of it It may be so and it 's all one to me whether you believe it to be true or no. You may think if you please that the Corollary is true still it will not hurt me Yet if you considered what had been said before you should have seen the reason viz. because the aggregate of the Impetus did not constitute a semiparabole but the complement of a semiparabola which is not â…” but â…“ of the Parallelogram The fift article hath the same faults with the fourth and runnes all upon the same mistakes The main foundation of all these continued errors was I told you the ignorance of what is proportion duplicate triplicate subduplicate subtriplicate c. Of three numbers in continuall proportion if the first be the lest the proportion of the first to the second is duplicate of what it hath to the third not subduplicate That was your opinion Cap. 13. § 16. of the Latine In the English you have retracted that error in part yet retaine all the ill consequences that followed from it Next you suppose the Aggregates of the Impetus increasing in the duplicate triplicate c. proportion of the times to be designed by the Parabola and Parabolasters as if their ordinates did increase in the duplicate triplicate c. proportion of their Diameters cujus contrarium verum est whereas you should have designed them by the complements of those figures But you aske me what line that complement is No Line good Sir but a Figure which with the figure of the Semiparabola c. doth compleare the Parallelogram You ought therefore as I then told you but you understood it not to have described your Parabola the other way that the convex not the concave of the parabolicall line should haue been towards the line of times AB so should the point K have fallen between N and F and the convex of the Parabola with AT the tangent and BI a parallel of the Diameter have contained the complement of that parabola whose diameter therefore must have been AC and its Ordinate CI. Next in pursuance of this error you make the whole velocity in these accelerations in duplicate triplicate c. proportion of the times to be â…” ¾ c. of the velocity of an uniforme motion with the greatest acquired Impetus because the Parabola and Parabolasters have such proportion to their Para â—lelograms whereas they are indeed but â…“ ¼ c. thereof for such is the proportion of the complements of those figures to their Parallelograms Now upon these false principles with many more consonant hereunto you ground not only the doctrine of the fourth and fifth Articles but also most of those that follow especially the thirteenth and thenceforth to the end of the Chapter which are all therefore of as little worth as these But enough of this The first five Articles therefore are found to be unsound and many ways faulty The sixth seventh and eighth Articles I did let passe for sound And you quarrell with me for so doing But I said withall you might have delivered as much to better purpose in three lines as there you did in five pages Beside such petty errors all along as it were endlesse every where to take notice of which gives you a new occasion to raile at Symbols After these three there is not one sound Article to the end of the Chapter and what those were before we have heard already The ninth article is this If a thing be moved by two Movents at once concurring in what angle soever of which the one is moved uniformely the other with motion uniformely acceleeated from rest till it acquâ—e an Impetus equall to that of the Vniforme motion the line in which the thing moved is carried will be the crooked line of a semiparabola Very good but of what semiparabola for hitherto we have nothing but a proporsion of Galilaeo's transcribed You tell us â—t shall be that Semiparabola whose Busis is the Impetus last acquired And this is the whole designation of your Parabola To this designation I objected many things First that the Basis of a Semiparabala is not an Impetus but a Line and therefore 't is absurd to talke of a Semiparabola whose Basis is an Impetus Secondly if it be said that an Impetus may be designed by a line I grant it a line may be the Symbol of an Impetus as well as a Letter but this line is what line you please for any Impetus may be designed by any line at pleasure so to say that It is a Semiparabola whose basis is that line which designes the Impetus is all one as to say it is a Semiparabola whose basis is what line you please So that we have not so much as the Basis of this Semiparabola determined Thirdly suppose that the Base had been determined as it is not yet it is a simple thing to think that determining the basis doth determine the Parabola For there may be infinite Parabola's described upon the same Base You doe not tell us what Altitude what Diameter nor what Inclination this Parabola is to have Now to this you keep a bawling but say nothing to the businesse You tell us that you had said what angle soever That is you supposed your Mevents to concurre in what angle â—soever but you sayd nothing of what was to be the angle of inclination in the Parabola You might have said indeed it was to be the same with that of the Movents But you did not and therefore I blam'd you for omitting it Then as to the Diameter you might have said but you did not that the line of the acccelerate motion would be the diameter 'T was another fault therefore not to say so for that had been requisite to the determining of the Parabola But when you had so said this had but determined the Position of the diameter not its magnitude it may be long or short at pleasure notwithstanding this Then as to the altitude of it this remaines as much undetermined as the rest You tell us neither where the Vertex is nor how farre it is supposed to be distant from the Base you might have said but you did not that the point of Rest where the two motions begunne was the vertex And t was your fault you did not say so in the latine as you have now done in the English But had you so said you had not thereby determined either the Altitude or the Diameters length You say The vertex and Base being given I had not the wit to see that the altitude of the Parabola is determined No truely nor have I yet But it seems you had so little wit as to think it was Had the vertex and Base been positione data I confesse it had been determined For then I had been told how farre off from the Base the Vertex had been But when the Base is only magnitudine data there is no
third of 1 to 8. This is that you would have had me learne But good Sir you have forgotten that since that time you have unlearned it your selfe For your 16. artic of Chap. 13. as it now stands corrected in the English teacheth us another doctrine viz. that if 1 2 4 8 bee continually proportionall 1 to 8 shall be as well triplicate though not bigger of 1 to 2 not this triplicate of that as 8 to 1 is of 2 to 1. The case is now altered from what it was in the Latine And therefore you are quite in a wrong box when in your English you cite Chapt. 13. Art 16 to patronize this absurdity For in so doing you doe but cut your own throat You must now learne to sing another song called Palinodia Well this is one of the faults of this article They that have a minde to see the rest of them may consult what I said before where I have noted a parcell of two dozen In the 11. Article you doe but undertake to demonstrate a proposition of Archimedes Your demonstration besides that it depends upon the second Article which is yet undemonstrated is otherwise also faulty as I then told you And therefore to say that I allow this to be demonstrated if your second bad been demonstrated is an untruth For I told you then that your manner of inferring this from that is very absurd The 12 Article like all the rest since the second beside their other faults depends upon the second and therefore till that be demonstrated this must fall with it In the 13. Art you undertake to demonstrate this Proposition of Archimedes that the Superficies of any portion of a Sphere is equall to that circle whose Radius is a streight line drawn from the pole of the portion to the circumference of its base Your demonstration I said was of no force but might as well be applyed to a portion of any Conoeid Parabolicall Hyperbolicall Ellipticall or any other as to the portion of a sphere By the truth of this say you let any man judg of your and my Geometry Content 'T is but transcribing your demonstration inserting the words Conoeid Vertex section by the Axis c. where you have Sphere Pole great Circle c. which termes in the Conoeid answer to those in the Sphere and the worke is done Let BAC in the seventh figure be a portion of a spheare or Conoeid Parabolicall Hyperbolicall Ellipticall c. whose Axis is AE and whose basis is BC and let AB be the streight line drawn from the Pole or vertex A to the base in B and let AD equall to AB touch the Great circle or Section made by a plain passing through the Axis of the Conoeid BAC in the Pole or vertex A. It is to be proved that a Circle made by the Radius AD is equall to the superficies of the portion BAC Let the plain AEBD be understood to make a revolution about the Axis AE And it is manifest that by the streight line AD a circle will be described and by the Arch or Section AB the superficies of a Sphere or Conoeid mentioned and lastly by the subtense AB the superficies of a right Cone Now seeing both the streight line AB and the Arch or Section AB make one and the same revolution and both of them have the same extreme points A B The cause why the Sphericall or Conoeidicall Superficies which is made by the Arch or Section is greater then the Conicall superficies which is made by the subtense is that AB the Arch or Section is greater then AB the subtense And the cause why it is greater consists in this that although they be both drawn from A to B yet the subtense is drawen streight but the arch or Section angularly namely according to that angle which the arch or Section makes with the Subtense which angle is equall to the angle DAB For the Angle of Contact whether of Circles or other crooked lines addes nothing to the angle at the segment as hath been shewn as to Circles in the 14 Chapter of the 16 article and as to all other crooked lines Lesson 3. pag. 28. lin ult Wherefore the magnitude of the angle DAB is the cause why the superficies of the portion described by the Arch or Section AB is greater than the superficies of the right Cone described by the Subtense AB Again the cause why the Circle described by the tangent AD is greater then the superficies of the right Cone described by the subtense AB notwithstanding that the Tangent and Subtense are equall and both moved round in the same time is this that AD stands at right angles to the axis but AB obliquely which obliquity consists in the same angle DAB Seeing therefore that the quantity of the angle DAB is that which makes the excesse both of the Superficies of the Portion and of the Circle made by the Radius AD above the superficies of the Right Cone described by the Subtense AB It followes that both the Superficies of the Portion and that of the Circle do equally exceed the Superficies of the Cone Wherefore the Circle made by AD or AB and the Sphericall or Conoeidicall Superficies made by the arch or Section AB are equall to one another Which was to be proved Shew me now if you can for you have pawned all your Geometry upon this one issue where the Demonstration halts more on my part then it doth on yours Or where is it that it doth not as strongly proceed in the case of any Conoeid as of a Sphere All that you can think of by way of exception and you have had time to think on 't ever since I wrote last amounts to no more but this which yet is nothing to the purpose you ask In case the crooked line AB were not the arch of a Circle whether do I think that the angles which it makes with the Subtense AB at the points A B must needs be equall I say that its possible that in some cases it may be so and J could for a need shew you where and therefore at least as to those cases you are clearely gone for you had nothing else to say for your selfe but this is nothing at all to the purpose whether they be or no For the angle at B what ever it be comes not into consideration at all nor is so much as once named in all the demonstration So that its equality or inequallity with that at A makes nothing at all to the businesse And therefore your exception is not worth a straw Think of a better against the next time or else all your Geometry is forfeited And they are like to have a great purchase that get it are they not At the 14. Article having before Art 4. undertooke to teach the way of drawing and continuing those curve lines by points and directed us for the word
require doth not please you for that end to take mean proportionalls you now tell us how that may be done viz. by these curve lines first drawn I asked whether this were not to commit a circle You tell me No. But mean while take no notice of that which was the main objection viz. That this constructiō of yours was but going about the bush for upon supposition that we had those lines already drawn the finding of mean proportionalls by them might be performed with much more ease than the way you take And I shewed you How But that which sticks most in your stomach is a clause in the close of this Chap. I told you that some considerable Propositions of this Chapter and I could have told you which were true though you had missed in your demonstration however you came by them But that I was confident they were none of your own and you know I guessed right And least you should think I dealt unworthily to intimate that you had them elsewhere unlesse I could shew you where I told you that I did no worse than those that a while before had hanged a man for stealing a horse from an unknown person There was evidence enough that the horse was stolen though they did not know from whom So though I knew not whence you had taken them yet I have ground enough to judge they were not your own And since that time and before that book was fully printed I found whence you had them namely out of Mersennus as I told you then pag. 132 133 134. And to take them out of Mersennus was all one as to rob a Carrier for there were at lest three men had right to the goods and some of them if they had been asked would scarce have given way that you should publish their inventions in your own name Des Chartes Fermat and Robervall And perhaps a fourth had as much right as any one of these and that is Cavallerio who though I then did not know it hath contrary to what you affirme that they were never demonstrated by any but you selfe and that as wisely as one could wish demonstrated those propositions in a Tractate of his De usu Indivisibilium in potestatibus Cossicis But though the thing be true enough and you cannot deny it yet you doe not like the Comparison And would have me consider who it was was hanged upon Hamans Gallows And truly J could tell you that too for a need The first letter of his name was H. But enough of this SECT XII Concerning his 18 19 20. Chapters WELL We have made pretty quick work with the 17 Chapter With the 18 we shall be yet quicker The charge against this Chapter was that it was all false And you confesse it Not one true Article in the whole But you tell us in the English 't is all well It is now so corrected in the English as that I shall not be able if I can sufficiently imagine motion that is if I can be giddy enough to reprehend Very well 'T is a good hearing when men grow better They that have a mind to believe it may I am not bound to undeceive them We have had experience all along that you have a speciall knack at mending as sowr Ale doth in summer You grant that I have truly demonstrated what was before to be all false You would have me do so again would you Very good When I have nothing else to doe I 'le consider of it They that think it worth the while may take the pains to examine it a second time For my part I think I have bestowed as much pains upon it already as it deserves and somewhat more And all the amendment that I find is this that whereas before wee had three false articles now we have but two and the number of true ones just as many as we had before viz. never a one In the 19 Chapter there were faults enough in conscience for a matter of no greater difficulty than that was I noted some of them and left the Reader to pick up the rest Two or three of the lighter touches about method you take notice of and make a businesse to justify or excuse them and the main exceptions as you use to do you passe over with a light touch and a way I told you in the beginning of it that your Chapters hang together like a rope of sand And 't is true enough for they have no connexion at all There are so few hooked atomes that a man cannot tell how to tacke them together Next that having in your 24 Chapter undertaken to shew us what is the Angle of Incidence and what the Angle of Reflexion and that the Angles of Incidence and of Reflexion are equall you do in pursuance of that assertion in this 19 chapter shew us the consequences thereof Upon this I asked why not either this after that or that before this You tell me that think I what I will you think that method still the best to set the Cart before the Horse Then you tell us that I say you define not here Nay that 's false I did not say so and 't is not the first time that I have taken you tripping in this kind but many Chapters after that I said I do confesse and you know 't is true what an Angle of Incidence and what an Angle of Reflexion is And then talk against hast and oversight But if your selfe had not been over hasty or rather willfully perverted my words you might have seen and you know it well enough that I blamed you here and two or three times before not so much for using words before you had defined them for this fault as J remember J mentioned but once and there you took it patiently but for defining words so long after you had used them For when words for two or three chapters together have been supposed and frequently so used as of known signification whether they had been before defined or not it is ridiculous for a Mathen atician to come dropping in with definitions of them at latter end as your fashion is like mustard after meat For these definitions should either have come in due time or else not at all The two first Articles are very triviall And yet as if it were impossible for you be the way never so plaine not to stumble there wants at least in the English a determination in the second Corollary and yet as if that were to make amends for t'other there 's one too much If upon any point say you between B and D fig. 2. yes or any where else upon the same streight line produced either way though not between those points there fall from the point A you should have said a streight line as AC whose reflected line is CH this also produced beyond C will fall upon F. Here I say that limitation between B and D is redundant and that from the point A is