Due correction for Mr Hobbes· Or Schoole discipline, for not saying his lessons right. In answer to his Six lessons, directed to the professors of mathematicks. / By the professor of geometry.

SECT. VIII. Concerning his 11, and 13 Chapters. (Book 8)

WEE shall next consider what you have to say in defense of your 11 and 13 Chapters, concerning Proportion.

And here after a freak; and then a rant against Euclide; you have a large discourse about Proportion; p. 15, 16. The summe of which, so farre as is to the purpose, is this, That there betwo kinds of Proportion, (as the word is now adaies taken;) the one of which is called Arithmeticall Proportion; the other, Geometricall Proportion: And as the Quotient gives us a measure of the Proportion of the Dividend to the Divisor, in Geo∣metricall Proportion; so the Remainder, after subtraction, is the measure of Proportion Arithmeticall. Pag. 16. And thus much is both true and clear, and to the purpose. And had you but thus delivered your doctrine of Proportions, in your Book de Corpore, I should never have found fault with it. But you, not knowing (till you learned it out of my Elenchus,) that the Quotient did as well determine Geo∣metricall Proportion, (and give name to it) as the Re∣mainder doth Proportion Arithmeticall, were fain to blunder on as well as you could, without it: and put your selfe upon a great many unhandsome shifts, and which will not hold water, to give account, even of Geometricall Proportion, from the Remainder or difference, which was not to be done otherwise then by the Quotient, as you here clearly confesse; For the Measure, you say, of Geometri∣call progression, is (not the Remainder, whether absolutely or comparatively considered, but) the Quotient.

But before you come thus farre; you tell us by the way, That I say, that you make proportion to consist in the Remainder, and that I make it consist in the Quotient. As to the former of these, I did not then say, that you make proportion to consist in the Remainder; though if I had said so, I had said true e∣nough, for you doe so, more than once. Cap. 11. parag. 7. In ratione inaequalium, say you, ratio minoris ad majus, Defe∣ctus; ratio majoris ad minus Excessus dicitur. And again par. 5. Consistit ratio antecedentis ad consequens in differentia, &c. sive in majoris (dempto minore) Refiduo. And. soon after,

Ratio binarii ad quinarium est ternarius, &c. You cannot de∣ny but that these are your words, and that I blamed you for them, as a piece of non sense; all that you have to say is, that it was too hastily put: & therefore you labour in the Eng∣lish a little to disguise it. So cap. 12. art. 8. Cum Ratio inaequalium, per cap. praeced. art. 5. consistit in differe••tia ipsa∣rum, &c. and again, Ratio inaequalium, EG, EF, consistit in differentia EF, quae est quantitas, (yes, quantitas absoluta, for 'tis a line.) And these, because I did not particularly tell you of them, are yet uncorrected in your English; seeing (by the fifth Article of the precedent Chapter,) the proportion of two unequall magnitudes consists in their difference, &c. And again, the Proportion of unequalls EG, EF, is quantity; for the difference GF, in which it consists is quantity. Now when, you say in expresse words, as in the places cited, The pro∣portion of the antecedent to the consequent consists in the Diffe∣rence, or the Remainder; it had been no wrong if I had said, as you say I doe, that you make Proportion to consist in the Re∣mainder; and that absurdly enough. And then, J pray, to whom belong those reproaches, that are so oft in your mouth, as if somebody did affirme, that Proportion is a Num∣ber, an Absolute quantity, &c? is it not your selfe that af∣firme it so to be? And doth any body so beside your selfe? And is not then, that (by your own law p. 10,) in your selfe intolerable, which you cannot tolerate in another?

But you adde farther, that I say, that I make it to consist in the Quotient. And is not this abominably false? J neither say so, nor doe so, nor did J give any ground at all for any man (that is in his witts) to believe J did. My words were these, Videmus igitur Rationis aestimationem esse (secundum Te) penes Residuum, non penes Quotum, & Subductione, non Divisione quaerendam esse. (And what reason J had to say so, they that consult the place will see.) Now could any man (who had not a great confidence that his English Reader understands no Latine) be so impudent as to say, that in those words, I say, you make Proportion to consist in the Remainder; and I, in the Quotient? Can any man, that un∣derstands, though but a little Latine, (if he be not either out of his witts, or halfe a sleep,) think that these words Rationis aestimatio est penes Quotum, (that is, the Proportion is to be estimated according to the Quotient, or, to use your own words, the quotient gives us the measure of the proportion,)

could be thus Englished, proportion consists in the quotient? And that then you should raile at us, quite through your Book, for saying that Proportion is a certain quotient, that it is a number, that it is an absolute quantity, &c. as if we had been so ridiculous as to speak like you. For, that you have so spoken you cannot deny, (and therefore the absurdity what ever it be, lights upon your selfe:) But, to say, that I said so, or any thing to that purpose, till you can shew where I said it, J take to be, (so farre as a word of your mouth can be) a manifest slander. J neither say so, nor think so.

Now some men perhaps may wonder, there should be so great a cry and so little wooll; they would think perhaps, by what you say, that J had somewhere said in expresse termes, that Proportion is a Quotient, or that it consists in the Quotient, or that it is a number, or an ab∣solute quantity, or that the quotient is the proportion, or that a Proportion is the double of a Number, but not of a propor∣tion, or somewhat that sounds like somewhat of these, when they hear me thus charged, again and again, many a time, and oft; and not that the whole ground of the accusation had been but this, that I said, The proportion is to be estimated by the quotient. And truly 'tis somewhat hard to give a good account of it: yet wee'l try what may be done.

J was told, some years a goe, of a man that had told a lye so often, and with so much confidence, that at length he began to believe it himselfe. And J am almost of opinion, that M. ••obs having now said it so often over, doth, by this time, be∣gin to think, that J had indeed said, somewhere, that the quotient was the proportion. And truly there is some reason why he should: For if he had heard any other man so oft and so confidently affirme it, he would no doubt have be∣lieved him: and why should he not as well believe him∣selfe.

But moreover; It did perhaps runne in his mind, that he had somewhere read some such words as these, Consistit au∣tem Ratio antecedentis ad consequens, in Differentia, hoc est in ea parte majoris qua minus ab eo superatur; sive in majoris (dem∣pto minore) Residuo. Or such as these, Ratio binarii ad qui∣narium est ternarius. Or else this, Ratio minoris ad majus, Defectus; ratio majoris ad minus, Excessus dicitur. (And well it might: for they are all his own words, Cap 11. parag. 3. & 5. and Cap. 12. parag. 8.) And he might think, that to

say thus, was all one, as to affirme Proportion to be a Number, or an Absolute quantity: (And truly I think so too.) And that therefore the expression was very absurd; (For so I had intimated to him in my Elenchus, upon this occasion.) And therefore (forgetting, perhaps, that they were his own words, and not mine.) he doth (like the Woman that called her daughter Bastard, not minding that in so doing shee called her selfe Whore,) exclaim against his own words, as most ridiculous non-sense. And who might doe it better?

Or else, to use his own comparison, like Women of poor and evill education, when they scold; amongst whom the readiest dis∣gracefull word is Whore; because, when they remember them∣selves, they think that reproach the likeliest to be true; at least, if they be called Whore themselves, though never so truly, they will be sure to call Whore again at all adventures, hit or misse. So M. Hobbs, finding himselfe to have been so absurd, as to make Proportion a Number, or Absolute quan∣tity, and that I had blamed him for it; thought, perhaps, it was possible I might, sometime or other, have been as carelesse in my language: and therefore, however, hee'l say so, ('tis easy to say it) and let me disprove it.

If any man, notwithstanding all this, be not satisfied that M. Hobs had reason to say as he doth; truly I cannot help it; he must speak for himselfe: These were the best reasons I could think of▪ And so wee'l goe on.

In your 11 Chap. parag. 3. you gave us in the Latine, (for in the English there be some things altered,) this de∣finition of Proportion; Proportion is nothing else but the aequa∣lity or in equality of the Antecedent, compared with the consequent, according to magnitude. With this Explication, As for exam∣ple, the proportion of Three to Two, is nothing else, but, that Three, is greater then Two, by One: and the proportion of Two to Five, is nothing else, but that Two, is lesse than Five by Three: And therefore in the proportion of Ʋnequalls, the proportion of the Lesse to the Greater is called the Defect; and that of the Greater to the Lesse, the Excesse. And this is your generall definition of Proportion, with the Explication of it; and nor a particular definition of Arithmeticall Proportions, (nor is it at all by you pretended so to be.) And there∣fore should have been so ordered, as at least to take in Geo∣metricall Proportion; For Geometricall proportion, and simply

proportion, are by your selfe made equivalent termes (Less. 2. p. 16. l. 25.) and this, you say, is onely taken notice of by the name of Proportion: And, so the word is constantly used in Euclide, and elsewhere: (And therefore you need not wonder as you doe p. 18. l. 7, that J should say, If Arith∣meticall Proportion, ought to be called Proportion; implying that though now that phrase be common, yet that it is a depart∣ing from the former use of the word; and that, according to Euclides use of the word Proportion, Arithmeticall Proportion cannot be so called.) Now your Definition and Explication of Proportion, doth wholly leave out Geome∣tricall Proportion altogether, (which yet is, if not the only, yet the more principall kind of Proportion.) For it takes no cognizance of the Quotient at all, but only of the Difference, the excesse or defect. And according to your doctrine the Proportion of 3 to 2, is + 1, the excesse of 1; and of 2 to 5, is -3, the defect of three.

From this I inferred, that if the proportion of one quan∣tity to another, be nothing else, but the excesse or defect of this to that, (as you teach,) then where ever the excesse or de∣fect is the same, there the proportion is the same; and so 3 to 2, must have the same proportion that 5 hath to 4; (You say, p. 17. True, the same Arithmeticall Proportion Very good: But J added farther, of which you did not think fit to take notice,) and on the contrary, where there is not the same defect or the same excesse there is not the same proportion, and conse∣quently, there is not the same proportion of 3 to 2 and of 6 to 4. To this you have nothing to say, and therefore say nothing, (but recite halfe my sentence, and leave out the other halfe:) For though, there be not the same Arithmeticall Proportion (as you speak) of 3 to 2, and of 6 to 4; (that is, not the same excesse,) yet there is the same Geometricall Pro∣portion; and that you cannot deny to be Proportion, though it doe not come, within your definition.

Now it's true, (but that's another fault, not an excuse) that you do not hold to this sense alwaies, for in the same page art. 5. (in the Latine, I mean) you do clearly con∣tradict what you had but now said in art. 3. The proportion, say you, of the Antecedent to the consequent consists in the Dif∣ference, or Remainder, not simply (yes simply, if that be true which you said before; for if it be nothing else but the diffe∣rence, that is it the difference simply: But if not simply;

how then?) but as compared with one of the termes related, &c. For though there be the same difference between 2 and 5, that there is between 9 and 12, yet not the same Proportion. And why not? as well as the same proportion between 3 and 2, and between 4 and 5? as we heard you reply but now. May not we as well say here, as you there, (Les. 2. p. 17.) Is there not the same Arithmeticall Proportion? And is not A∣rithmeticall proportion, proportion? But it seems, by this time, you had forgotten your former exposition, whereby in the same page, your definition of Proportion must be so under∣stood, as will agree to none but Arithmeticall proportion; now it must bear such a sense as can agree to none but Geo∣metricall.

In the English, I confesse, your Translator hath a little mended the matter, and but a little, ('tis but Coblers work at the best;) But however, 'tis good to hear folks mend, though it be but a little: it may come to something in time.

But now of those two senses, which you have given, of the Definition of Proportion, (opposite enough in con∣science one to another, though, I suppose, you did not in∣tend therein to contradict your selfe,) neither of them will serve your turn. For the Proportion here defined, and so explicated as we have heard, is a Genus, which is, in the beginning of your 13 Chapter, to be distributed into its two Species; Proportion Arithmeticall, and Proportion Geometricall. Now take your definition of Proportion in generall, according to which of your two expositions you please, it cannot be thus distributed. For if Propor••••on (as you say chap. 11, ••art. 3.) be nothing else but the excesse or defect, &c. as 3 is lesse then 2 by 1; then it cannot agree to Geometricall proportion, for that is somewhat else. If it be such a comparative difference, as you mention cap. 11. art. 3. it will not agree to Arithmeticall proportion; for according to that sense, you say, 2 to 5, and 9 to 12, are not in the same proportion. I say therefore, that neither of those two expositions, do agree to that generall notion of Pro∣portion, which shall be common to both Arithmeticall and Geometricall. And when I aske, which of the two expo∣sitions you are willing to stand to. Whether that of Cap. 11. art. 3. or that of Cap, 11. art. 5. (shewing withall that neither of them will serve your turne, for neither of them

will take in both Arithmeticall and Geometricall Progres∣sion,) you fall a raving in the beginning of your third Lesson, something at Euclide, and something at us, but nothing to the purpose. And then tell us, that when you say the Difference is the Proportion, by Difference, we might if we would, have understood, the act of Differing. That is, wee might understand, as madly as you speak. Your words were these, Cap. 11. art. 5. Consistit autem Ratio in Dif∣ferentia, sive Residuo, &c. ita ratio binarii ad quinarium est ter∣narius, &c. Would you have us understand Residuum, and Ternarius, to be the Act of Differing? And C. 12. art. 8. Ratio inaequaliū (EG, EF) consistit in differentia GF. Would you have us understand that line GF, to be the act of differing? You say, we might if we would. But you'ld think us very simple if we should. To as good purpose is it, that you tell your English Reader (for you think you may tell him any thing,) that •• say, that (thus much of) your Definition, Ch. 11. Art. 1. [Pro∣portion is the Comparison of two Magnitudes one to another,] a∣grees neither with Arithmeticall nor Geometricall proportion. For I said nothing of any such words, good or bad. And 'twere much if I should: for I can find no such words there.

At the second Article (chap. 13.) I note, you say, for a fault in method, that after you had used the words, Antecedent, and Consequent of a Proportion, in the precedent Chapters, you now define them. 'Tis true, I did take notice of it, but I said withall, that this was but a small fault in comparison of many others. But what if I did? You do not believe, you say that I spake this against my knowledge. No; why should you for you know 'tis true. Have you not used the words ma∣ny times before in the precedent chapters? And doe you not define them here? And is not this a fault in Method? Do Mathematicians use, when they have taken a Terme for two or three chapters together, to be of a known significa∣tion, and sufficiently understood, come at length to define it? you say, you had before defined it chap. 11. art. 3. 'Tis true you had there defined the Antecedent and Consequent of Cor∣relatives; (which definitions might have served well enough for the Antecedent and consequent in Proportions too, for those are Correlatives, and you need not have brought any new ones.) But where was my oversight? Did I deny this? I did not blame you for using the words before you had defined them,

(nor would I have blamed you, if they had not been de∣fined at all;) But for defining them after you had thus long used them. For, if they had now, ever since the beginning of the 11 Chapter, been taken for words of a known significati∣on, and as such frequently used, (which you do not deny, and your definitions at that place do but aggravate, not ex∣tenuate, this charge,) then, I say, it was immethodicall and superfluous to define them in the 13 chapter. Nor was it my oversight to say so. And the like impertinent answer you give p. 51. where I blamed you (not for omitting in the 19 chapter, but) for defining in the 24 chapter, those termes which were of frequent use in the 19 chapter. But wee go on.

You tell us, Chap. 13. art. 3. That the proportion of Ine∣quality is Quantity, but that of Equality is not. Which I said was very absurd; and that the one did no more belong to the Praedicament of Quantity than the other; and that it is to bee, of both equally, either denied or affirmed: And that your argument for it, (That One equality is not greater or lesse then another; but of proportions of inequality, one may be more or lesse unequall:) might as well conclude that Oblique angles, be quantities, but not Right angles, for these be all equall, and equally Right; but not those. For answer to this, you fall a ranting at Aristotle, at Praedicaments, and the L••gick Schooles, &c. And then you tell us the Greater and Lesser cannot be attributed to Right Angles, because a Right Angle is a Quantity determined, (as though the quantity of the Proportion of Equality were not so too.) What you alledge out of Mersennus, was but his mistake. Composition of Pro∣portion is a work of Multiplication, not of Addition, as appears by the definition of it 5 d 6. and to argue, that Proportion of equality is as Nothing, because in composition of Proportions it doth not increase or diminish another proportion; is but as to conclude that, 1, a Ʋnity, is Nothing, because in Multiplication it doth neither increase nor diminish the quantity multiplyed thereby. But of this mistake of Mersennus, I have spoken already in the end of another Treatise, already Printed, against Meibomius; and vindicated Clavius suffici∣ently from what both Mersennus and Meibomius allege a∣gainst him.

To the fourth Article, where you define Greater and Lesser Proportion; I said nothing (because it were endlesse

to note all the faults I see) though those definitions are liable enough to censure. Greater Proportion, you say, is the proportion of a greater Antecedent to the same Consequent, or of the same Antecedent to a lesse Consequent. And Lesse Propor∣tion, is the proportion of a lesse Antecedent to the same Consequent, or of the same Antecedent to a greater consequent. Yet we know, that the proportion of an Ell to a Yard, is lesse then that of a Pottle to a Pint, (and this therefore greater then that,) though neither the Antecedents nor the Consequents, be ei∣ther the same, or Equall, or Homogeneous.

To the 5 and 6 Articles, where you define the same Pro∣portion. I said First, that, had Proportion been well defi∣ned before, you might have spared these definitions of the same proportion. For having before defined (as well as you could) what is Proportion (both Arithmeticall, and Geome∣tricall;) and withall told us, art. 4. that by the same pro∣portion was meant Equall proportions; and having also defined before (after your fashion) what are Equalls chap. 8. and what is the Same chap. 11. Why should you think (if those definitions were such as they should have been) that wee needed another definition of the Same, or Equall Proportions? But, since you were resolved to doe works of Supereroga∣tion; I ask why, having defined the same Arithmeticall pro∣portion, art: 5. by the Equality of the Differences; you did not also define the same Geometricall Proportion, art. 6, by the Equality of the Quotients? For by the Same, you say, you mean Equall, art. 4. Now universally all quantities are Equall, that are measured by the same number of the same Measures (Less: 1 p: 4.) and therefore those are the same or equall Proportions, which have the same or equall Measures: And you know now (though perhaps you did not then) that as the Quotient gives us a measure of the Proportion in Geometricall Proportion, so the Remainder is the Measure of Proportion Arithmeticall. (Les: 2. p. 16.) And therefore, as, in the one, you define the same or equall proportion, by the Equality of the Remainder; so you should in the other, by the equality of the Quotient, (that is, in both places by the equality of its measure:) And not have brought us such an imbrangled definition as this. viz: One Geometricall progressi∣on is the same with another, when a cause in equall times tro∣ducing equall effects, determining the proportion, may be assigned the same in both, or as your English hath it, when the same

cause producing equall effects in equall times, determines both the proportions. So that, to prove, that 4 to 2, and 6 to 3, are in the same Geometricall proportion, we must call in the help of Time, and Motion, and Velocity, and Ʋniformity, &c. which are wholly extrinsecall to it; and why, but because, forsooth, there is no effect in Nature which is not produced in Time by Motion, (as though some Motion, in some Time or other, had made this to be a true Proposition, that 4 is the double of 2: and therefore if we cannot find what motion did make it so, we must imagine some that might have made it.) I need not tell you, that, if this be a good reason, you should upon the same account, have found out as bad a de∣finition for the same Arithmeticall proportion: (for that 8 to 6, and 12 to 10, are in the same Arithmeticall proportion, is, doubtlesse, as much as that other of Geometricall pro∣portion, an effect which nature hath at some Time or other pro∣duced by Motion.) But, since you have waved this considera∣tion of nature in the definition of the same Arithmeticall pro∣portion, which you define by the equality of the Remainders; I said, it might have been expected, that you might have done so in the definition of the same Geometricall proportion••, and accordingly defined it, by the Equality of the Quotients. But you are very angry with me, for saying, It might have been expected. And truly I could almost find in my heart to confesse that this was a fault. For though it might have been expected from another man; yet it was not to be expe∣cted from M. Hobs; for his witt is not like the witt of o∣ther men, He is the First (he tells us) that hath made the grounds of Geometry firm and coherent. But why was it not to be exspected? Because, you say, It is impossible to define (Geo∣metricall) proportion universally by comparing Quotients. (Im∣possible, I confesse, is a hard word; but yet, I hope, it may be.) But why is it impossible? more than it is impossible to define Arithmeticall proportion universally by comparing of Remainders? Because, forsooth, In quantities incommensura∣ble there may be the same proportion, where neverthelesse there is no Quotient: (Very good! But why no quotient?) for quo∣tient there is none but in Aliquot parts. (Gooder, and gooder!) But, I pray, is not A / B as good a Quotient, as A-B is a Remainder? whether the quantities be commensurable, or Incommensurable? No, you say; For setting their Symbols

one above another with a line between, doth not make a Quotient. But why not? as well, as setting their Symbols one after another, with a line between, makes a Remainder? For, if the quantities be incōmensurable, the Remainder is no more explicable in Rationall numbers, then is the quotient. If from 3 you subduct √2, the Remainder is but 3 − ••2. If you divide 3 by √2, the quotient is 3/√2;. And is not his as much a Quotient, as that a Remainder? and as well designed? Yet this is all you have to say to the businesse: The rest is but Ranting, or vapouring. But, however, we are much deceived, you tell us, if we think, with pricking of Bladders to let out their vapour; for we see, you say, we make them swell more then ever. What? till they bu••st? I hope not so. (Crepent licet, modo non Rumpantur.) I have heard, I con∣fesse, that a Toad would swell the more for being pricked; but I never knew that a Bladder would, till now.

The next thing that troubles you, is, that I said, that the Corollaries of these two Articles taught us nothing new. (There be as I recon five and nine; fourteen in all.) Yes, you say, the ninth Corollary of the sixth Article is new: (No; it is not. We are taught the same by the second of the fifth of Euclid; and by the converse of the eleventh prop. of the sixth chapter of M. Oughtred's Clavis;) and the rest were never before exactly demonstrated. What? none of them? That's much. You mean, I suppose not all. And that I am content to believe: For they are not all true. As for exam∣ple; The second Corollary of the fifth Article, is thus de∣livered Universally, If there be never so many magnitudes A∣rithmetically proportional, (whether in continuall or interrup∣ted proportion; for you doe not limit it to either, more then you had done that next before it, which you cannot deny to be understood of both) the summe of them all will be equall to the product of halfe number of Terms, multiplied by the summe of the extremes. And then that we may be sure it is not intended only of cōtinual proportion, you give instance in proportion discontinued, For (say you) if A. B∷C. D∷E. F. be Arithmetically proportionall (though but discontinued, for so your Symbols import, both in the Latine and the English, least we might think it had been the Printers fault, and not the Authors;) the couples A + F, B + E, C + D, (you say) will be equall to one an other. This,

though it be true of continued Arithmeticall proporti∣on, yet of discont••nued proportion, as you here affirme it, it is notoriously false. For how doth it appeare, that C+D, is equall to A + F. For instance, let the termes be these 2. 1 ∷ 20. 19 ∷ 3. 2. in arithmeticall proportion. is 20+19, equall to 2 + 2? or to 1 + 3? It's no marvell then that this was never before exactly demonstrated. But we are taught nothing new by this. For though this be new and be years, yet we cannot learn it. Wee'l go on therefore: and see what you say next of the thirteenth Article.

Wee began, as I said, with slighter skirmishings; about Definitions &c. The skirmish now growes hotter; when I charge you with false propositions and demonstrations; and that you be touched to the quick, we may guesse by the loud out-cry; In objecting against the thirteenth, and sixteenth Articles, we doe at once bewray both the greatest Igno∣rance, & the greatest Malice, &c (and so on, for a whole leafe or more;) Now this Ignorance h••wrayd, was your own, viz. that you had given us false demonstrations &c. and then is it not spightfully done of us to discover them? Well; let's see what 'tis that makes you cry out so fiercely.

The proposition is this, Of three quantities that have pro∣portion to one another, (suppose AB, AC, AD; or 6, 3, 1;) the proportion of

[illustration]

the first to the second, and of the second to the third taken together, are equall to the proportion of the first to the third. That is, said I, The proper∣tion compounded of that of the first to the second, (suppose 6 to 3. which is double,) and that of the second to the third (viz. 3 to 1, which is treble,) is equall to that of the first to the third, (viz. 6 to 1, which is sextuple.) And was not this your meaning? (I am su••e 'tis either thus or worse) This composition, I said, was such as Euclide defines 5 d 6; which is done by multiplying the quantities of the propor∣tions: viz. 6/3 × 3/1 = 6/1, (not by adding them; for so 6/3 + 3/1 = 2/1 + 3/1 = 5/1.) Did I not explaine your meaning right? I••meant no hurt in saying this was your meaning; for the meaning was a good meaning; and the proporsion so meant, is a good proporsion; (but, if you mean otherwise, the proposition is false:) and, doubtlesse, 'twas a good meaning too, when you meant

to demonstrate it; (all the mischiefe was, you could not do what, you meant to doe.) If this be your meaning (as J am sure it is or should be,) what is it that troubles you? You doe not like the word Composition: that's one thing. Well then let it be called Addition for once, J told you then, J would not content for the name; (but you know 'tis such an Additon of Proportions, as is made by multiply∣ing of the quantities; as appeares by the very words of the definition 5 d 6) Then you doe not like that J should say the proportion of 6 to 3. is double; and that of 3 to 1, treble. Tell me (say you) egregious Professors, How is 6 to 3 double proportion? The answer is easy, (though perhaps you will not like it;) The proportion of 6 to 3, or 2 to 1, is that which is commonly called Double; and that of 3 to 1, is is commonly called Treble; And if you will not believe me, pray believe your own words, Corp. pag. 110. l. 5, 6. Ratio 2 ad 1. vocatur Dupla; et 3 ad 1 Tripla. You tell us then, We may observe that Euclide never distinguisheth be∣tween Double and Duplicate (no more then other Greek writers do between 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 and 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉.) one word (you say, serves him every where for either. You might as well bid us put out our eyes; or else believe that 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 and 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉▪ are the same words. Perhaps you thought so when you wrote your booke in Latine; but, since that time you have been better instructed, and have learned at length to distinguish between Double and Duplicate, as we shall heare anon. But let's goe on. All this hitherto hath been but scuffling, and little to the purpose, though there you make the greatest out cry, (like a lapwhing, when shee's furthest off her nest.) we are now comming to a close grapple. (and 'tis like to prove as had as a Cornish hugge.) Your de∣monstration, I said, was false (and that greeves you.) The strength of it, as I told you, lyes in this, The difference of AB, AC, (be they Lines or Times, chuse you whether, for by construction the times and lines are made proportio∣nall,) together with the difference of AC, AD, taken together, are equall to

[illustration]

the difference of AB, AD; therefore the proportion of AB, to AC, and of AC, to AD, taken together is equall to that of AB to AD. That

this is the strength of your demonstration you doe not de∣ny. Now that consequence I denyed; affirming that from that equality of the difference, you could not inferre the equality of Geometricall proportion; (and, of Arithme∣ticall, the question is not; nor is pretended to be.) And J gave this instance to the contrary, to shew the weaknesse of your Argument; Taking between A and B, any point at pleasure suppose a; you may as well conclude the pro∣portion of aB to aD, aS of AB to AD. to be compounded of that of AB to AC, and of AC to AD. For, (in your own words.) the difference of AB, AC, with that of AC, AD, are equall to the difference (not only of AB, AD, but even of) aB, aD; and therefore the proportions of those, to that of these. Now all that you have to say against it, (for I doe suppose, as you would have me, the motion to be equally swift all the way,) is this, The difference of AB, AC, ••annot

[illustration]

be the same with the difference of aB, aC, except AB and aB are equall. And here we joyne issue. The difference of AB, AC, say I, is BC; and the difference of aB, aC, is the same BC; though AB, aB, are not equall. The case is ripe for a verdict. Let the Jury judge. And now you may, if you will, go on to rant at Ignorance and Malice, at Sym∣bols and Gambols, at double and duplicate, at asses and eares, at Cla••ius, Orontius, and too learned men, or whom you will; haeret lateri lethalis arundo. But thus 'tis, when men will needs have Geometricall proportion, to be estimated by Differences, and not by Quotients.

(I told you moreover that your demonstration was but Petitio principii, and shewed wherein, with some other faults which you take no notice of, because you had no∣thing to say to them. And shewed you how your 13, 14, and 15, articles with all their Corollaries, (which fill up a matter of 4 pages.) might have been to better purpose de∣livered in so many lines. But this is no great fault with you, who think the farthest way about, the nearest way home.)

At the 16 Article the case is as bad or worse. The cry goes on still. This is all Ignorance and Malice too. And a huge out cry against Quotients, and Symbols, and a loud On••ethmus as you call it. But not a word to the purpose of what was objected; (except only one clause wherein you

tell us how absurd you mean to be by and by.) The busi∣nesse is this, Euclide (10 d 5) defines Duplicate, and Tri∣plicate proportion, &c. in this manner, If three magnitudes be in continuall proportion, the first to the last hath duplicate pro∣portion of what it hath to the second; if four, triplicate; &c. (and that indifferently whether the first or last be the bigger.) Now you (that you might shew your selfe wiser then Eu∣clide, and be the first that ever made the grounds of Geometry firm and coherent,) thought it was to be limited to this case only, when the first quantity is the greatest. And therefore thus define, The proportion of a greater quantity to a lesse (very warily) is said to be multiplied by a number, when other propor∣tions equall to it, be added. And therefore if the quantities (con∣tinued in the same proportion) be three; the proportion of the first to the last is Double, of what it hath to the second; if four, Treble, &c. (which most men, you say, call duplicate, tripli∣cate, &c.) But if the proportion be of the lesse to the greater (of which Euclide, it seems, was not aware) and there be an ad∣dition of more proportions equall to it, it is not properly said to be multiplied, but submultiplied (that is, divided; which yet you tell us, by and by, is to be done by taking mean proportio∣nalls.) So that of three quantities (so continued) the propor∣tion of the first to the last, is halfe of what it hath to the second; if four, a third part, &c. which are commonly called subdupli∣cate, subtriplicate, &c. Now this, I told you, was foul great mistake, and such a one as should not have procee∣ded from a Reformer of the Mathematicks. And, to use your own distinction (Less. 2. p. 9.) 'tis a fault not of Neg∣ligence, but of Ignorance, or want of understanding principles: and therefore an ill favoured fault, and, by your own rule, to be attended with shame. I shewd you there (and you be∣lieve me now) that in the numbers 1, 3, 9, 27, &c. the proportion of 1 to 9, though lesse, was not subduplicate to that of 1 to 3, but duplicate, as truly as the proportion of 9 to 1 is duplicate to that of 3 to 1; and that of 1 to 27 was triplicate, not subtriplicate, of that of 1 to 3; Of which I gave you this demonstration, (though it seems, you did not understand it, and therefore say, I bring no Argument.) Because 1/9 = ⅓ × ⅓, and 1/27 = ⅓ × ⅓ × ⅓, as well as 9•• = 2/1 × 3/1, add 27/1 = 3/1 × 3/1 × 3/1. And the subduplicate of 1 to 3, is not, as you suppose, that of 1 to 9, but of 1 to

√3. Now this was so unlucky a mistake, or Ignorance, in a thing so fundamentall, that (as I then told you, and you have since found to be true) an hundred to one, but it would doe you a deal of mischief all along. And it was the touching upon this fore place, that gawled you so much but now, and put you beside your patience.

But let's see now how you behave your selfe. A loud rant we have, as if it were grievous doctrine I had taught, and your own had been much better. But not a word to the purpose save only this 'Tis absurd to say, that taking the same quantity twice, should make it lesse. But though you say so, you doe not think so. For when you have done your rant, you goe slyly, (without saying a word of it, or ac∣knowledging any error,) and put out that whole sixteenth Article, which we had in the Latine, giving us in the English another instead of it, quite of another tenour, and quite contrary to what you had before. And now a proportion of the lesse to the greater, (as well as of the greater to the lesse,) being twice taken, shall be duplicate, (not subduplicate as before;) and thrice taken, (not subtri∣plicate, but) triplicate. Now (because you say it,) it is not ab∣surd to say, that taking the same quantity twice, should make it lesse; (though when I said it, it was absurd.) Now A pro∣portion is said to be multiplyed by number, not submultiplyed, when it is so often taken as there be unities in that number. (Whether it be of the greater to the lesse, or of the lesse to the greater;) And if the proportion be the greater to the lesse, then shall also the quantity of the proportion be increased by the multiplication; but when the proportion is of the lesse to the grea∣ter, then as the number increaseth, the quantity of the proportion diminisheth; For it is no absurdity now, to say that taking the same quantity twice makes it lesse. And truly now, methinks, thou sayst thy lesson pretty well; I could find in my heart to spit in thy mouth and make much of thee, hadst thou not railed at him that taught thee; which is but a trick of an ungratefull schollar: But let's goe on, and see whether this good fit will hold? As in these numbers, 4, 2, 1. the proportion of 4 to 1, is not only the duplicate of 4 to 2, but also twice as great. (Nay that is good againe; he hath learned that there is a difference between Duplicate and twice as great. Surely this is not he, (or else the world's well

amended with him,) that laughed at the distinction of Duplicate and Double. Well, let's heare some more of it.) But, inverting the order of those numbers thus, 1, 2, 4, the pro∣portion of 1 to 2, is greater than that of 1 to 4; and therefore though the proportion of 1 to 4, be the duplicate of 1 to 2, yet it is not twice so great as that of 1 to 2, but contrarily the halfe of it. In good truth; a prety apt Schollar: for one of his inches; He says just as I bid him. Well, well! the world's well amended with T. H. The••'s hopes he may come to good. Yee see he learnes apace. He may be a Mathematician in time; though I say't that should not say't. I confesse he hath his faults still, as well as other men, (you must not think he can mend all at once,) The whole article is not so good throughout, at this bit at the beginning. He hath got a naughty trick of saying The proportion of equality is no quan∣tity, (but he hath been whipt for already;) He makes it stand for a Cyphar, (but that's a thing of nothing: It should have been but 1, and that's not much more.) And he tells us that the proportion of 9 to 4 is not onely duplicate, of 9 to 6, but also the Double, or twice as greate. And again, that the propor∣tion of—4 to—6, is double to the proportion of—4 to—9, &c. which would have deserved whipping at another time; but because he said the rest so well, I'le spare him for this once. He doth, it seems, believe there is a difference between double and duplicate, though he doe not yet know what it is; he will learn against next time. And to the like purpose is that which follows; If there be more quantities then three (it's no matter how many) as A, B, C, D, in conti∣nued proportion, what ever the proportion be, so that A be the least; it may be made appeare that the proportion of A to B, is triple magnitude, though subtriple in multitude, to the propor∣tion of A to D. But however he shall be spared for this bout; because I said so; and I will be as good as my word.

We have but one touch more, and I have done with this Chapter. 'Tis at the Corollary of the 28 Article. Here you find fault first with the word aliquot; and ask whe∣ther I think that partes aliquot can be numero infinitae? And I think they may. Where there are more then one, there be at least aliquot, whether few or many. What I objected a∣gainst that Corollary, was not against the truth of it, for it is obvious and facile; but that it needed not so much a doe, as to be ushered in with three teadious Propositions, art:

26. 27. 28. Which, I said, (and you do not deny it,) seems to be put in only in order to this Corollary. In Art 28. you were come thus farre, If from a line (AB) be cut off a part (suppose AC) and between that and the whole, be taken two meanes, the one Geometricall, the other Arithmeticall (AD, AE;) the greater the part

[illustration]

is that is cut off AC (and con∣sequently the remainder CB, the lesse,) the lesse will be the difference between those two meanes (AD, AE.) Hence at length you come to this Corollary, That if the line (AB) be divided into equall parts infinite in number, (and so the part remaining CB, one of them, infinitely small,) the difference between the Arithmeticall and Geometricall means, will be infi∣nitely little. This I said might be proved universally, not only of these two, the Arithmeticall and Geometricall Meanes, but of any two Meanes whatsoever. For supposing any two lines AD,

[illustration]

AE, which are each of them greater then AC, but lesse then AB, (as all means must be) your points DE, (whether D, or E, stand first, it matters not) must needs fall between C and B: And there∣fore the difference DE, cannot be bigger than CB, but rather a part thereof, and so lesse. Now the whole CB is supposed infinitely little, and therefore its part DE can∣not be bigger. And is not this as well proved as if I had premised in order to it, three whole Articles, and spent three pages about it? You say, (and that's all you object against it) that I doe not prove, that BE the Arithmeticall difference is lesse than BD the Geometricall difference. No; Nor ought I so to doe. For the thing to be proved was not which of the two is greatest, but that (whether soever were the greater) the difference between the two is infinitely small: which is done sufficiently without that other. And are not you then a wise Mathematician to make such an ob∣jection?