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N^{o} 1672.
 
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March last7); in such words as he thought fittest to expresse what he did apprehend to be their concurrent sense. Nor do I see any necessity of receding from it save yt where he speakes of the Termination of converging Series, in stead of ye Termination, by a mistake it is sayd the Summe8). Nor is the Royal Assembly (that I know of) further concerned in it. Soon after; those in France, in their Journal des Sçavans, published an opinion of Monsieur Huygens9), to whom, it seems, it had been referred to consider & deliver an opinion of it. Hee there takes notice yt those of England had in ye generall given it a favourable character, (to which he addeth a like of his own;) but that they had sayd nothing as to that particular whether it were therein demonstrated that it is impossible Analytically to square ye Circle & Hyperbola; (that is, by Addition, Subduction, Multiplication, Division, & Extraction of Roots, which he calls Analytical operations)10). And delivers his opinion in the negative; and that supposing all to bee true which is demonstrated in his 11th proposition (where that demonstration is supposed to ly) it proves no more but that it cannot be performed by his methode; not that it cannot be done at all: unless it be supposed that the termination of a converging series can be no other way formed but by his methode; or at lest, yt if it may be formed any other way, it may be formed this way allso, which is not (he sayd) demonstrated. To which Mister Gregory published an Answere11), inserted in the Transactions of Julie last. Which were both made publike before I had seen either of them. Which were both sent to mee, & my opinion desired concerning them. My answere was; that, (beside some other particulars of lesse moment) the proposition mentioned, did not so much as affirm, yt ye circle could not analytically be squared; & therefore it was not be expected it should be there demonstrated; nor was ye Demonstration to be blamed for not proving what ye Proposition did not assert. And to ye same propose I wrote myself12) to Monsieur Huygens. I confess, My Lord, I had not all this while observed (nor was I told it) that Mister Gregory did pretend, therein, to haue demonstrated yt ye circle could not be analytically squared; which made mee give such answere. And though, in his Preface, page 5, he intimate something of it; yet when hee presently adds, verum certè est me hanc demonstrationem integram ad phrasem geometricam non  
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reduxisse; nam ut hoc perficiatur, opus est non parvo volumine, &c. I did not think yt I was to expect a formed demonstration of it; but onely an intimation of some principles from whence such a demonstration (hee supposed) might be formed. But presently after, hee sent mee word, that he did not onely Affirm it, but did thinke hee had Demonstrated, that it was not possible Analytically to square ye cercle in any methode whatsoever, pressing yt I would positively assent to it, or give my reasons why I did not. By which I perceived yt Monsieur Huygens (who, it seems, knew this before, though I did not,) had more reason to make yt Exception to ye Demonstration, than I was aware of, who knew not of that pretense. I haue severall times since signified to him, yt although I was satisfyed, as to my own judgement, that it could not be done; yet I was not satisfied, that is was by him demonstrated. And haue given him severall reasons (though, it seems, not such as satisfy him,) why I was so unsatisfyed. As, that there be many Lemmata or Suppositions which he doth either postulate or silently take for granted, which, though they may be true, yet are not so clear but that there is reason they should be proved, before his Demonstration can be judged full & perfect. That ye exception made by Monsieur Huygens, is not yet removed: which is as much as to say, that, although his 10^{th} proposition be demonstrated, the converse of it is not: And, consequently, it proves onely that ye converging series cannot his way be terminated, not that it can no way be terminated analytically. In summo, That hee hath no where proved this consequence, That if ye Sector be not in his way analytically composed of its Triangle & Trapezium (which is the whole of his 11^{th} proposition,) then ye Circle can no way be analytically squared. And haue desired him to giue a cleare demonstration of this consequence; Presuming that in applying himself so to do, hee would either shew mee more light than yet I see, or else meet with such insuperable difficulty as would discover to himself that I had reason not to be hasty in affirming his Demonstration to be full and perfect. And I haue ye more reason to insist upon ye proof of that consequence, because, though other less objections could be all removed, this one great one seems to mee insuperable; That his 11^{th} proposition, though ever so well demonstrated, shews onely yt ye Sector indefinitely considered can not be so compounded as is there sayd: Or, (which is equivalent) not every Sector. Notwithstanding which, it might well inough be possible, that some Sector (if not all) might be Analyticall to its Triangle or Trapezium: (And I think he doth allow it so to bee, or even commensurable, page 29. Respondeo hoc esse verissimum,13) &c.). Like as, in this Equation, for ye Trisection of an Arch 3r^{2}a  a^{3} = r^{2}c indefinitely taken; the Root a, is not Analyticall with r and c, (that is, ye Proportion of ye Chord of ye Single Arch, to that of ye triple arch ye Radius, cannot be universally designed  
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by those he calls Analyticall operations; or ye value of a analytically compounded of r and c, as he speakes; that is, it cannot be designed by commensurable numbers & surd Roots:) as Charles14), Schoten15), & others agree. Yet in some cases (though not universally) it may happen to be not onely Analytical, but even commensurable, (as, for instance, if c = 2r be ye Subtense of a Semicircle, a may be equal to r or to  2r.) Now if but some one Sector (though not all, or ye Sector indefinitely taken,) be found Analytical with his Triangle, or Trapezium; there be many ways, (as, by its proportion to ye whole, by its center of gravity, &c.) by ye help of this one, to square ye whole circle. I might adde allso (though this may more safely be avoided by altering his construction,) that his whole processe concerns onely such Sectors as are lesse than a Semicircle. For if it be a Semicircle (to say nothing of those yt be greater) the two tangents will never meet in F (as his figure supposeth) to make his Trapezium. And therefore it proves nothing directly as to ye Semicircle, much lesse as to ye circle itself. Now though we cannot, Analytically, trisect some lesser Archs; yet ye Semicircle wee cann. And though we cannot, Analytically, assign ye center of Gravity of a Sector; yet we can of a circle. To proue therefore, yt lesser Sectors (at lest some of them) are not Analyticall to their Triangles, or to ye Square of ye Diameter; doth not presently prove, yt ye Semicircle, or Circle, are not so. But on this Objection I lay lesser weight. For though it shew a fault in ye Demonstration, yet it is onely such a fault as may bee amended: which, I doubt, ye former cannot bee. Hee hath attempted answere, (first in writing, & since in Print16)) to some of those Reasons aboue mentioned: which he calls Objections against his Doctrine: (Hee should rather haue called them Objections against his Demonstration; For I did not object against his Doctrine at all; as having many years since demonstrated the same myself, though he take no notice of it, in my Arithmetica Infinitorum, propositio 190 with ye Scholium annexed:) But such as do no more satisfy mee, than my Reasons did him. Hee is yet every earnest to haue me satisfied yt his Demonstration is good; & to haue a like approbation from ye Royal Society: thinking yt he hath hard measure, yt having (as he is confident) the truth on his side, not onely those of France have declared against him; but those of this Royal Society seem at lest tacitely so to do. Which hee complains of with some regrett, both in Letters to mee, and in his printed preface16) to his Exercitations. Professing that he desires no more but a fair character granting what he hath done, shewing in what he hath failed, and  
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proposing what yet rests to do the work, that so if he or any man else can adde what is wanting it may be supplyd. I shall therefore represent to your Lordship (as to a very competent judge) how ye state of his Demonstration stands to my apprehension. As well that he may be satisfied that I haue considered, & do understand (in some measure) ye strength of his Demonstration and not reject it unconsidered. As that your Lordship allso may judge, whether it be Obstinacy or Reason yt holds mee yet unsatisfied. First, therefore, I take it as evident & confessed; that there is not in his Book any such Proposition formally layd down to be demonstrated, as That ye Circle cannot Analytically be squared: Nor any Demonstration which doth in terms conclude any such Proposition. All therefore that wee are to inquire after is but, whether from what he hath demonstrated, such a proposition may bee directly inferred. I could have wished therefore, that he would himself have drawn up his demonstration into form, & not left us onely to seeke materialls for it as they ly scattered up & down: that there might be no occasion for him to complain yt I haue not represented the strength of his demonstration to ye best advantage. But since he hath not done it; and yet would haue it thought that the thing is fully demonstrated, though ye demonstration be not put into form: I shall lay down ye severall branches of that demonstration in ye best order I can, & shew which of them I judge to be proved, & which not. 1. A Sector (lesse than a Semicircle) indefinitely taken, is the Termination of a Converging Series infinitely continued; beginning with ye Inscribed Triangle & circumscribed Trapezium, (& so onward, by continuall Bisection, with Inscriptions & Circumscriptions respectively;) whose respective converging terms are continually in ye same manner analytically compounded of those next foregoing; and so continually approaching as that at length they become coincident each with other; & with ye Sector. 2. And, in particular; the two first (or any two respective antecedents) being a, b; ye two next consequently will be √ab, 2ab / a + √ab. Both which are proved in Scholium propositionis 5. from ye Antecedent propositions, with yt which next follows. 3. These component terms a, b, (the Triangle & Trapezium) supposing ye chord Analytical with ye Radius, are Analyticall each to other. Which is necessary to this business, may be easyly proved. 4. And, consequently, what is Analytical to either of them, is Analyticall to both which may be proved from Definitions 6. 7. and Petition 1. 5. Now in any such converging series, if there can be found a quantity which may in ye same manner be analytically compounded (without introducing any ex  
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trinsick quantity) of ye two first, & of ye two second, converging terms; by help of this quantity, that Series may be Analytically terminated: That is, the termination thereof may by Analytical operations be compounded of the two first converging Terms. This is proved by Proposition 10. (as it is now reformed & explained in his Reply17) to Monsieur Huygens;) But not ye converse of it; because ye converse of that proposition is not proved. Which is ye exception of Monsieur Huygens18). 6. And, consequently, if the two first converging terms be Analytical each to other; the Termination will be Analyticall to both of them. Which may be proved from Petition 1. 7. If therefore any quantity can be found analytically composed of a, b (the two first terms of ye converging series proposed,) and, in ye very same manner, of (the two next terms) √ab, 2ab / a + √ab, (without ye intermission of any other quantity in either of ye compositions) then may this series be analytically terminated. This (but not ye converse of it) follows from § 5. 8. And consequently (when as a, b, be analyticall each to other, viz. § 3) the Termination thereof, (yt is, ye Sector indefinitely taken,) will be analytical with both of them. which follows from § 6. 7. 9. And consequently, every such Sector with its respective Triangle & Trapezium. For such Sector indefinitely taken, is any such Sector whatsoever. 10. Now of some Sectors ye chord is analytical with ye radius as, for instance, of ye Quadrantal Sector; and consequently ye Triangle & Trapezium are Analyticall with ye square of ye Radius, or of ye Diameter. As is easyly proved. 11. And therefore their Sectors will then be so; & therefore may be analytically squared: That is, their proportion to ye Square of ye Radius or ye Diameter, may be designed by Analyticall operations, or by commensurable quantities & surd Roots. Which may be proved from Definitions 6, 7, and Petition 1. 12. Now to some at lest of those Sectors, the Circle is Analyticall; and particularly to ye Quadrantall; As being ye quadruple thereof. 13. And will therefore be Analyticall to their Triangles, Trapezia, and Squares of ye Radius and diameter; (and so may be analytically squared.) Which may be proved from Definitions 67. and Petition 1. 14. But, on the contrary, if in such a converging series no such quantity can be found, (as is mentioned § 5) which can, in ye same manner, be analytically compounded of the two first, & of the two second, converging terms: then cannot this series be by that process analytically terminated. For that process supposeth such a quantity; at § 5 &c. 15. And if not by that processe; then not at all. Which is yet to be proved.  
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16. If therefore no such quantity can in ye same manner be analytically compounded of a, b, and of √ab, 2ab / a + √ab, then cannot this series, by that process be analytically terminated. For that processe supposeth such a quantity, at § 7 &c. Or it may be proved from § 14. 17. And, consequently, the Circle cannot by that process be analytically squared. For that process supposeth such a termination, at § 8. &c. 18. And if not by this process, then not at all. Which is to be proved. 19. Now if the first terms were a^{3} + a^{2}b, ab^{2} + b^{3}, and ye second terms (in like manner compounded of ye first, as in ye series proposed) ba^{2} + b^{2}a, 2b^{2}a: No such quantity could in ye same manner be analytically compounded of the two first, & of ye two second, converging terms. Which is proved at Proposition 11.
20. And therefore, not if ye two first be a, b, & ye two second √ab, 2ab / a + √ab, which connexion deserves to be cleared. 21. Therefore this series cannot be by that process analytically terminated. By § 14. 22. And therefore can not at all be analytically terminated. To be proved by § 15. 23. Therefore ye Sector indefinitely taken is not analyticall with its Triangle & Trapezium. For ye Sector indefinitely taken is this termination. 24. Therefore not every Sector. For if every Sector, then any sector whatsoever; that is, the sector indefinitely taken. 25. Therefore no Sector: At lest no Sector which is analyticall with ye whole Circle, and whose Triangle & Trapezium are analyticall with ye square of ye Radius or of the Diameter. Which consequence, I doubt, will hardly be made good. 26. Therefore ye Circle cannot be by that process analytically squared. For that process supposeth some such Sector which shall be analyticall with the Circle, & with its own Triangle & Trapezium, and these with ye Square of ye Radius or Diameter, at § 10, 12. 27. Therefore ye Circle can in noe manner be Analytically squared. Which is to be proved from § 18.
This, my Lord, I take to be the true Anatomy of that demonstration, which from his principles should prove that ye circle cannot at all be analytically squared. Which to mee, I confesse, seemes somewhat lame at § 15, 18, 25, and those which depend on these. Especially at § 15, & 25. The former of which is that which Monsieur Huygens excepts against as not proved: The latter seems to mee as much or more considerable. Till this be supplyed, his argument seemes to mee  
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to beare such a forme as this; If the Sector be so compounded, the circle may be analytically squared; But the Sector is not so compounded; Therefore the circle cannot be analytically squared. Which Syllogism is peccant in form, though ye propositions be true; & therefore ye conclusion follows not: Unlesse we suppose not onely ye consequence of ye major proposition to be demonstrated; but ye converse of that consequence; which is not done. I shall onely adde two things; The first is that however I am not satisfyed that this demonstration is full & perfect, yet this hinders not but that divers other things in that Book may be very ingenious & well demonstrated. For this proposition, be it true or false, demonstrated or not demonstrated, doth not at all influence ye rest of the Book, or enervate ye other demonstrations. The other is, that this Author need not be very solicitous for ye supplying of what is defective in this demonstration; because the work is done allready, the thing itself being proved long since in my Arithmetica Infinitorum; proposition 190. with ye Scholium annexed to it. Where it is proved, that what was before demonstrated to be ye true proportion between ye Circle & ye Square of its Diameter or Radius, or between ye Diameter & ye Perimeter; cannot be expressed either by Rational Numbers or Surd Rootes (or, as this Author speakes, is not Analyticall;) without supposing an odde number to be equally divided into two integers; and a forming of Equations between ye Laterall & ye Quadratick, between ye Quadratick & ye Cubick, &c.; that is, which shall have more then one Root but fewer then two, and more then two but fewer then three, &c. which are impossible. Notwithstanding all which, My Lord, if Mister Gregory shall supply these defects; or otherwise make it evident to ye Royall Society that these consequences are allready proved though I have not been so quicksighted as yet to see it: I shall very willingly consent that ye Royall Society shall give as full an attestation19) thereof as hee can desire. Having had no other design in all this but to satisfy his importunity, which I could hardly avoyd without being uncivill. I am
My Lord
Your Honours very humble servant John Wallis. 
