Due correction for Mr Hobbes· Or Schoole discipline, for not saying his lessons right. In answer to his Six lessons, directed to the professors of mathematicks. / By the professor of geometry.

SECT. XI. Concerning his 17. Chapter. (Book 11)

THE Reader by this time may perhaps be weary, as well as J; and think it but dull work to busy him∣selfe upon such an inquiry, where the result is but this, That M. Hobbs his Geometry is nothing worth; which (if he had any himselfe) he knew before. To save him therefore, and myselfe the labour, wee'l make quicker work in what's behind.

In the 17. Chapter, some of the Propositions are true and good; (and truely I wondred at first where you had them, but since I know:) But the demonstrations are foolish and ridiculous. The Propositions therefore are your own (you know where you stole them;) and the Demonstrations are of your own making; (for there be scarce such to be found any where else.)

What you say to the first Article comes to this result; that I should say, It is well known, that, in Proportion, Double is one thing, and Duplicate another. And you aske, To whom it is known? (it seems it was not known to you:) And tell us, that they are words that signify the same thing; and, that they differ (in what subject soever) you never heard till now. It's very possible that this may be true; that you did never know the difference between those two words till I taught you. (But this was your ignorance not my fault.) But now, you know there is a difference. And

therefore (contrary to what you had affirmed in the Latin) you tell us in your English, Chap. 13. art. 16. p. 121. l. 7. &c. and p. 122. l. 26. &c. that the proportion of 4 to 1, to that 4 to 2 &c. is not only Duplicate, but also double or twise as great. But on the contrary, the proportion of 1 to 4, to that of 1 to 2, &c. though it be duplicate, it is not the double, or twise as great, but contrarily the halfe of it; and that of 1 to 2, to that of 1 to 4, &c. is Double you say, and yet not duplicate but subduplicate. Now if you never heard of such a difference till you heard it from me, then you are indebted to me for that peece of knowledge: and have no reason to quarrell with me, as you use to doe, for saying you did not understand what was duplicate and subdupli∣cate proportion; for you confesse you did not, but tooke it to be the same with double and subduple, and never heard that they did differ till now.

In the second Article, because it is fundamentall to those that follow, I took the paines first to shew how un∣handsomely the proposition and 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 were contrived; and then to shatter your demonstration all to pieces; and shewed it to be as simple a thing as ever was put together, (unlesse by you, or some such like your selfe.)

As to the first, you tell us, that, to proceed which way you pleased was in your own choice. And I take that for a suffici∣ent answer. You did it, as well as you could; and they that can do better may.

As to the Demonstration, you keep a vapouring (no∣thing to the purpose,) as if it were a good demonstration. and not confuted. Yet, when you have done, (because you knew it to be naught) you leave it quite out in the Eng∣lish, and give us another (as bad) in stead of it. That is, you confesse the charge. Your fundamentall Proposition was not demonstrated; and so this whole chapter comes to nothing.

But however, 'tis to be hoped, that your new demon∣stration is a good one; is it not? No, 'Tis as bad as the other. Only 'tis not so long: And of a bad thing, (you know,) the lesse the better. It begins thus, The proportion of the complement BEFCD, (fig. 1.) to the deficient figure ABEFC, is all the proportions of DB to OE, and DB to QF, and of all the lines parallell to DB, terminated in the line BEFC, to all the parallells to AB terminated in

the same points of the line BEFC. Now for this (besides that it is a piece of non-sense) you send us for proof to the second Article of the 15. Chapter, where there is nothing at all to that purpose. Then you go on. And seeing the propor∣tion of DB to OE, and of DB to QF, &c, are every where triplicate to the proportion of AB to GE, and of AB to HF &c. the proportions of HF to AB, and of GE to AB, &c. are triplicate (no, but subtriplicate) of the proportions of QF to DB, and of OE to DB &c. Now this is but the same Bull that hath been baited fo often. viz. because the diame∣ters (DB, OE, QF, &c. that is CA, CG, CH,) are in the triplicate proportion of the Ordinates (AB, GE, HF,) therefore these Ordinates are in the triplicate proportion of those diameters. You might as well have sayd, seeing that 6 is the triple of two therefore 2 is the triple of 6. But let's hear the rest, for there is not much behind.) And therefore the deficient fig. ABEFC, which is the aggregate of all the lines HF, GE, AB, &c. is triple to the complement BEFCD, made of all the lines QF, OE, DB, &c. A very good consequence! Because the Ordinates are in triplicate proportion to the diameters (yet that is false too, for they are in subtriplicate) there∣fore the figure is triple to its complement? But how doe you prove this consequence? Nay, not a word of proof. We must take your word for it. Well then, of this last Enthy∣mem, (which was directly to have concluded the question,) the Antecedent is false▪ and the consequence at lest not proved (I might have said false also, for so it is.) And this is your new demonstration.

The third article, I sayd, falls with the second; for having no other foundation but that, (nor do you pretend to other) that being undemonstrated (for your former demonstration your selfe have thrown away, and your new one we have now shewen to be nothing worth,) this must be undemonstrated too.

In the fourth Article, you attempt the drawing of these Curve lines, by point; and to that purpose require the finding of as many mean proportionalls as one will, (like as you had before done Cap. 16. 6. 16. for the finding out an arbitrary line to be taken at pleasure: Which I told you was simply done, because that without such mean pro∣portionalls, (that is, without the effection of solid & Lineary problems,) it might have been done by the Geometry of

Plains, that is with Rule and Compasse. And I shewed you how. To which you have nothing to reply, but, that I made use of one of your figures (to save my selfe the labour of cutting a new one,) that is, I made better use of your figure then you could doe.

The fifth proposition (beside that it is built upon the se∣cond, and therefore falls with it,) is inferred only from the Corrollary of the 28. article of the 13. Chapter, (nor doth your English produce any other proof,) where, sayd I, there is not a word to that purpose. And you confesse it.

The 6, 7, 8, & 9. Art. do not pretend to other foundation than the second; & therefore till that be proved, fall with it.

The 10. Article is a sad one, as may be seen by what I did object against it, as you say, for almost three leaves toge∣ther. One fault amongst the rest you take notice of, and you would have your Reader think that's all; though there be above twenty more. 'Tis this, Because (in fig. 6.) B C is to BF for so your words are, though your Lesson mis-recite them, in triplicate proportion of CD, to FE; therefore, inver∣ting, FE, to CD is in triplicate proportion of BF to CB. And doe you not take this to be a fault? No, you say, this I did object then (Yes and doe so still, as absurd enough:) But now, you say, you have taught me; (what a hard hap have I, that I cannot learn;) That of three quantities, (you should rather have taken foure; but however three shall serve for this turne,) beginning at the lest, (suppose 1, 2, 8,) if the third to the first (8 to 1) be in triplicate proportion of the second to the first (of 2 to 1) also, by conversion, the first to the second (1 to 2) shall be in triplicate proportion of the first to the third, of 1 to 8. This is that you would have had me learne. But, good Sir, you have forgotten that, since that time, you have unlearned it your selfe. For your 16. artic. of Chap. 13. as it now stands corrected in the English, teacheth us another doctrine; viz. that if 1, 2, 4, 8, bee continually proportionall, 1 to 8 shall be as well triplicate (though not bigger) of 1 to 2, (not this triplicate of that,) as 8 to 1 is of 2 to 1. The case is now altered from what it was in the Latine. And therefore you are quite in a wrong box, when, in your English, you cite Chapt. 13. Art. 16, to patronize this absurdity. For in so doing you doe but cut your own throat. You must now learne to sing another song; called Palinodia. Well, this is one of

the faults of this article. They that have a minde to see the rest of them, may consult what I said before; where I have noted a parcell of two dozen.

In the 11. Article, you doe but undertake to demonstrate a proposition of Archimedes. Your demonstration (besides that it depends upon the second Article which is yet un∣demonstrated) 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 there∣fore, till that be demonstrated, this must fall with it.

In the 13. Art. you undertake to demonstrate this Proposi∣tion 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 de∣monstration, I said, was of no force; but might as well be ap∣plyed to a portion of any Conoeid, Parabolicall, Hyperboli∣call, 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 Geo∣metry. Content, 'Tis 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 pro∣ved that a Circle made by the Radius AD, is equall to the su∣perficies 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 Se∣ction

AB, make one and the same revolution; and both of them have the same extreme points A & B: The cause why the Sphe∣ricall 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 an∣gularly; namely, according to that angle which the arch or Se∣ction makes with the Subtense; which angle is equall to the an∣gle 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 Por∣tion, and of the Circle made by the Radius AD, above the su∣perficies 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 Cono∣eidicall Superficies made by the arch or Section AB, are e∣quall 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 Demon∣stration 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 equa∣lity 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. under∣tooke 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 objecti∣on; viz. That this constructiō of yours was but going about the bush; for, upon supposition that we had those lines al∣ready 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 de∣monstration,) 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 un∣worthily to intimate that you had them elsewhere: un∣lesse 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 evi∣dence 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 be∣fore

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 Caval∣lerio, 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.