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. X. Concerning his 16 Chapter. (Book 10)

IN the 16 Chapter, I said, there were 20 Articles; you say, but 19. 'Tis easily reconciled. There be twenty in my book; and there were 20 in yours too, before the last was cut or torne out: now, it seems, in yours there are but nineteen.

Well; but, be they twenty, or be they nineteen; twenty to one but the greatest number of them be naught. I do confidently affirme, you say, that all but three are false. Nay, that's false, to begin with. I said, that, all but three were unsound. Some of them be non-sense, or absurd; some be false; some undemonstrated; all unsound; at least, within three: And I have already proved them so to be. But you (you say) do affirme, that they are all true, and truly demon∣strated. And that's answer enough to all my arguments. What need you say any more? If that be true, doubtlesse you have the better on't. But let's trie a little, if we cannot find one unsound amongst them.

Your first Proposition as it stands yet in the Latine, you say, is this, The velocity of any Body moved, during any Time, is so much, as is the product of the Impetus in one point of Time, mul∣tiplied into the whole Time. Well, I hope at least the first is sound, is it not? In one Point, you say; but which one? Is it any one? or some one? Nay 'tis but some one, not any one; but, which one, you tell us not. What say you to this? Is it sound? This, you confesse, without supplying what is want∣ing, is not intelligible. Very good! Habemus confitentem re∣rum. To the first•• Article as it is uncorrected in the Latine, •• ob∣ject, you say, That meaning by Impetus, some middle impetus, and assigning none, you determine nothing▪ Well what say you to that? you say, 'tis true. And then you rant at us for not mending it, (as though we were bound to mend your faults) yet look again, and you'l find J did. J told you what you should have said; as well as what you said amisse. But e∣nough of this. Here's one fault confessed.

In the same Article; you would have the Impetus applied ordinately to any streight line, making an angle with it. J asked, How an impetus can be ordinately applied to a Line? or make an Angle with it? Absurdly, you say; and that's the answer. And J told you how this should have been mended too.

You tell me that Archimedes and others say, Let such a line be the Time, and again p. 36. l. 16. Let the line AB be the Time. Very likely! just as when we say, Let the Time be A. That is, Let it be so designed; or, Let the Line AB, or the letter A, be the Symbole of the Time. What then? Doth it therefore follow, that either Lines or Letters be homoge∣neous to Time? No such matter. Their Symbols may be Homogeneous though the Things be not. You say farther, in the same Article: If the Impetus increase uniformely, the whole velocity of the motion shall be represented by a Triangle, one side whereof is the whole Time, and the other the greatest Impe∣tus, (Well! & what shall be the third side? or what angle shall these contain? Do you think that the assigning of two sides, without an Angle, will sufficiently determine the bignesse of a triangle? But lets go on.)

Or else &c. Or lastly by a Parallelogramme having for one side a mean proportionall between the greatest Impetus and the halfe thereof. Well, but what for the other side? And, what Angle? Is a Parallelogramme, said J, sufficiently deter∣mined, be the assignement of but one side, and never an angle? what think you? is this sound? It was indeed a very great o∣versight, you confesse, to designe a Parallelogram by one only side. And is not all this sufficient to prove the first Article unsound? if it be not, wee'l go on, for there be more faults yet.

For, say you, these two parallelograms are equall both each to other, and to the (fore mentioned) Triangle (without ha∣ving any consideration of Angles at all) as is demonstrated in the Elements of Geometry. This, I say, is notoriously false: For a Triangle of which nothing is determined but two sides: and a Parallelogramme, of which the sides only are determined, but nothing concerning the Angles: can never by any Geometry, be demonstrated to be equall. This therfore is not only unsound, but false. And all this J told you before. What an impudence then is it, when you knew all this, to affirm, that they be, all true and all truly demonstra∣ted,

when the very first of them is thus notoriously faulty! But we have not done yet.

It might be hoped, that this confessed oversight is, at lest mended in the English: (especially since you tell us that one from beyond sea hath taught you how to mend it) No such matter. For the Amendment is as bad or worse then what we had before. For now it runs thus. The whole velocity shall be represented by a Triangle &c. (as be∣fore) or else by a Parallelogram, one of whose sides is the whole Time of motion; and the other, half the greatest Impetus: Or lastly, by a Parallelogram, having for one si••e a mean pro∣portionall between the whole Time and the halfe of that Time; and for the other side the halfe of the greatest impetus. For both these Parallelograms are equall to one another, and seve∣rally equall to the Triangle which is made of the whole line of Time, and the greatest acquired impetus. As is demonstrated in the Elements of Geometry. Now this, you shall see, is pit∣tifully faise. Let the time be T; and the greatest impetus, I: and let the Angles be supposed all Right Angles (for such your Figures represent, though your text says nothing of them.) The Altitude therefore of the triangle, is T, (the whole time:) the Basis I, (the greatest impet••s:) and consequently the Area thereof is one halfe of T × I: that is ½ IT. Again the Altitude of the former Paral∣lelogram, T, (the whole time,) its Basis, ½ ••, (half the greatest impetus,) and therefore the area T × ½ I, or ½ IT; equall to that of the Triangle Lets see now whe∣ther the last Parallelogram be equall to either of these, as you affirm. The Altitude you will have to be a mean pro∣portionall between the whole Time and its halfe: that is, between T&½ T; It is therefore the root of T × ½ T, that is the root of ½ Tq, that is √½ Tq, or T√½: The Basis you will have to be one half of the greatest Impetus, that is ½ I: And con∣sequently, the Area must be ½ I × √½ Tq, or ½ I × T √½, or ½ IT√½. But ½ IT√½ is not equall to ½ IT: Therefore this Parallelogram is not equall either to the former, or to the Triangle. 'Tis false therefore which you affirmed. Quod erat demonstrandum.

Now what do you think of the businesse? is not the matter well amended? 'Twas bad before, now 'tis worse. When you told us but of one side, and left us to guesse the other, 'twas at our perill if we did not guesse right, and 'twas to be hoped, you meant well, though you forgot to set it down. But, now you tell us, what you meant, we find that you neither said well, nor meant well: For what you now say is clearly false. The two Parallelograms which you affirm to be equall, are no more equall then the Side and the Diagonall of a Square; but just in the same propor∣tion; viz. as √½ to 1. Nay was it not a pure piece of wisdome in you, that, when you had been taught from be∣yond Sea, as you tell us, how it should have been mended, you had not yet the wisdome to take good counsell; but, trusting to your own little wit, have made it worse than it was? it falls out very unluckily, you see, that when you af∣firmed so confidently, that they are all true, and all truly de∣monstrated, the very first of them should be so wretchedly faulty. But enough of this. Wee'l try whether the next will prove better.

In the second Article you give us this Proposition. In every uniform motion, the lengths passed over are to one another, as the product of the ones Impetus multiplied into its time, to the product of the others Impetus multiplied into its time. And why not, said J, (without any more adoe) as the time to the time? Which needed no other demonstration than to cite the definition of Ʋniform motion, (viz. which doth in e∣quall or proportionall times, dispatch equall or proportionall lengths.) What need had you to cumber the Proposition with Impetus and Multiplication, and Products, when they might as well be spared? and then put your selfe to the trouble of a long and needlesse demonstration, when the bare citing of a definition would have served the turne? You answer, That the product of the Time and ••mpetus, to the product of the Time and Impetus, is also as the Time to the Time. and therefore the Proposition is true. Yes doubtlesse; and therefore I did not find fault with it, as false; but as foo∣lish, to make such a busle to no purpose. For, by your own confession, the proportion of the lengths dispatched, is as well designed by the termes alone, as by those multiplica∣tions and products.

But there is another fault which J f•••••• with your pro∣position;

•• told you that, instead of, in every uniforme motion, you should have said, (and, that you might have said it safely, as the rest of the wordsly,) in all uniform motions; for you make use of this proposition afterwards, not only in comparing divers parts of any the same uniforme moti∣on, but in comparing divers motions one with another. But at this you are highly offended, that J should under∣stand to what purpose this Proposition is brought, better than your selfe; and that J should presume to tell you, what you ought to have said. (And, on the other hand, when J do not do so, you blame mee, that J do not to my reprehension adde a correction: So that, it seems, you are nei∣ther well, full nor fasting: J must neither do it, nor let it alone.) And then you go on to rant, after your fashion, at Wit and Mystery, and times and wayes, and steddy brains, at rea∣ding thoughts, and noise of words, at step and stumble, &c. And yet, for all the anger, (when the heats over) you think best to take my counsell; and therefore say in the English, just as J said it should have been in the Latine.

The proposition then being thus to be understood, (though at first, ill worded,) the demonstration, I said, would not hold. For though it will doe well enough (yea more then enough; for you might have spared halfe of it;) in comparing severall parts of the same motion, and in comparing severall motions of the same swiftnesse; yet for the comparing of uniforme motions in generall, it will not serve by no meanes; for you do assume at the first dash, that the motions compared have the same Impetus. Now this must not be allowed. For it's very possible (as you now know, since, J told you, though before you see∣med to be ignorant of it, as J then convinced you;) that two motions may be both uniforme; and yet not have both the same Impetus. Your proposition therefore (as it was to be understood) was not truly demonstrated.

Now, because this was very evident, and not to be de∣nied; therefore you thought it best to make no words of it, but mend it as well as you could. And so, in the Eng∣lish, you have mended the proposition, as J bid you; and given us a new demonstration, which is pretty good; But not yet without fault. For in stead of the length AF (fig. 1.) you should have said, the length DG: for the length should have been taken in the line DE, which, according to

your construction, is the line of Lengths; not in the line AB, which is, by construction, the line of Times. So im∣possible a thing is it, for you to mend one fault and not to make another.

But if all these faults be not enough to make this Article unsound, there is yet another, before we leave. Since there∣fore you say, in uniforme motion, the Lengths dispatched are to one another, as the Times in which they are dispatched; it will also be, by permutation, as time to length, so time to length. This consequence I denied; because Permutation of pro∣portion hath place only in Homogenealls, no•• in Hetero∣genealls; (and referred you for farther instruction concer∣ning it, to what Clavious hath on the 16. Prop. of the 15. of Euclide.) You tell me, that I think, line and time are He∣terogeneous. Yes, and you think so to if you be not a foole. If not, pray tell me how many yards long is an hour? Or, How much line will make a day?

Well, lets try a third Article. (For the two first you see be nought, that's a bad begining.)

Art. 3. In motion uniformely accelerated from rest, (that is, when the Impetus increaseth in proportion to the times) the length run over in one time, is to the length run over in an other time. (In the English for Impetus, you have put mean Impetus, and so in some other propositions; but that neither mends nor mars the businesse.)

To this, first you dream of an objection, and then think of an answer to it. I object you say, that the Lengths run over, are in that proportion which the Impetus hath to the Impetus. Prithee tell me, where I made that objection to this ar∣ticle; and i'le confesse 'twas simply done. But 'till then, i'le say 'tis done like your selfe, to say so however. (For 'tis lawfull with you to say any thing, true or false) Your English Reader, perhaps, may think 'tis true.

Next, You aske, you say, where it is that you say or dreame, that the lengths run over are in proportion of the Impetus to the Times? But prithee, why dost thou aske me such a questi∣on? Am I bound to give an account of all thy dreames? Perhaps you dreamed that I had charged you with such a saying; But, look again, and you'l find that's but a dreame as well as the rest.

That which I said was this, The parallell line FH, BI, (fig. 1.) do shew what proportion the Impetus at F hath to

the Impetus at B; to wit, the same with the time AF, to the time AB: (And is not this your meaning, when you say the Impetus increaseth in proportion to the times?) But, though those (and other parallell lines) do define what proportion the severall Impetus have to each other; yet they do not designe (by permutation of proportion, as you fancied in the Corollary of the precedent article) what proportion the severall Impetus have to the Times; be∣cause they be Heterogeneous, and do not admit of that permutation. And these are the words, which gave occa∣sion to those your two dreames. And then (as if between sleeping and waking) you ask, if it be you or •• that dream? Had you been well awake, you needed not have asked the question.

The objections that I made to it, were these.

First, that in stead of motu accelerato, (accelerate moti∣on,) you should have said, motibus acceleratis (accelerate motions,) because you speake of more than one. You say, there is no such matter: and bid mee give an instance. J will so, and that without going farther then your present 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉. Let AB (say you) represent a Time &c. Againe, let AF represent another time &c. And in each of these times you suppose a Motion, which motions this proposi∣tion compares. Therefore, say I, there must be at least two Motions, because two Times; unlesse you will say, that one and the same motion may be now, and anon too.

I objected farther, that the demonstration doth nor prove the proportion; except only in one case, to which you do not restraine it. For the whose stresse of your demonstration, (in the Latine) lyes upon this, that the triangles ABI, AFK, be like triangles (where you inferre, that the space dispatched in the latter time AK, is to that of the former time AB, as the triangle ABI, to the trian∣gle AFK, that is in the duplicate proportion of the times AB, AF.) Which supposeth that the second motion in the time AF, doth acquire the same Impetus which the first motion had acquired in equall time. Whereas it is possi∣ble, that, of two motions, each of them uniformly accele∣rated, the one of them may in half the time acquire as great a swiftnesse, as the other doth in the whole time; If there∣fore the latter motion in the same AF, do acquire a swift∣nesse equall to that of the former in the time AB, (which

may very well be, for the words uniformly accelerated, doe imply only the manner of acceleration, not the degree of cele∣rity; as your selfe now discern, though then you did not,) the triangles will be, not ABI, AFK, but ABI, AFH; which are not like triangles, but unlike; and so the de∣monstration falls. You should have provided in your pro∣position, not only that the two motions, (the one in the time AB, the other in the time AF,) be each of them u∣niformly accelerated, but that they be both equally swift. Which when you have neglected to take care of, you af∣firm that universally, which will hold only in one case.

But the truth is, 'tis evident enough, by this and divers other Articles, that you took the manner of acceleration, (viz. if in the same, in the duplicate, or triplicate, &c. proportion to the times,) had sufficiently determined the speed also. And therefore took it for granted, that the motion in the time AF, if uniformly accelerated, must needs attain precisely the same degree of the celerity, that the other motion in the time AB, uniformly also accele∣rated, had attained in equall time. (Which to be a very great mistake, you now doe apprehend.) Otherwise you would not have let these Articles ly so naked without such provision; nor would you, (as in the 13 Article, and those that follow,) undertake, by the manner of acce∣leration, and the last acquired Impetus, to determine the time of motion. Whereas, in the same manner of accele∣ration (whether uniformly, or in the duplicate, or tri∣plicate, or quadruplicate proportion;) any assignable im∣petus or degree of celerity, may be attained in any assigna∣ble time whatever.

I objected farther, that because, as hath been shewed, the Triangle AFK, or AFH, is not necessarily like to the triangle ABI, therefore it doth not follow that the length passed over, will be in duplicate proportion to the time. For unlesse the triangles be alike, the proportion of them will not be duplicate to that of their homologous sides.

Now these two Objections were clear and full, (and did destroy your whole demonstration;) and this you dis∣cerned well enough, though you did not think fit to make any reply or confession; (but invent some other objecti∣ons, which I never made, that you might seem to answer to somewhat.) And therefore in the English, without

making any words of it, you mend it. And instead of those words in the Latine, As the triangle ABI, to the triangle AFK, that is, in duplicate proportion of the time AB to AF: you say in the English. As the triangle ABI, to the tri∣angle AFK, that is, if the triangles be like in the duplicate proportion of the time AB, to the time AF; but, if unlike in the proportion compounded &c. (which is a clear confession of all those objections. But let's go on. Compounded of what?) of AB, to Bi, and of AK, to AF. No such mat∣ter; of AF to FK, (that's it you would have said:) not, of AK to AF. There's one fault therefore; but thats not all. Of AB to AF, and of BI to FK; thats it you should have said: for AB to BI, the Time to the Impetus, hath no proportion at all; but are Heterogeneous, as I have of∣ten told you. There's a second fault therefore in your e∣mendation. And is not this Tinker-like, to mend one hole and make two? Nay there is a third yet, which is the worst of all.

In the mending of this fault, (though you had not mis∣sed in it,) you have discovered another, which you did your endeavour, but now, to hide. I said in the proposi∣tion for motion, you should have said motions; because it was intended of more than one compared. You tell me, there's no such matter; meaning, I suppose, the latter motion in the time AF, was but part of that former motion in the time AB: But if, as you now confesse, the triangle AFK, be not necessarily alike triangle to ABI, (but that the point K may fall either within or without the line AI,) then must this be not only another, but an unlike motion to the former: viz. either faster or slower, though uniformly accelerated as that was. Do not you know that old rule; Oportet esse memorem. But this 'tis, when men will commit faults, and then deny them. And yet presently after, by going about to mend them, betray themselves.

Much such luck you have in mending the Corollary. You had said in the Latine, In motion uniformly accelerated, the lengths transmitted are in the duplicate proportion of their times. This, I said, was true in one case, (viz. in equall celeri∣ties,) but not universally. Therefore you, to mend the matter, in the English make it worse; In motion uniformly accelerated, say you, the proportion of the lengths transmitted, to that of their Times, (No, but the proportion of the length

transmitted, one to the other,) is compounded of the propor∣tions of the Times to the Times, and Impetus to Impetus.

There be more faults in this Article; but I am weary of the businesse; let's go to the next.

The fourth Article hath all the faults that the third hath, (which are enough as wee have seen already,) and some more.

First, for motu accelerato, you should have said motibus acceleratis; because you compare more motions then one.

Secondly, the Motion performed in the time AF, (Fig. 2.) though accelerate according to the duplicate propor∣tion of the times, as well as that in the time AB; yet may that be either swifter or slower than this; (because as we have often said, the manner of acceleration doth not de∣termine the degree of celerity;) And therefore the point K which determines its greatest Impetus, doth not necessa∣rily fall in the Parabolicall line, but may fall either within or without it: according as the celerity is lesse or more.

Thirdly, And therefore it doth not follow, that the Lengths dispatched by such motion, are in triplicate pro∣portion to their Times. For this only depends upon sup∣position that the point K in the second motion, must needs fall in the Parabola AI, designed by the first motion.

Now these two latter faults, in the former Article, you did endeavour to amend in the English: But because, it seems, here it was harder to doe, you have left them as they were before. That these were faults, you were clear∣ly convinced of; and do as good as confesse, by your at∣tempt to mend them in the third Article. But because you saw it was impossible for one of your capacity to think of mending all; you resolve to give over mending, and (which is the easier of the two) resolve to try the strength of your brow.

But, as if there were a necessity of growing worse and worse; beside those, common to this and the third article, here is an addition of more faults, as foul as any of them.

In your demonstration; your stresse lyes upon this argu∣ment, Seeing the proportion of FK to BI, is supposed dupli∣cate to that of AE to AB, (which yet is a false supposi∣tion; for the ordinate lines in a Parabola are not in du∣plicate, but in subduplicate proportion to the diameters:

But, suppose it true, what then?) that of AB to AF, will be duplicate to that of BI to FK. That is, Because the Or∣dinate lines in a Parabola, are in duplicate proportion to the Dia∣meters; therefore those Diameters are in duplicate proportion to those Ordinate lines. Which if it be not absurd enough, I would it were. First, the proportion of the Ordinates, must be duplicate to that of the Diameters, (because M. Hobs will have it so;) and then (by the virtue of Hocus Pocus) this must be duplicate to that.

To this you make no reply: but inslead thereof, disguise the matter in your Lesson, by putting double for duplicate, as if they were all one; (though yet Chap. 13. art. 16. wee have, in the English, a long harangue of your own to shew the difference between them;) and then raile at those that first brought up the distinction; and tell us, (which is notoriously false) that Euclide never used but one word for Double and Duplicate; (that is 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, and 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, are with M. Hobs but one and the same word.) But what is all this to the ra∣king off that absurdity with which you are here char∣ged?

Next I shewed you, that your whole argument was grounded upon a false supposition; viz. that the veloci∣ty of the motion in hand, was to be designed by the Semi∣parabola AKB; and that the ordinate lines in that Semi∣parabola, (by which you would have the increasing Impe∣tus to be designed) did increase in duplicate proportion to their Diameters (by which you designe the Times▪) both which are false. For, these ordinate lines, are well known (to all but M. Hobs) to increase in the Subdupli∣cate (not the Duplicate) proportion of the Diameters: And consequently that Semiparabola can never expresse the Aggregate of the Impetus thus increasing.

I did farther demonstrate, that the point K, ought to fall within the Triangle ABI, not without it; and there∣fore not in the Parabolicall line by you designed. The de∣monstration was easy. For if the time AF be one halfe of AB, that is, as 1 to 2: the Impetus increasing in dupli∣cate proportion to the times, must be as 1 to 4; and there∣fore FK will be but a quarter of BI. But because AF is halfe of AB, therefore FN will be halfe of BI. And

consequently FK (a quarter) will be lesse then FN, which is the halfe of BI. Which because you saw too evident to be contradicted▪ you thought it best (as your usuall cu∣stome is in such cases,) to raise at it in stead of answe∣ring it.

I shewed you farther, that the Aggregate of all the Impe∣tus in a motion thus accelerated, or the whole Velocity, was not ⅔ of the Parallelogram AI, but only ⅓ of it. For this aggregate is not to be designed by a Semiparabola, but by the complement of a Semiparabola. And many o∣ther mistakes▪ consequent thereunto. And indeed so many, as that dispairing of mending them all, you resolved to let them stand as they were.

Yet I shewed you withall the chiefe ground of all these mistakes, and how they might have been mended. But it appeares you had not the wit to understand it, and there∣fore durst not venture upon it. But have left this whole article such an Hodge podge of errors, as would turne a quea••ie stomach, but to examine it.

And your Corollarys are false also.

In the first Corollary, 'Tis false which you affirme, that the proportion of the parabola ABI to the parabola AFK, is triplicate to the proportion of the times, AB to AF, (as it is in the English.) or of the Impetus BI to FK, (as it is in the Latine.) This exception you confesse to be just, yet leave it uncorrected in the English; because you know not how to mend it; without giving your selfe the ly in the rest. For as badde as it is, it follows, with the rest of your doctrine. It must all stand or fall together.

The second Corollary, (at least, if understood of the Parabola,) is also false; for the segments of a parabola (of equall height) successively from the Vertex, are not as the numbers 7, 19, 37, &c. the difference of the Cubes 1. 8, 27, 64, &c. but us the differences of these surd nūbers 1, √8, √27, √24. &c. That which you alledge to justify your selfe; that the parts of the Parabola cut off are as the cubes of their bases; is but a repetition of the same error. They are not as the Cubes of their Bases, but as the square roots of such Cubes.

The third Corrollary is wholly false, A motion so acce∣lerated doth not dispach two thirds; but one third, of

what a uniform motion would have done, with an Impetus equall to the greatest of those so increasing. You say, I give no demonstration 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 aggre∣gate of the Impetus did not constitute a semiparabole, but the complement of a semiparabola, which is not ⅔ but ⅓ 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 increa∣sing 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, tripli∣cate, &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 com∣pleare 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 to∣wards 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 Diame∣ter,) 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, tripli∣cate

&c. proportion of the times) to be ⅔, 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 ••lelograms) whereas they are indeed but ⅓, 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 conso∣nant 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. (Be∣side such petty errors all along as it were endlesse every where to take notice of) which gives you a new occasi∣on 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 Mo∣vents at once, concurring in what angle soever, of which the one is moved uniformely, the other with motion uniformely accele∣eated from rest, till it acqu••e an Impetus equall to that of the Ʋniforme 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, ••t shall be that Semiparabola, whose Busis is the Impetus last ac∣quired; And this is the whole designation of your Para∣bola.

To this designation I objected many things.

First, that the Basis of a Semiparabala is not an Impetus but a Line: and therefore 'tis absurd to talke of a Semi∣parabola 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 Im∣petus,

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 Semipa∣rabola, whose basis is what line you please. So that we have not so much as the Basis of this Semiparabola deter∣mined.

Thirdly, suppose that the Base had been determined, (as it is not) yet it is a simple thing to think that determi∣ning 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 so∣ever. That is, you supposed your Mevents to concurre in what angle ••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 omit∣ting 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. 'Twas 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 unde∣termined 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 twas 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 true∣ly; 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 such thing determined. For a base of such a bignesse, may be within an Inch, and it may be above an E••l from the Vertex, according as the Parameter is greater or lesse. Now you doe not pretend any other designation of the Base, then that it be equall to such an Impetus; which de∣termines only the bignesse of it, not the distance from the vertex. So that the altit••de, notwithstanding this flamme, remaines undetermined. (And must do so, whatever you think, till you do determine the degree of celerity, which answers to the Parameter of the Parabola; as well as the manner of acceleration, which only determines that it is a Parabola, but not what Parabola. The proposition there∣fore is extreamly imperfect; nor doth determine that which it did undertake to determine.

The figure is yet worse. You suppose the line AB, (fig. 6.) by uniforme motion, to have dispatched the length AC, or BD, and so ly in CD; in the same time that the line AC, by motion uniformely accelerated, dispatcheth, the length AB, or CD, to come and lye on CD. That is, (because AB, according to your figure, it about twise the length of AC,) the motion accelerated doth, in the same time, dispatch about twise the length of what is dispatched by the uniforme motion. But it is evident, the accelerate motion is all the way, to the very last point, slower than the uniforme, (for by supposition, it doth not till the last point, attain to that Impetus or swiftnesse, with which the uniform motion was carryed all the way.) Therefore according to you, a slower motion doth, in the same time, dispatch a a greater length then the swifter, Which is absurd enough: And to which you make no reply.

The demonstration also (saving what you have from Galileo) I then shewed you to be faulty; and you reply nothing in its vindication and therefore I need not repeat it. You have in the English a little disguised the proposition, but to little purpose. The Parabola which you undertake to determine, remaines as undetermined as it was before. And the figure the same with all its faults: And the de∣monstration no whit mended. So much of this Article as

yo•• tooke out of Galileo was good, before you spoild it; but the next is all naught.

Your tenth Article doth but repeat all the faults of the ninth, and you have nothing more to say in the vindicati∣on of this then of that. The Parabolaster here, remaines as undetermined as the Porabola there; your Figure (fig. 6.) makes the flower motion in the same time to dispatch the greater length; your demonstration is faulty as that was. Nay you have not here, so much as disguised it in your English, as you did the former; but left it as it was in the Latine. So that this falls under the same condemnation with the former.

I hinted also, that we have here a great talke of Para∣bolasters which are not to be defined till the next Chapter. But that's a small fault. Your English helps it, by sending us thither for the definiton.

Your eleventh Article undertakes to give us a generall rule, to find what kind of line shall be made by the motion of a body carryed by the concurse of any two Movents, the one of them Ʋniformely, the other with acccleration, but in such proportion of Spaces and Times as are explicable by Numbers, as Duplicate, Triplicate &c. or such as may be designed by any broken num∣ber whatsoever. Your rule for this, sends us to the Table of Chap, 17▪ art. 3. to seek there a Fraction whose Denominator is to be the summe of the Exponents of Length and Time; and its Numerator, the exponent of the Length. Upon this I proposed you a case which falls within your proposition, but not within your Rule: (to shew that your Rule did not performe what you undertook to performe by it.) Let the motions, sayd I, be, the one, uniforme; the other acce∣lerate, so as that the spaces be in subduplicate proportion to the Times; or, in your language, as 1 to 2. We are therefore, by your Rule, to seeek in the Table the fraction ⅓. But there's no such fraction to be found (nor any lesse then ½.) Your rule therefore doth not serve the turne. Well▪ let's heare what you have to say for your selfe. Did I not see (you aske) that the Table is only of those figures which are de∣scribed by the concourse of a motion Ʋniforme, with a motion accelerated. Yes I did, see that the table is only of such: Nay more, I saw (which is more to your purpose) that the pro∣position is only of such; (though yet if need be, I could

shew you how the same figures might be described by mo∣tion retarded as well as motion accelerated,) & therefore I proposed such a case; viz. an acceleration in the subduplicate proportion of the times, that is after the rate √1▪ √2. √3. √4. &c. which is the subduplicate rate of 1, 2, 3, 4, &c. I had no reason therefore, say you, to look for ⅓ in thae Takle. That is, I had no reason to expect, that your Rule should performe what you undertake. But why no reason to expect it? For my case is of motion uniforme concurring with motion retarded. No▪ such matter, (nor be you so simple to think so, whatever you here pretend;) for √1. √2. √3. √4. &c. is no decreasing progression, but increasing: for √2, is more then √1 & √3, & more then √2. & so on. But why should you think it is not so? Because forsooth. I do not make the pro∣portion of the spaces to that of the times duplicate, but subduplicate. Very good 〈◊〉〈◊〉 But if times be proposed in a series increasing as 1, 2, 3, 4, &c. will not the subduplicate rate be increasing also, as well as the duplicate? that is, doe not the Rootes of these numbers continually increase, as well as their Squares? Think againe and you'l see they doe. Well, but however, though this table will not serve the turne, yet the ••ase may be solved, you tell us, another way. No doubt of it. I could have told you so before. (For though you knew not how to resolve it; I did; and therefore directed▪ you to the 64. Prop of my Arithmetica Infinitorum; where you have the case resolved more universally then it is by you proposed; viz. where the exponent of the rate of accele∣ration is not explicable by numbers; but even by surd ro••tes, or other irrationall quantities.) But what be∣comes of your rule in the mean while, which sent us to that Table for solution? where, you now tell us, (for I had told you so before) it is not to be hard? This eleventh Article therefore, is like the rest. Nor is it at all amended in the English.

Your twelfth proposition, I said, was wretchedly false; And I say so, still. But, you say, you have left it standing un∣altered; (& yet that's false too; for your English hath a con∣siderable alteration from what was in the Latine, though not much for the better) Your words were these▪ If motion be made by the concourse of two movents, whereof one is moved

uniformely, the other with any acceleration whatever (for which you say in the English, the other beginning with Rest in the Angle of concurse, with any acceleration whatsoever) the movent which is moved uniformely shall put forward the thing so moved, in the severall parallel spaces, lesse, than if both motions had been Ʋniforme. I gave instance to the contrary, (in fig. 5.) The streight line AND, may be described by a compound of two uniforme motions; and the parabolick line AGD, by a motion compounded of two, the one uniforme and the other accelerated, (neither of which you can deny, for you affirme both, at art. 8, and 9.) But within the Paralells AC, EF, the thing moved (contrary to your assertion) is more put forward by this, than by that motion (for EG, is greater then EN,) The p••oposition therefore, in this case is false. Yonr answer is, that other Geometricians find no fault with it. It may be so. But is there any Geometrician (who hath well examined it) will say 'tis true? and that, in all cases? In some cases I told you, it may happen to be true; and in in other cases it will be certainely false: (And I told you also, when, and where.) And I did in the case proposed prove it so to be; and you can say nothing to the demonstration. You would indeed tell me of another case wherein, you think it is true. But what's that to the purpose? When I give instance to the contrary of a uni∣versall proposition, you must allow me to lay the case as I think good (so as it be within the limites of that univer∣sall) and not as you would have me. The proposition therefore is demonstrated to be false. And you have no∣thing to say in vindication of it.

The thirteenth Article doth propose a Problem as ridi∣culous as a man would desire to read. 'Tis this Let AB (fig. 8.) be a Length transmitted with uniforme motion in the Time AC: And let it be required to find another length which shall be transmitted in the same time with motion uniformly accelerated, so as the Impetus (or, as in the English, the line of the impetus) last acquired be equall to the streight line AC. The Answer say J to this Probleme, is what length you please. (And you might as well have propounded, A quantity being assigned which is equall to its foure quarters; let it be required to find another quantity which is equall to its two halves. Or thus A parallelogram being proposed of a known Base and Altitude; let it be required to find what may be the

altitude of a triangle on the same base. Where, what quantity you will, doth serve for answer to the former: And, what alti∣tude you will for the latter. And, what length you will, is the answer to your Problem.) For there is no length assignable, which may not, in any assignable Time, be dispatched by a motion uniformly accelerated, whose last Impetus shall be what you please. And 'tis but as if you should have asked; What may be the height of that Parabola, or Triangle, whose Basis is equall to AC?

The Problem being thus ridiculous, it cannot be expe∣cted that the construction or demonstration should be bet∣ter. And truly 'tis pittifull stuffe all of it: as J then shewed. And you do not so much as attempt any thing by way of answer, to justify either your construction or demonstra∣tion.

You ask here, (for you have no more witt then to pro∣pose such a question,) granting that a Parabola may be de∣scribed upon a Base given; and yet have any Altitude, or any Di∣ameter one will: (which you say who doubts?) How it will hence follow, that when a Parabolicall line is described (is to be described, you should have said; for the Problem is of somewhat to be done, not, of somewhat done already,) by two motions, the one uniform, the other uniformly accelerated from rest; That the determining the Base, doth not also deter∣mine the whole Parabola? J answer. Because every Parabola may be so described; (which if you did not know before, you may now learn of me:) And therefore, since that, up∣on a Base given, a Parabola may be described of any altitude (as you grant;) and that every Parabola may be so descri∣bed: the determining of the Base, doth not determine the Altitude of a Parabola so to be described; more then the Altitude of a Parabola simpliciter.

But if you would have done any thing to acquit your selfe of the charge in this Article, (of proposing a Ridicu∣lous, Nugatorious Problem:) You should have assigned some Length, which by a motion so accelerated, and acquiring such an Impetus, could not have been dispatched in a Time assigned. Till then; I say, it may dispatch what length you please: And therefore your Problem is as ridiculous as a man could wish.

There be divers other petty faults, that J took notice of by the way; as that those words, so as the Impetus acquired

be equall to a Time (as if heterogeneous things could be e∣quall.) And, those words, as duplicate proportion is to single proportion, so let the line AH be to the line AI. (which is as pure nonsense as need to be:) As if there were one certain Proportion of the Duplicate proportion, to the single Pro∣portion. You tell us, upon second thoughts, in your Eng∣lish, cap. 13. art. 16. that Duplicate proportion is sometime greater then the single; and that it is sometimes lesse: And yet you would here have us think that it is alwaies as 2 to 1. The proportion of 9 to 1, is duplicate of that of 3 to 1: And the proportion of 4 to 1, is duplicate of that of 2 to 1. But there is not the same proportion of the proportion 2/1 to the propor••ion ••/1, that there is of the pro∣portion 4/1 to the proportion 2/1▪ but that is triple this double: (for nine times as many, is the triple of three times as many; and four times as many, is but the double of twice as many.) But this you cannot understand, and there∣fore call for help from somebody that is more ready in Sym∣bols. It seems a man must speak to you in words at length, and not in figures. And truly, all's little enough to make you understand it.

The 14, 15, and 16 Articles are just like the 13: and as ridiculous as it. What was there objected, you confesse, may as well be objected to these. But that hath been proved to be ridiculous: and therefore so are these. Any length being gi∣ven, which, in a Time given, is dispatched with uniform motion; To find out what length will be dispatched in the same time with motion so accelerated, as that the Lengths dispatched be conti∣nually in triplicate proportion to that of their times. (so Art. 14.) or quadruplicate, quintuplicate, &c. (ibidem.) or as any number to any number. (so Art. 15. 16.) and the Impetus last acquired equall to the Time given. That's the Problem. The Solution should have been; What length you please. Take where you will you cannot take amisse. If you say, 'tis an Inch, you say true: If you say, 'tis an Ell, you say true: And if you say 'tis a thousand miles, no body can contradict you. For it may be what you please.

And is it not a wise thing of you then, for the design∣ing of an Arbitrary Quantity, a What-you-will, to bring a parcell of Constructions, and Demonstrations, with finding of Mean Proportionalls, as many as one please; for a mat∣ter

of two leaves together? And, when you have done all, 'tis but, (as you were,) What you will.

J noted farther that in all these Articles 13, 14, 15, 16, as in those before Art. 9, 10, 11. & those following 17, 18, 19. You doe every where make the slower motion, in the same time, dispatch the greater length. Which I did clearly demonstrate. To this you reply nothing to the purpose: But cavill, that you might seem to say something. You say, I corrupt your Article by putting Movens for Mobile. But there's no such matter; for in the place alleadged (Art. 1.) Movens is your own word, not mine. You say, 'tis no mat∣ter whether AB or AC (in the fifth figure) be the greater. Yes it is; it's impossible that AB, according to your sup∣position, should be so bigge as AC; and yet, you have made it almost twice as big. You say, you speak of the concurse of two movents; very true. But each of those movents have their se∣verall pace assigned them; & therefore you should not have made the slower movent to rid more ground. And then you would tell mee, what I think; and then talk of hard speculations, of edge and wit and malice &c. But nothing to the purpose. For when you have all done, its evident, and you cannot deny, that in your 5 and 6 and 11 Figures, AB is made welnigh twice as long as AC; and so again in your 8, 9, and 10, Figures AH much longer then AB; and yet these longer lines designe the length, dispatched by the slower motions in the same time. For the motion accele∣rate, which doth not till its last moment attain the swiftnesse, with which the uniform motion proceeds all the way, must needs be slower then that uniform motion. But this was a fault which I might safely have let passe; for these Articles were ridiculous enough before.

In the 17 Article, I shewed first, that the Proposition, as it was proposed, was not perfect sense. Then, that, the sense being supplied, the Proposition was false. And lastly, that your Demonstration had at lest fourteen faults, and most of them such, as that any one was sufficient to over∣throw the Demonstration.

The Proposition was this, If in a time given, a Body run over two lengths, one with Ʋniform, the other with accelerated motion, in any proportion of the length to the time, And again in a part of that time, it run over parts of those lengths with the same motions; the excesse of the whole longitude above the whole

(to what?) is the same proportion with the excesse of the part a∣bove the part, to what? Is this good sense? No; you con∣fesse there was somewhat left out in that Proposition, but say, it was absurdly done to reprehend it. Very good! It seems you must have the liberty to speak non-sense without controll.

Well; but how is the sense to be supplied? we made two or three essays the last time, and found never a one would hold water, but which way soever we turned it, the Proposition was false. We have two proportions de∣signed only by their Antecedents, and we are to guesse at the consequents. The best conjecture I could make was this; As the excesse of the whole above the whole, is to one of those wholes; so is the ex••esse•• of the part above the part, to one of those parts, (respectively.) That is (calling the greatest whole G, and its part g: and the lesser whole L, and its part l.) as G − L, to G; so g − l, to g. Or se∣condly thus; as G − L, to L; so g − l, to l. But both these are found false. My next conjecture was from the 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, (but there I was fain to leave my proposition quite, and take up new Antecedents, as well as seek new conse∣quents,) and that directs me to such an Analogisme (p. 140. l. 39.) I say that as AH to AB, so AB, to AI; but this is ambiguous, because, AB comming twice, once as a whole, and another time as a part, sit doth not appear which is which; therefore here be two conjectures more; viz. a third thus, as the whole to the whole, so the part to the part; (that is G. L∷ g. l.) Or fourthly thus, as the whole to its part, so the whole to its part. (that is G. g∷ L. l.) But these two are both false also. My next attempt was from the 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, (but here also I must desert the proposition too, and seek new antecedents as well as consequents,) where I find it thus (p. 141. l. 7, 9.) as AH to AB, so is the excesse of AH above AB, to the excesse of AB above AI: which was to be demonstrated. that sends me to a fifth, sixth, seventh and eighth analogisme (because it doth not appear which AB is the whole, and which the part;) the fifth thus, as the whole to the whole, so the excesse of the whole above the whole, to the excesse of the part above the part, (taking AB in the two first places for the whole) that is G. L∷ G − L. g − l. The sixth thus, as the whole to the whole, so the excesse of the whole above its part, to the excesse of the

whole above its part (taking AB in the first and last place, for the whole,) that is G. L∷ G − g. L − l. The se∣venth thus, as the whole to its part, so the excesse of the whole above the whole, to the excesse of the part above the part, (ta∣king AB in the first and last place for a part,) that is G. g∷ G − L. g − l. The eighth thus, as the whole is to its part, so the excesse of the whole above its part, to the excesse of the other whole above its part, (taking AB in the two first places, for the part,) that is G. g∷ G − g. L − l. But these four be all false likewise, as well as those before. Now all these eight conjectures are of equall probability (though all false) it cannot be said which of them is more like to be the sense intended than the other. And yet, forsooth, when, by talking non-sense, you leave us at this uncertainty of conjecture, it is (you say) absurdly done to reprehend it. I confesse, if any one of these Analogismes had been true, we might have guessed that to be your meaning: but when they be all equally probable, and equally false, which should we take? Well, but 'tis to be hoped, that now you will tell us. You tell us therefore (Less. p. 38.) it should be thus, as the excesse of the whole above its part, to the excesse of the other wh••le above its part, so that whole, to this whole: which affords us a ninth analogisme, G − g. L − l∷G. L. which is coincident with my sixth conjecture. And yet again (Less. p. 39.) you tell us, that the proposition is now made (in the English) according to the demonstration (that is; both false,) and there we find it thus, the whole to the whole, as the part to the part; that is G. L∷g. l. which allso is coincident with our third conjecture. But which soever of all these analogismes you take, the Propo∣sition is false, and therefore the demonstration must needs be so too.

Now to prove that this Proposition is false, which way soever you turne it, (either as it was before, or as it is now,) I made use of the figure of your first article, and proceeded to this purpose. Let the whole time (fig. 1.) be AB, an hour, (that is, because I would not have you mi∣stake mee, as you doe Archimedes, let the line AB repre∣sent an hour, or, be the symboll of an hour; for I would not have you think that I take a line to be an hour; but to represent an houre; and the letters AB to represent that line, not to be that line; like as at another time we take

a letter, without a line, to represent an houre:) and part of that time AF, halfe an houre. Let also the continued Impetus of the Uniform motion (I mean the Symboll of it) be AC, or BI: which BI also is to be (the Symbol of) the last acquired Impetus of the motion accelerated. And this acceleration we will suppose at present (as your selfe do in your 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉) to be uniform acceleration. The velocity therefore of the whole uniform motion, will be represented by the Parallelogram ACIB; (by the first article;) and of it's part, ACHF; (by the same ar∣ticle;) The velocity of the whole uniformly accelerated motion, will be the Triangle AIB; and of its part, AKF; (by the same article.) Since therefore the lengths dispatched be proportionall to those velocities; the whole length of uniform motion, to the whole of the accelerate, will be as the Parallelogram ACIB, to the Triangle AIB, that is, as 2 to 1. (viz. the length of the uniform motion, bigger than that of the accelerate; whereas your figure and demonstration, do all the way suppose the con∣trary;) so that if the uniform motion do in an houre di∣spatch 16 yards, the accelerate will in the same time di∣spatch 8 yards, (that is G = 16, & L = 8.) Again, the length dispatched by the uniform motion in the whole time; to that in half the time, is as the Parallelogram ACIB, to the Parallelogram ACHF; that is, as 2 to 1; so that if G (as before) be 16, then is g = 8. Lastly; the length dispatched by the accelerate motion in the whole time, to that in halfe the time, is as the Triangle AIB, to the Triangle AKF; that is as 4 to 1, (for the sides AB, to AF, being as 2 to 1, and the triangles in duplicate proportion to their sides, the triangles will be as 4 to 1:) So that if L (as before) be 8, then is l = 2. Now ha∣ving thus found the measures of these four lengths; (viz. G = 16. L = 8. g = 8. l = 2.) You shall see that those Analogismes are all false; not one true amongst them. The first is this, G − L. G ∷ g − l. g. that is 16 − 8 = 8. 16 ∷ 8 − 2 = 6. 8. or 8. 16 ∷ 6. 8. But this is false. The second this, G − L. L ∷ g − l. l. that is 16 − 8. 8 ∷ 8 − 2. 2. or 8. 8 ∷ 6. 2. But this is false also The third this, G. L ∷ g. l. that is 16. 8 ∷ 8. 2. and this also is false. The fourth this, G. g ∷ L. l. that is 16. 8 ∷ 8. 2. and this is also as false as the other. The

fifth is this, G. L∷G − L. g − l. that is 16. 8∷16 − 8. 8 − 2. or 16. 8∷8. 6. which is also false. The sixth this, G. L∷G − g. L − l. that is 16. 8∷16 − 8. 8 − 2, or 16. 8∷8. 6. which is like the rest. The se∣venth is this G. g∷G − L. g − l. that is, 16. 8∷16 − 8. 8 − 2 or 16. 8∷8. 6. false also. The eighth is this. G. g∷ G − g. L − l. that is 16. 8∷16 − 8. 8 − 2. or 16. 8∷8. 6. which is also false. The ninth is this, G − g. L − l∷G. L. that is 16 − 8. 8 − 2∷16. 8. or 8. 6∷16. 8. The tenth is like the third, G. L∷g. l, that is 16. 8∷8. 2. all false. The proposition therefore, turne it which way you will, is a false Proposition. And yet you have the Impudence to tell us (though you knew this before, for I told it you last time, and brought the same demonstration, to which you have not replied one word) that 'tis all true, and truly demonstrated.

Do you think 'tis worth while after all this, to examine your demonstration? 'Tis a sad one, I confesse; but tis yours, and therefore it may perhaps be beautifull in your eye. The last time we looked upon it, we found it had at least fourteen grosse faults: (and most of them such, as were singly enough to destroy it:) enough in conscience for one poor demonstration. (And had you not been good at it, a man would have wondred how you could have made so many ex tempore.) Since that time, 'tis quite defunct. And there is a young one start up in stead of it. But 'tis of the same breed, and tis not two pence to choose, whether this or that. Your new demonstration runs it self out of breath at the first dash. You had told us (Art. 3. Co∣roll. 3) In motion Ʋniformly accelerated from rest, (such as is one of these) the length transmitted (as here AH, fig 8.) is to another length (viz. AB,) transmitted uniformly in the same time, but with such Impetus as was acquired by the acccle∣rated motion in the last point of that time (just the case in hand) as a Triangle to a Parallelogram which have their alti∣tude and base common, that is, as 1 to 2, for the Parallelogram is double of the Triangle. So that AH, in your figure, should be but just half as bigge as AB; and you have made it all∣most twise as big. And upon this foundation depends the whole demonstration. For if that fault were mended, your whole construction comes to nothing. And is not

this demonstration then well amended? especially when you had faire warning of it the last time.

And then you send us to the demonstration of the 13 Article for confirmation of this, whereas that Article hath been cashiered long agoe, and the demonstration with it. But thus 'tis when men will not take warning.

At length you fall to raating, (as you use to do when you be vexed;) about skill, and diligence, and too much trusting; about discretion, Hyperbole's, and Sir H. Savile, &c. And tell us that when a beast (Joseph Scaliger) is slain by a Lion (Clavius) 'tis easy for any of the fowles of the aire (Sir H. S.) to settle upon, and peck him. And Vespasian's law, no doubt, will bear you out in all this. Only this I must tell you, that Sir H. Savile, had confuted Joseph Scaliger's Cyclometry, as well as Clavius; and, I suppose, before him. Which if you have not seen, I have.

In the 18 Article, we have this Proposition. If, in any Parallelogram, (suppose ACDB, fig. 11.) two sides contain∣ing an angle be moved to the sides opposite to them, (as AB to CD, and AC to BD,) one of them (AB) with uniform motion, the other (AC) with motion uniformly accelerated: that side which is moved uniformly (AB) will effect as much, with its concurse through the whole length, as it would do if the other motion were also uniform, (or were not at all. For what e∣ver the other motion be, the motion of AB to CD, car∣rieth the thing moved with it from side to side, and that's all. What point of the opposite side it shall come to, de∣pends upon the other transverse motion, not upon this at all. And this is so easy that no body would deny it. If you mean any thing more then this, that it shall carry it just to the opposite side & no farther, your demonstration doth not at all reach it▪ But you go on) and the length transmitted by it in the same time, a mean proportionall between the whole and the halfe, of what? Till you tell us of what? I say, as I said before, that these words have no sense.

The construction and demonstration of this propositi∣on, I remember, we made sad work with, the last time we had to doe with them, as well as with those of the for∣mer Article; which will be now too long to repeat. The whole weight of the Demonstration lies, severally, upon at lest these three Pillars, of which if any one do but fail, the whole demonstration falls. First, upon the strength

of the 13 Article, which we have destroyed long agoe. Secondly upon the 12 Article, which we have also long since proved to be false. Thirdly, upon this learned as∣sertion, the streight line FB will be the excesse by which the (lesser) length transmitted by AC with motion uniform∣ly accelerated, till it acquire the impetus BD, will exceed the (greater) length transmitted by the same AC in the same time with uniform motion, and with the Impetus every where equall to BD. Which destroys it selfe. For if the accele∣rated motion, as is supposed, do not till its last moment acquire that speed with which the uniforme motion is moved all the way; then that must needs be slower than this; and consequently dispatch a lesser length in the same time: whereas you according to your discretion, make the length dispatched by that slower motion, to be more then that of the swifter in the same time, and tell us the excesse is FB. And then to helpe the matter, when I presse you with this absurdity, you tell us you speak of motions in concurse: as though in concurse, the slower motion did in the same time, caeteris paribus, dispatch a greater length than the swifter, though out of concurse the swifter motion did dispatch a greater length than the slower▪ Now either of these three, much more all of them, doth wholly destroy the strength of your demon∣stration. Yet they that desire to see more may consult what I sayd before.

The ninteenth Article doth not pretend to any other strength than that of the eighteenth. And therefore falls with it.

The twentieth Article I did before prove to be false and frivolous. (it depended upon Chap. 14. Art. 15. Corol. 3. which Corrollary I have there consuted.) You say nothing by way of vindication of what I excepted against; only passe your word for it, that it is true. Yet withall confesse, there is a great error; and that error say I, though there were nothing else, would make that article unsound. But this article you say, was never published (yet 'tis as good as most of those that were in this Chapter; for i'le undertake for it, there he above a dozen worse;) and therefore it was inhuman∣ly done, you say, to take notice of it. Truly, if the proposi∣tion were a good proposition, as you say it was. J think J

did you a courtesy to publish it for you, that you may have the credit of it; yet J should not have done it, had it not been publike before. If you would not have it ta∣ken notice of, you should have taken care not to send it abroad. For it hath been commonly sold with the rest of your book (to many more persons beside my selfe;) they that would, might teare it out (as some did) and they that would, might keep it in, as J did.

Well, (be the number of articles 20, or be they 19,) before the sixth there was none sound, (but either in whole or in part unsound,) and from the eighth there hath been none sound; therefore there have not been above three sound at the most. Quod erat demonstrandum.