Plain is generated cannot you apply the same generation to any other Plain But you object that the Plain of a Circle may be generated by the motion of the Radius whose every point describeth not a straight but a crooked Line wherein you are deceived for you cannot draw a Circle though you can draw the perimeter of a Circle but in a Plain already generated For the motion of a straight Line whose one Point resting describeth with the other Points severall perimeters of Circles may as well describe a Conique Superficies as a Plain The Question therefore is how you will in your Definition take in the Plain which must be generated before you begin to describe your Circle and before you know what Point to make your Center This objection therefore is to no purpose and besides that it reflecteth upon the perfect definitions of Euclide before the eleventh Element it cannot make good his Definition which is nothing worth of a Plain Superficies before his first Element In the next place you reprehend briefly this Corollarie that two Plaines cannot inclose a Solid I should indeed have added with the base on whose extreams they insist But this is not a fault to be ashamed of For any man by his own understanding might have mended my expression without departing from my meaning But from your Doctrine that a Superficies has no thickness 't is impossible to include a Solid with any Number of Plains whatsoever unless you say that Solid is included which nothing at all includes At the third Article where I say of crooked lines some are every where crooked and some have parts not crooked You ask me what crooked Line has parts not crooked and I answer it is that Line which with a straight Line makes a rectilineall Triangle But this you say cannot stand with what I said before namely that a straight and crooked line cannot be coincident which is true nor is there any contradiction for that part of a crooked line which is straight may with a straight line be coincident To the fourth Article where I define the Center of a Circle to be that Point of the Radius which in the description of the Circle is unmoved You object as a contradiction that I had before defined a Point to be the body which is moved in the description of a Line Foolishly As I have already shown at your objection to Chap. 8. Art 12. But at the sixth Article where I say that crooked and incongruous Lines touch one another but in one Point you make a cavill from this that a Circle may touch a Parabola in two Points Tell me truely did you read and understand these words that followed a crooked Line cannot be congruent with a straight line because if it could one and the same line should be both straight and crooked If you did you could not but understand the sense of my words to be this when two crooked lines which
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