Chapter X (5)

My demonstration is this, _in the parallelogram A B C D, (Fig. 11). Let the side A B be conceived to be moved uniformly till it lie in C D; and let the time of that motion be A C, or B D. And in the same time let it be conceived that A C is moved with uniform acceleration, till it lie in B D._ To which you object, _that then the acceleration last acquired must be far greater than that wherewith A B is moved uniformly: else it shall never come to the place you would have it in the same time_. What proof bring you for this? None here. Where then? Nowhere that I remember. On the contrary I have proved (Art. 9 of the chapter) that the line described by the concourse of those two motions, namely, uniform from A B to C D, and uniformly accelerated from A C to B D, is the crooked line of the semiparabola A H D. And though I had not, yet it is well known that the same is demonstrated by Galileo. And seeing it is manifest that in what proportion the motion is accelerated in the line A B, in the same proportion the impetus beginning from rest in A is increased in the same times (which impetus is designed all the way by the ordinate lines of the semiparabola), the greatest impetus acquired must needs be the base of the semiparabola, namely B D, equal to A C, which designs the whole time. I cannot therefore imagine what should make you say without proof, that the greatest acquired impetus is greater than that which is designed by the base B D. Next you say, “you see not to what end I divide A B in the middle at E.” No wonder; for you have seen nothing all the way. Others would see it is necessary for the demonstration; as also that the point F is not to be taken arbitrarily; and likewise that the thirteenth article, which you admit not for proof, is sufficiently demonstrated, and your objections to it answered. By the way you advise me, where I say _percursam eodem motu uniformi, cum impetu ubique_, &c. to blot out _cum_; because the _impetus_ is not a _companion_ in the way, but the _cause_. Pardon me in that I cannot take your learned counsel; for the word _motu uniformi_ is the ablative of the _cause_, and _impetu_ the ablative of the _manner_. But to come again to your objections, you say, I make “_a greater space run over in the same time by the slower motion than by the swifter_.” How does that appear? _because there is no doubt, but the swiftness is greater where the greatest impetus is always maintained, than where it is attained to in the same time from rest_. True, but that is, when they are considered asunder without concourse, but not then when by the concourse they debilitate one another, and describe a third line different from both the lines, which they would describe singly. In this place I compare their effects as contributing to the description of the parabolical line A H D. What the effects of their several motions are, when they are considered asunder, is sufficiently shown before in the first article. You should first have gotten into your minds the perfect and distinct ideas of all the motions mentioned in this chapter, and then have ventured upon the censure of them, but not before. And then you would have seen that the body moved from A, describeth not the line A C, nor the line A B, but a third, namely the semiparabolical line A H D.

Again, where I say, _Wherefore, if the whole A B be uniformly moved to C D, in the same time wherein A C is moved uniformly to F G_; you ask me “_whether with the same impetus or not?_” How is it possible that in the same time two unequal lengths should be passed over the same impetus? “_But why_,” say you, “_do you not tell us with what impetus A C comes to F G?_” What need is there of that, when all men know that in uniform motion and the same time, impetus is to impetus, as length to length? Which to have expressed had not been pertinent to the demonstration. That which follows in the demonstration, _rursus suppono quod latus A C_, &c. to these words, _ut ostensum est_, _Art. 12_, you confute with saying you have proved that article to be false. But you may see now, if you please, at the same place that I have proved your objection to be frivolous.

After this you run on without any argument against the rest of the demonstration, showing nothing all the way, but that the variety and concourse of motions, the speculations whereof you have not been used to, have made you giddy.

To the nineteenth article you apply the same objection which you made to the eighteenth. Which having been answered, it appears that from the very beginning of your Elenchus to this place all your objections (except such as are made to three or four mistakes of small importance in setting down my mind), are mere paralogisms, and such are less pardonable than any paralogism in Orontius, both because the subject as less difficult is more easily mastered, and because the same faults are most shamefully committed by a reprehender than by any other man.

I had once added to these nineteen articles a twentieth, which was this: “_If from a point in the circumference there be drawn a cord, and a tangent equal to it, the angle which they make shall be double to the aggregate of all the angles made by the cords of all the equal arches into which the arch given can possibly be divided_.” Which proposition is true, and I did when I writ it think I might have use of it. But be it, or the demonstration of it true or false, seeing it was not published by me, it is somewhat barbarous to charge me with the faults thereof. No doctor of humanity but would have thought it a poor and wretched malice, publicly to examine and censure papers of geometry never published, by what means soever they came into his hands. I must confess that in these words, _in such kind of progression arithmetical_ (that is, which begins with 0) _the sum of all the numbers taken together, is equal to half the number that is made by multiplying the greatest into the least_, there is a great error; for by this account these numbers, 0, 1, 2, 3, 4, taken together, should be equal to nothing. I should have said they are equal to that number which is made by multiplying half the greatest into the number of the terms. There was therefore, if those words were mine (for truly I have no copy of them, nor have had since the book was printed, and I have no great reason, as any man may see, to trust your faith) a great error in the writing, but not an erroneous opinion in the writer. The demonstration so corrected is true. And the angles that have the proportions of the numbers 1, 2, 3, 4, are in the table of your _Elenchus_ , fig. 12, the angles G A D, H D E, I E F, K F B. And if the divisions were infinite, so that the first were not to be reckoned but as a cypher, the angle C A B would be double to them altogether. This mistake of mine, and the finding that I had made no use of it in the whole book, was the cause why I thought fit to leave it quite out. But your professorships, could not forbear to take occasion thereby, to commend your zeal against _Leviathan_ to your doctorships of divinity, by censuring it.

==========

OF THE FAULTS THAT OCCUR IN
DEMONSTRATION.

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

LESSON V.

At the seventeenth chapter, your first exception is to the definition of proportional proportions, which is this: “_Four proportions are then proportional, when the first is to the second, as the third to the fourth_.” The reader will hardly believe that your exception is in earnest. You say, I mean not by proportionality the “_quantity of the proportions_.” Yes I do. Therefore I say again, that _four proportions are then proportional, when the quantity of the first proportion, is to the quantity of the second proportion, as the quantity of the third proportion, to the quantity of the fourth proportion_. Is not my meaning now plainly enough expressed? Or is it not the same definition with the former. But what do I mean, you will say, by the quantity of a proportion? I mean the determined greatness of it, that is, for example, in these numbers, the quantity of the proportion of two to three, is the same with the quantity of the proportion of four to six, or six to nine; and again, the quantity of the proportion of six to four, is the same with the quantity of the proportion of nine to six, or of three to two. But now what do you mean by the quantity of a proportion? You mean that two and three, are the quantities of the proportion of two to three (for so Euclid calls them) and that six and four are the quantities of the proportion of six to four, which is the same with the proportion of three to two. And by this rule, one and the same proportion shall have an infinite number of quantities; and consequently the quantity of a proportion can never be determined. I call one proportion double to another, when one is equal to twice the other; as the proportion of four to one, is double to the proportion of two to one. You call that proportion double where one number, line, or quantity absolute, is double to the other; so that with you the proportion of two to one is a double proportion. It is easy to understand how the number two is double to one, but to what, I pray you, is double the proportion of two to one, or of one to two? Is not every double proportion double to some proportion? See whether this geometry of yours can be taken by any man of sound mind for sense. “_But it is known_,” you say, “_that in proportions, double is one thing, and duplicate another_;” so that it seems to you, that in talking of proportion men are allowed to speak senselessly. “_It is known_,” you say. To whom? It is indeed in use at this day to call _double duplicate_, and _triple triplicate_. And it is well enough; for they are words that signify the same thing, but that they differ (in what subject soever) I never heard till now. I am sure that Euclid, whom you have undertaken to expound, maketh no such difference. And even there where he putteth these numbers, one, two, four, eight, &c. for numbers in _double_ proportion (which is the last proposition of the ninth element) he meaneth not that one to two, or two to one, is a _double_ proportion, but that every number in that progression is _double_ to the number next before it; and yet he does not call it _analogia dupla_, but _duplicate_. This distinction in proportions between _double_ and _duplicate_, proceeded long after from want of knowledge that the proportion of one to two is _double_ to the proportion of one to four; and this from ignorance of the different nature of proportions of _excess_, and proportions of _defect_. And you that have nothing but by tradition saw not the absurdities that did hang thereon.

In the second article I make E K, (fig. 1) the third part of L K, which you say is false; and consequently the proposition undemonstrated. And thus you prove it false: “_Let A C be to G C, or G K to G L, as eight to one_ (_for seeing the point G is taken arbitrarily, we may place it where we will, &c._)” and upon this placing of G arbitrarily, you prove well enough that E K is not a third part of L K. But you did not then observe, that I make _the altitude A G, less than any quantity given_, and by consequence E K to differ from a third part by a less difference than any quantity that can be given. Therefore as yet the demonstration proceedeth well enough. But perceiving your oversight, you thought fit (though before, you thought this confutation sufficient) to endeavour to confute it another way; but with much more evidence of ignorance. For when I come to say, _the proportion therefore between A C and G C is triple, in arithmetical proportion, to the proportion between G K and G E, &c._ you say, “_the proportion of A C to G C is the proportion of identity, as also that of G K to G E.”_ But why? Does my construction make it so? Do not I make G C less than A C, though with less difference than any quantity that can be assigned? And then where I say, _therefore E K is the third part of L K_, you come in, by parenthesis, with (_or a fourth, or a fifth, &c._). Upon what ground? Because you think it will pass for current, without proof, that a point is nothing. Which if it do, geometry also shall pass for nothing, as having no ground nor beginning but in nothing. But I have already in a former lesson sufficiently showed you the consequence of that opinion. To which I may add, that it destroys the method of _indivisibles_, invented by Bonaventura; and upon which, not well understood, you have grounded all your scurvy book of _Arithmetica Infinitorum_; where your indivisibles have nothing to do, but as they are supposed to have quantity, that is to say, to be _divisibles_. You allow, it seems, your own nothings to be somethings, and yet will not allow my somethings to be considered as nothing. The rest of your objections having no other ground than this, “_that a point is nothing_,” my whole demonstration standeth firm; and so do the demonstrations of all such geometricians, ancient and modern, as have inferred any thing in the manner following, viz. _If it be not greater nor less, then it is equal. But it is neither greater nor less. Therefore, &c. If it be greater, say by how much. By so much. It is not greater by so much. Therefore it is not greater. If it be less, say how much, &c._ Which being good demonstrations are together with mine overthrown by the nothingness of your _point_, or rather of your understanding; upon which you nevertheless have the vanity of advising me what to do, if I demonstrate the same again; meaning I should come to your false, impossible, and absurd method of _Arithmetica Infinitorum_, worthy to be gilded, I do not mean with gold.

And for your question, why I set the base of my figure upwards, you may be sure it was not because I was afraid to say, that the proportions of the ordinate lines beginning at the vertex were triplicate, or otherwise multiplicate of the proportions of the intercepted parts of the diameter. For I never doubted to call double duplicate, nor triple triplicate, &c., or if I had, I should have avoided it afterwards at the tenth article of the same chapter. But because when I went about to compare the proportions of the ordinate lines with those of their contiguous diameters, the first thing I considered in them was in what manner the base grew less and less till it vanished into a point. And though the base had been placed below, it had not therefore required any change in the demonstration. But I was the more apt to place the base uppermost, because the motion began at the base, and ended at the vertex. To proceed which way I pleased was in my own choice; and it is of grace that I give you any account of it at all.

To the third article, together with its table, you say, “_it falls in the ruin of the second; and that the same is to be understood of the sixth, seventh, eighth, and ninth_.” For confutation whereof I need to say no more, but that they all stand good by the confutation of your objections to the second.

To the fourth article you say, “_the description of those curvilineal figures is easy_.” True, to some men; and now that I have showed you the way, it is easy enough for you also. For the way you propound is wholly transcribed out of the figure of the second article, which article you had before rejected. For seeing the lines H F, G E, A B, &c. are equal to the lines C Q, C O, C D; and the lines Q F, O E, B D, equal to the lines C H, C G, C A; the proportion of D B to O E, will be triple (that is, triplicate) to the proportion of C O to G E; and the proportion of D B to Q F, triple to the proportion of C D to C Q; and consequently, because the complement B D C F E B is made by the decrease of A C in triple proportion to that of the decrease of C D, it will be (by the second article) a third part of the figure A B E F C A. So that it comes all to one pass, whether we take triple proportion in decreasing to make the complement, or triple proportion in increasing to make the figure; for the proportion of H F to B A, is triple to the proportion of C H to C A. Wherefore you have done no more but what you have seen first done, saving that from your construction you prove not the figure to be triple to the complement; perhaps because you have proved the contrary in your _Arithmetica Infinitorum_. But your way differs from mine, in that you call the proportion subtriplicate, which I call triplicate; as if the divers naming of the same thing made it differ from itself. You might as well have said briefly, the proposition is true, but ill proved, because I call the proportion of one or two triple, or triplicate of that of one to eight; which you say is false, and hath infected the fourth, fifth, ninth, tenth, eleventh, thirteenth, fourteenth, fifteenth, sixteenth, seventeenth, and nineteenth articles of the sixteenth chapter. But I say, and you know now, that it is true; and that all those articles are demonstrated.

Lastly you add, “_Tu vero, in presente articulo, &c. id est, you bid find as many mean proportionals as one will, between two given lines; as if that could not be done by the geometry of planes, &c._” You might have left out _Tu vero_ to seek an _Ego quidem_. But tell me, do you think that you can find two mean proportionals (which is less than as many as one will) by the geometry of planes? We shall see anon how you go about it. I never said it was impossible, and if you look upon the places cited by you more attentively, you will find yourself mistaken. But I say, the way to do it has not been yet found out, and therefore it may prove a solid problem for anything you know.

The fifth article you reject, because it citeth the corollary of the twenty-eighth article of the thirteenth chapter, where there is never a word to that purpose. But there is in the twenty-sixth article; which was my own fault, though you knew not but it might have been the printer’s.

To the tenth you object for almost three leaves together, against these words of mine, _because_, in the sixth figure, _B C is to B F in triplicate proportion of C D to F E, therefore inverting, F E is to C D in triplicate proportion of B F to B C_. This you objected then. But now that I have taught you so much geometry, as to know _that of three quantities, beginning at the least, if the third be to the first in triplicate proportion of the second to the first, also by conversion the first to the second shall be in triplicate proportion of the first to the third_; if it were to do again, you would not object it.

My eleventh article you would allow for demonstrated, if my second had been demonstrated, upon which it dependeth. Therefore seeing your objections to that article are sufficiently answered, this article also is to be allowed.

The twelfth also is allowed upon the same reason. What falsities you shall find in such following propositions as depend upon the same second article, we shall then see when I come to the places where you object against them.

To the thirteenth article you object, “_that the same demonstration may be as well applied to a portion of any conoeides, parabolical, hyperbolical, elliptical, or any other, as to the portion of a sphere_.” By the truth of this let any man judge of your and my geometry. Your comparison of the sphere and conoeides, so far holds good, as to prove that the superficies of the conoeides is greater than the superficies of the cone described by the subtense of the parabolical, hyperbolical, or elliptical line. But when I come to say, that _the cause of the excess of the superficies of the portion of the sphere above the superficies of the cone, consists in the angle D A B, and the cause of the excess of the circle made upon the tangent A D, above the superficies of the same cone, consists in the magnitude of the same angle D A B_, how will you apply this to your conoeides? For suppose that the crooked line A B (in the seventh figure) were not an arch of a circle, do you think that the angles which it maketh with the subtense A B, at the points A and B, must needs be equal? Or if they be not, does the excess of the superficies of the circle upon A D above the superficies of the cone, or the excess of the superficies of the portion of the conoeides above the superficies of the same cone, consist in the angle D A B, or rather in the magnitude of the two unequal angles D A B, and A B A? You should have drawn some other crooked line, and made tangents to it through A and B, and you would presently have seen your error. See how you can answer this; for if this demonstration of mine stand firm, I may be bold to say, though the same be well demonstrated by Archimedes, that this way of mine is more natural, as proceeding immediately from the natural efficient causes of the effect contained in the conclusion; and besides, more brief and more easy to be followed by the fancy of the reader.

To the fourteenth article you say that I “_commit a circle in that I require in the fourth article the finding of two mean proportionals, and come not till now to show how it is to be done_.” Nor now neither. But in the mean time you commit two mistakes in saying so. The place cited by you in the fourth article is, in the Latin, p. 215, line 26, in the English, p. 255, line 24. Let any reader judge whether that be a requiring it, or a supposing it to be done; this is your first mistake. The second is, that in this place the proportion itself, which is, “_If these deficient figures could be described in a parallelogram exquisitely, there might be found thereby between any two lines given, as many mean proportionals as one would_,” is a theorem, upon supposition of these crooked lines exquisitely drawn; but you take it for a problem.

And proceeding in that error, you undertake the invention of two mean proportionals, using therein my first figure, which is of the same construction with the eighth that belongeth to this fourteenth article. Your construction is, “_Let there be taken in the diameter C A, (fig. 1) the two given lines, or two others proportional to them, as C H, C G, and their ordinate lines H F, G E (which by construction are in subtriplicate proportion of the intercepted diameters). These lines will show the proportions which those four proportionals are to have._” But how will you find the length of H F or G E, the ordinate lines? Will you not do it by so drawing the crooked line C F E, as it may pass through both the points F and E? You may make it pass through one of them, but to make it pass through the other, you must find two mean proportionals between G K and G L, or between H I and H P; which you cannot do, unless the crooked line be exactly drawn; which it cannot be by the geometry of planes. Go shew this demonstration of yours to Orontius, and see what he will say to it.

I am now come to an end of your objections to the seventeenth chapter, where you have an epiphonema not to be passed over in silence. But because you pretend to the demonstration of some of these propositions by another method in your _Arithmetica Infinitorum_, I shall first try whether you be able to defend those demonstrations as well as I have done these of mine by the method of motion.

The first proposition of your _Arithmetica Infinitorum_ is this lemma: “_In a series, or row of quantities, arithmetically proportional, beginning at a point or cypher, as 0, 1, 2, 3, 4, &c. to find the proportion of the aggregate of them all, to the aggregate of so many times the greatest, as there are terms_.” This is to be done by multiplying the greatest into half the number of the terms.

The demonstration is easy. But how do you demonstrate the same? “_The most simple way_,” say you, “_of finding this and some other problems, is to do the thing itself a little way, and to observe and compare the appearing proportions, and then by induction to conclude it universally_.” Egregious logicians and geometricians, that think an induction, without a numeration of all the particulars sufficient, to infer a conclusion universal, and fit to be received for a geometrical demonstration! But why do you limit it to the natural consecution of the numbers, 0, 1, 2, 3, 4, &c? Is it not also true in these numbers, 0, 2, 4, 6, &c. or in these, 0, 7, 14, 21, &c? Or in any numbers where the difference of nothing and the first number is equal to the difference between the first and second, and between the second and third, &c.? Again, are not these quantities, 1, 3, 5, 7, &c. in continual proportion arithmetical? And if you put before them a cypher thus, 0, 1, 3, 5, 7, do you think that the sum of them is equal to the half of five times seven? Therefore though your lemma be true, and by me (Chap. XIII. art. 5) demonstrated; yet you did not know why it is true; which also appears most evidently in the first proposition of your _Conic Sections_ , where first you have this, “_that a parallelogram whose altitude is infinitely little, that is to say, none, is scarce anything else but a line_.” Is this the language of geometry? How do you determine this word _scarce_? The least altitude, is somewhat or nothing. If somewhat, then the first character of your arithmetical progression must not be a cypher; and consequently the first eighteen propositions of this your _Arithmetica Infinitorum_ are all nought. If nothing, then your whole figure is without altitude, and consequently your understanding nought. Again, in the same proposition, you say thus: “_We will sometimes call those parallelograms rather by the name of lines than of parallelograms, at least when there is no consideration of a determinate altitude; but where there is a consideration of a determinate altitude (which will happen sometimes) there that little altitude shall be so far considered, as that being infinitely multiplied it may be equal to the altitude of the whole figure._” See here in what a confusion you are when you resist the truth. When you consider no determinate altitude, that is no quantity of altitude, then you say your parallelogram shall be called a line. But when the altitude is determined, that is, when it is quantity, then you will call it a parallelogram. Is not this the very same doctrine which you so much wonder at and reprehend in me, in your objections to my eighth chapter, and your word _considered_ used as I used it? It is very ugly in one that so bitterly reprehendeth a doctrine in another, to be driven upon the same himself by the force of truth when he thinks not on it. Again, seeing you admit in any case those infinitely little altitudes to be quantity, what need you this limitation of yours, “_so far forth as that by multiplication they may be made equal to the altitude of the whole figure_?” May not the half, the third, the fourth, or the fifth part, &c. be made equal to the whole by multiplication? Why could you not have said plainly, _so far forth as that every one of those infinitely little altitudes be not only something but an aliquot part of the whole_? So you will have an _infinitely little_ altitude, that is to say, _a point to be both nothing and something and an aliquot part_. And all this proceeds from not understanding the ground of your profession. Well, the lemma is true. Let us see the theorems you draw from it. The first is (p. 3) “_that a triangle to a parallelogram of equal base and altitude is as one to two_.” The conclusion is true, but how know you that? “_Because_,” say you, “_the triangle consists as it were_ [_as it were_, is no phrase of a geometrician] _of an infinite number of straight parallel lines_.” Does it so? Then by your own doctrine, which is, that “_lines have no breadth_,” the altitude of your triangle consisteth of an infinite number of no altitudes, that is of an infinite number of nothings, and consequently the area of your triangle has no quantity. If you say that by the parallels you mean infinitely little parallelograms, you are never the better; for if infinitely little, either they are nothing, or if somewhat, yet seeing that no two sides of a triangle are parallel, those parallels cannot be parallelograms. I see they may be counted for parallelograms by not considering the quantity of their altitudes in the demonstration. But you are barred of that plea, by your spiteful arguing against it in your _Elenchus_ . Therefore this third proposition, and with it the fourth, is undemonstrated.

Your fifth proposition is, “_the spiral line is equal to half the circle of the first revolution_.” But what spiral line? We shall understand that by your construction, which is this: “_The straight line M A_ [in your figure which I have placed at the end of the fifth lesson] _turned round (the point M remaining unmoved) is supposed to describe with its point A the circle A O A, whilst some point, in the same M A, whilst it goes about, is supposed to be moved uniformly from M to A, describing the spiral line_.” This therefore, is the spiral line of Archimedes; and your proposition affirms it to be equal to the half of the circle A O A; which you perceived not long after to be false. But thinking it had been true, you go about to prove it, “_by inscribing in the circle an infinite multitude of equal angles, and consequently an infinite number of sectors, whose arches will therefore be in arithmetical proportion_;” which is true. “_And the aggregate of those arches equal to half the circumference A O A_;” which is true also. And thence you conclude “_that the spiral line is equal to half the circumference of the circle A O A_;” which is false. For the aggregate of that infinite number of infinitely little arches, is not the spiral line made by your construction, seeing by your construction the line you make is manifestly the spiral of Archimedes; whereas no number, though infinite, of arches of circles, how little soever, is any kind of spiral at all; and though you call it a spiral, that is but a patch to cover your fault, and deceiveth no man but yourself. Besides, you saw not how absurd it was, for you that hold a point to be absolutely nothing, to make an infinite number of equal angles (the radius increasing as the number of angles increaseth) and then to say, “_that the arches of the sectors whose angles they are, are as_ 0, 1, 2, 3, 4, &c.” For you make the first angle 0, and all the rest equal to it; and so make 0, 0, 0, 0, 0, &c. to be the same progression with 0, 1, 2, 3, 4, &c. The influence of this absurdity reacheth to the end of the eighteenth proposition. So many are therefore false, or nothing worth. And you needed not to wonder that the doctrine contained in them was omitted by Archimedes, who never was so senseless as to think a spiral line was compounded of arches of circles.

Your nineteenth proposition is this other lemma: “_In a series, or a row, of quantities, beginning from a point, or cypher, and proceeding according to the order of the square numbers, as_ 0, 1, 4, 9, 16, _&c. to find what proportion the whole series hath to so many times the greatest_.” And you conclude “_the proportions to be that of 1 to 3_.” Which is false, as you shall presently see. First, let the series of squares with the prefixed cypher, and under every one of them the greatest 4 be (0 . 1 . 4)/(4 . 4 . 4). And you have for the sum of the squares 5, and for thrice the greatest 12, the third part whereof is 4. But 5 is greater than 4, by 1, that is, by one twelfth of 12; which quantity is somewhat, let it be called A. Again, let the row of squares be lengthened one term further, and the greatepm divst set under every one of them as (0 . 1 . 4 . 9)/(9 . 9 . 9 . 9). The sum of the squares is 14, and the sum of four times the greatest is 36, whereof the third part is 12. But 14 is greater than 12 by two unities, that is, by two twelfths of 12, that is, by 2 A. The difference therefore between the sum of the squares, and the sum of so many times the greatest square, is greater, when the cypher is followed by three squares, than when by but two. Again, let the row have five terms, as in these numbers (0 . 1 . 4 . 9 . 16)/(16 . 16 . 16 . 16 . 16) with the greatest five times described, and the sum of the squares will be 30, the sum of all the greatest will be 80. The third part whereof is 26(2)/(3). But 30 is greater than 26(2)/(3) by 3(1)/(3), that is, by three twelfths of twelve, and (1)/(3) of a twelfth, that is, by 3(1)/(3) A. Likewise in the series continued to six places with the greatest six times subscribed, as ( 0 . 1 . 4 . 9 . 16 . 25)/(25 . 25 . 25 . 25 . 25 . 25) the sum of the squares is 55, and the sum of the greatest six times taken is 150, the third part whereof is 50. But 55 is greater than 50 by 5, that is, by five-twelfths of 12, that is by 5 A. And so continually as the row groweth longer, the excess also of the aggregate of the squares above the third part of the aggregate of so many times the greatest square, growing greater. And consequently if the number of the squares were infinite, their sum would be so far from being equal to the third part of the aggregate of the greatest as often taken, as that it would be greater than it by a quantity greater than any that can be given or named.

That which deceived you was partly this, that you think, as you do in your _Elenchus_ , that these fractions (1)/(12) (1)/(18) (1)/(24) (1)/(30) (1)/(36) &c. are proportions, as if (1)/(12) were the proportion of one to twelve, and consequently (2)/(12) double the proportion of one to twelve; which is as unintelligible as school-divinity; and I assure you, far from the meaning of Mr. Ougthred in the sixth chapter of his _Clavis Mathematica_, where he says that 4(3)/(7) is the proportion of 31 to 7; for his meaning is, that the proportion of 4(3)/(7) to one, is the proportion of 31 to 7; whereas if he meant as you do, then 8(6)/(7) should be double the proportion of 31 to 7. Partly also because you think (as in the end of the twentieth proposition) that if the proportion of the numerators of these fractions (1)/(12) (1)/(18) (1)/(24) (1)/(30) (1)/(36) to their denominators decrease eternally, they shall so vanish at last as to leave the proportion of the sum of all the squares to the sum of the greatest so often taken, (that is, an infinite number of times), as one to three, or the sum of the greatest to the sum of the increasing squares, as three to one; for which there is no more reason than for four to one, or five to one, or any other such proportion. For if the proportions come eternally nearer and nearer to the subtriple, they must needs also come nearer and nearer to subquadruple; and you may as well conclude thence that the upper quantities shall be to the lower quantities as one to four, or as one to five, &c. as conclude they are as one to three. You can see without admonition, what effect this false ground of yours will produce in the whole structure of your _Arithmetica Infinitorum_; and how it makes all that you have said unto the end of your thirty-eighth proposition, undemonstrated, and much of it false.

The thirty-ninth is this other lemma: “_In a series of quantities beginning with a point or cypher, and proceeding according to the series of the cubic numbers, as O. 1. 8. 27. 64, &c. to find the proportion of the sum of the cubes to the sum of the greatest cube, so many times taken as there be terms_.” And you conclude that “_they have a proportion of 1 to 4_;” which is false.

Let the first series be of three terms subscribed with the greatest (0. 1. 8.)/(8. 8. 8.); the sum of the cubes is nine; the sum of all the greatest is 24; a quarter whereof is 6. But 9 is greater than 6 by three unities. An unity is something. Let it be therefore A. Therefore the row of cubes is greater than a quarter of three times eight, by three A. Again, let the series have four terms, as (0. 1. 8. 27)/(27. 27. 27. 27); the sum of the cubes is 36; a quarter of the sum of all the greatest is twenty-seven. But thirty-six is greater than twenty-seven by nine, that is, by 9 A. The excess therefore of the sum of the cubes above the fourth part of the sum of all the greatest, is increased by the increase of the number of terms. Again, let the terms be five, as (0. 1. 8. 27. 64)/(64. 64. 64. 64. 64), the sum of the cubes is one hundred; the sum of all the greatest three hundred and twenty; a quarter whereof is eighty. But one hundred is greater than eighty by twenty, that is, by 20 A. So you see that this lemma also is false. And yet there is grounded upon it all that which you have of comparing parabolas and paraboloeides with the parallelograms wherein they are accommodated. And therefore though it be true, that the parabola is (2)/(3) and the cubical paraboloeides (3)/(4) of their parallelograms respectively, yet it is more than you were certain of when you referred me, for the learning of geometry, to this book of yours. Besides, any man may perceive that without these two lemmas (which are mingled with all your compounded series with their excesses) there is nothing demonstrated to the end of your book: which to prosecute particularly, were but a vain expense of time. Truly, were it not that I must defend my reputation, I should not have showed the world how little there is of sound doctrine in any of your books. For when I think how dejected you will be for the future, and how the grief of so much time irrecoverably lost, together with the conscience of taking so great a stipend, for mis-teaching the young men of the University, and the consideration of how much your friends will be ashamed of you, will accompany you for the rest of your life, I have more compassion for you than you have deserved. Your treatise of the _Angle of Contact_ , I have before confuted in a very few leaves. And for that of your _Conic Sections_ , it is so covered over with the scab of symbols, that I had not the patience to examine whether it be well or ill demonstrated.

Yet I observed thus much, that you find a tangent to a point given in the section by a diameter given; and in the next chapter after, you teach the finding of a diameter, which is not artificially done.

I observe also, that you call the _parameter_ an imaginary line, as if the place thereof were less determined than the diameter itself; and then you take a mean proportional between the intercepted diameter, and its contiguous ordinate line, to find it. And it is true, you find it: but the parameter has a determined quantity, to be found without taking a mean proportional. For the diameter and half the section being given, draw a tangent through the vertex, and dividing the angle in the midst which is made by the diameter and tangent, the line that so divideth the angle, will cut the crooked line. From the intersection draw a line (if it be a parabola) parallel to the diameter, and that line shall cut off in the tangent from the vertex the parameter sought. But if the section be an ellipsis, or an hyperbole, you may use the same method, saving that the line drawn from the intersection must not be parallel, but must pass through the end of the transverse diameter, and then also it shall cut off a part of the tangent, which measured from the vertex is the parameter. So that there is no more reason to call the parameter an imaginary line than the diameter.

Lastly, I observe that in all this your new method of conics, you show not how to find the _burning points_, which writers call the _foci_ and _umbilici_ of the section, which are of all other things belonging to the conics most useful in philosophy. Why therefore were they not as worthy of your pains as the rest, for the rest also have already been demonstrated by others? You know the focus of the parabola is in the axis distant from the vertex a quarter of the parameter. Know also that the focus of an hyperbole, is in the axis, distant from the vertex, as much as the hypotenusal of a rectangled triangle, whose one side is half the transverse axis, the other side half the mean proportional between the whole transverse axis and the parameter, is greater than half the transverse axis.

The cause why you have performed nothing in any of your books (saving that in your _Elenchus_ you have spied a few negligences of mine, which I need not be ashamed of) is this, that you understood not what is _quantity_, _line_, _superficies_, _angle_, and _proportion_; without which you cannot have the science of any one proposition in geometry. From this one and first definition of Euclid, “_a point is that whereof there is no part_,” understood by Sextus Empiricus, as you understand it, that is to say misunderstood, Sextus Empiricus had utterly destroyed most of the rest, and demonstrated, that in geometry there is no science, and by that means you have betrayed the most evident of the sciences to the sceptics. But as I understand it for _that whereof no part is reckoned_, his arguments have no force at all, and geometry is redeemed. If a line have no latitude, how shall a cylinder rolling on a plane, which it toucheth not but in a line, describe a superficies? How can you affirm that any of those things can be without quantity, whereof the one may be greater or less than the other? But in the common contact of divers circles the external circle maketh with the common tangent a less angle of contact than the internal. Why then is it not quantity? An angle is made by the concourse of two lines from several regions, concurring, by their generation, in one and the same point. How then can you say the angle of contact is no angle? One measure cannot be applicable at once to the angle of contact, and angle of conversion. How then can you infer, if they be both angles, that they must be homogeneous? Proportion is the relation of two quantities. How then can a quotient or fraction, which is quantity absolute, be a proportion? But to come at last to your _Epiphonema_ , wherein, though I have perfectly demonstrated all those propositions concerning the proportion of parabolasters to their parallelograms, and you have demonstrated none of them (as you cannot now but plainly see), but committed most gross paralogisms, how could you be so transported with pride, as insolently to compare the setting of them forth as mine, to the act of him that steals a horse, and comes to the gallows for it. You have read, I think, of the gallows set up by Haman. Remember therefore also who was hanged upon it.

After your dejection I shall comfort you a little, a very little, with this, that whereas this eighteenth chapter containeth two problems, one, “_the finding of a straight line equal to the crooked line of a semi-parabola_;” the other, “_the finding of straight lines equal to the crooked lines of the parabolasters, in the table of the third article of the seventeenth chapter_;” you have truly demonstrated that they are both false; and another hath also demonstrated the same another way. Nevertheless, the fault was not in my method, but in a mistake of one line for another and such as was not hard to correct; and is now so corrected in the English as you shall not be able (if you can sufficiently imagine motions) to reprehend. The fault was this, that in the triangles which have the same base and altitude with the parabola and parabolaster, I take for designation of the mean uniform impetus, a mean proportional, in the first figure, between the whole diameter and its half, and, in the second figure, a mean proportional between the whole diameter and its third part; which was manifestly false, and contrary to what I had shown in the sixteenth chapter. Whereas I ought to have taken the half of the base, as now I have done, and thereby exhibited the straight lines equal to those crooked lines, as I undertook to do. Which error therefore proceeded not from want of skill, but from want of care; and what I promised (as bold as you say the promise was), I have now performed.

The rest of your exceptions to this chapter, are to these words in the end: “_There be some that say, that though there be equality between a straight and crooked line, yet now, they say, after the fall of Adam, it cannot be found without the especial help of divine grace_.” And you say you think there be none that say so. I am not bound to tell you who they are. Nevertheless, that other men may see the spirit of an ambitious part of the clergy, I will tell you where I read it. It is in the _Prolegomena_ of Lalovera, a Jesuit, to his Quadrature of the Circle, p. 13 and 14, in these words: “_Quamvis circuli tetragonismus sit_ φύσει _possibilis, an tamen etiam_ πρός ἡμᾶς, _hoc est, post Adæ lapsum homo ejus scientiam absque speciali divinæ gratiæ auxilio, possit comparare, jure merito inquirunt theologi, pronunciantque; hanc veritatem tanta esse caligine involutam ut illam videre nemo possit, nisi ignorantiæ ex primi parentis prævaricatione propagatas tenebras indebitus divinæ lucis radius dissipet; quod verissimum esse sentio_.” Wherein I observed that he, supposing he had found that quadrature, would have us believe it was not by the ordinary and natural help of God (whereby one man reasoneth, judgeth and remembereth better than another), but by a special (which must be a supernatural) help of God, that he hath given to him of the order of Jesus above others that have attempted the same in vain. Insinuating thereby, as handsomely as he could, a special love of God towards the Jesuits. But you taking no notice of the word _special_, would have men think I held, that human sciences might be acquired without any help of God. And thereupon proceed in a great deal of ill language to the end of your objections to this chapter. But I shall take notice of your manners for altogether in my next lesson.

At the nineteenth chapter you see not, you say, the method. Like enough. In this chapter I consider not the cause of reflection, which consisteth in the resistance of bodies natural; but I consider the consequences, arising from the supposition of the equality of the angle of reflection, to that of incidence; leaving the causes both of reflection, and of refraction, to be handled together in the twenty-fourth chapter. Which method, think what you will, I still think best.

Secondly, you say I define not, here, but many chapters after, what an angle of incidence, and what an angle of reflection is. Had you not been more hasty than diligent readers, you had found that those definitions of the angle of incidence, and of reflection, were here set down in the first article, and not deferred to the twenty-fourth. Let not therefore your own oversight be any more brought in for an objection.

Thirdly, you say there is no great difficulty in the business of this chapter. It may be so, now it is down; but before it was done, I doubt not but you that are a professor would have done the same, as well as you have done that of the _Angle of Contact_ , or the business of your _Arithmetica Infinitorum_ . But what a novice in geometry would have done I cannot tell.

To the third, fourth, and fifth article, you object a want of determination; and show it by instance, as to the third article. But what those determinations should be, you determine not, because you could not. The words in the third article, are first these, _if there fall two straight lines parallel, &c._ which is too general. It should be, _if there fall the same way two straight lines parallel, &c._ Next these, _their reflected lines produced inwards shall make an angle, &c._ This also is too general. I should have said, _their reflected lines produced inwards, if they meet within, shall make an angle, &c._ Which done, both this article and the fourth and fifth are fully demonstrated. And without it, an intelligent reader had been satisfied, supplying the want himself by the construction.