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. VII. Concerning the Eighth Chapter in M. Hobs his Book of Body. (Book 7)

HItherto we have tryed your skill and valour in point of Assault: And found, that, though you charge as furiously as if you meant to look us dead; yet you come off as poorly as a man could wish. J am apt to think,

that your weapons were not well made, and that your Musket was of a bad bore, (for it hath done no executi∣on, save only in the recoile;) or else you held it by the wrong end, (like the Jack-an-Ape that peep'd in the gunns mouth to see the bullet come out,) for though it made a great noyse, yet it hath hurt no body but your selfe. My Colleague and I, are both of us alive, and live-like; and Eu∣clide sleeps as securely as he did before.

Wee'l try now, how good you are in point of Defense; and see how you can defend your Corpus against my Elen∣chus. Perhaps you may have better luck at that.

But, mee thinks, it begins unluckyly. Before you fall to work with Elenchus; you traverse your ground, that you may take it to the best advantage: and distinguish, between faults of Ignorance, and faults of Negligence, (pag. 9.) you tell us that from right Principles to draw false Conclusions (which you are very good at) are but faults of negligence and humane frailty, and such as are not attended with shame, &c. That 'tis only as being lesse awake, &c. (and yet think much to be told, that you discourse as if you were halfe a sleep:) And much more your preface to that purpose. As if the first consideration to be had, in the choice of your ground, were, whence you might with best advantage runne away; (a businesse of ill Omen in the beginning of a Combate;) that when you shall be forced to quit your ground, you may, at least, shew a fair pair of heeles.

My Elenchus, as I then told you, begins at first with some lighter skirmishes, shewing how unhandsome some of your Definitions and Distributions are, giving instance in a few; which though faults had enough, yet are but small ones in comparison of those greater which follow, in false Propositions and Demonstrations.

I begin with that of Chap. 8. § 12. Where you define a line, a length, a point, in this manner. If when a Body is mo∣ved, its magnitude (though it alwaies have some) be not all con∣sidered, the way it makes is called a Line, or one single dimensi∣on; the space through which it passeth is called Length; and the Body it selfe a Point. But what if a Body be not moved? i•• there then neither Point, nor Line, nor Length? A Point there may be, which is not a Body, much lesse A Body moved: and a Line, or Length, through which no body passeth: And therefore the definitions are not good, because not

reciprocall. The Axis of the Earth, is a Line, and that line hath its Length; yet doe I not believe that any Body doth, or ever did, passe directly from the one to the other Pole, to describe that Line. The notion therefore of Motion or Body moved, I then said, was wholly extrinsecall and acci∣dentall to the notion of Line, or Length, or of a Point; no waies essentiall or necessary to it, or to the understanding of it: and that therefore it was not convenient, to clog the definitions of these, with the notion of that.

To this you answer, (having waved first, what you at∣tempted, as from the example of Euclide,) That, how ever it may be to others, it was fit for you to define a Line by Motion. And I doe acquiesse in that Answer. For, though it would not become any man else so to define it; yet it becomes M. Hobs very well; as well agreeing with his accuratenesse in other things.

I said farther, That the distance of two points though resting, was a Length, as well as the measure of a passage, (and there∣fore the notion of a body moved, not necessary to the definition of Length.) To which you answer, that the distance of the two ends of a thread wound up into a Clew, is not the length of the thread. Much to the purpose.

I asked, Whoever defined a Line to be a Body? And you tell mee, you take it for an honour to be the first that doe so. And you may, for ought I know, have also the honour to be the last. And as to that long rant against Euclide; That if a Point have no parts, and so no magnitude; A Line can have no breadth, nor can be drawn (mechanically you mean;) and then there is not in Euclide one Proposition demonstrated, or de∣monstrable. We doe not think, that your asseveration a suf∣ficient argument, more than we take a word of your mouth to be a slander; but desire some better proofe of that con∣sequence before we assent to it. You tell us else where, that A Point is to Magnitude, as a ciphar is to Number (cap. 16 art. 20.) And yet I suppose you will not say that, unlesse a Ciphar have some multitude, as well as a Point some Magni∣tude, there is not in Euclide any one Proposition demonstrated. And to the same purpose is that Cap. 14. § 16. An angle of contingence, if compared with an angle simply so called how little so ever, hath such proportion to it, as a point to a Line, that is, (neque rationem, neque quantitatem ullam,) no proportion, nor a∣ny quantity at all. Which how well it agrees with your o∣ther

doctrines, it concerns you to see to, (for if a Point to a Line, have no proportion nor any quantity at all, then is it not a Part thereof;) and how little this comes short of what you so often rant at, as making a Point to be no∣thing.

Again, whereas in the place cited (both in Latine and English) you thus define; The Way (of the Body so mo∣ved) is called a Line, or one single Dimension; and the Space through which it passeth, is called Length. I argued, that Length, doubtlesse, was one single dimension; and therefore, if one single dimension, as in your definition, be the same with Line; then Length will be a Line, and not therefore need a se∣cond definition. Now, to help the matter, in your Lessons; you define thus, The Way is called a Line; and the space gone over by that motion, Length or one single dimension. Whence my argument is yet farther inforced, If one single dimension signify the same with Line, (as in your Book;) and also the same with Length, (as in your Lesson;) then Line and Length signify with you the same thing; & therefore with you, should not have had two distinct and different definitions. Which I take to be ad hominem, a good argument. You an∣swer, that to say Line is Length, proceeds from want of under∣standing English. It may be so. But what's this to the clear∣ing of your Definitions? where those two words are made equivalent. Yet farther, chap. 12. parag. 1. there are, say you, three dimensions, Line (or Length,) Superficies, and Solid. where again Line and Length are made the same. Now whether or no Line be Length, or whether it be for want of understanding English that you affirme it, it concernes you to cleare; for 'tis you, not I, that affirme it so to be.

Your next definition is of Equall Bodies; which you thus define, Equall Bodies, are those which may possesse the same place. Against which definition J objected, That you should rather define a thing, by what it is, then by what it may be: That the notion of Place, was wholly extrinsecall to the notion of Equality; for Time, Tone, Numbers, Pro∣portions, and many other quantities are capable of Equali∣ty, without any connotation of Place; and the notion of Equality in them, is the same notion with that of Equality in Bodies; (else how can you say, that two Equall Num∣bers, and two Equall Bodies, are in the same Proportion;)

And therefore, That one good definition of Equality, or E∣qualls, in generall; had been much better, then so many particulars, of Equall Bodies, Equall Magnitudes, Equall Motions, Equall Times, Equall Swiftnesse, &c. as you here bring; and yet, when you have all done, there be a great many more Equalls, which you leave undefined: (And your bare assertion, That there is no Subject of Quantity, or of Equality, or of any other Accident, but Body, doth not help the matter at all; for we are not bound to take your word for it:) That, if you would needs mention place, you should rather have defined them by the place they have, then what they may have; & so, defined those bodies to be Equall, which do possesse Equall places, rather then, which may possesse the same place: That a Pyramid, remaining a Py∣ramid, may be Equall to a Cube; yet cannot, remaining a Pyramid, possesse the place of that Cube: Or, if you will, That a Pyramidall Atome, though so Adamantine as to be in∣capable of any transmutation, (as those who teach the do∣ctrine of Atomes doe maintain,) may yet be equall to a Cu∣bicall Atome, though not possesse the place thereof: That you might as well have defined a Man, to be one who may be Prince of Transilvania, as to define Equall Bodies, to be those which May possesse the same place. (with much more, of which you take no notice.)

To that last particular, you answer, that 'tis wittily obje∣cted, as I count witt, but impertinently. And why imperti∣nently? Is not that definition of a Man; as good as yours of Bodies Equall? You think not, Because if so, J must be of opinion, That the possibility of being Prince of Transilvania, is no lesse essentiall to a Man; then the possibility of being in the same Place, is essentiall to Equall Bodies. And truly J am of that opinion. J think it every way as Possible for for any man living, to be Prince of Transilvania; as for the Arctick and Antartick Circles, (or the Segments of the Sphere which they cut off,) be they never so Equall, to possesse the same place. Nor is that possibility lesse essentiall, than this.

You adde, That there is no man (beside such Egregious Geo∣metricians as we are) that inquires the Equality of two bodies, but by measure: And, as for liquid bodies, &c. by putting them one after another into the same vessell, that is to say, into the same place; And, as for hard bodies, they inquire their Equality

••y weight. To which I shall reply nothing at all; because you speak therein so like a Geometrician.

I objected farther, That it is not yet agreed amongst Phi∣losophers (and your authority will not decide the contro∣versy,) whether or no, the same body may not, by Rarefa∣ction and Condensation, (words understood by other men, though you understand them not,) sometimes possesse a bigger, some times a lesser place. We see, that the same Air in the head of a weather-glasse, doth sometimes possesse a bigger, sometimes a lesser part of the glasse, according as the Wea∣ther is cold or hot, and you cannot deny, (what ever o∣thers may) but that both are filled; for you doe not allow any Vacuum at all. We know, that into a Wind-gun, though it were full (you say) before, yet much more Air may be forced in. And into the Artificiall Fountain, (which you mention Cap. 26. fig 2.) though full of Air, may be for∣ced also a great quantity of Water. Now how to salve these Phaenomena, (with many others of the like kind) with∣out either allowing Vacuum, which you deny; or Condensa∣tion, which you laugh at; (one of which others use to as∣signe) because you find it too hard a task for you to under∣take, (as well you may,) you leave to a m••lius inquirendum p. 144. l. 27. (or in the English, p. 316. l. 34.) Now if it be true, that the same body doth, or possible, that it may, possesse, some time a bigger, sometime a lesse space, (as those who deny Vacuum doe generally affirme,) then, by your definition, the same body (I doe not say may possibly become, but) at present is both bigger, and lesse, and equall to it selfe: Because it hath at present a possibility of posses∣sing hereafter both a larger place, by Rarefaction, and a les∣ser place, by condensation, than now it doth. And so you, by determining the equality or inequality of Bodies, not by the place they have, but by such place as possibly they may have (upon any supposed metamorphosis or transmu∣tation,) doe confound Bigger, and Lesse, and Equall, and so take away the whole foundation of Mathematicks: For if there be no difference between Bigger, Lesse, and Equall, there is no roome either for Mathematicks or Measure. But, whether that opinion of Rarefaction and Condensation be true or not: yet since you cannot deny, but that it is at least a considerable controversy, and, by men as wise, and as good Philosophers as M. Hobs, maintained against you;

yea and a Controversy not belonging to Mathematicks but Physicks, or Naturall Philosophy, and there to be determined; it was not wisdome to hang the whole weight of Mathema∣ticks, upon so slender a thread, as the decision of that con∣troversy in Naturall Philosophy, which whether way it be determined, is wholly impertinent to a Mathematicall De∣finition.

To which you reply onely this, (which is easy to say) that Rarefying and Condensing, are but empty words; and that (of which we have spoken already) Mathematicall Defini∣tion, is not a good phrase.

To that definition you had annexed this also; Eadem ratione, magnitudo magnitudini, &c. Ʋpon the same account one Magnitude is equall, or greater, or lesser, then another, when the bodies whose they are, are greater, equall, or lesse. These words, I said, must bear one of these two ••enses, either, that E∣quall Bodies, or Bodies equally great, are of equall greatnesse, (which is no very profound notion:) or else, that the mag∣nitudes, towlt the lines, superficies, &c. or at least, the length, bredth, &c. of Equall bodies, is Equall, (taking the words for a definition of Equall Lines, Equall Superficies, &c) and this, I said, was manifestly false: for no bodies may be e∣quall, whose length, breadth, superficies, &c. are unequall. You say now, that you meant the former, (and I cannot contradict it, for you know your own meaning best, yet you must give me leave to think:) and so leave us without any definition of Equall Lines, Plaines, or Superficies Which yet, considering how oft you are afterwards to. make use of, might have been as worthy of a definition, as some of those equalls that you have defined.

In the next Paragraph, Cap. 8. parag. 14. you undertake to prove, that one and the same Body, is alwaies of one and the same magnitude, and not bigger at one time then another, or at one time fill a bigger place, than it doth at another time. Let's heare how you prove it (for, by what we heard but now, you are much concerned to make good proofe of it, because if there be a possibility of possessing at any time a bigger or lesse place than now it doth, than it is, by your definition, at present bigger or lesse than it selfe.) Your proofe is in these words, For seeing a Body, and the Magni∣tude, and the Place thereof, cannot be comprehended in the mind otherwise than as they are coincident, (observe therefore, that

this argument doth no more prove, that a Body cannot change its Magnitude, than that it cannot change its Place, for you make Place as much coincident with Body, as you doe Magnitude, and the argument proceeds equally of both:) if any Body be understood to be at rest, that is, to remain in the same place during some time, and the Magnitude thereof be in one part of the time greater, and in another part lesse, that Bodies place, which is one and the same, will be coincident some∣time with greater, sometime with lesse magnitude, that is, the same Place will be greater and lesse than it selfe, which is im∣possible. This is your whole proof to a word. Now this, I told you, is no sufficient proof, because it proves only that a Body doth not change its quantity so long as it is at rest, and doth precisely keep the same place; (which no body doth af∣firme.) And, pray look upon the Argument once again: doth it prove any more than so? But that which you un∣dertook to prove was, that it doth never change its magni∣tude, but hath alwaies the same, as well when its place is altered, as when it remains in the same place: (for, J sup∣pose, you will not deny, but that a Body may change its place.) Those that hold the contrary opinion, doe not say that a body doth change its greatnesse while it doth precisely keep the same place; but that, with change of place, it may change its dimensions too: And to this, if you would have said any thing, you should have applied your argument. And is not this then a just exception to your argument? Will this argument hold, think you, Be∣cause a Body doth not change its magnitude so long as it keeps precisely the same place: Therefore, it never changeth its mag∣nitude, but hath alwaies the same? This argument hath no appearance of consequence, but only upon this supposi∣tion, that a Body doth alwaies keep precisely the same place. And, then, I confesse, the Argument looks like an Argu∣ment, in this forme, So long as a body keeps precisely one and the same place, it hath precisely one & the same Magnitude: But a Body doth alwaies keep precisely one and the same Place: There∣fore it hath alwaies one and the same Magnitude. And if this be your argument, we allow the form, but deny the matter of it, and say, the Minor ought to be proved. For we are of opinion, that it is possible, for the same Body, not to be alwaies in the same place. If you think otherwise, pray

prove it. For 'till that be proved, your present argument is to no purpose.

Sed rem ita per se manifestam, demonstrare opus non esset, &c. But, say you, a thing of it selfe so manifest, would need no De∣monstration at all, (a fine facile way of Demonstration, that which you know not how to prove needs no demonstra∣tion.) but that you see there are some, whose opinion concerning Bodies and their magnitude, is, that Body may exist separated from its magnitude, (no not so, but that it may change its magnitude, For they doe no more believe that it can ex∣sist without Magnitude, than that it can exsist without a Figure: It cannot be but that a finite Body must have al∣waies some figure, though not alwaies the same: and so al∣waies some Magnitude, but whether alwaies the same or no, you should have proved if you could:) and have grea∣ter or lesse magnitude bestowed upon it; (as well as different figures:) Making use of this principle for the explication of the nature of Rarum, and Densum. Since therefore you know there are that do so; why did not you, (at least in your English Editition, after you had notice of the weaknesse of your Latine Argument) bring some good Argument to overthrow that opinion; and not content your selfe to say that it is so manifest of it selfe, as that it needed no demonstrati∣on. Especially, (as I then told you) since you doe not allow that Euclide may assume to himselfe gratis without demonstra∣tion, That the whole is greater than its part; (those were my words, though you recite them a little otherwise.)

But you say, I know this to be untrue, that is, I lye: My words were these; Non interim Euclidi permittis, ut citra de∣monstrationem hoc sibi gratis assumat, Totum esse majus sua par∣te: that is, You do not allow it Euclide, that he may without Demonstration assume to himselfe, or challenge, That the whole is greater then its part. Now let your own words be judge, who is the lyar, you or I. Cap. 6. artic. 12, 13. The whole method of Demonstration, you say, is Syntheticall,— beginning with Principles, or primary Propositions. Now such Principles are nothing but Definitions,—And, Besides Definitions, there is no other Proposition that ought to be called Primary or (si paulo severius agere volumus) be received in∣to the number of Principles. For those Axioms of Euclide, seeing they may be demonstrated, are no Principles of Demonstration. And accordingly art. 16. you define Demonstration, to

be a syllogisme, or series of syllegismes, derived and continued from the Definitions of names, to the last conclusion. And parag. 17. You require to a Demonstration, That, the premi∣ses of all Syllogismes be demonstrated from the first Definitions. (And the like cap. 20. parag. 6. diverse times.) So that these Axioms, being no Definitions, nor any Principles of Demonstration, no Demonstration can take rise from them, nor can they be otherwise assumed in demonstration, than as they are themselves deduced or demonstrated from Definitions. And doth not this come home to what I said? And cap. 8. parag. 25. Of which Axioms (omitting the rest) I will only (say you) demonstrate this one, The whole is grea∣ter then any part thereof. To the end that the Reader may know, that those Axioms are not indemonstrable, and therefore not Principles of Demonstration. And yet again Less. 1. p. 4. As for the commonly received third sort of Principles, called Common Notions, they are Principles only by permission of him that is a Disciple; who being ingenuous, and coming not to cavill but to learn, is content to receive them (though demonstrable) with∣out their demonstration. And again pag. 9. you exclude those common notions called Axioms, from the number of Princi∣ples, as being demonstrable from the definitions of their termes, acknowledging no other Principles, but Definitions, and Postulata, (those the only principles of Demonstration; these of Construction.) If therefore they be no Principles of Demonstration; if only principles by permission of the Disciple, and only in curtesy; then, though your selfe possibly may he so gracious or liberall, as to admit of them without their demonstration; Yet the Teacher cannot, without this fa∣vour, assume to himselfe, or require them to be granted, as he may doe Principles, without Demonstration. 'Twas not I therefore was the lyar, when I said, You doe not allow that Euclide may assume to himselfe gratis, or require to be granted, without de∣monstration, That the whole is greater than its part. For 'tis but in courtesy, if you grant it him, as you may any other true Proposition, and only upon supposition that it may be demonstrated: upon which supposition, you may also al∣low all the Propositions in Euclide, for they may be all de∣monstrated.

And thus much concerning your eight Chapter.