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. IX. Concerning his 14. and 15 Chapters. (Book 9)

IN Your 14 Chapter, Art: 2. I found fault with your definition of a Plain, to be that which is described by a streight line so moved as that every point of it describe a streight line. I told you, it is not necessary, much lesse es∣sentiall, to be so described, (and you confesse it;) and ma∣ny plains there are which are not so described. The defi∣nition therefore is not good.

Again. You had said in the first Article: Two streight lines cannot include a superficies. (Right,) And then Art: 2. Two plain superficies cannot include a solid. No, said I, nor yet Three. 'Twas simply done then to name but two. And you confesse it to be a fault; but not a fault to be ashamed of.

Again, you had said Art: 1. That a streight line and a croo∣ked, cannot be coincident, no not in the least part. And then Art: 3. You tell us of some crooked lines which have parts that are not crooked. This I noted for a contradiction; because with those parts not crooked, a streight line may be coin∣cident. And you cannot deny it. Therefore in the Eng∣lish, instead of crooked, in the former place, you put perpe∣tually crooked; which though it be but a botch, helps the matter a little.

In the fourth Art. In the description of a circle, by car∣rying round a Radius; you define the Center to be that point which is not moved. Now a Point you had before defined cap. 8. art. 12. to be a Body moved &c. So that to say, the Point which is not moved, is as much as to say, the Body moved &c. which is not moved. Which seems to me a contradicti∣ction. To this objection, you say only that which I must say to your answer, viz: It is foolish.

Art: 6. you say, If two streight lines touch one another in any one point, they will be contiguous through their whole length. No, nor alwaies;

[illustration]

The streight lines AB, BD, touch in one point B, and in that only. And the streight lines AC, BD, are contiguous only in their common part BC, not through their whole length.

Yet are they such contiguous lines as your proposition meanes, viz. such as meeting in some one point, will not cut one other though never so much produced.

You said farther, Crooked incongruous lines cannot touch each other, save only in one point. Yes, said I, a Circle may touch a Parabola in two points. And you confesse it. But say, you meant that each contact is not in a line, but only in one point. Perhaps you meant so, (though yet I question whether you did then think of more con∣tacts then one:) but why then did you not say so? (I mean, in the Latine? for in the English, upon this notice it is a little mended) But I reply, Yes, if those incongru∣ous curve lines, have but some parts which are not crooked, (as even now you told us,) they may touch in a line. Yea & incongruous lines continually crooked, may in some pasts of them agree, though not congruous all the way, and there∣fore touch in a line. And therefore even yet, it is not ac∣curate.

But you'l say (as pag. 10.) Such faults as these, are not attended with shame, unlesse they be very frequent. What you mean by very frequent, I cannot tell; but, mee thinks, 'tis very ugly to have them come thus thick.

Art 7. you divide a superficiall Angle, into an Angle simply so called, and an Angle of contingence. Which you define in this manner; Two streight lines applied to each other, and contiguous in their whole length, being separated or pulled open in such manner, that their concurrence in one point remains; If it be by way of circular motion, whose center is the point of concurrence, and the lines retain their streightnesse; the quan∣tity of this divergence is an angle simply so called: If by conti∣nuall flexion in every imaginable point; an angle of contingence▪ I asked; to which of these two you referre the angle made by a right line cutting a circle? or whether you doe 〈◊〉〈◊〉 take that to be a superficiall angle. You say, to an angle 〈◊〉〈◊〉 so called, that is, as we heard but now, to an angle made by two lines which retain their streightnesse, (though one 〈◊〉〈◊〉 them be crooked.) And then, you tell us that Rectilin•••••• and Curvilincall hath nothing to doe with the nature of an angle simply so called: When yet your definition requires, that the lines retain their streightnesse. I will ask, you say, (yes I do ask; and do you give a wise answer if you can;) How can that angle which is generated by the divergence of two streigh

lines, [whose streightnesse remains,] be other then Rectili∣neall? You say, A house may remain a house, though the carriage of the timber cease. Much to the purpose! How do you ap∣ply the similitude? Even so, the lines retain their streightnesse, though they be crooked, is that it? Or is it thus, Even so, the Angle remains an angle made by lines retaining their streight∣nesse, when they be crooked? Perhaps you mean thus, The Angle being once made by the divergence of streight lines, re∣mains an Angle though one or both of those lines be afterwards made crooked. Very good! but doth it remain the same An∣gle? the same quantity of divergence? (for so you define an angle,) doth not (in your account,) the bowing of one of the lines (the other remaining as it was) alter the quanti∣ty of divergence, (measurable by the Arch of a circle, as you determine) from what it was before such bowing? though yet that very bowing alone, by your doctrine, be enough to make an Angle of it selfe? Well, let it be so for once, (though it should not be so, by your principles.) But how∣ever, though this should be allowed, yet at least, so long as the Angle is in making, the lines must be streight. Tell me then, J prithee, how a Sphericall Angle comes to be an An∣gle simply so called. Is a sphericall Angle made by the diver∣gence of streight lines or of cooked? Can it be made a sphericall Angle so long as the lines retain their streightnesse? It seemes so: for an Angle properly so called, that is, an Angle made by the divergence of streight lines, whose streightnesse re∣mains, is distributed into Plain and others, (as though all Right lined angles, were not Plain Angles;) and then again into Rectilineall, Curvilineall, and mixt; as though these were, species of Rightlined Angles. Do you think it possible to make an Angle Sphericall, Curvilineall, or mixed, so long as the lines retain their streightnesse? do you think these things will ever hold together? or is this to make the prin∣ciples of Geometry firm and coherent? You were better say, as the truth is, that when you formed that definition of an Angle simply so called, you had your eye only upon a Right∣lined Angle, and fitted your definition thereunto; but when afterward, under the same name, you took in curvi∣lineall and mixt angles, you should have altered the defi∣nition, but neglected it: And then apply your ordinary apology▪ That it was indeed a fault, but not such an one

as you need be ashamed of. But, to goe about to defend it, is more ridiculous then the thing it selfe.

At the ninth Article, I had shewed how simply you de∣fined the quantity of an Angle, your definition as you call it, is this: The quantity of an Angle, is an Arch of a circle de∣termined by its proportion to the whole perimeter. An Angle was before defined to be the Quantity of Divergence; That which you define now is the quantity of an angle, that is, the quantity of the quantity of divergence. Very handsomely! Then in stead of, the quantity of an Angle is measured by an Arch; you say, the quantity of an Angle is an Arch. Again, it is, you say the Arch of a circle: But what Arch? and of what circle? for you determine neither. You mean, I sup∣pose, that Circle whose center is the Angular point; but you doe not say so: and, you mean also, the Arch of that circle intercepted between the two streight lines contain∣ing the angle; But then you should have said so, as well as meant so. For, as the definition now runs, neither Arch, nor circle, is determined. Next you say, that this quanti∣ty is to be determined (for so the words must be constru∣ed to make sense of them) by the proportion of that Arch to the whole Perimeter: That is, what proportion that inter∣cepted Arch hath to the whole perimeter; such proportion hath that Angle to—what? you do not tell us, to what. As for instance, suppose the Arch be a quadrant or quarter of the whole perimeter; the Angle is then a quar∣ter of—somewhat no doubt; but you doe not tell us of what, Is it a quarter of an Angle? or a quarter of an Arch? or a quarter of a Circle? No; 'tis a quarter of four right Angles. 'Tis that, you should have said. Now are not these faults enough for one poor definition? They are but Negligences, you'l say: but they be scurvy ones; and there be enough of them, for lesse then two lines. But whether to commit so many negligences, in lesse then two lines, be so very frequent, as that they be attended with shame, I leave for others to judge. You should have said thus, as I then told you, (but I see you are not alwaies willing to learne;) The quantity of a Rectilineall angle, in proportion to four Right angles, is determined by the proportion of an Arch of a Circle (whose center is the Angular point) intercepted between the two streight lines containing that angle, to the whole circumference. But, it seems, you had ra∣ther

keep your own definition, with all its faults, then seem to be taught by mee: Though yet you have nothing to say in defence of any one of them; and therefore (as you use to doe in such cases) take no notice of them in your answer at all; as if no such exceptions had been made.

The like exceptions, I said, ly against the 18 Article. And you take the like care neither to mend them, nor to take notice of them.

At the 12 Art. I shewed, what a pittifull definition you had brought of Parallells; and that the Consectary from it was false, and the Demonstration thereof a sad one. You confesse all: But are not pleased that I should triumph.

Your emendation which you intimate, by inserting the same way; will do some good in the consectary, but will not make good the definition. Your new definition in the English, is little better then that of the Latine. The con∣sectary, as it is now mended in the English, is true; but the demōstration of it hath many of the same faults, though not all, that I noted in the Latine: and doth not at all conclude the truth of the consectary, from that definition. As ap∣pears by what I objected formerly. What you attempt to prove of two lines, you should have proved universally of any two; for so much your definition requires.

At the 13 Art. you bring a sorry argument to prove The Perimeters of Circles to be proportionable to their Semidiame∣ters. The strength of the argument lies in this, The bignesse of the Perimeter is determined by its distance from the center; and the length of the Semidiameter is determined likewise by the same distance; therefore, since the same cause determines both effects, the Perimeters are proportionall to their Semidiame∣ters. This consequence I deny; because, not only the big∣nesse of the Perimeter, but of the circle also is determined by the same cause; as also the superficies and the solid con∣tent of a spheare. For that distance of the circumference and Center, determines the greatnesse of all these. And therefore, by your argument, circles, and spheares, &c. must be proportionall to their semidiameters: which is absurd. To which retort, because you can answer nothing; you d••e, according to your usuall Rhetorick, fall to ranting.

At the 14 Article, I said, that your argument was but

petitio principii. You say, There was a fault in the figure, (that it was not exactly drawn) which is now amended. True; but there is a worse fault in the demonstration, which is not amended yet. For though you have altered your Figure, and your demonstration too; yet the fault remaines. And 'twas this, not the figure, which I found fault with. For you do not prove that BH, BI, BC, (fig. 6.) are proportio∣nall to AF, AD, AB, but upon supposition that FG, DE, BC, were so: which was the thing at first to be pro∣ved. You say, that AF, FD, DE, are equall by construction. (True.) And, that FG, DK, BH, KE, HI, IC, are equall by Parallelism. But this is not true. The Parallelism proves that FG, DK, BH, are equall; and that KE, HI, are also equall; but not that either of these two, are equall to either of those three, (or to IC:) unlesse you first suppose that DE, is the double of DK, or FG, as AD is the double of AF, which is the very thing to be proved.

You tell me; There was another fault (yes, three or four for failing) which I might have excepted against. But the weight of the demonstration did not ly there; and I did not intend to trouble the Reader with every petty fault; (for then I should never have done:) especially in this and the next Article; where I did not then repeat your Figure at all; and therefore did briefly intimate where the fault lay: which had been direction enough for an intelligent man to have ••ound it out: But because J did not point with a festcue to every letter, you had not the wit to under∣stand it.

In like manner Art. 15. when I told you the third Corol∣lary was false, and shewed you briefly the ground of your mistake; because J did not, with a festcue point from letter to letter, you were not able to spell out the meaning; but, as being lesse awake, thought it had been a dream. You had told us, that (in your 7 figure) the angles KBC, GCD, HDE, &c. were as 1, 2, 3, &c. And 'tis true. Thence you undertake in your third Corollary to give account of the bending of a streight line into the circumference of a circle; namely, by its fraction continually increasing according to the sayd numbers 1, 2, 3, &c. But how so? For, say you, the streight line KB being broken at B according to any angle, as that of KBC, and again at C according to the double of that Angle, and at D according to the treble &c. 'twill containe

a rectilineall figure; But if the parts so broken be considered as the least that can be, that is, as so many points, 'twill be a cir∣cumference. This, I said was false, and that the ground of your mistake was, that for the Angle BDE and its Remain∣der HDE, you took CDE and its remainder.

And J need not say more; verbum sapienti a word for a wise man, had been enough; but, for you it seemes, it was not. You, like a man halfe a sleep, took it to be a dreame. Therefore, if you please to rub up your eyes a little, and take a festcue I will, for your better noddification, point to the letters as we goe along, and teach you to spell it out. The tangent line BK, continued indefinitely both ways, being broken at B, according to the Angle KBC, will lye in BCG: Now this line BCG being broken at C, ac∣cording to the Angle GCD which is the double of KBC, its part CG, will lye in CD continued, CDδ And hi∣therto you be right. But this continuation of CD, is not DH, as you seem to suppose, but Dδ which will fall be∣tween DH, DE. When therefore this line CDδ comes to be broken againe at D, that its continuation may lye in DE, the faction will not be according to the Angle HDE (which indeed is the triple of KBC) but accor∣ding to the angle δDE: which will be lesse then HDE, because it is evident that CD cuts BH, And indeed the very same Angle of fraction with that at C; For seeing the angle CDE, is equal to BCD, by construction, the subtenses being taken equall; the adjacent angles (anguli 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉) must be equall also, that is δDE = GCD And therefore the angle of fraction at D, precisely equall with that at C; not as 3 to 2, as you suppose. And by the same reason the angle of fraction at E must be equall to that at D; not as 4 to 3, as you suppose. And so the Angles of fraction at C, D, E &c are not as 2, 3, 4, &c. but are all equall. You see therefore, if you be yet awake, that it was not a dreame of mine, but a reall mistake of yours, to take HDE for the angle of fraction of CD. And conse∣quently that your proposition was false. And this fas••ho••d was the occasion of another falsehood in the 20. Article of the 16. Chapter. (which since you have blotted out.) for there you cite this proposition as the foundation of that: And whereas you say, You cannot guesse what that propo∣sition was, (and yet are very sure that it was true,) for that

you have no coppy of that article either printed or written. If you have not, J am sure you may have, for there be enough that have. For your book sold in sheets unbound, had commonly that article amongst the rest, and by that meanes it came to me. And, rather then you should be farre to seek for it, I have recited that whole article verbatim, yea to a letter, in its due place in my Elenchus; and proved it to be false.

Against your opinion concerning the Angle of Contact, (in the 16. Article,) J said little; because J think it needs no refutation. Your opinion is this, That the angle PAD, (Fig. 2. Sect. 3) is bigger then the angle PAE, as being divi∣ded by the line AE. But the angle EAC, is not bigger then the angle DAC, nor is divided by the line DA, but both of them equall as well to each other, as to the angle PAC, and also to the angle GAC. That this is your opinion, is evident. They that like it may imbrace it, for all me: And I hope, they that like it not may leave it.

The rest of what concernes this businesse, is considered before in its proper place.

At the 18. Art beside what is common to this and the seventh, J noted for a fault, and you doe not deny it so to be, that you deliver it as Euclide's opinion, that a Solid angle is but an Aggregate of plain Angles. Jt may be your opinion; but surely 'twas none of Euclide's. If you had thought it had; you should have here if you could, pro∣duced somewhat out of Euclide where he declares such an opinion.

At the 19. Article All the ways by which two lines respect one an other, or all the variety of their position, seem, you sayd, to be comprised under four kinds; For they are etiher Parallells; or (if produced at least) make an angle; or (if bigge enough) be Contingents; or lastly are asymptotes. By Asymptots you mean (not all such as never meet, for then Prallells would fall under this kind; but) such as will come always nearer and nearer together, but never touch one another (you might have added this other character; that they doe so approach each other, as that at length their distance will be lesse than any assignable quantity. But it seems you allow your Asymptotes a greater latitude: And doe in your English, determine your meaning so to be: And that, I suppose, because you had neglected to put in, that limitation, in the latine;

and therefore were not willing upon my intimation to mend it in the English. For none else that I knew, speak of any other lines under the name of Asymptotes, but such as doe not only eternally approach, but do approach also infi∣nitely neare, And, I have reason to believe, from your sim∣ple objection Less. 5. p. 48. l. 23. that you thought those two must needs go together, viz. that whatsoever quanti∣ties doe eternally approach, must needs at last come infinitely neare. But however wee'l be content, if you would have it so, to take Asymptotes at what latitude you will give it them.) You say now, that I am offended at the word it seems. No, Sir, no offence at all. I am not at all angry, that, to you, it should seem so. I said but, that to mee, it seemed otherwise; (And, I hope you are not offended that all things did not seem to me, as they did to you: For I perceive, that by this time, it seems otherwise to you also. Which hath made you in the English, to give us this Article new moulded.) I shewed you then, ma∣ny other positions of lines, which doe not agree to any of your four kinds. And you confesse it. And some of them such, as will not be salved with your new botch. As they that please to compare them will soon find.

J touched at some other faults; as, That the definition of points alike situate, (art. 31.) seemed very uncouth.

That the word Figure, which is defined art. 22. had been oft used long before it was defined; (which though it be, with you a small fault, yet a fault it is.) And you con∣fesse it.

That by your definition a solid spheare, and a spheare made hollow within, is the same figure. (For your defini∣tion takes notice of no superficies, but that within which they are included: your words are, intra quam solidum includitur. You say, It is my shall••wnesse, to think, those points which are in the concave superficies of a hallowed sphear not to be contiguous to any thing without it, because that whole concave superficies is within the whole spheare. It may be my shallownesse perhaps; but it is I confesse, my opinion, that this concave superficies being, as you say all within a spheare, (and therefore may be contiguous to somewhat within the spheare,) is not con∣tiguous to any thing without it, (if it be, tell me to what? and how it can be contiguous when the whole thicknesse

of the spheare is between? unlesse you think it can touch at a distance:) Nor, is that superficies intra quam sphaera in∣cluditur: For if, as you say, that whole superficies be within the whole sphear, how can the spheare be within that superficies? You should rather have confest, as the truth is, that you did not think of a solides being contained by two or more superficies, not contiguous to one an another: and 〈◊〉〈◊〉, had not provided for that case.

I excepted likewise against your definition of Like t••••ngs, cited here out of Cap. 9. art. 2. Those things, you de••ine to be Like, which differ only in magitude. They do not, I say, al∣waies differ in this; for it is possible like things may be equall (And therefore if they differ in nothing else, they differ not ut all.) And sometimes again they may differ in somewhat else; at least in position. Else what needs your next definion, of similia similiter posita? if it were not possible for similia to be dissimiliter posita? To which exception (because you had nothing to say) you say nothing

So your definition of like figures alike placed, I said was false: you confesse it is so, (and therefore amend it in the English.) You confesse you say, there wants something which should have been added; but call we Foole for taking notice of it: Or else, you call your selfe Foole, for not supplying it; For you say, that it might easily be supplyed by any student in Geometry, that is not otherwise a Foole. But, rather then fall out for it, wee'l divide the Foole between us; and cry Ambo. 'Twas I, like a foole, took notice of that to be wanting, which you like a Foole, omitted, when you should have supplyed it.

The 15. Chapter, because it contained but little Mathe∣maticall, I did but touch at; leaving that for my worthy Collegue to take to taske, with the rest of your Philosophy. Which he hath done to purpose. Yet some few things J noted as a rast of the rest.

J noted that (contrary to others who define Time to be the measure of motion) you determine Motion to be the measure of time; And yet (contrary to your own deter∣mination) you do frequently make Time the measure of motion; measuring both motion, and its affections (swift∣nesse, slownesse, uniformity, &c.) by Time. You confesse it to be so: But raile at us for minding Books, more than Clocks and hour-glasses. And then (contrary to both) you

tell us, that time and motion have but one dimension which is a line. And at last would perswade your English Reader, that I would have you measure swiftnesse and slownesse, by longer and shorter motion: But they that understand Latine, can find nothing to that purpose: I only told you what you did, (and how absurd that was,) not, what I would have you do▪

Then, because it still runnes into you mind, that I had some where said, That a point is nothing (though no body can tell where;) you fall againe upon that. For my part, though I oft affirm that a Mathematicall point, hath no parts▪ yet J never denyed it to stand for as much at least, as a cy∣phar doth in numbers; and you allow it noe more, (c. 16; art. 20.) your words are these Punctum inter quantitates nihil est, ut inter numeros cyphra. Is it then J, or you? that say a point is nothing?

You told us soon after, that All endeavour (for even that is motion) whether strong or weak, is propogated to infinite distance. As if (said J) the sk••pping of a Flea did propagate a motion as farre as the Indies. You ask, how we know it? If you meane, How we know that it is so; Truely, J doe not know that at all. If you meane, how we know that it follows from what you affirme; It is so evident a conse∣quence from the words alleadged, that you need not aske; Or, if those words be not enough those that follow be yet fuller, Procedit ergo omnis conatus, sive in Va••uo, sive in plano, non modo ad distantiam quantamvis, sed etiam in tempore quantulocunque, id••est, in instanti. That ••s, All endeavour of mo∣tion whether the space be Full or Emty, is continued, not only to as great a distance as is imaginable, but in as little a time, that is, in an instant. But if your meaning be, what do I say to the contrary? Truely I say nothing to the contrary. They that have a minde to believe it, may.

Then you goe on to catechise us; What is your name? Are you Philosophers? or Geometricians? or Logicians? &c. (Nay, never aske that question, we know you are good at giving names, without asking) I hope, the next question will be, Who gave you that name? And truely as to many of the names you give us, a man might easily believe, yourself were the Godfather, you call us so often by your own names.

Lastly, Of two things moving with equall swiftnesse, that, say you, strikes hardest which is bignesse. No, say I, but that

which is heaviest. A bullet of Lead, though but with equall speed, strikes harder then a blown Bladder. If any man think otherwise let him try.