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. III. Concerning Euclide: and the Principles of Geometry. (Book 3)

WE have seen your Elegances already, in the first Section, and then your Critsicismes in the second. It's time now to look upon your Geometry. And I should here begin with your first Lesson; but that, by what we heard even now, you will not allow me to call it Geometricall, or any peece of Geometry, consisting, as it doth, of Definitions. And yet, what ever the matter is, me thinks you come pretty neer it: for you call them Principles of Geometry. But you'l say, perhaps, they be Principles of Geo∣metry, but not Geometricall Principles, (for to call any De∣finitions Geometricall, were as bad as to call them Mathema∣ticall, which were a marke of ignorance unexcusable.) Acute∣ly resolved!

But, whatever else they be, Principles they are without doubt. For, as you define p. 4. A Principle, is, the beginning of something: And no man can deny, but that the first Les∣son is a beginning of something: And therefore, a Principle. Now contra principia, we know, non est disputandum. I must take heed therefore, what I say here.

In this Lesson, you take Euclide to task, and give him his

Iurry: (And when you have lesson'd him, it is to be hoped, wee will not think much to be lesson'd by you:) And with∣all intermingle some Principles of your own, for his and our correction and instruction: such as these,

That 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 can have no place in solid bodies. p. 2. (because you know not how to distinguish between a Me∣chanicall and a Mathematicall 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, as knowing no other way of measuring but by the Yard and the Bushell, or at least by the Pound. p. 4. & 13.) And yet you tell us by and by. p. 3. that there may be in bodies, a Coincidence in all points (which coincidence, had it been Greek, would have been as hard a word as 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉,) and that this may pro∣perly be called 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉: and yet presently p. 4. you tel us again, that 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 hath no place in solids; nay more, nor in circular, or other crooked lines; (as though you did not know, that two equall arches of the same circumference, would 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.)

That the length of T••me, is the length of a Body. p. 2. (As though he had not spoken absurdly, that said, Profecto vide, bam fartum, tam Diu, pointing to the length of his arme.)

That an Angle hath quantity, though it he not the Subject of quantity. p. 3. (for there be octo modi habendi.)

That the quantity of an Angle, is the quantity of an Arch. p. 3. (And why not as well of a Sector, since Sectors, as well as Archs, in the same circle, be proportionall to their corre∣spondent Angles.)

That 'tis a wonder to you, that Euclide hath not any where defined, what are Equalls, at least, what are equall Bodies. p. 4. (As though every body did not, without a definition, know what the word meanes. Any Clown can tell you, that those bodies are Equall, which are both of the same bignesse.)

That Homogeneous quantities are those which may be compared by 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, or application of their measures to one another. p. 4. (And consequently, two solids cannot be Homogeneous; because, you say, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 hath no place in solids p. 2. & 4. And also, that incommensurable quantities, cannot be homogene∣ous; because by 1 d 10▪ they have no common measure.)

That the quantity of Time, and Line are Homogeneous, p. 4. Because Time is to be measured by the Yard; (or, in your

own words, because the quantity of Time, is measured by appli∣cation of a line to a line;) But why not, by the Pint? For you know Time may be measured by the Hour-glasse, as well as by the Clock. And though the Hand of a Clock or Diall, determine a Line, yet the sand of an Hour-glasse fills a vessell.

That, Line and Angle have their quantity homogeneous, be∣cause their measure is an Arch or Arches of a Circle applicable in every point to one another. p 4. (As though you had forgot, that you told us but now, that 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, or application, hath no place in circular or crooked lines.)

And All hitherto, you say p. 5. is so plain and easy to be un∣derstood that we cannot without discovering our ignorance to all men of reason, though no Geometricians, deny it. Nay more, 'Tis new, 'Tis necessary, and 'Tis yours. very good! Now have at Euclide.

Euclid's first definition, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, &c. A Marke is that of which there is no part; is, you say, to be candidly construed, for his meaning is, that it hath parts, and that a good many. For a marke, or as some put instead of it, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, which is a marke with a hot iron, is Visible; if visible then it hath quanti∣ty; and consequently may be divided into parts innumerable p. 5. (A witty argument! 'Tis visible, therefore 'tis divisible, But could you not as well have said, That A Marke consists of two Nobles? For that is as much to the businesse, as a marke with a hot iron.) Nay more Euclids definition, you say is the same with yours, which is, A point is that Body whose quan∣tity is not considered. Lay them both together and look else. A marke is that of which there is no part. A point is that Body, whose quantity is not considered. Just the same to a cow's thumb. They begin both with the letter. As like, as an Apple and a Oyster.

But by the way, how comes a Point on a suddaine to be a Body? you told us just before, in the same page, p. 5. that a Point is neither Substance, nor Quality, and therefore it must be Quantity or else 'tis Nothing. If it be no Substance, how can it be a Body in your language?

But we have not done yet. Prithee tell me, good Tho. (before we leave this point) who twas told thee, that 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 was a marke with a hot iron? for 'tis a notion I never heard till now, (and doe not believe it yet.) Ne∣ver

believe him againe, that told thee that lye; for, as sure as can be, he did it to abuse thee. 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 signifies a distin∣ctive point in writing, made with a pen or quill, not a mark made with a hot Iron, such as they used to brand Rogues and Slaves with; (And accordingly 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, distinguo, interstinguo, inter••ung••, &c. are oft so used;) It is also used of a Mathematicall Point; or somewhat else that is very small: As 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, a moment, or point of time, and the like. What should come in your cap, to make you think, that 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 signifies a mark or brand with a hot iron? I perceive where the businesse lies. 'Twas 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 run in your mind, when you talked of 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, and, because the words are some∣what alike, you jumbled them b••••h together, according to your usuall care and accuratenes•• 〈◊〉〈◊〉 as if they had been the same. (Just as when, in Euclide 〈◊〉〈◊〉 you would have us be∣lieve that 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 & 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 〈◊〉〈◊〉 is but one word.) Do you not think now, that a boy 〈◊〉〈◊〉 Westminster Schoole would have been soundly whipt for such a fault? Me thinks I heare his Master ranting it at this rate; How now Sirrah! Is 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 and 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, all one with you? I'le shew you a difference presently. Take him up Boyes. I'le shew you how 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 may be made without a hot Iron, I warrant you. And after a lash or two, thus goes on: 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, is a Point made with a Pen, quoth he (with a lash) will you remember that? 'Tis 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, is a mark with a hot Iron, (lashing again,) think upon that too. Henceforth, quoth he, (setting him down,) Remember the difference between 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 and 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.

The second definition. A line is length which hath no breadth; you would have to be candidly interpreted al∣so. If a man, you say, have any ingenuity, he will understand it thus, A line is a body &c. very likely!

The Fourth definition, is this, A streight line is that which lies evenly between its own points. p. 6. Well; how is this to be understood? Nay, this definition is inexcusable. Say you so? let it passe then, and shift for its selfe as well as it can. It hath made a pretty good shift hitherto; perhaps it may outlive this brunt also. But, because you are willing to lend

it a helping hand, you say, He meant, perhaps, to call a streight line, that which is all the way from one extreme to another, equal∣ly distant from any two or more such lines, as being like and equall have the same extremes. It may be so. Many strange things are possible. But it would have been a great while before I should have thought this to be the meaning of those words.

The seventh definition, you say hath the same faults. Then let that passe too; and answer for it selfe as well as it can.

The eighth, is the Definition of a plain Angle. Against which you object onely this of your own, That by this Defi∣nition, two right angles taken together are no Angle. And 'tis granted. Euclide did not intend to call an aggregate of two right angles, by the name of an Angle: And therefore gave such a definition of an 〈◊〉〈◊〉, as would not take that in. Where's the fault then?

The thirteenth definit••••••, A Terme or Bound, is that which is the extreme of any thin•• 〈◊〉〈◊〉 you say, is exact, (very good?) But, that it makes against 〈◊〉〈◊〉 doctrine. What doctrine of mine? viz. that a point is nothing. Who told you, that this is my doctrine? I have said, perhaps, that a Point hath no hignesse; or, that a Point hath no parts, (and so said Euclide in his first definition,) but when or where did I say, it is nothing? But how do you prove hence, that a point hath parts? Because, you say, The extremes of a line are Points. True. What then? A point therefore, you say, is a part. It doth not follow. How prove you this consequence, If an extreme, then a part? But, say you, what in a line is the extreme, but the first or last part? I answer; A Point, which is no part. Have you any more to say?—If you have no more to say, then heare mee. A point is the extreme of a line: Therefore it hath no parts. I prove it thus; because, if that point have parts; then, either all its parts are extreme, and bound the line, or some one, or more: Not all: For they cannot be all utmost; but one must stand beyond another: if onely some, or one; then not the Point, but some part of it, bounds the line, which is con∣trary to the supposition. You see, therefore, the Definition doth not make against my doctrine.

The fourteenth Definition of Euclide, you would have abbreviated thus. A figure is quantity every way determined, and then tell us, it is in your opinion as exact a definition of a Figure as can possibly be given. But I am not of your opinion;

For by this Definition of yours, a streight line (of a deter∣minate length) must as well be a Figure, as a circle. For such a line, having no other dimension but length, if its length be determined, it is every way determined; that is, accor∣ding to all the dimensions it hath. (If you object, that it hath no determinate breadth; I answer, the breadth of a streight line is as much determined, as the thicknesse of a Circle, or other plain figure.) And, by the same reason, A Pound, a Pint, a Hundred, an Hour, &c. must be Figures, because they are Quantities every way determined, viz. ac∣cording to all the dimensions that those words import. This Definition of Euclide,— (stay a while, the Defini∣tion mentioned is not Euclides, nor equivalent to it His 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, imports more then your determined. 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. should be rendred A figure, is that which is every way encompassed by some bound, or boundes. Which can be only in such a quan∣tity as hath locall extension; and that, finite.) But The Defini∣tion, you say, (whose soever it be) cannot possibly be im∣braced by us who carry double, namely Mathematicks and Theo∣logy;) but by you it seems, it may, who carry simple, and care not how destructive your principles are to Theology.) Your Definition, we (whether Theologers or Mathemati∣cians) cannot admit; for the reason by us already assigned. But it seems you have a farther reach in it: Lets hear what it is. For this determination, say you, is the same thing with circumscription. A locall determination, intended by Eu∣clide, is so. But what then? And whatsoever is any where (ubicunque) Definitivè, is there also Circumscriptivé. How do you prove this? or how doth this follow from the other? —You cannot but know this is generally denyed. Have you any thing to offer by way of proof?—Not a word. Well; but what is it you drive at? You offer no∣thing of proofe, for what you affirme (by your own con∣fession) against all Divines, or as you call them Theologers. But lets see what you would gather from it. By this means, you say, the distinction is lost, by which Theologers, when they deny God to be in any place, save themselves from being accused of saying he is nowhere; for that which is nowhere is nothing. 'Tis true, that Divines do ••ay, (and I hope you'l say so too) that God is not bounded, or circumscribed, within the limits of any

place; because they say, and do believe, there is no place where he is not. And he that saies the latter, must needs say the former. For to say that God, who is every where, & fills all places; is yet bounded within certain limits; were a con∣tradiction. For, to be concluded within certain limits, is to be excluded from all places without those limits; And therefore not to be every where. And if this be not your opinion too, speak out, if you can for shame, that the world may see what you are. Do you believe, that what thing so∣ever is at all any where, (not excepting God himselfe) must needs be circumscribed within some certain bounds, so as not to be without or beyond them? And that whatsoever is not, in any place so circumscribed, is no where, and therefore nothing? If so; then whether of the two do you affirme? That God is so circumscribed or concluded within certain limits, and excluded from all others at the same time? Or, That he is not so concluded, and therefore no where, and so nothing? If you say the first, you deny God to be Infinite: If the se∣cond you deny him to bee. And, either way, you may with∣out injury be affirmed to maintain horrid opinions concerning God. As for that distinction of Definitivè and Circumscrip∣tivè, with which you say the Theologers think to save them∣selves: You are wholly out in the businesse: Theologers use not that distinction in this case. It's true, that, in the case of Angells, and the Soules of men, there are that affirme them to be in loco definitivè, but not circumscriptivè: because though they be not bodies, and so locally extended per po∣sitionem partis extra partem; yet neither are they infinite, or every where, but have a definite, determinate existence, as to be here, and not at the same time elsewhere. But as to God, we neither affirme him to be circumscribed, nor to be confined within any bounds; but to be Infinite and every where. And if any be so absurd as to affirme that God is determined within some place, so as not to be at the same time without or beyond it, whether by Circumscription or Definition, we shall without scruple, (notwithstanding that we carry double,) reject the distinction so applied, and your opinion with it, without fear of being cast out from the so∣ciety of all Divines.

But in the mean while, I wonder how this Definition of Euclide comes to have any thing to doe with this businesse. A Figure, saith Euclide, is that which is incompassed within

some bound or bounds. Well, what then? Will you assume But God is a figure? and then conclude, That, if God be at all any where, he must be so concluded within bounds? If you do, you argue profanely enough, and deserve as bad Epithites as any have been yet bestowed upon you. We should ra∣ther, admitting Euclides definition, argue thus, A figure is concluded within certain bounds; But God is not so concluded, (as being infinite, and so without bounds;) Therefore God is not a Figure: And be neither in danger of being cast out of the Mathematick Schooles, nor yet, from the Society of Schoole-Divines.

The Fifteenth Definition, which is, of a Circle, you grant to be true.

And skip over the rest to the five and twentieth, which is, of Parallell streight lines. This Definition you think to be lesse accurate, and think your own to be better: But of this it will be time enough, if need be, to consider in its proper place.

After this, you let all the Definitions passe untouched, till the third of the Fift Book. Saving that you touch by the way, on the Fourth of the Third Book, which you grant to be true: and the first of the Fift Book, which, you say, may passe for a Definition of an Aliquot part, as was by Euclide intended.

But, the Third Definition of the Fift Book (the Definition of 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, Ratio,) you say, is intollerable. Yea 'tis as bad as any thing was ever said in Geometry by D. Wallis. (Because for∣sooth, you can make nothing of it, but this, that Proportion is a what-shall-I call it asnesse or sonesse of two magnitudes &c.) Yet this definition hath hitherto been permitted to passe, and may do still. And when you understand it a little bet∣ter, perhaps you may think so too. But of this I have dis∣coursed more at large, in a peculiar Treatise against Meibo∣mius: and shall therefore forbear to examine it here.

Against the fourth definition, you object nothing, but that the sixt might be spared.

The Fourteenth, you say is good. And tell us farther, that the composition here defined, is not the same composition which he defineth in the fourth def. before the sixth book. And you say true; for this is a composition by Addition, and that is composition by Multiplication. And therefore do not

think much if hereafter I shall say, that there be two com∣positions of proportion.

To the rest of his definitions you give a generall appro∣bation. His Postulata you allow also: and so give over Lessoning of Euclide: But tell us before you part, that A man may easily perceive, that Euclide did not intend, That a point should be (without parts, which you call) nothing; or a line, without latitude; or a Superficies, without thicknesse: though it be evident that he hath defined them so to be. But why must we not think, he meant as he saith? (Because, say you, Lines are not drawn but by Motion, and Motion is of Body only. A pretty argument, and worth Marking! like that above, of 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, a Mark or brand with a hot Iron.