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
n and let the inteâ—â—â—ction of the two Lines a c and d o be in r which being done the Triangles m n T r c T will be like and equall Therefore m n and r c are equall and â—…sequently the stâ—aight Line I m T U shall pass through c. Dividing therefore a c in the midst at t and S N in the midst at l and joyning t N L l the Lines L l t N and c T produced with all meet i 〈◊〉 one and the same Point of B S produced suppose at q. Therefore the Point q being given by the two known Points T and I the Lines drawn from q through equall parts of the Sine of the Arch B I for example through the Points P Q R of the Sine M I shall cut off equall Arches as B L L N N O O I. And this is enough to make good that Probleme as to your objection The straight Line therefore B U for any thing you have said is pâ—oved equall to the Arch B I and the division of any Angle given into any proportion given the Quadrature of any Sector and the Construction of any equilaterall Pâ—lygon is also given And though in this also I should have erred yet it cannot be denyed but that I have used a more natural a more Geometrical and a more peâ—spicuous method in thes search of this so difficult a Probleme then you have done in your Arithmetica Infinitorum For though it be true that the aggregate of all the mean Proportionals between the Radius together with an infinitely little part of the same and the Radius wanting an infinitely little part of the same and again between the Radius together with two infinitely little parts and the Radius wanting two infinitely little parts and so on eternally will be equall to the Quadrant a thing which every mean Geometrician knew before yet it was absurd to think those Means could be calculated in Numbers by Interpoling of a Symbole especially when you make that Symbole to stand for a Numbet neither true nor surd as if there were a number that could neither be uttered in words nor not be uttered in words For what else is surd but that which cannot be spoken To the fifth Article though your discourse be long you object but two things One is that Whereas the Spirall of A chiâ—edes is made of two Motions one straight the other circular both uniform I taking the Motion compounded of them both for one of those that are compounded conclude falsely that the generation of the Spirall is like to the generation of the Parabola What heed you use to take in your repâ—ehensions appears most manifestly in this objection For I say in that demonstration of mine that the velocity of the point A in describing the Spirall enâ—reaseth continually in Proportion to the Times For seeing it goes on uniformly in the Semidiameter it is impossible it should not pass into greater and greater Circles proportionally to the Times and consequently it must have a swifter and swifter Motion circular to be compounded with the uniform Motion in every Point of the Radius as it turneth about This objection therefore is nothing but an effect of a Will without cause to contradict The other objection is that Granting all to be true hithâ—rto yet because it depends upon the finding of a straight Line equall to a Parabolicâ—ll Line in the 18 Chapter where I was deceived I am also deceived here True But because in the 18 Chapter of this English Eâ—ition I have found a straight Line equall to a Parabolicall Line I have also found a straight Line equall to the Spirall Line of Archimeâ—â—s I must here pâ—t you in minde that by these words in your objections to the fifth Article at your Number 2 Quatenus verum â—st c. we have demonstrated Prop. 10 11 13. Arithmet Infinit you make it appear that you thought your Spirall made of Aâ—ches or Circles was the true Spirall of Archimedes which is fully â—s absurd as the Quadrature of Joseph Scaliger whose Geometry you so much dâ—spise To the sixth Article which is a Digression concerning the Analytiques of Geometâ—iciaâ—s you deny that the Efficient cause of the Construction ought to be contained in the Demonstration As if any Probleme could be known to be truly done otherwise then by knowing first how that is to say by what Efficient Cause and in what manner it is to be done Whatsoever is done without that knowledge cannot be demonstrated to be done as you see in your computation of the Parabola and Parabolocides in your Arithmetica Infinitorum And whereas I said that The ends of all straight Lines drawn from a straight Line and passing through one and the same Point if their parts be Proportionall shall be in aâ—…aight Line is true and accurate as also If they begin in the Circumference of a Cirâ—le they 〈◊〉 also be in the Cirâ—umference of another Circle And so is this If the Proportion be duplicate they shall be in a Parabola All this I say is true and accurately spoken But this was no place for the demonstration of it Others have done it And I perceive by that you put in by Parenthesis intelligiâ— credo inter â—duâ—s Peripheriâ—â— concentricâ—s that you understand not what I mean Hitherto reach your objections to my Geometry For the rest of your Book it containeth nothing but a collection of lies wherewith you do what you can to extenuate as vulgar and disgrace as false that which followeth and to which you have made no speciall objection I shall therefore only add in this place concerning your Analytica per Potestates that it is no Art For the Rule both in Mr. Ougthred and in Des Cartes is this When a Probleme or Question is propounded suppose the thing required done and then using a fit ratiocination put A or some other vowell for the magnitude sought How is a man the better for this Rule without another rule How to know when the ratiocinatión is fit There may therefore be in this kind of Analysis more or less naturall prudence according as the Analyst is more or less wisâ— or as one man in chusing of the unknown Quantity with which he will begin or in chusing the way of the consequences which he will draw from the Hypothesis may have better luck then another But this is nothing to Art A man may sometimes spend a whole day in deriving of consequences in vain and perhaps another time solve the same Probleme in a few minutes I shall also add that Symboles though they shorten the writing yet they do not make the Reader understand it soonâ—r then if it were written in words For the conception of the Lines and Figures without which a man learneth nothing must proceed from words either spoken or thought upon So that there is a double labour of the mind one to reduce your Symboles to words which are also Symboles another to attend
are incongruous or a crooked and a straight line touch one another the contact is not in a Line but only in one Point and then your instance of a Circle and a Parabola was a wilfull cavill not befitting a Doctor If you either read them not or unstood them not it is your own fault In the rest that followeth upon this Article with your Diagram there is nothing against me nor any thing of use novelty subtilty or learning At the seventh Article where I define both an Angle simply so called and an Angle of Contingence by their severall generations namely that the former is generated when two straight Lines are coincident and one of them is moved and distracted from the other by circular motion upon one common Point resting c. You ask me to which of these kinds of Angle I refâ—r the Angle made by a straight Line when it cuts a crooked Line I answer easily and truly to that kind of Angle which is called simply an Angle This you understand not For how will you say can that Angle which is generated by the divergence of two straight Lines be other then Rectilineall O how can that Angle which is not comprehended by two straight Lines be other then Curvilineall I see what it is that troubles you namely the same which made you say before that if the Body which describes a Line be a Point then there is nothing which is not moved that can be called a Point So you say here If an Angle be generated by the motion of a straight Line then no Angle so generated can be Curvilineall Which is as well argued as if a man should say the House was built by the carriage and motion of Stone and Timber therefore when the carriage and that motion is ended it is no more a house Rectilineall and Curvilineall hath nothing to do with the nature of an Angle simply so called though it be essentiall to an Angle of Contact The measure of an Angle simply so called is a circumference of a Circle and the measure is alwayes the same kind of Quantity with the thing measured The Rectitude or Curvity of the Lines which drawn from the Center intercept the Arch is accidentary to the Angle which is the same whether it be drawn by the motion circular of a streight line or of a crooked The Diameter and the Circumference of a Circle make a right Angle and the same which is made by the Diameter and the Tangent And because the point of Contact is not as you think nothing but a line unreckoned and common both to the Tangent and the Circumference the same Angle computed in the Tangent is Rectilineall but computed in the Circumference not Rectilineall but mixt or if two Circles cut one another Curvilineall For every Chord maketh the same Angle with the Circumference which it maketh with the line that toucheth the Circumference at the end of the Chord And therefore when I divide an Angle simply so called into Rectilineall and Curvilineall I respect no more the generation of it then when I divide it into Right and Oblique I then respect the generation when I divide an Angle into an Angle simply so called and an Angle of Contact This that I have now said if the Reader remember when he reads your objections to this and to the nineth Article he will need no more to make him see that you are utterly ignorant of the nature of an Angle and that if ignorance be madness not I but you are mad and when an Angle is comprehended between a straight and a crooked Line if I may compute the same Angle as comprehended between the same straight Line and the Point of Contact that it is consonant to my definition of a Point by a Magnitude not considered But when you in your treatise de Angulo Contactûs Chap. 3. Pag. 6. Lin. 8. have these words Though the whole concurrent Lines incline to one another yet they form no Angle any where but in the very point of concourse You that deny a Point to be any thing tell me how two nothings can form an Angle or if the Angle be not formed neither before the concurrent Lines meet not in the Point of concourse how can you apprehend that any Angle can possibly be framed But I wonder not at this absurdity because this whole treatise of yours is but one absurdity continued from the beginning to the end as shall then appear when I come to answer your objections to that which I have briefly and fully said of that Subject in my 14. Chapter At the twelfth Article I confess your exception to my universall definition of Parallels to be just though insolently set down For it is no fault of ignorance though it also infect the demonstration next it but of too much security The Definition is this Parallels are those Lines or Superficies upon which two straight Lines falling and wheresoever they fall making equall Angles with them both are equall which is not as it stands universally true But inserting these words the same way and making it stand thus Parallel Lines or Superficies are those upon which two straight Lines falling the same way and wheresoever they fall making equall Angles are equall it is both true and universall and the following Consectary with very little change as you may see in the translation perspicuously demonstrated The same fault occurreth once or twice more and you triumph unreasonably as if you had given therein a very great proof of your Geometry The same was observed also upon this place by one of the prime Geometricians of Paris and noted in a Letter to his friend in these words Chap. 14. Art 12. the Definition of Parallels wanteth somewhat to be supplyed And of the Consectary he says it concludeth not because it is grounded on the Definition of Parallels Truely and severely enough though without any such words as savour of Arrogance or of Malice or of the Clown At the thirteenth Article you recite the Demonstration by which I prove the Perimeters of two Circles to be Proportionall to their Semidiameters and with Esto fortasse recte omnino noddying to the severall parts thereof you come at Length to my last inference Therefore by Chap. 13. Art 6. the Perimeters and Semidiameters of Circles are Proportionall which you deny and therefore deny because you say it followeth by the same Ratiocination that Circles also and Spheres are Proportionall to their Semidiameters For the same distance you say of the Perimeter from the Center which determines the magnitude of the Semidiameter determines also the magnitude both of the Circle and of the Sphere You acknowledge that Perimeters and Semidiameters have the cause of their determination such as in equal times make equall spaces Suppose now a Sphere generated by the Semidiameters whilst the Semicircle is turned about There is but one Radius of an infinite number of Radii which describes a great Circle all the
of any one Proposition in Geometry From this one and first Definition of Euclise a Point is thâ—â— whereof there is no pâ—â—t understood by Sextus Empiricus as you understand it that is to say mis-understood Sextus Empiricus hath utterly destroyed most of the rest and Demonstrated that in Geometry there is no Science and by that means you have betrayed the most evident of the Sciences to the Sceptiques But as I understand it for that whereof no part is reckoned his Arguments have no force at all and Geometry is redeemed If a Line have no Latitude how shall a Cylinder rowling on a Plain which it toucheth not but in a Line describe a Superficies How can you affirm that any of those things can be without Quantity whereof the one may be greater or less then the other But in the common Contact of divers Circles the externall Circle maketh with the common Tangent a less Angle of Contact then the internall Why then is it not Quantity An Angle is made by the concourse of two Lines from severall Regions concurring by their generation in one and the same Point How then can you say the Angle of Contact is no Angle One measure cannot be applicable at once to the Angle of Contact and Angle of Conversion How then can you infer if they be both Angles that they must be Homogeneous Proportion is the Relation of two Quantities How then can a Quotient or Fraction which is Quantity absolute be a Proportion But to come at last to your Ephiphonema wherein though I have perfectly demonstrated all those Propositions concerning the Proportion of Parabolasters to their Parallelogâ—ams and you have demonstrated none of them as you cannot now but plainly see but committed most 〈◊〉 Paralogisms How could you be so transported with pride as insolently to compare the setting of them forth as mine to the Act of him that steals a horse and comes to the gallows for it You have read I think of the gallows set up by Haman Remember therefore also who was hanged upon it After your dejection I shall comfort you a little a very little with this that whereas this 18 Chapter containeth two Problems one the finding of a straight Line equall to the â—rooked Line of a Semiparabola The other the finding of straight Lines equall to the crocked Lines of the Parabolasters in the table of the third Article of the 17 Chapter You have â—…uly demonstrated that they are both false and another hath also Demonstrated the same another way Nevertheless the fault was not in my method but in a mistake of one Line for another and such as was not hard to correct and is now so corrected in the English as you shall not be able if you can sufficiently imagine Motions to reprehend The fault was this That in the Triangles which have the same Base and Altitude with the Parabola and Parabolaster I take for designation of the mean uniform Impetus a mean Proportionall in the first Figure between the whole Diameter and its half and in the second Figure a mean Proportionall between the whole Diameter and its third part which was manifestly false and contrary to what I had shewn in the 16 Chapter Whereas I ought to have taken the half of the Base as now I have done and thereby exhibited the Straight Lines equall to those crocked Lines as I undertook to do Which error therefore proceeded not from want of skill but from want of care and what I promised as bold as you say the promise was I have now performed The rest of your exceptions to this Chapter are to these words in the end there be some that say that though there be equality between a straight and crooked Line yet now they say after the full of Adam it cannot be found without the especiall help of divine grace And you say you think there be none that say so I am not bound to tell you who they are Nevertheless that other men may see the Spirit of an ambitious part of the Clergy I will tell you where I read it It is in the Prolegomena of Lalovera a Jesuite to his Quadrature of the Circle pag. 13 14 in these worâ—s Quamvis circuli tetragonismus fit 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 possibilis an taman etiam sit 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 hoc est post Adae lapsum homo ejus scientiam absque speciali diâ—inae gratiae auxilio possit comparare jure merito inquirunt Theologi pronunciantque hanc veritatem tanta esse caligine involutam ut illam videre nemo possit nisi Ignorantiae ex primi parentis praevaricatione propagatas tenebras indebitus divina lucis radius dissipet quod verissimum esse sentio Wherein I observed that he supposing he had found that Quadrature would have us believe it was not by the ordinary and Naturall help of God whereby one man reasoneth judgeth and remembreth better then another but by a Special which must be a Superturall help of God that he hath given to him of the order of Jesus above others that have attempted the same in vain Insinuating thereby as handsomely as he could a Speciall love of God towards the Jesuites But you taking no notice of the word Speciall would have men think I held that humane Sciences might be acquired without any help of God And thereupon proceed in a great deal of ill language to the end of your objections to this Chapter But I shall take notice of your Manners for altogether in my next Lesson At the nineteenth Chapter you see not you say the Method Like enough In this Chapter I consider not the Cause of Reflection which consisteth in the resistance of Bodies naturall but I consider the consequences arising from the supposition of the equality of the Angle of Reflection to that of Incidence leaving the causes both of Reflection and of Refraction to be handled together in the 24 Chapter Which Method think what you will I still think best Secondly you say I define not here but many Chapters after what an Angle of Incidence and what an Angle of Reflection is Had you not been more hasty then diligent Readers you had found that those Definitions of the Angle of Incidence and of Reflection were here set down in the first Article and not deferred to the 24. Let not therefore your own oversight be any more brought in for an objection Thirdly you say there is no great difficulty in the business of this Chapter It may be so now 't is down but before it was done I doubt not but you that are a Professor would have done the same as well as you have done that of the Angle of Contact or the business of your Arithmetica Infinitorum But what a novice in Geometry would have done I cannot tell To the third fourth and fifth Article you object a want of Determination and shew it by instance as to the third Article But what those Determinations should be you
suppose that the crocked Line AB in the seventh Figure were not an Arch of a Circle do you think that the Angles which it maketh with the Subtense AB at the Points A and B must needs be equall Or if they be not does the excess of the Superficies of the Circle upon AD above the Superficies of the Cone or the exceis of the Superficiâ—s of the Portion of the Conoeides above the Superficies of the same Cone consist in the Angle DAB oâ— rather in the â—…tude of the two unequall Angles DAB and ABA You should have drawn some other crooked Line and made Tangents to it through A and B and you would presently have seen your error See how you can answer this for if this Demonstration of minestand firm I may be bold to say though the same be well Demonstrated by Arâ—… that this way of mine is more naturall as proceeding immediately from the naturall efficient causes of the effect contained in the conclusion and besides more brief and more easie to be followed by the fancy of the Reader To the fourteenth Article you say that I commit a Circle in that I require in the fourth Article the finding of two mean Proportionals and come not till now to show how it is to be done Nor now neither But in the mean time you commit two mistakes in saying so The place cited by you in the fourth Article is in the Latine Pag. 149. Lin 9. in the English Pag. 188. Lin. 3. Let any Reader judge whether that be a requiring it or a supposing it to be done this is your first mistake The second is that in this place the Preposition itself which is If those Deficient Figures could be described in a Parallelâ—gram exquisitely there might be found thereby between any two Lines given as many mean Proportionals as one would is a Theoreme upon supposition of these crooked Lines exquisitely drawn but you take it for a Probleme And proceeding in that error you undertake the invention of two mean Proportionals using therein my first Figure which is of the same construction with the eighth that belongeth to this fourteenth Article Your construction is Let there be taken in the Diameter CA Figure 1 the two given Lines or two others Proportionall to them as CH CG and their â—â—â—nate Lines HF GE which by construction are in subtriplicate Prâ—pâ—rtion of the intercepted Diameters These Lines will shew the Proportions which those four Proportionals are to 〈◊〉 But how will you find the Length of HF or GE the ordinate Lines Will you not do it by so drawing the crooked Line CFE as it may paâ—s through both the Points F and E You may make it pass through one of them but to make it pass through the other you must finde two mean Proportionals between GK and GL or between HI and HP Which you cannot do unless the crooked Line be exactly drawn which it cannot be by the Geometry of Plunes Go show this Demonstration of yours to Orontius and see what he will say to it I am now come to an end of your objections to the seventeenth Chapter where you have an Epiphonema not to be passed over in silence But becausâ— you pâ—etend to the Dâ—…tration of some of these Propositions by another Method in your Arithmetica Infinitorum I shall first try whether you be able to defend those Demonstrations as well as I have done theie of mine by the Method of Motion The first Proposition of your Arithmetica Infinitorum is this Lâ—mma In a Sâ—ries or Row of Quantities Arith â—tically Proportionall beginning at a Poiâ—t or Cypâ—â—r as 0 1 2 3 4 c. to finde the Proportion of the Aggregate of them all to the Aggregate of so many times the greatest as there are Terms This is to be done by multiplying the greatest into half the Number of the Terms The Demonstration is easie But how do you demonstrate the same The most simple way say you of finding this and some other Problemes it to do the thing it self a little way and to observe and compare the appearing Proportions and then by Induction to conclude it universally Egregious Logicians and Geometricians that think an Induction without a Numeration of all the particulars sufficient to infer a Conclusion universall and fit to be received for a Geometricall Demonstration But why do you limit it to the naturall consequution of the Numbers 0 1 2 3 4 c Is it not also true in these Numbers 0 2 4 6 c. or in these 0 7 14 21 c Or in any Numbers where the Diffâ—rence of nothing and the first Number is equall to the difference between the first and second and between the second and third c Again are not these Quantities 1 3 5 7 c. in continuall Proportion Arithmaticall And if you put before them a Cypher thus 0 1 3 5 7 do you think that the sum of them is equall to the half of five times seven Therefore though your Lemma be true and by me Chap. 13. Art 5. demonstrated yet you did not know why it is true which also appears most evidently in the first Proposition of your Conique-sections Where first you have this That a Parallelogram whose Altitude is infinitely little that is to say none is scarce any thing else but a Line Is this the Language of Geometry How do you determine this word scarce The least Altitude is Somewhat or Nothing If Somewhat then the first character of your Arithmeticall Progression must not be a cypher and consequently the first eighteen Propositions of this yâ—ur Arithmetica Infinitorum are all naught If Nothing then your whole figure is without Altitude and consequently your Understanding naught Again in the same Proposition you say thus We will sometimes call those Parallelograms rather by the name of Lines then of Parallelograms at least when there is no consideration of a determinate Altitude But where there is a consideration of a determinate Altitude which will happen sometimes there that little Altitude shall be so far considered as that being infinitely multiplyed it may be equall to the Altitude of the whole Figure See here in what a confusion you are when you resist the truth When you consider no determinate Altitude that is no Quantity of Altitude then you say your Parallelogram shall be called a Line But when the Altitude is determined that is when it is Quantity then you will call it a Parallelogram Is not this the very same doctrine which you so much wonder at and reprehend in me in your objections to my eighth Chapter and your word considered used as I used it 'T is very ugly in one that so bitterly reprehendeth a doctrine in another to be driven upon the same himself by the force of truth when he thinks not on 't Again seeing you admit in any case those infinitely little altitudes to be quantity what need you this limitation of yours so far forth as that by
multiplication they may be made equall to the Altitude of the whole figure May not the half the third the fourth or the firth part c. be made equall to the whole by multiplication Why could you not have said plainly so far forth as that every one of those infinitely little Altitudes be not only something but an aliquoâ—part of the wholâ— â—o you will have an infinitely little Altitude that is to say a Point to be both nothing and something and an aliquot part And all this proceeds from not understanding the ground of your Profession Well the Lemma is true Let us see the Theoremes you draw from it The first is Pag. 3. that a Triangle to a Parallelogram of equall Base and Altitude is as one to two The conclusion is true but how know you that Because say you the Triangle consists as it were as is were is no Phrase of a Geometrician of an Infinite Number of straâ—ght Parallel Lines Does it so Then by your own doctrine which is that Lines have no breadth the Altitude of your Triangle consisteth of an infinite Number of no Altitudes that is of an infinite Number of Nothings and consequently the Area of your Triangle has no Quantity If you say that by the Parallels you mean infinitely little Parallelograms you are never the better for if infinitely little either they are nothing or if somewhaâ— yet seeing that no two sides of a Triangle are Parallel those Parallels cannot be Parallelograms I see they may be counted for Parallelograms by not considering the Quantity of their Altitudes in the Demonstration But you are barred of that Plea by your spightfull arguing against it in your Elenchus Therefore this third Proposition and with it the fourth is undemonstrated Your fifth Proposition is The Spirall Line is equall to half the Circle of the first Revolution But what Spirall Line We shall understand that by your construction which is this The straight Line MA in your Figure which I have placed at the end of the fifth Lesson turned round the Point M remaining unmoved is supposed to describe with its Point A the Circle AOA whilst some Point in the same MA whilst it goes about is supposed to be moved uniformly from M to A describing the Spirall Line This therefore is the Spirall Line of Archimedes and your Proposition affirms it to be equall to the half of the Circle AOA which you perceived not long after to be false But thinking it had been true you go about to prove it by inscribing in the Circle an infinite multitude of equall Angles and consequently an infinite Number of Sectors whose Arches will therefore be in Arithmeticall Proportion Which is true And the Aggregate of those Arches equall to half the Circumference AOA Which is true also And thence you conclude that the Spirall Line is equall to half the Circumference of the Circle AOA Which is false For the Aggregate of that infinite Number of infinitely little Arches is not the Spirall Line made by your construction seeing by your construction the Line you make is manifestly the Spiral of Archimedes whereas no Number though infinite of Arches of Circles how little soever is any kind of Spirall at all and though you call it a Spirall that is but a patch to cover your fault and deceiveth no man but your self Besides you saw not how absurd it was for you that hold a Point to be absolutely nothing to make an infinite Number of equall Angles the Radius increasing as the Number of Angles increaseth and then to say that the Arches of the Sectors whose Angles they are are as 0 1 2 3 4 c. For you make the first Angle 0 and all the rest equall to it and so make 0 0 0 0 0 c. to be the same Progression with 0 1 2 3 4 c. The influence of this absurdity reacheth to the end of the eighteenth Proposition So many are therefore false or nothing worth And you needed not to wonder that the Doctrine contained in them was omitted by Archimedes who never was so senseless as to think a Spiâ—â—ll Line was compounded of Arches of Circles Your nineteenth Proposition is this other Lemma In a Series or a Row of Quantities beginning from a Point or Cypher and proceeding according to the order of the square Numbers as 0 1 4 9 16 c. to finde what Proportion the whole Series hath to so many times the greatest And you conclude the Proportion to be that of 1 to 3. Which is false as you shall presently see First let the Series of Squares with the prefixed Cypher and under every one of them the greatest 4 be And you have for the sum of the Squares 5 and for thrice the greatest 12 the third part whereof is 4. But 5 is greater then 4 by 1 that is by one twelfth of 12 which Quantity is somewhat let it be called A. Again let the Row of Squares be lengthened one term surther and the greatest set under every one of them as The sum of the Squares is 14 and the sum of four times the greatest is 36 whereof the third part is 12. But 14 is greater then 12 by two unities that is by two twelfths of 12 that is by 2 A. The difference therefore between the sum of the Squares and the sum of so many times the greatest Square is greater when the cypher is followed by three Squares then when by but two Again let the Row have five terms as in these Numbers with the greatest five times subscribed and the sum of the Squares will be 30 the sum of all the greatest will be 80. The third part whereof is 26 â…” But 30 is greater then 26â…“ by 3â…“ that is by three twelfths of twelve and â…“ of a twelfth that is by 3â…“ A. Likewise in the Series continued to six places with the greatest six times subscribed as the sum of the Squares is 55 and the sum of the greatest six times taken is 150 the third part whereâ—f is 50. But 55 is greater then 50 by 5 that is by five twelfths of 12 that is by 5. A. And so continually as the Row groweth longer the excess also of the aggregate of the Squares above the third part of the aggregate of so many times the greatest Square groweth greater And consequently if the Number of the Squares were infinite their sum would be so far from being equal to the third part of the aggregate of the greatest as often taken as that it would be greater then it by a Quantity greater then any that can be given or named That which deceived you was partly this that you think as you do in your Elenchus that these Fractions c. are Proportions as if 1 12 were the Proportion of one to twelve and consequently 2 12 double the Proportion of one to twelve which is as unintelligible as School-Divinity and I assure you far from the meaning of Mr. Ougthred in the sixth
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
determine not because you could not The words in the third Article are first these If there fall two straight Lines Parallel c. which is too generall It should be if there fall the same way two straight Lines Parallel c. Next these their Reflected Lines produced inwards shall make an Angle c. This also is too generall I should have said their Reflected Lines produced inwards if they meet within shall make an Angle c. Which done both this Article and the 4 and 5 are fully demonstrated And without it an intelligent Reader had been satisfied supplying the want himself by the construction To the eight you object onely the two great Length and labour of it because you can do it a shorter way Perhaps so now as being easie to shorten many of the Demonstrations both of Euclide and other the best Geometricians that are or have been And this is all you had to say to my 19 Chapter Before I proceed I must put you in mind that these words of yours Adducis malleum ut occiâ—as muscam are not good Latine Malleum affers Malleum adhiâ—es Malleo uteris are good When you speak of bringing Bodies animate Ducere and Adducâ—re are good for there to bring is to guâ—â—e or leaâ— And of Bodies inanimate Adducere is good for Attraâ—ere which is to draw to But when you bring a hammer will you say Adduco Malleum I lead a hammer A man may lead another man and a ninny may be said to lead another ninny but not a hammer Neverthelese I should not have thought fit to reprehend this fault upon this occasion in an English man nor to take notice of it but that I finde you in some places nibling but causelessly at my Latine Concerning the twentieth Chapter before I answer to the objections against the Propositions themselves I must answer to the exception you first take to these words of mine Quae de dimensione Circuli Angulorum pronuntiata sunt tanquam exacté inventa accipiat Lector â—â—nquam dicta Problematice To which you say thus we are wont in Geometry to call some Propositions Theoremâ—s other's Problems c. of which 〈◊〉 Theoreme is that wherein some assertion is propouâ—nded to be proved a Problem that wherein something is commanded to be done Do you mean to be done and not proved By your favour a Probleme in all ancient writers signifies no more but a Proposition uttered to the end to have it by them to whom it is uttered examined whether it be true or not true faiâ—able or not faisable and differs not amongst Geometricians from a Theoreme but in the manner of Propounding For this Proposition To make an equilaterall Triangle so propounded they call a Problem But if propounded thus If upon the ends of a straight Line given be described two Circles whose Radius is the same straight Line and there be drawn from the intersection of the Circles to their two Centers two straight Lines there will be made an equilateral Triangle then they call it a Theoreme and yet the Proposition is the same Therefore these words Accipiat Lector tanquam dicta Problematiâ—ey signifie plainly this that I would have the Reader take for pâ—opounded to him to examine whether from my construction the Quadrature of the Circle can be truly inferred or not and this is not to bid him as you inteâ—pret it to square the Circle And if you beliâ—ve that Problematicey signifies probably you have been very negligent in observing the sense of the anâ—ienâ— Gâ—eek Philosophers in the word Probleme Therefore your Solemus in Geometria c. is nothing to the pu pose nor had it been though you had spoken more properly and said Solâ—nt leaving out your selves My first Article hath this Title from a False suppâ—sition a false Quadrature of the Circle Seeing therefore you were resolved to shew where I erred you should have proved either that the Supposition was true and the Conclusion falsely inferred or contrarily that though the Supposition be false yet the Conclusion is true for else you object nothing to my Geometry but only to my Judgement in thinking fit to publish it which nevertheless you cannot justly do seeing it was likely to give occasion to ingenuous men they practise of it being so accurate to sense to inquire wherein the Fallacy did consist And for the Probleme as it was first printed but never published and consequently ought to have passed for a private paper stoln out of my study your publique objecting against it in the opinion of all men that have conversed so much with honest company as to know what belongs to civill conversation was sufficiently barbarous in Divines And seeing you knew I had rejected that Proposition it was but a poor Ambition to take wing as you thought to do like Beetles from my egestions But let that be as it will you will think strange now I should resume and make good at least against your objection that very same Proposition So much of the Figure as is needfull you will finde noted with the same letters and placed at the end of this 5 Lesson Wherein let B I be an Arch not greater then the Radius of the Circle and divided into four equall parts in L N O. Draw S N the Sine of the Arch B N and produce it to T so as S T be double to S N that is equall to the Chord B I. Draw likewise a L the Sine of the Arch B L and produce it to c so as a c be quadruple to a L that is equall to the two Chords B N N I. Upon the Center N with the Radius N I draw the Aâ—ch I d cutting B U the Tangent in d. Then will B N produced cut the Arch I d in the midst at o. In the Line B S produced take S b equall to B S then draw and produce b N and it will fall on the Point d. And B d S T will be equall and d T joyned and produced will fall upon o the midst of the Arch I d. Joyn I T and produce it to the Tangent B U in U. I say that the stâ—aight Line I T U shall pass through c. For seeing B S S h are equall and the Angle at S a right Angle the straight Lines B N and b N are also equall and the Triangles B N b d N o like and equall and the Lines d T T o equall Draw o i Parallel to d U cutting I U in i and the Triangles d T v o T i will also be like and equal Produce S T to the Arch d o I in e and produce it further to f so that the Line e f be equall to T c and then S f will be equall to a c. Therefore f c joyned will be Parallel to B S. In c f produced take f g equall to c f and draw g m Parallel to d U cutting I U in m and d o in