I Have, according to your re­quest, perused the small Ge­ometrical Tract lately pub­lished by Mr. Thomas White: whereof accordingly I give you here this brief account. The in­tent and occasion of his present Writing, as on this Subject, [Page 4] (having scarce hitherto appeared in that kinde) is (as is manifest by the Title) to make known to the world the great light he hath received particularly from God in that noble Science of Geome­try; as having never studied it, nor much applied himself that way: that thereby other men, conceiving this so excellent a Piece must needs have been con­veighed to him by particular light from Heaven, may learn thence more to prize and esteem, then hitherto they have done, his Works already published. For so he tells his Reader in the end of his Preface, that the things he is now to declare, ought to be sufficient to give esteem to all his former Labours. For if (saith he) they came from the Author, and from that force and vigour of Wit, by which he is able to per­form many more equal to this, [Page 5] then his precedent Works are not to be contemned, as pro­ceeding from such a Father, quia de tali exorta sunt Patre: But if it come from Heaven, then much more are his other Works wor­thy consideration, to wit, as co­ming also from Heaven before it.

Now that they come not from himself, he openly avouches, as having never had any Master in Geometry, nor much applied himself to that Science, or read so much as Euclide. Yea, he free­ly acknowledges himself so little versed in Geometry, that he plainly affirms, no man will call him a Geometrician, if he be one himself: Intuere me hominem, quem nemo Geometram salutet, mo­dò ipse sit. Whence he concludes, that the things he is now to de­liver, must assuredly come from Heaven: Talis cum sim, non à me haec habes, sed ab eo, qui ex legibus [Page 6] Providentiae suae, ea gubernationi Ecclesiae suae, in hoc rerum ar­ticulo, opportuna & fecit, & vi­dit. Being I am such a one; (to wit, so little vers'd in Geometry) thou hast not these things from me, but from him, who accord­ing to the Laws of his Provi­dence, both saw and made them fit for the Government of his Church, in this present conjun­cture: give him the thanks, &c. Wherefore he exhorts the Rea­der, not to neglect his own good, nor contemn a wonder fallen to him from Heaven. Quod superest, tibi consule, & ostentum a caelo ad te delapsum ne contemnito.

All this with some other such expressions hath Mr. White in his Preface; whereby you clearly see, how highly he values this his Tract, as fallen from Hea­ven, and accordingly desires the like esteem should be framed of [Page 7] all his other Works: that so his Readers considering and weigh­ing with themselves, that it is im­possible so learned and subtil a work should come from one that never studyed Geometry, must necessarily conclude that it came particularly from Heaven: and by consequence also have a better esteem, then hitherto, of his for­mer Labours, as undoubtedly coming from the same place.

This is the aim and drift of Mr. White in this his Tutela; wherein truly he seems, by so far expressing himself, to have given a very great advantage to who­soever will impugne his former Writings. For now his Adver­sary hath no more to do, but to shew (as easily he may) that this Geometry never came from Hea­ven, and by consequence, that neither any of his former Works, (whereof Mr. White would have [Page 8] this to be a pattern according to which they are to be measured) ever came from thence. This I say he will easily make manifest; for it is impossible, that ever so weak a piece, as this is, and with so many Patent and open er­rors against Geometry (as we shall presently see) should ever come from any Geometrician, much less from Heaven. Which that it may appear, we will brief­ly run over the Propositions, as they are in the Book, to see which of them may deserve to be thought to have particularly de­scended from Heaven: and then note onely some more patent and obvious errors, such as you your self may easily conceive: by which you may guess at the rest, and what can be here expected.

His Treatise then contains in all thirteen Propositions; of which the two first onely expli­cate [Page 9] the tearms he is to use. The two next are taken out of his Brother, Mr.Richard Whites Book called Hemisphaerium dissectum, as he also acknowledges; so that cer­tain it is, that these first four came not from Heaven. The five fol­lowing aim at the Quadratura Circuli, but perform no more then a hypothetical, or conditio­nate Quadrature: that is, if such or such a proportion were known, it were possible to square a circle; but of such Quadratures as these, we have enough alrea­dy, and books are every where full of them. For the rest, I finde nothing in these propositions truly demonstrated, that may not be found in other Authors: so that in these nine first Propositi­tions, we have nothing that may be thought a wonder fallen from Heaven, as was promised.

In the Annotation before the [Page 10] tenth Proposition he endeavours to demonstrate, that a Spiral line of the first revolution is precise­ly equal to half the circumfe­rence of the including circle. For example that the Spiral, noted here by the pricked line EABCMGHID, is equal to the half circumference, DQR. This Demonstration indeed seems to have somewhat of the extraordinary in it: but yet it is neither new, nor true; and so impossible it should come from Heaven. For this self-same De­monstration was found out above thirty years ago by one Gulden a Jesuite: and is to be seen in his Book, called Centrobaryca, or, De centro Gravitatis; as Mr. White here also acknowledges. This Demonstration, I say, Gulden found out, and shews in the said Book, lib. 2. c. 2. prop. 6. But afterwards, before the Book was

[Page] [Page 11] printed, being advertised of the falsity found out by calculation, and perceiving it to be so, he pre­sently recalled the same in the next Chapter, and clearly shews, that not onely the Spiral line it self is bigger then the said half circle, to which before he thought it was equal; but also evidently shews, that the sides of an inscribed Polygone of twelve Angles is also bigger then the said half circumference, as I shall afterwards declare.

Of this Guldens recalling Mr. White (as himself testifies) was by some friends informed; but not being able, as it seems, to exa­mine Guldens calculation, nor to see the force of it, he presumed to print the said Demonstration, as his own, and to maintain it to be good, and evident, and Guldens calculation, or retractation, to be manifestly false, as we shall pre­sently see.

This Demonstration I say, Mr. White let pass to the print; yet conceiving, as it seems, that people would still think it to be taken out of Gulden, it being the very same with his, he thought good to joyn thereto another of his own, demonstrating the same assertion, in a different way from the former; which he performs in the tenth and eleventh Propo­sitions. And certain it is, that this Demonstration is wholly his own, that is, that it neither came from Heaven, nor from any other Geometrician; it being impos­sible, that so many and patent er­rors should come from any that ever studyed Geometry, or read so much as Euclide; or even knew but how to resolve a plain Triangle.

For in the tenth Proposition he affirms and pretends to de­monstrate, first, that if in a [Page 13] Spiral line of the first revolution, be inscribed a Polygone with e­qual angles, (as in the adjoyned Diagram, you see here inscribed the Polygone EABCMGHID, with eight equal angles) that then the sides of the said Polygone shall equally exceed each other: that is, as much as DI exceeds IH, so much pre­cisely shall IH exceed HG, and HG exceed GM; and so of the rest.

Secondly, he affirms this com­mon excess to be equal to the least side of all; viz. to the side EA. These two Assertions he puts in the Title of the said tenth Prop. which is this, Latera Poly­goni inscripti spirali per aequales angulos exuperant sese invicem per excessus minimo lateri aequales.

Thirdly he likewise affirms, that having let down from the points I, H, G, M, &c. Perpen­diculars [Page 14] to the opposite Semidi­ameters, (as here you see let down the Perpendiculars IK, HF, GL, &c.) that then the parts of the said Semidiameters, inter­cepted between the Perpendicu­lars and the Spiral, that is, the parts KD, IF, GL, and so of the rest, shall be equall. This third he inferres (though falsely, as presently we shall see) about the middle of the said tenth Prop. in these words, Aequales itaque sunt rectae KD, IF, & HL. all which three Assertions are evidently false, as I shall briefly shew.

For if we put the Semidiame­ter ED (which according to the construction of the Spiral, is here supposed to be divided into eight equal parts) to contain 800 equal parts; the next EI will contain 700: and EH will have 600, and EG 500, and so forward: [Page 15] So that the first EA will have 100, and EB 200, &c.

Whereby we have now in every Triangle EDI, EIH, EHG, &c. two sides known, to­gether with the angle compre­hended.

For example, in the Triangle EDI, we have the side ED 800, and E I 700, together with the comprehended angle DEI of 45 degrees. So likewise in the Triangle EIH, we have E I 700, EH 600, and the angle IEH 45 degrees as before; and so of all the rest. Which being known, we may presently by re­solving the said Triangles, finde the two last sides of the inscribed Polygone, to wit DI, and IH, to contain the one 581, and the other 505, whose difference or excess is 76. But if in the same manner we resolve the first Tri­angle EAB, we shall finde the [Page 16] second side AB to be onely 147; from whence being taken the first side EA 100, shews the dif­ference between the first and se­cond side to be onely 47. And so in like manner will the diffe­rence between the second and third, AB and BC be onely 65. Now these three differences or excesses, 76, 47, and 65, are far from being equal, as Mr. White would have them. Wherefore in this he must needs confess him­self quite mistaken, and his de­monstration thereof to be false.

Neither is his errour less noto­rious in affirming the said excess, (which he also falsely supposes to be common to all) to be equal to the least side, that is, to the side EA: for EA being 100, is bigger then any of them all, as we have seen. Yea, he is so inex­cusable in this, that his very eyes might have discovered the error.

His third Assertion is also as false and unexcusable, to wit, that the lines KD, FI, LH, &c. are all equal. For resolving the Triangles EIK, EHF, and EGL (in which you have a side with all the Angels) you will finde EK 495. EF 424. and EL 353 which being respe­ctively substracted from ED 800 EI 700. and EH 600, leave KD 305, FI 276, and LH 247. which three numbers are also far (as you see) from being equal, as Mr. White pretends to have de­demonstrated. Wherefore we must needs here conclude, that such Demonstrations as these never came from Heaven, as Mr. White perswades himself, and would have us believe. And [...]ruly whosoever reads this his tenth Prop. will clearly see his want of Principles, and that he was fallen upon a business he un­derstood [Page 18] not: wherein he was so puzzelled, that he quite forgot what he had said he would prove, to wit, that the said com­mon excess was equal to the least side: for of this, after he had put it in the title, he makes no more mention, nor once goes about to prove it.

Now out of so weak and false a ground as this of the tenth Prop. he demonstrates in the eleventh (at least he thinks so) that the Spiral line EABCMGHID, is equal to the half circumference DQR: but a­gain performs it so unskilfully, that although the ground now laid in the tenth were true, yet follows not his intent. For by shewing onely that it is not big­ger, he inferrs it to be equal; which is no consequence, al­though the Antecedent were true: but both the one and the [Page 19] other are false, as we shall pre­sently see.

Now after so weak and false a Demonstration, by which he thinks he hath concluded the said Spiral and half circumference to be equal, he proceeds in the twelfth Prop. to refute Gulden; who recalling (as was said) this very Demonstration, (which now Mr. White pretends to be his own, and maintains to be good) clearly shews, that not onely the Spiral it self is bigger then the said half circumference; but also an inscribed Polygone, for example of twelve equal an­gles, is considerably bigger.

To confute this assertion of Gulden, Mr. White puts his twelfth Prop. wherein he pre­tends to demonstrate against Gul­den, that the sides of such a Po­lygone being added together are [Page 20] less then the said half circumfe­rence: but truly with a Demon­stration like the rest, that is most false and frivolous. For having put the line EC in the Dia­gram of the said twelfth Prop (which Diagram I have here ad­joyned) to be the greatest side of a Polygone inscribed with twelve equal angles, he supposes that the same longest side EC being taken six times, will be equall to all the sides of the said Polygone added together: for so he writes a little after the be­ginning of the aforesaid twelfth Prop. Clarum est itaque, EC sexies repititam, hoc est figuram in­scriptam spirali, esse majorem, &c. Whereby you see that according to Mr. White it is all one, to take the longest side of such a Poly­gone six times, or to take the whole inscribed figure, that is,

[Page] [Page 21] all the twelve sides added toge­ther: which notwithstanding is most false. For if we put the Semidiameter DB, or DC to be 12000, we shall have in the Triangle DEC two sides known, to wit DC 12000, and DE 11000, together with the angle EDC grad. 30. From whence is evidently concluded EC to have 6031. (I omit here alwayes the Fraction as making not to our present purpose) which number taken six times makes onely 36186: whereas all the twelve sides, found out in the same manner and added together, make 40003, as appears in the Table here adjoyned. So that in this supposition Mr. White is quite out.

Yet notwithstanding this false supposition, he goes forward to demonstrate against Gulden, that [Page 22] the sides of such a Polygone ad­ded together are less then the half circumference; which he per­forms so confusedly, and unskil­fully, that it is impossible to infer any thing to his purpose out of such a discourse. But be the dis­course what it will, at last he strongly concludes against Gul­den, that the sides of the said Po­lygone are less then the half cir­cumference.

But this his conclusion is most false, as Gulden hath evidently shewn, lib. 2. c. 3. prop. 1. and may be so apprehended by any man that knows but how to re­solve a plain Triangle. For by finding every of the twelve sides of the Polygone in such manner, as we now found the twelfth or longest side EC to be 6031. we shall have all their numbers, as appears in the here adjoyn­ed [Page 23] Table;

all which added together, make, as you see, 40003. Whereas if according to Archi­medes, you number the said half Cir­cumference, by ta­king the said Se­midiameter ED 12000 thrice with its seventh part, we shall finde the said half cir­cumference to contain at most on­ly 37715: which is far less then 40003. And by consequence the sides of a twelve angled Poly­gone inscribed in a Spiral, are absolutely longer then half the circumference of the first circle, as Gulden truly and learnedly shews against Mr. Whites so weak [Page 24] a Demonstration for the contra­ry, as we have seen.

By these discoveries of so ma­ny undeniable errours in his Ageometricall Demonstration, one would judge that Mr. White had the least reason of all others to censure any one; yet such is his passion, that he falls bitterly upon Gulden, censures, vilifies, and reviles him insufferably, calling his Computation unskilfull, and that he hath not a jot of Mathe­matick or Geometry in him; terming him one of those half Schollars, who stealing divers excellent things out of other Learned mens Writings, endea­vour to make them seem their own. This bitter invective hath Mr. White against Gulden, a man who never had in the least of­fended him, perhaps never heard of him, being dead many years [Page 25] since, and so not able now to an­swer for himself. Take Mr. Whites own words at the end of his twelfth Proposition. Calculus itaque Guldenianus imperitus est, & qualem ab ipso acceptari (neque enim vel talem ipse instruxit) dece­bat: Homine prorsus Amathema­tico, ut legenti ipsius scripta pro­num est patere. And a little af­ter having taxed his want of hu­mility and candor, he concludes him to be, Hominem officij Geo­metricij prorsus ignarum; & ex eo semidoctorum genere, qui cum ex magnorum virorum scriptis egre­gia multa depeculati fuerint, ut sua faciant, additis quibusdam levibus, justi voluminis ostentatione se vul­go discentium ostentant, &c. This gall, whilest Mr. White flourish­ed amongst his admirers with his new Demonstration, might have affixed some seeming ble­mish upon Gulden, amongst such [Page 26] Ageometricians as Mr. White is, but now appearing by what is said, to proceed from so unskilful a hand, it cannot tend to the dis­grace of any, save the censurer, who condemns that which he understands not. For certainly no Geometrician would or durst have said so much; the Compu­tation being performed accord­ing to the 47. 1. Euclidis, by the extraction of the square Root; then which there can be none more exact and manifest. As for that he calls him Semidoctus, a half Schollar, one utterly void of all Mathematick; that he hath stolen out of other mens works; and all this immediate­ly after so many errors commit­ted by himself, he hath put the lash into the hands of such, who if they please, will quickly know to use it; especially being so justly provoked by seeing one of [Page 27] their own Order so wrongfully abused; and will not fail to re­tort upon him all that he impo­ses upon Gulden. And truly who­soever shall read this Geometri­cal Treatise (which Mr. White esteems the master-piece of all that ever he hath writ) and Gul­dens Book called Centrobaryca, will finde so main a difference, that Mr. White without any pre­judice, by what appears in his, may be scarce thought fit to be Guldens Schollar.

And whereas he calls Gulden one of those half Schollars, who steal out of other Books, they will easily make it appear, that, whatsoever it be of Gulden, cer­tain it is, that Mr. White hath stolen that Demonstration out of Gulden. For even by his own confession it came not from him­self; Non à me haec habes, &c. and to say it came from Heaven, [Page 28] were a blasphemy, it being mani­festly false, as we have seen: Wherefore it must necessarily be concluded, that Mr. White took it out of Gulden, who printed it many years ago, as a particular invention of his own; neither can any other Authour be cited, who published it before him.

Truly a man would think Mr. White to have already said more then enough in so vilifying, and even trampling upon this Au­thor, especially there appearing no cause for such bitterness. But he is not satisfied to have thus disgraced him, as much as lies in his power, with the note of igno­rance in the Science he professes; but he falls upon his Moral Ver­tues, taxing him of Vanity, want of Humility, Candour, and the like, affirming him to be so vain, that although he thought he had committed an errour, (to wit, in his [Page 29] Demonstration of the Spiral) yet he could by no means be induced to co­ver it, by blotting it out, or candid­ly to confess the same, but goes on, framing excuses, as if in the very errour he had carried himself gal­lantly. Master Whites words are these in the place now cited: Et (quod faedissimum est) tantae va­nitatis est, ut cum erravisse se pu­taverat, neque delendo tegere, ne­que candidè confiteri sustinuerit; sed excusationes texere, quasi in ipso errore egregiè se gesserit, ostentare pergat, &c.

O most unworthy and false ca­lumny! when I had read these in Mr. white, and compared them with what Gulden sayes in recal­ling the said Demonstration of the Spiral, I was amazed, how Mr. White did not even blush when he writ so foul and evident an untruth. For of all that, which he so maliciously here imputes [Page 30] unto this man, there is not one word to be seen in Gulden, nor the least ground or shadow in his writings; yea, the just contrary to what is here so shamefully avouched, doth manifestly ap­pear, as any man may see in his Book called Centrobaryca, cited by Mr. white. Where lib. 2. c. 3. re­tracting the said Demonstrati­on, he plainly tells the occasion of it; viz. that being informed that a certain Mathematician had by Calculation discovered an errour in his Demonstration, al­though at first it made no great impression in him, for he thought himself so secure, that he hoped sooner to finde a thousand errors in that Mathematicians Calcu­lation, then one in his own De­monstration: Mille potius spera­bam me in Calculo hujus examinis, inventurum errores, quàm vel uni­cum in meis inventis; sed contra [Page 31] quasi accidit, &c. But I found, sayes he, just the contrary. For having examined the said Calcu­lation, I clearly saw the errour, and was forced to confess it, Victus debui dare manus. Where­upon he presently retracts it, and is so far from excusing the error, or refusing to confess it, or brag­ging as if he had carried himself gallantly therein, (as Mr. White most falsely and injuriously im­poses upon him) that he plainly and candidly confesses it, saying, that he had rather follow the ex­ample of other worthy Authors, who in like case have, to their own praise and profit of others, revoked their errours, then of such as had rather accuse Archi­medes, Euclide, yea, Geometry it self, then once acknowledge the errours of which they were con­vinced.

As to that whereof Mr. White [Page 32] most wrongfully taxes him (in those words neque delendo tege­re) for printing the said Demon­stration, although he thought it to be false, Gulden gives there also the reason, why he printed it: to wit, that others seeing how he had erred in a Demon­stration, which at first sight seem­ed so currant, might beware of the like fallacy: Ut sciant sibi ca­vere a scopulis.

By all which is most evident, that it was ignorance, and pas­sion, and neither knowledge, nor reason, which extorted these ugly censures from Mr. White against Gulden: and how far that Au­thor was from that vanity and stubbornness in maintaining what he had once asserted, though he thought it to be false, as Mr. White would make the world believe. For I dare main­tain, that there is not an Author [Page 33] to be found, who in the like case hath carryed himself more mo­destly and candidly then this man hath done: as any, who shall read the said third As you may see at the end of this Letter. chapter, will, to Guldens praise and Mr. Whites confusion, plainly disco­ver. And God grant Mr. White may but with as much humility recall and acknowledge what he hath written amiss in matters of more concern, as this man does retract his Mathematical error.

Wherefore in this so much vilifying of Gulden, he hath again put the lash into his adver­saries hands, who may use it at their pleasure, and make known unto the world, that no man that had any worth in him, con­science, or moral honesty, would ever so unworthily have carried himself as Mr. White hath in this. Yea, they may, if they please, re­tort all that he so wrongfully [Page 34] layes upon Gulden, most justly upon Mr. white: making it ap­pear, that he is rather to be taxed of vanity, as having gotten one­ly some few Tearms of Geome­try, (and yet more then he knows well how to use) would fain have the glory of a Mathemati­cian. For although with the one hand he seems to drive it away, yet with the other he draws it to him, as any man but reading his Preface will clearly see. For al­though he tell his Reader, that he is no Geometrician; and that these so great things (as he fan­cies them) come not from him­self but from God; Non à me haec habes, &c. yet he would have him withall to take notice and well understand, that he is also able even by the force of his natural wit, to perform as great things as these are. For speaking of himself and what he is to deli­ver [Page 35] in the said Geometrical Tract, he writes thus: Author vel suâ industriâ perfecit quae of­fort, vel privilegio magnae Pro­videntiae accepit. Si à se, & inge­nij eâ virtute, qua plura ejusmodi conficere in parato habeat, certè is est, ut non sint contemnenda illa cae­tera, quae in publicum usum elabo­ravit, &c. Whereby you see, he plainly tells his Reader, that he hath now already in store di­vers other things, as good as these. Plura ejusmodi in parato ha­bet. And these also found out by the vigour and strength of his own wit. Eâ ingenij virtute, qua, &c. Yea although he tells his Reader, that he most onely thank God for these wonderfull things; and that in thanking the Author he shall do him injury, and lay a burden on his shoulders more then he is able to bear: Mi­hi si grataris, injuriarum te postulo, [Page 36] quod plus in me oneris aggeras, quam cui sim ferendo. Notwith­standing he plainly shews by what you have heard, that he is ready and able to bear more thanks, then I believe his Rea­der will give him: especially when he shall perceive himself deluded in the Preface, with ex­pectation of wonders from Hea­ven, and when all is done, find­ing nothing worth the reading.

But Mr. White is not content with so much depressing this Au­thor, but passes further, branding him with the badge of an Here­retick, or worse; intimating him to be one of that pernicious Sect of Pedants, who by their pra­ting, labour and endeavour to de­stroy not onely all humane Sci­ences, but even Christian Faith it self, by taking all certainty from them. For giving a reason why he so much enveighs against [Page 37] a man wholly unknown to him, he presently adds, Quantumvis operae pretium erat, lectorem moni­tum reddere de exitiali hac sciolo­rum secta, quae sub professione fa­cultatis garriendi, omnem certitu­dinem, tum è scientiis, tum ex fide Christianâ tollere molitur. Here Mr. White stops; and truly it was time: for having forgot what he first intended, to wit, to draw out a perfect picture of Gulden, he hath mistaken the colours, and goes on drawing forth his own, as any man that ever knew them both, will evidently discover.

Now if you ask me what was the main cause, that moved Mr. White to this height of passion, he himself tells you, to wit, that he was forced and compelled to utter those censures. And why? Because the shadow (as he sayes) of Guldens great Tome did hin­der his Scholars from embra­cing [Page 38] the truth (he should have said the falsity) he proposed to them. For so he writes in the place before cited. Haec coactus sum de homine caeteroqui ignoto prodere, quia umbra Tomi illustris, per opinionem consequam, officiebat veritati, quam ejusdem studiosis offerebam. In which words I should rather think Mr. White to have wronged his Scholars, in making them such as should be frighted with a shadow. But it seems more probable, that his Scholars better understood the force of Guldens Computati­on, then their Master either would or could, and saw clearly that it did conclude. Howso­ever it is most strange, that any wise man for so frivolous a toy as this, should so highly offend both Almighty God and his Neighbour, and so evidently expose his own reputation to the [Page 39] unavoidable stain of a notorious Detractour.

Truly, as it seems to me, in this the particular hand of God shews it self, as well for his own good, (if he will make use of it) as for the good of others: in per­mitting Mr. White so to cross his own designs, that whereas he thought in this Tract to advance himself and his former writings, in the repute of every one, he should finde the quite contrary. For whereas he thought thereby to have got the name of a great Mathematician, he hath clearly shewed that he is none; and that he is indeed onely furnished with such general Terms and common Notions in the Mathematicks, as being with confidence and boldnesse pronounced in the company of such, as do no more thorowly understand them then himself, are apt to produce in [Page 40] their mindes, an opinion, that the pronouncer is certainly a learned man, & understands exactly what they hear so strongly assevered by him: whereas if some learned Mathematician should perhaps over hear him he would smile to hear so much Geometrical Non­sense. Nay, whereas he assured himself to conciliate an immove­able authority to all his former Dictates amongst his admirers, by this unparalleld Demonstra­tion, even some of them (as I am certainly informed) have disco­vered the weakness of it, and both blush to see it, and labour to hide it. In like manner, whereas (by vertue of his said Tutela) he aimed to be accounted a person whom Almighty God particularly designed to use as his Instrument for the governing of his Church in this present con­juncture; and to this effect, to [Page 41] have received great light and In­fused knowledge from him, as we have heard him speak in his Preface; he hath now given such a Character of himself, that it is impossible, that any man should be so simple as to think, that the wisdom of God would particu­larly make choice of such an In­strument for so high a Work; to which men of far greater Chari­ty and Perfection of Vertue then he can with any reason or ground be supposed to have, are wont to be called.

This unworthy proceeding of Mr. White had made me almost forget to refute his Quadratura Circuli, pretended to be shewen in the first nine Propositions: which I deferred to the last, because he in his thirteenth and last Propo­sition hath put the last hand thereunto, and so confirmed as he thinks the ground thereof, that [Page 42] he supposes it now as evident (to use his own phrase)as that a Boat is a Boat.

Wherefore in this thirteenth Prop. he again affirms, what he had before averred in the sixth; to wit, that the Segments, or Por­tions of unequal circles, having the same Chord, (so that they be less then a Semicircle) are propor­tional to their Axes. Portiones circulorum inaequalium, semicircu­lo minores, quarum subtensae sunt oequales, sunt in ratione suorum axi­um. For example, in the adjoyn­ed Diagram, he affirms, that the greater Segment BFCB hath the same proportion to the les­ser Segment BDCB, that the greater Axis F G hath to the lesser DG. This is the ground of his Quadratura, which we will now shew to be most false, and by consequence, the whole building to fall. Which

[Page] [Page 43] that it may the more clearly ap­pear, let us suppose EB or EF the Semidiameter of the greater Segment BFCB, to be 1000, and his angle at the Center, BEC, to be 120 degrees, or the third part of the whole circle BFC: which being supposed we have.

• 1. EG 500, as being the Sine of the angle EBC 30 de­grees, and by consequence the greater Axis GF is also 500.
• 2. By the usual proportion of the Diameter to the circumfe­rence we shall finde the Sector E B F C, being the third part of the whole circle, to contain 10476191/21.
• 3. By the Perpendicular EG 500, and the half base GB 866, or Sine of 60 degree. We shall finde the Triangle EBC to contain 433000: which being subtracted from 10476191/21 the [Page 44] whole Sector, leaves 614619 [...] for the greater Segment BFCB.

In like manner, if we put AB the Semidiameter of the lesser Segment BDCB to be 2000, that is, double to EB, we shall finde

• 1. By the 47. 1. Eucl. AG 18027/9 proximè: which taken from A 2000, leaves 1972/9 for the lesser Axis GD.
• 2. By what is known in the Triangle ABE we shall finde the angle BAE, whose double shews the whole angle of the Sector ABDC, to be 51 degr. 19′ 30″ from whence by pro­portion thereof to 360 degr. is found the Sector ABDC to contain 1792126.
• 3. By the Perpendicular AG 18027/9, and the half base BG 866, is found the Triangle ABC to contain 15612051/9 which being taken from the [Page 45] whole Sector 1792126, leaves 2309204/9 for the lesser Segment BDCB.

So that now we have the said Segments and their Axes, all four in numbers, to wit, the greater Axis 500, the lesser 1972/9 the greater Segment 6146191/21 and the lesser 2309204/9 which four numbers are by no means pro­portional, as they should be, if Mr. Whites Demonstration were true. For by saying as 500 to 1972/9 so 6146191/21 to a fourth, there will not be found (as was expected) 2309204/9 but another number far bigger, to wit, 2424331/3 the difference being (as you see) 115128/9. Which great difference shews evidently the falsity of Mr. Whites Assertion. Yea, if we put the greater Seg­mentto want but very little of a Semicircle, for example onely one Minute, or one Second, &c. [Page 46] the errour will be yet more noto­rious, and the proof more easie. For then the greater Axis will be 1000 Proximè, and the greater Segment will be 1571428: the lesser Axis will be 268: and the lesser Segment 363238. which four numbers are yet far more disproportional: for by saying as 100 to 268: so 1571428 to a fourth, we shall finde 411142, which is greater then 363238 by 47904, almost an eighth part of the lesser Segment. So Mr. Whites Demonstration of the Quadratura comes to just no­thing. But this is like the rest: for with him Demonstrations are nothing but stout and undaunt­ed asseverations, proved by a company of Terms (that make a shew of learning to the unlearn­ed) jumbled at a venture toge­ther.

Some perhaps, to excuse this [Page 47] so gross an errour of Mr. Whites, will say, that by Portiones Circu­lorum he meant not the Seg­ments, as we have said, but one­ly the Arches or Circular Lines BFC and BDC. But this explication will not suffice: for neither had this been to his pur­pose for the Quadratura: nor in it self is it true. For neither are these Arches proportional to the said Axes; the one BFC being 20955/21 and the other BDC be­ing 17921/2 which numbers have by no means the same proportion that 500 hath to 1972/9 as a blinde man may see. Wherefore Mr. White must be content to lay up this Errour with the rest.

And thus much, Honoured Sir, concerning such things as Mr. White pretended here to demon­strate, but hath not performed. But if you ask me what he hath in this little Treatise truly and [Page 48] clearly demonstrated, I can one­ly answer, that he hath demon­strated, first that he hath a great deal of vanity: secondly, that he hath very little or no Geometry: and thirdly, that he hath as little or less charity. For the rest, I have no more to say at present, but hoping you will rather reflect upon what is here said, then up­on the rude and unpolisht Style by which it is exprest, I remain