Chapter X (4)

The nineteenth article of mine in this fourteenth chapter, is this: “_All respect or variety of position of two lines, seemeth to be comprehended in four kinds_. For they are either _parallel_, or (_being if need be produced_) _make an angle_; or, (if drawn out far enough) _touch_; or, lastly, they are _asymptotes_”; in which you are first offended with the word _It seems_. But I allow you, that never err, to be more peremptory than I am. For to me it seemed (I say again seemed) that such a phrase, in case I should leave out something in the enumeration of the several kinds of position, would save me from being censured for untruth; and yet your instance of two straight lines in divers planes, does not make my enumeration insufficient. For those lines, though not parallels, nor cutting both the planes, yet being moved parallelly from one plane to another, will fall into one or other of the kinds of position by me enumerated; and consequently, are as much that position, as two straight lines in the same plane, not parallel, make the same angle, though not produced till they meet, which they would make if they were so produced: for you have nowhere proved, nor can prove, that two such lines do not make an angle. It is not the actual concurrence of the lines, but the arch of a circle, drawn upon that point for centre, in which they would meet if they were produced, and intercepted between them, that constitutes the angle.

Also your objection concerning asymptotes _in general_ is absurd. You would have me add, that _their distance shall at last be less than any distance that can be assigned_; and so make the definition of the _genus_ the same with that of the _species_. But because you are not professors of logic, it is not necessary for me to follow your counsel. In like manner, if we understand one line to be moved towards another always parallelly to itself, which is, though not actually, yet potentially the same position, all the rest of your instances will come to nothing.

At the two-and-twentieth article you object to me the use of the word _figure_, before I had defined it: wherein also you do absurdly; for I have nowhere before made such use of the word _figure_, as to argue anything from it; and therefore your objection is just as wise as if you had found fault with putting the word figure in the titles of the chapters placed before the book. If you had known the nature of demonstration, you had not objected this.

You add further, that by my definition of _figure_, a solid sphere, and a sphere made hollow within, is the same figure; but you say not why, nor can you derive any such thing from my definition. That which deceived your shallowness, is, that you take those points that are in the concave superficies of a hollowed sphere, not to be contiguous to anything without it, because that whole concave superficies is within the whole sphere. Lastly, for the fault you find with the definition of _like figures in like positions_, I confess there wants the same word which was wanting in the definition of parallels; namely, _ad easdem partes_ (_the same way_) which should have been added in the end of the definition of like figures, &c., and may easily be supplied by any student of geometry, that is not otherwise a fool.

At the fifteenth chapter, art. 1, number 6, you object as a contradiction, that _I make motion to be the measure of time; and yet, in other places, do usually measure motion and the affections thereof by time_. If your thoughts were your own, and not taken rashly out of books, you could not but, (with all men else that see time measured by clocks, dials, hour-glasses, and the like), have conceived sufficiently, that there cannot be of time any other measure besides motion; and that the most universal measure of motion, is a line described by some other motion; which line being once exposed to sense, and the motion whereby it was described sufficiently explicated, will serve to measure all other motions and their time: for time and motion (time being but the mental image or remembrance of the motion) have but one and the same dimension, which is a line. But you, that would have me measure _swiftness_ and _slowness_ by longer and shorter motion, what do you mean by _longer_ and _shorter motion_? Is _longer_ and _shorter_ in the motion, or in the duration of the motion, which is time? Or is the motion, or the duration of the motion, that which is exposed, or designed by a line? Geometricians say often, _let the line A B be the time_; but never say, _let the line A B be the motion_. There is no unlearned man that understandeth not what is time, and motion, and measure; and yet you, that undertake to teach it (most egregious professors) understand it not.

At the second article you bring another argument (which it seems in its proper place you had forgotten), to prove that a point is not quantity not considered, but absolutely nothing; which is this, _That if a point be not nothing, then the whole is greater than its two halves_. How does that follow? Is it impossible when a line is divided into two halves, that the middle point should be divided into two halves also, being quantity?

At the seventh article, I have sufficiently demonstrated, that all motion is infinitely propagated, as far as space is filled with body. You allege no fault in the demonstration, but object from sense, _that the skipping of a flea is not propagated to the Indies_. If I ask you how you know it, you may wonder perhaps, but answer you cannot. Are you philosophers, or geometricians, or logicians, more than are the simplest of rural people? or are you not rather less, by as much as he that standeth still in ignorance, is nearer to knowledge, than he that runneth from it by erroneous learning?

And, lastly, what an absurd objection is it which you make to the eighth article, where I say that _when two bodies of equal magnitude fall upon a third body, that which falls with greater velocity, imprints the greater motion_? You object, _that not so much the magnitude is to be considered as the weight_; as if the weight made no difference in the velocity, when notwithstanding weight is nothing else but motion downward. Tell me, when a weighty body thrown upwards worketh on the body it meeteth with, do you not then think it worketh the more for the greatness, and the less for the weight.

==========

OF THE FAULTS THAT OCCUR IN
DEMONSTRATION.

TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN
THE UNIVERSITY OF OXFORD.

LESSON IV.

Of twenty articles which you say (of nineteen which I say) make the sixteenth chapter, you except but three, and confidently affirm the rest are false. On the contrary, except three or four faults, such as any geometrician may see proceed not from ignorance of the subject, or from want of the art of demonstration, (and such as any man might have mended of himself) but from security; I affirm that they are all true, and truly demonstrated; and that all your objections proceed from mere ignorance of the mathematics.

The first fault you find is this, that I express not (art. 1) what _impetus_ it is, which I would have to be multiplied into the time.

The last article of my thirteenth chapter was this, “_If there be a number of quantities propounded, howsoever equal or unequal to one another; and there be another quantity which so often taken as there be quantities propounded, is equal to their whole sum; that quantity I call the mean arithmetical of them all_.” Which definition I did there insert to serve me in the explication of those propositions of which the sixteenth chapter consisted, but did not use it here as I intended. My first proposition therefore as it standeth yet in the Latin, being 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, multiplied into the whole time_;” to a man that hath not skill enough to supply what is wanting, is not intelligible. Therefore I have caused it in the English to go thus: “_the velocity of any body in whatsoever time moved, hath its quantity determined by the sum of all the several_ (impetus) _quicknesses, which it hath in the several points of the time of the body’s motion_. And added, _that all the_ impetus _together taken through the whole time is the same thing with the mean_ impetus (which mean is defined (Chapter XIII. art. 29) _multiplied into the whole time_.” To this first article, as it is uncorrected in the Latin, you object, _that meaning by_ impetus _some middle_ impetus, _and assigning none, I determine nothing_. And it is true. But if you had been geometricians sufficient to be professors, you would have shewed your skill much better, by making it appear that this middle _impetus_ could be none but that, which being taken so often, as there be points in the line of time, would be equal to the sum of all the several _impetus_ taken in the points of time respectively; which you could not do.

To the _corollary_, you ask first how _impetus_ can be ordinately applied to a line; absurdly. For does not Archimedes sometimes say, and with him many other excellent geometricians, _let such a line be the time_? And do they not mean, that that line, or the motion over it, is the measure of the time? And may not also a line serve to measure the swiftness of a motion? _You thought_, you say, _only lines ought to be said to be ordinately applied to lines_. Which I easily believe; for I see you understand not that a line, though it be not the time itself, may be the quantity of a time. You thought also, all you have said in your _Elenchus_ , in your doctrine of the _angle of contact_, in your _Arithmetica Infinitorum_, and in your _Conics_ , is true; and yet it is almost all proved false, and the rest nothing worth.

Secondly, you object, that _I design a parallelogram by one only side_. It was indeed a great oversight, and argueth somewhat against the man, but nothing against his art. For he is not worthy to be thought a geometrician that cannot supply such a fault as that, and correct his book himself. Though you could not do it, yet another from beyond sea took notice of the same fault in this manner, “_He maketh a parallelogram of but one side_; it should be thus: _vel denique per parallelogrammum cujus unum latus est medium proportionale inter impetum maximum (sive ultimo acquisitum) et impetus ejusdem maximi semissem; alterum vero latus, medium proportionale, inter totum tempus, et ejusdem totius temporis semissem_.” Which I therefore repeat, that you may learn good manners; and know, that they who reprehend, ought also, when they can, to add to their reprehension the correction.

At the second article, you are pleased to advise me, instead of _in omni motu uniformi_, to put in _in omnibus motibus uniformibus_. You have a strange opinion of your own judgment, to think you know to what end another man useth any word, better than himself. My intention was only to consider motions uniform, and motions from rest uniformly, or regularly accelerated, that I might thereby compute the lengths of crooked lines, such as are described by any of those motions. And therefore it was enough to prove this theorem to be true in all uniform or uniformly accelerated _motion_, not _motions_; though it be true also in the plural. It seems you think a man must write all he knows, whether it conduce, or not, to his intended purpose. But that you may know that I was not (as you think), ignorant how far it might be extended, you may read it demonstrated at the same article in the English universally. Against the demonstration itself you run into another article, namely, the thirteenth, which is this problem: “_the length being given, which is passed over in a given time by uniform motion, to find the length which shall be passed over by motion uniformly accelerated in the same time, so as that the_ impetus _last acquired be equal to the time_.” Which you recite imperfectly, thereby to make it seem that such a length is not determined. Whether you did this out of ignorance, or on purpose, thinking it a piece of wit, as your pretended mystery which goes immediately before, I cannot tell, for in neither place can any wit be espied by any but yourselves. To imagine motions with their times and ways, is a new business, and requires a steady brain, and a man that can constantly read in his own thoughts, without being diverted by the noise of words. The want of this ability, made you stumble and fall unhandsomely in the very first place (that is in Chap. XIII. art. 13), where you venture to reckon both motion and time at once; and hath made you in this chapter to stumble in the like manner at every step you go. As, for example, when I say, _as the product of the time, and impetus, to the product of the time and impetus, so the space to the space when the motion is uniform_; you come in with, _nay, rather as the time to the time_; as if the parallelograms A I, and A H, were not also as the times A B, and A F. Thus it is, when men venture upon ways they never had been in before, without a guide.

In the corollary, you are offended with the permutation of the proportion of times and lines, because you think, (you that have scarce one right thought of the principles of geometry), that line and time are heterogeneous quantities. I know time and line are of divers natures; and more, that neither of them is _quantity_. Yet they may be both of them _quanta_, that is, they may _have quantity_; but that their quantities are heterogeneous is false. For they are compared and measured both of them by straight lines. And to this there is nothing contrary in the place cited by you out of Clavius; or if there were, it were not to be valued. And to your question, what is the proportion of an _hour_ to an _ell_? I answer, it is the same proportion that _two hours_ have to _two ells_. You see your question is not so subtle as you thought it. By and bye you confess that in times and lines there is _quid homogeneum_ (this _quid_ is an infallible sign of not fully understanding what you say); which is false if you take it of the lines themselves; though if you take it of their quantities, it is true without a _quid_. Lastly, you tell m”e how I might have expressed myself so as it might have been true. But because your expressions please me not, I have not followed your advice.

To the third article, which is this: “_In motu uniformiter a quiete accelerato_,” _etc._ “_In motion uniformly 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 another time, as the product of the impetus multiplied by the time, to the product of the impetus multiplied by the time_;” you object, “_that the lengths run over are in that proportion which the impetus hath to the impetus; not that which the impetus hath to the time, because impetus to time has no proportion, as being heterogeneous_.” First, when you say the impetus, do you mean some one impetus designed by some one of the unequal straight lines parallel to the base B I? That is manifestly false. You mean the aggregate of all those unequal parallels. But that is the same thing with the time multiplied into the mean impetus. And so you say the same that I do. Again, I ask, where it is that I say or dream that the lengths run over are in the proportion of the impetus to the times? Is it you or I that dream? And for your heterogeneity of the quantities of time and of swiftness, I have already in divers places showed you your error. Again, why do you make B I represent the lengths run over, which I make to be represented by D E, a line taken at pleasure; and you also a few lines before make the same B I to design the greatest acquired impetus? These are things which show that you are puzzled and entangled with the unusual speculation of time and motion, and yet are thrust on with pride and spite to adventure upon the examination of this chapter.

Secondly, you grant the demonstration to be good, supposing I mean it, as I seem to speak, of one and the same motion. But why do I not mean it of one and the same motion, when I say not in _motions_, but in _motion_ uniform? _Because_, say you, _in that which follows, I draw it also to different motions_. You should have given at least one instance of it; but there is no such matter. And yet the proposition is in that case also true; though then it must not be demonstrated by the similitude of triangles, as in the case present. And therefore the objections you make from different impetus acquired in the same time, and from other cases which you mention, are nothing worth.

At the fourth article, you allow the demonstration all the way (except the faults of the third, which I have already proved to be none) till I come to say, “_that because the proportion of F K to B I is double to the proportion of A F to A B, therefore the proportion of A B to A F is double to the proportion of B I to F K_.” This you deny, and wonder at as strange, (for it is indeed strange to you), and in many places you exclaim against it as extreme ignorance in geometry. In this place you only say, “_no such matter; for though one proportion be double to another, yet it does not follow that the converse is the double of the converse_.” So that this is the issue to which the question is reduced, whether you have any or no geometry. I say, if there be three quantities in continual proportion, and the first be the least, the proportion of the first to the second is double to the proportion of the first to the third; and you deny it. The reason of our dissent consisteth in this, that you think the doubling of a proportion to be the doubling of the quantity of the proportion, as well in proportions of defect, as in proportions of excess; and I think that the doubling of a proportion of defect, is the doubling of the defect of the quantity of the same. As for example in these three numbers, 1, 2, 4, which are in continual proportion, I say the quantity of the proportion of one to two, is double the quantity of the proportion of one to four. And the quantity of the proportion of one to four, is half the quantity of the proportion of one to two. And yet deny not but that the quantity of the defect in the proportion of one to two is doubled in the proportion of one to four. But because the doubling of defect makes greater defect, it maketh the quantity of the proportion less. And as for the part which I hold in this question, first, there is thus much demonstrated by Euclid, El. v. prop. 8; that the proportion of one to two, is greater than the proportion of one to four, though how much it is greater be not there demonstrated. Secondly, I have demonstrated (Chap, XIII. art. 16); that it is twice as great, that is to say, (to a man that speaks English), double. The introducing of _duplicate_, _triplicate_, &c. instead of _double_, _triple_, &c. (though now they be words well understood by such as understand what proportion is), proceeded at first from such as durst not for fear of absurdity, call the half of any thing double to the whole, though it be manifest that the half of any defect is a double quantity to the whole defect; for want added to want maketh greater want, that is, a less positive quantity. This difference between _double_ and _duplicate_, lighting upon weak understandings, has put men out of the way of true reasoning in very many questions of geometry. Euclid never used but one word both for _double_ and _duplicate_. It is the same fault when men call half a quantity _subduplicate_, and a third part _subtriplicate_ of the whole, with intention (as in this case) to make them pass for words of signification different from the _half_ and the _third part_. Besides, from my definition of proportion (which is clear, and easy to be understood by all men, but such as have read the geometry of others unluckily) I can demonstrate the same evidently and briefly thus. My definition is this, _proportion is the quantity of one magnitude taken comparatively to another_. Let there be therefore three quantities, 1, 2, 4, in continual proportion. Seeing therefore the quantity of four in respect of one, is twice as great as the quantity of the same four in respect of 2, it followeth manifestly that the quantity of 1 in respect of 4, is twice as little as the quantity of the same 1 in respect of 2; and consequently the quantity of 1 in respect of 2, is twice as great as the quantity of the same 1 in respect of 4; which is the thing I maintain in this question. Would not a man that employs his time at bowls, choose rather to have the advantage given him of three in nine, than of one in nine? And why, but that three is a greater quantity in respect of nine, than is one? Which is as much as to say, three to nine hath a greater proportion than one to nine; as is demonstrated by Euclid, El. v. prop. 8. Is it not therefore (you that profess mathematics, and theology, and practise the depression of the truth in both) well owled of you, to teach the contrary? But where you say “_that the point K_ (in the second figure of the table belonging to this sixteenth chapter) _is not in the parabolical line whose diameter is A B, and base B I, but in the parabolical line of the complement of my semiparabola_ (_as I may learn from the twenty-third proposition of your_ Arithmetica Infinitorum) _whose diameter is A C, and base I C_.” What line is that? Is it the same line with that of my semiparabola, or not the same? If the same, why find you fault? If not the same, you ought to have made a semiparabola on the diameter A C, and base I C, and following my construction made it appear that K is not in the line wherein I say it is; which you have not done, nor could do.

Then again, running on in the same blindness of passion, you pretend I make the proportion of B I to F K double to that of A B to A F, and then confute it; when you knew I made the proportion of F K to B I, double to that of F N, to B I, that is, of A F to A B; and this was it you should have confuted. That which followeth is but a triumphing in your own ignorance, wherein you also say, “_that all that I afterwards build upon this doctrine is false_.” You see whether it be like to prove so or not. As for your _Arithmetica Infinitorum_, I shall then read to you a piece of a lesson on it when I come to your objections against the next Chapter. In the mean time let me tell you, it is not likely you should be great geometricians, that know not what is quantity, nor measure, nor straight, nor angle, nor homogeneous, nor heterogeneous, nor proportion, as I have already made appear in this and the former lessons.

To the first corollary of this fourth article your exception I confess is just, and (which I wonder at) without any incivility. But this argues not ignorance, but security. For who is there that ever read any thing in the Conics, that knows not that the parts of a parabola cut off by lines parallel to the base, are in triplicate proportion to their bases? But having hitherto designed the time by the diameter, and the impetus by the base; and in the next chapter (where I was to calculate the proportion of the parabola, to the parallelogram) intending to design the time by the base, I mistook and put the diameter again for the time; which any man but you might as easily have corrected as reprehended.

To the second corollary, which is this, _that the lengths run over in equal times by motion so accelerated, as that the impetus increase in double proportion to their times, are as the differences of the cubic numbers beginning at unity, that is, as seven, nineteen, thirty-seven, &c._ you say it is false. But why? “_Because_” say you “_portions of the parabola of equal altitude, taken from the beginning, are not as those numbers seven, nineteen, thirty-seven, &c._” Does this, think you, contradict any thing in this proposition of mine? Yes, because, you think, the lengths gone over in equal times, are the same with the parts of the diameter cut off from the vertex, and proportional to the numbers one, two, three, &c. Whereas the lengths run over, are as the aggregates of their velocities, that is, as the parts of the parabola itself, that is, as the cubes of their bases, that is, as the numbers one, eight, twenty-seven, sixty-four, &c., and consequently the lengths run over in equal times, are as the differences of those cubic numbers, one, eight, twenty-seven, sixty-four, whose differences are seven, nineteen, thirty-seven, &c. The cause of your mistake was, that you cannot yet, nor perhaps ever will, contemplate time and motion (which requireth a steady brain) without confusion.

The third corollary you also say is false, “_whether it be meant of motion uniformly accelerated_ (as the words are) _or_ (_as perhaps_, you say, _I meant it_) _of such motion as is accelerated in double proportion to the time_.” You need not say perhaps I meant it. The words of the proposition are enough to make the meaning of the corollary understood. But so also you say it is false. Methinks you should have offered some little proof to make it seem so. You think your authority will carry it. But on the contrary I believe rather that they that shall see how your other objections hitherto have sped, will the rather think it true, because you think it false. The demonstration as it is, is evident enough; and therefore I saw no cause to change a word of it.

To the fifth article you object nothing, but that it dependeth on this proposition (Chap. XIII. art. 16): “_That when three quantities are in continual proportion, and the first is the least, as in these numbers, four, six, nine, the proportion of the first to the second, is double to the proportion of the same first to the last_;” which is there demonstrated, and in the former lessons so amply explicated, as no man can make any further doubt of the truth of it. And you will, I doubt not, assent unto it. But in what estate of mind will you be then? A man of a tender forehead after so much insolence, and so much contumelious language grounded upon arrogance and ignorance, would hardly endure to outlive it. In this vanity of yours, you ask me whether I be angry, or blush, or can endure to hear you. I have some reason to be angry; for what man can be so patient as not to be moved with so many injuries? And I have some reason to blush, considering the opinion men will have beyond sea, (when they shall see this in Latin) of the geometry taught in Oxford. But to read the worst you can say against me, I can endure, as easily at least, as to read any thing you have written in your treatises of the _Angle of Contact_, of the _Conic Sections_, or your _Arithmetica Infinitorum_.

The sixth, seventh, eighth articles, you say are sound. True. But never the more to be thought so for your approbation, but the less; because you are not fit, neither to reprehend, nor praise; and because all that you have hitherto condemned as false, hath been proved true. Then you show me how you could demonstrate the sixth and seventh articles a shorter way. But though there be your symbols, yet no man is obliged to take them for demonstration. And though they be granted to be dumb demonstrations, yet when they are taught to speak as they ought to do, they will be longer demonstrations than these of mine.

To the ninth article, which is this, “_If a body be moved by two movents at once, concurring in what angle soever, of which, one is moved uniformly, the other, with motion uniformly accelerated from rest, till it acquire an impetus equal to that of the uniform motion, the line in which the body is carried, shall be the crooked line of a semiparabola_,” you lift up your voice again, and ask, _what latitude? what diameter? what inclination_ of the diameter to the ordinate lines? If your founder should see this, or the like objections of yours, he would think his money ill bestowed. When I say, _in what angle soever_, you ask, _in what angle?_ When I say _two movents, one uniform, the other uniformly accelerated, make the body describe a semiparabolical line_; you ask, _which is the diameter?_ as not knowing that the accelerated motion describes the diameter, and the other a parallel to the base. And when I say _the two movents meet in a point, from which point both the motions begin, and one of them from rest_, you ask me _what is the altitude?_ As if that point where the motion begins from rest were not the vertex; or that the vertex and base being given, you had not wit enough to see that the altitude of the parabola is determined? When Galileo’s proposition, which is the same with this of mine, supposed no more but a body moved by these two motions, to prove the line described to be the crooked line of a semiparabola, I never thought of asking him what altitude, nor what diameter, nor what angle, nor what base, had his parabola. And when Archimedes said, let the line A B be the time, I should never have said to him, _do you think time to be a line_, as you ask me whether I think impetus can be the base of a parabola. And why, but because I am not so egregious a mathematician, as you are. In this giddiness of yours, caused by looking upon this intricate business of motion, and of time, and the concourse of motion uniform, and uniformly accelerated, you rave upon the numbers 1, 4, 9, 16, &c. without reference to any thing that I had said; insomuch as any one that had seen how much you have been deceived in them before, in your scurvy book of _Arithmetica Infinitorum_, would presently conclude, that this objection was nothing else but a fit of the same madness which possessed you there.

My tenth article is like my ninth; and your objections to it are the same which are to the former. Therefore you must take for answer just the same which I have given to your objection there.

To the eleventh, you say first, you have done it better at the sixty-fourth article of your _Arithmetica Infinitorum_. But what you have done there, shall be examined when I come to the defence of my next chapter. And whereas I direct the reader for the finding of the proportions of the complements of those figures to the figures themselves, to the table of art. 3, Chap, XVII., you say that if the increase of the _spaces_, were to the increase of the times, as one to two, then the complement should be to the parallelogram as one to three, and say you find not (1)/(3) in the table. Did you not see that the table is only of those figures which are described by the concourse of a motion uniform with a motion accelerated? You had no reason therefore to look for (1)/(3) in that table; for your case is of motion uniform concurring with motion retarded, because you make not the proportions of the spaces to the proportions of the times as two to one, but the contrary; so that your objection ariseth from want of observing what you read. But I “_may learn_” you say, “_these, and greater matters than these, in your twenty-third and sixty-fourth propositions of your_ Arithmetica Infinitorum.” This, which you say here is a great absurdity; but if you mean I shall find greater there, I will not say against you. This (1)/(3) you looked for, belongs to the complements of the figures calculated in that table; which because you are not able to find out of yourselves, I will direct you to them. Your case is of (1)/(3) for the complement of a parabola. Take the denominator of the fraction which belongs to the parabola, namely three, and for numerator take the numerator of the fraction which belongs to the triangle, namely one, and you have the fraction sought. And in like manner for the complement of any other figure. As, for example, of the second parabolaster, whose fraction hath for denominator five, take the numerator of the fraction of the same triangle which is one, and you have (1)/(3) for the fraction sought for; and so of the rest, taking always one for the numerator.

The twelfth article, which you say is miserably false, I have left standing unaltered. For not comprehending the sense of the proposition, you make a figure of your own, and fight against your own fancied motions, different from mine. Other geometricians that understand the construction better, find no fault. And if you had in your own fifth figure drawn a line through N parallel to A E, and upon that line supposed your accelerated motion, you would quickly have seen that in the time A E, the body moved from rest in A, would have fallen short of the diagonal A D; and that all your extravagant pursuing of your own mistake had been absurd.

My thirteenth article you say is ridiculous. But why? “_The impetus last acquired cannot_” you say, “_be equal to a time_.” But the quantity of the impetus may be equal to the quantity of a time, seeing they are both measured by line. And when they are measured by the same described line, each of their quantities is equal to that same line, and consequently to one another. But when I meet with this kind of objection again, since I have so often already shown it to be frivolous, and no less to be objected against all the ancients that ever demonstrated any thing by motion, than against me, I purpose to neglect it.

Secondly, you object “_that motion uniformly accelerated does no more determine swiftness, than motion uniform_.” True; you needed not have used sixteen lines to set down that. But suppose I add, as I do, so as the last acquired impetus be equal to the time. _But that_, you say, _is not sense_; which is the objection I am to neglect. But, you say again, supposing it sense, this limitation helps me nothing. Why? _Because_, you say, _a parabola may be described upon a base given, and yet have any altitude, or any diameter one will_. Who doubts it? But how follows it from thence, that when a parabolical line is described by two motions, one uniform, the other uniformly accelerated from rest, that the determining of the base does not also determine the whole parabola? But fifthly, you say, _that this equality of the impetus to the time does not determine the base_. Why not? _Because_, you say, _it is an error proceeding from this, that I understand not what is_ ratio subduplicata. I looked for this. I have shown and inculcated sufficiently before, but the error is on your side; and therefore must tell you, that this objection, and also a great part of the rest of your errors in geometry, proceedeth from this, that you know not what proportion is. But see how wisely you argue about this duplication of proportion. For thus you say _verbatim_. “_Stay a little. What proportion has duplicate proportion to single proportion? Is it always the same? I think not for example, duplicate proportion_ (4)/(1) = (2 in 2)/(1 in 1) _is double to the single_ (2)/(1). _Duplicate proportion_ (9)/(1) = (3 in 3)/(1 in 1) _is triple to its single_ (3)/(1).” Let any man, even of them that are most ready in your symbols, say in your behalf (if he be not ashamed) that the proportion of nine to one is triple to the proportion of three to one, as you do.

In the fourteenth, fifteenth, and sixteenth articles, you bid me repeat your objections to the thirteenth. I have done it; and find that what you have objected to the thirteenth, may as well be objected to these; and consequently, that my answer there will also serve me here. Therefore, if you can endure it, read the same answer over again.

But you have not yet done, you say, with these articles. Therefore (after you had for a while spoken perplextly, conjecturing, not without just cause, that I could not understand you) you say that to the end I may the better perceive your meaning, I should take the example following. “_Let a movent (in the first figure of this chapter) be moved uniformly in the time A B, with the continual impetus A C, or B I, whose whole velocity shall therefore be the parallelogram A C I B. And another movent be uniformly accelerated, so as in the time A B it acquire the same impetus B I. Now as the whole velocity, is to the whole velocity, so is the length run over, to the length run over._” All this I acknowledge to be according to my sense, saving that your putting your word _movens_ instead of my word _mobile_ hath corrupted this article. For in the first article, I meddle not with motion by concourse, wherein only I have to do with two movents to make one motion; but in this I do, wherein my word is not _movens_ but _mobile_; by which it is easy to perceive you understand not this proposition. Then you proceed: “_But the length run over by that accelerated motion is greater than the length run over by that uniform motion._” Where do I say that? You answer, “_in the ninth and thirteenth article, in making A B (in the fifth figure) greater than A C; and A H (in the eighth figure) greater than A B; and consequently, the triangle A B I, greater than the parallelogram A C I B_.” That consequently is without consequence; for it importeth nothing at all in this demonstration, whether A B, or A C in the fifth figure be the greater. Besides I speak there of the concourse of two movents, that describe the parabolical line A G D; where the increasing impetus (because it increaseth as the times) will be designed by the ordinate lines in the parabola A G D B. And if both the motions in A B and A C were uniform, the aggregate of the impetus would be designed by the triangle A B D, which is less than the parallelogram A C D B. But you thought that the motion made by A C uniformly, is the same with the motion made uniformly in the same time by the motions in A B and A C concurring; so likewise, in the eighth figure, there is nothing hinders A H from being greater than A B, unless I had said that A B had been described in the time A C with the whole impetus A C maintained entire; of which there is nothing in the proposition, nor would at all have been pertinent to it. Therefore all this new undertaking of the thirteenth, fourteenth, fifteenth, and sixteenth articles, is to as little purpose as your former objections. But I perceive that these new and hard speculations, though they turn the edge of your wit, turn not the edge of your malice.

At the seventeenth article, you show again the same confusion. Return to the eighth figure: “_if in a time given a body run over two lengths, one with uniform, the other with accelerated motion_”; as for example, if in the same time A C, a body, run over the line A B with uniform motion, and the line A H with motion accelerated; “_and again in a part of that time it run over a part of the length A H, with uniform motion, and another part of the same with motion accelerated_;” as for example, in the time A M it run over with uniform motion the line A I, and with motion accelerated the line A B. _I say the excess of the whole A H above the part A B, is to the excess of the whole A B above the part A I, as the whole A H to the whole A B._ But first you will say, that these words _as the whole A H to the whole A B_, are left out in the proposition. But you acknowledge that it was my meaning; and you see it is expressed before I come to the demonstration. And therefore it was absurdly done to reprehend it. Let us therefore pass to the demonstration. Draw I K parallel to A C, and make up the parallelogram A I K M. And supposing first the acceleration to be uniform, divide I K in the midst at N; and between I N, and I K, take a mean proportional I L. _And the straight line A L, drawn and produced, shall cut the line B D in F, and the line C G in G_ (which lines C G, and B D, as also H G and B F, are determined, though you could not carry it so long in memory, by the demonstration of the thirteenth article). _For seeing A B is described by motion uniformly accelerated, and A I by motion uniform in the same time A M; and I L is a mean proportional between I N (the half of I K) and I K; therefore by the demonstration of the thirteenth article, A I is a mean proportional between A B and the half of A B, namely A O. Again, because A B is described by uniform motion, and A H by motion uniformly accelerated, both of them in the same time A C, B F is a mean proportional between B D and half B D, namely B E; therefore by the demonstration of the same thirteenth article, the straight line A L F produced will fall on G; and the line A H will be to the line A B, as the line A B to the line A I. And consequently as A H to A B, so H B to B I; which was to be demonstrated._ And by the like demonstration the same may be proved, where the acceleration is in any other proportion that can be assigned in numbers, saving that whereas this demonstration dependeth on the construction of the thirteenth article, if the motion had been accelerated in double proportion to the times, it would have depended on the fourteenth, where the lines are determined. Which determinations being not repeated, but declared before, in the thirteenth article, to which this diagram belongeth, you take no notice of, but go back to a figure belonging to another article, where there was no use of these determinations. But because I see that the words of the proposition, are as of four motions, and not of two motions made by twice two movents, I must pardon them that have not rightly understood my meaning; and I have now made the proposition according to the demonstration. Which being done, all that you have said in very near two leaves of your _Elenchus_ comes to nothing; and the fault you find comes to no more than a too much trusting to the skill and diligence of the reader. And whereas after you had sufficiently troubled yourself upon this occasion, you add, “_that if Sir H. Savile had read my Geometry, he had never given that censure of Joseph Scaliger, in his lecture upon Euclid, that he was the worst geometrician of all mortal men, not exceptioning so much as Orontius, but that praise should have been kept for me_.” You see by this time, at least others do, how little I ought to value that opinion; and that though I be the least of geometricians, yet my geometry is to yours as 1 to 0. I recite these words of yours, to let the world see your indiscretion in mentioning so needlessly that passage of your founder. It is well known that Joseph Scaliger deserved as well of the state of learning, as any man before or since him; and that though he failed in his ratiocination concerning the quadrature of the circle, yet there appears in that very failing so much knowledge of geometry, that Sir H. Savile could not but see that there were mortal men very many that had less; and consequently he knew that that censure of his in a rigid sense (without the license of an hyperbole) was unjust. But who is there that will approve of such hyperboles to the dishonour of any but of unworthy persons, or think Joseph Scaliger unworthy of honour from learned men? Besides, it was not Sir H. Savile that confuted that false quadrature, but Clavius. What honour was it then for him to triumph in the victory of another? When a beast is slain by a lion, is it not easy for any of the fowls of the air to settle upon, and peck him? Lastly, though it were a great error in Scaliger, yet it was not so great a fault as the least sin; and I believe that a public contumely done to any worthy person after his death, is not the least of sins. Judge therefore whether you have not done indiscreetly, in reviving the only fault, perhaps that any man living can lay to your founder’s charge; and yet this error of Scaliger’s was no greater than one of your own of the like nature, in making the true spiral of Archimedes equal to half the circumference of the circle of the first revolution; and then thinking to cover your fault by calling it afterwards an aggregate of arches of circles (which is no spiral at all of any kind) you do not repair but double the absurdity. What would Sir Henry Savile have said to this?

The eighteenth article is this, “_in any parallelogram, if the two sides that contain the angle be moved to their opposite sides, the one uniformly, the other uniformly accelerated; the side that is moved uniformly, by its concourse through all its longitude, hath the same effect which it would have if the other motion were also uniform, and the line described were a mean proportional between the whole length, and the half of the same_.”

To the proposition you object first, “_that it is all one whether the other motion be uniform or not, because the effect of each of their motions, is but to carry the body to the opposite side_.” But do you think that whatsoever be the motions, the body shall be carried by their concourse always to the same point of the opposite side? If not, then the effect is not all one when a motion is made by the concourse of two motions uniform and accelerated, and when it is made by the concourse of two uniform or of two accelerated motions.

Secondly, you say that these words, _and the line described were a mean proportional between the whole length, and the half of the same_, have no sense, or that you are deceived. True. For you are deceived; or rather you have not understanding enough distinctly to conceive variety of motions though distinctly expressed. For when a line is gone over with motion uniformly accelerated, you cannot understand how a mean proportional can be taken between it and its half; or if you can, you cannot conceive that that mean can be gone over with uniform motion in the same time that the whole line was run over by motion uniformly accelerated. Yet these are things conceivable, and your want of understanding must be made my fault.