Querela Geometrica: OR, GEOMETRY'S Complaint Of the Injuries lately received from Mr. THOMAS WHITE In his late Tract, Entitled, Tutela Geometrica. In the end you have some Places at large out of Mr. White's TUTELA, and Gulden's CENTROBARYCA, Reprinted, and faithfully Translated into English. LONDON, Printed by R. W. 1660. The Publisher TO THE READER. HAving in the following Letter from my Friend, received a brief account of Mr. Whites so much famed and expected Geometrical Treatise, I thought fit to publish it for satisfaction of many, very desirous to understand the success he has had therein. Know then, that the Letter consists of two parts. The first demonstrates the many and gross Errors against Geometry, committed by Mr. White in his Tutela Geometrica: which yet he terms his Chrysaspis, or Golden Shield, wherewith to defend all his other Works. The latter part lays open his most unworthy proceeding against a famous Mathematician, by charging him with many and evidently false imputations, on purpose to disgrace him. From all which he is here fully vindicated, as well in the ensuing Letter, as in the Additionals. Querela Geometrica: OR, Geometry's Complaint Of the Injuries received lately from TUTELA GEOMETRICA. OR, The Copy of a Missive, etc. SIR, 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, (having scarce hitherto appeared in that kind) 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 conveyed to him by particular light from Heaven, may learn thence more to prise 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, aught 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, than 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 Euclid. 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 Providentiae suae, ea gubernationi Ecclesiae suae, in hoc rerum articulo, opportuna & fecit, & vidit. Being I am such a one; (to wit, so little versed 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, etc. Wherefore he exhorts the Reader, not to neglect his own good, nor contemn a wonder fallen to him from Heaven. Quod superest, tibi consul, & 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 all his other Works: that so his Readers considering and weighing with themselves, that it is impossible so learned and subtle a work should come from one that never studied 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 impugn his former Writings. For now his Adversary hath no more to do, but to show (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 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 only some more patent and obvious errors, such as you yourself may easily conceive: by which you may guests at the rest, and what can be here expected. His Treatise then contains in all thirteen Propositions; of which the two first only explicate the terms 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 than 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 find 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. 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 join 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 studied Geometry, or read so much as Euclid; 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 Spiral line of the first revolution, be inscribed a Polygone with equal angles, (as in the adjoined 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 DIEGO 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, etc. Perpendiculars to the opposite Semidiameters, (as here you see let down the Perpendiculars IK, HF, GL, etc.) 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 equal. This third he infers (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 show. 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 EGLANTINE 500, and so forward: So that the first EA will have 100, and EBB 200, etc. Whereby we have now in every Triangle EDI, EIH, EHG, etc. 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 JEH 45 degrees as before; and so of all the rest. Which being known, we may presently by resolving the said Triangles, find the two last sides of the inscribed Polygone, to wit DIEGO, 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 find the second side AB to be only 147; from whence being taken the first side EA 100, shows the difference between the first and second side to be only 47. And so in like manner will the difference between the second and third, AB and BC be only 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 error 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 than 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, etc. are all equal. For resolving the Triangles EIK, EHF, and EGL (in which you have a side with all the Angels) you will find EKE 495. OF 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 persuades 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 not: wherein he was so puzzled, 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 showing only that it is not bigger, he infers it to be equal; which is no consequence, although the Antecedent were true: but both the one and the 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 shows, that not only the Spiral itself is bigger than the said half circumference; but also an inscribed Polygone, for example of twelve equal angles, is considerably bigger. Yet notwithstanding this false supposition, he goes forward to demonstrate against Gulden, that the sides of such a Polygone added together are less than 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 than the half circumference. But this his conclusion is most false, as Gulden hath evidently shown, 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 adjoined Table; 12 6031 11 5520 10 5011 9 4505 8 4001 7 3501 6 3006 5 2522 4 2053 3 1614 2 1239 1 1000 Summa lat 40003 Semiperiph 37715 Differentia 2288 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 find the said half circumference to contain at most only 37715: which is far less than 40003. And by consequence the sides of a twelve angled Polygone inscribed in a Spiral, are absolutely longer than half the circumference of the first circle, as Gulden truly and learnedly shows against Mr. Whites so weak a Demonstration for the contrary, as we have seen. By these discoveries of so many undeniable errors 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 unskilful, and that he hath not a jot of Mathematic or Geometry in him; terming him one of those half Scholars, who stealing divers excellent things out of other Learned men's 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 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 candour, 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, etc. This gall, whilst Mr. White flourished amongst his admirers with his new Demonstration, might have affixed some seeming blemish upon Gulden, amongst such 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; than which there can be none more exact and manifest. As for that he calls him Semidoctus, a half Scholar, one utterly void of all Mathematic; that he hath stolen out of other men's 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 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 masterpiece of all that ever he hath writ) and Guldens Book called Centrobaryca, will find so main a difference, that Mr. White without any prejudice, by what appears in his, may be scarce thought fit to be Guldens Scholar. And whereas he calls Gulden one of those half Scholars, 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, etc. and to say it came from Heaven, 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 Author 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 Virtues, 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 error, (to wit, in his 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 error 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, etc. O most unworthy and false calumny! when I had read these in Mr. white, and compared them with what Gulden says 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 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 error in his Demonstration, although at first it made no great impression in him, for he thought himself so secure, that he hoped sooner to find a thousand errors in that Mathematicians Calculation, than one in his own Demonstration: Mille potius sperabam me in Calculo hujus examinis, inventurum errores, quàm vel unicum in meis inventis; sed contra quasi accidit, etc. But I found, says he, just the contrary. For having examined the said Calculation, I clearly saw the error, 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 errors, then of such as had rather accuse Archimedes, Euclid, yea, Geometry itself, then once acknowledge the errors of which they were convinced. As to that whereof Mr. White 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 to be found, who in the like case hath carried 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 lays upon Gulden, most justly upon Mr. white: making it appear, that he is rather to be taxed of vanity, as having gotten only some few Terms of Geometry, (and yet more than 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, etc. yet he would have him withal 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 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, etc. 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, etc. Yea although he tells his Reader, that he most only thank God for these wonderful things; and that in thanking the Author he shall do him injury, and lay a burden on his shoulders more than he is able to bear: Mihi si grataris, injuriarum te postulo, quod plus in me oneris aggeras, quam cui sim ferendo. Notwithstanding he plainly shows by what you have heard, that he is ready and able to bear more thanks, than 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 only all humane Sciences, but even Christian Faith itself, by taking all certainty from them. For giving a reason why he so much inveighs against 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 says) of Guldens great Tom did hinder his Scholars from embracing 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 gulden's Computation, than 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 unavoidable stain of a notorious Detractor. Truly, as it seems to me, in this the particular hand of God shows itself, 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 find the quite contrary. For whereas he thought thereby to have got the name of a great Mathematician, he hath clearly showed that he is none; and that he is indeed only furnished with such general Terms and common Notions in the Mathematics, as being with confidence and boldness pronounced in the company of such, as do no more thoroughly understand them then himself, are apt to produce in their minds, 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 authority to all his former Dictates amongst his admirers, by this unparallelled 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 hid it. In like manner, whereas (by virtue 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 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 Virtue than 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 shown 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 he supposes it now as evident (to use his own phrase) as that a Boat is a Boat. 1. EGLANTINE 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 find the Sector E B F C, being the third part of the whole circle, to contain 10476191/21. 3. By the Perpendicular EGLANTINE 500, and the half base GB 866, or Sine of 60 degree. We shall find the Triangle EBC to contain 433000: which being subtracted from 10476191/21 the 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 EBB, we shall find 1. By the 47. 1. Eucl. AGNOSTUS 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 find the angle BAE, whose double shows 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 AGNOSTUS 18027/9, and the half base BG 866, is found the Triangle ABC to contain 15612051/9 which being taken from the 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 shows evidently the falsity of Mr. Whites Assertion. Yea, if we put the greater Segmentto want but very little of a Semicircle, for example only one Minute, or one Second, etc. the error will be yet more notorious, and the proof more easy. 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 find 411142, which is greater than 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 show of learning to the unlearned) jumbled at a venture together. Some perhaps, to excuse this so gross an error of Mr. Whites, will say, that by Portiones Circulorum he meant not the Segments, as we have said, but only 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 itself 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 blind man may see. Wherefore Mr. White must be content to lay up this Error 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 clearly demonstrated, I can only 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 expressed, I remain Your humble Servant. The Publisher TO THE READER. HAving Printed the precedent Letter, and understanding before the publishing of it, that Mr. Whites Tutela so often mentioned, was suppressed, no more Copies to be had, whereby the Reader might receive satisfaction in conferring what is here said, with the Authors own words: I thought good to annex such places, verbatim out of Mr. Whites Book, as are touched in the said Letter. To which purpose I have also adjoined Guldens Retractation of his Spiral, taken out of his Book, de Centro Gravitatis, the Book itself being dear, and scarce to be got. I have put both these, first in Latin, as they were written by the Authors: next in English, because the precedent Letter mentioning them, is also in English. And first I will set down Mr. Whites Preface to the Reader, there being first mention made of it in the precedent Letter. Which Preface is to be seen in the beginning of the said Book, to which he frames this Title, CHRYSASPIS, Seu Scriptorum suorum in Scientiis obscurioribus Apologiae vice propalata TUTELA GEOMETRICA Ad Lectorem Cordatum & Serium. ARistotelis (dicam, an Naturae?) pomaeria extenderat Digbaeus Eques, coactis in pellucida stativa Naturae partibus, quas turbidè miscuerat generationum necessitas. Solum, fundatura Substructiones, occupaverat quanti ipsa moles, rari densique suprà serpentium ludis tessellata. Proximo sese exposuit gradu fraterna elementorum acies, primis (ut appellant) armata qualitatibus. Haec obnixis in alternam internecionem frontibus, consanguineo cruore quanti aream, ad inexhaustam mixtorum ubertatem per admirandos & inscrutabiles plexus ebulliendam irrigant & faecundant. Ornabant mixta Phrygionatae secundarum qualitatum Texturae; accendebant Actionis & Passionis emicantia & humanum contuitum obtundentia lumina. Sed neque Electricorum assultus & resultus, neque Magneticorum in Homogeneo corpore mutabiles quasi consultò leges; neque Sympatheticorum ex insidiis dolosa & tenebricosa è longinquo Sagittatio, origines & semitas suas à tam acri vestigatore celare valuêrunt. Quin & ad superiora sedilia mixtorum capita plantae ascenderunt, & gradum ad animalia promoverunt. Hic se objecêrunt scrutino sensus & sensuum metae, venenerandi quodammodo naturae limites: & superati scrutatoris oculum in arcana Animae, & invisum orbem trajecêre. Substitit in hâc altitudine Digbaeus, materiam & materiatorum universitatem tanto à se intervallo in imum dissitam, non sine horrore despectans; & nobilissimo operi, cui de Immortalitate Animae nomen fecerat, columnas apposuit. Tantus erat Scientiae fulgor, ut lippitudini saeculi caecitatem adjiceret, & furebant vanitatis, quae in multiloquio efflorescit, amatores veritatis imaginem non sustinentes, & potioribus haec scientiis adversari jactiabant. Proptereà necessarius erat aliarum Disciplinarum consensus, & rudem acceptaverat ingentis illius opificii Author. Inventus sum, qui etsi Eloquentiae Decessoris impar & compendio natus, auderem desideratorum Epitomen aggredi, & contractis, quae fusius Digbaeus, & pro rerum qualitate disputaverat; adjectisque Metaphysicâ tum corporum tum incorporeorum delineatione, Institutiones Peripateticas conderem. Adjeci & Sacras, & opuscula (quamvis nihil meum opusculi molem excedat) nonnulla Philosophica, de Mundo Dialogum, & praefationem ante Latinam Editionem operis Digbaeani. Theologicam quoque Buccinam de Fidei & Theologiae Naturâ; & ejusdem defensionem adversus errorem cujusdam Regularis de Personali Infallibilitate Papae. Praeterea de Gratiae cum libertate consensu, & medio Animarum statu singula Commentariola. Non mirum, si haec duriùs excepta sint, quam Digbaeani labores. Cum & infaeliciori stylo sint exarata, & iter caecioribus obfessum scopulis & magis affectuum tempestatibus objectum terant. Sed idcircò maximè, quod in omnibus Physicam, Metaphysicam, & ipsam Theologiam, inaudito conamine ad severiores Disciplinas adjungere, & Architectonicam contignationem perspectabilem in toto processu, & dictorum consensum & consequentiarum fidem (nihilominus citra rigoris Geometrici ostentationem) in eas inducere tentaverim. Quare hunc desiderari suspicatus duos Euclidas, Physicum majorem natu, adolescentiorem Metaphysicum effudi, non vanae spei futuros vades. Verum enim verò etiam hanc evidentiam obstinatâ incredulitate opprimi sum expertus. Quid super mihi reliqueram? Memineram à Novatoribus fidei posci miracula. Sed ad ea quae suâ evidentiâ stabilienda erant, flagitare argumenta ultrà vim naturae posita propudiosum erat: attamen si quae in Scientiarum Thesauris admiranda laterent miraculis supparia, non immeritò ad difficiliorum fidem adhiberi consentaneum erat. Conjeci itaque oculos in Geometriam, cujus si qua dogmata hujusmodi veneratione consecrata laterent, ea neque alienis ad famam praesidiis indigerent, & suo munitis sigillo fidem conciliarent. Et adverti reservata quaedam ab ipsa usque Disciplinae infantiâ arcana, quae maximorum ingeniorum labores passa in impossibiliam transiverant classem. Pappus & plerique posteriores Geometrae tres Problematum ordines declaraverant; quorum infimus regulâ & circino perficiebatur; medius corporum sectilium vi; supremum non nisi fictitiis lineis subjiciebant. Et in posterioribus haec arcana recondiderant. Vieta etiam adjectis argumentis quaedam 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 demonstravisse visus est. Cartesius desperatam rem agnovit. Plerique proposito Problemati satisfecisse sibi visi sunt, si ad hoc redegissent, ut eo soluto monstrarent aliquod clausorum istiusmodi esse reseratum. Te testem invoco, Maxime Archimedes, in secundà secundi de Sphaerâ & Cylindro, nisi mendax imponat memoria, Hinc itaque captandam scriptis meis umbram censui. Tu modo apud temet in consilium se vocatum hoc pensi habeto. Author vel suâ industriâ perfecit quae offert, vel privilegio magnae Providentiae accepit. Si à se & ingeniieâ virtute, quâ plura ejusmodi conficere in parato habeat, certè is est, ut non sint contemnenda illa caetera, quae in publicum usum elaboravit; imo hoc nomine trutinâ acri àigna, quia de tali orta sunt patre. Sin ab exorti 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 vigilantiâ profectum hoc munus suspicaris, expende quanto fortiùs te ad reliquorum examinationem allegatum comperias. Me aspicis? Intuere me hominem, quem nemo Geometram salutet, modo ipse sit. Neque enim Geometrices plenitudinem vel appetivi. Prelectorem non audivi; studium non sum professus; magnorum Authorum nullum perlegi, non saltem Euclidem. Aliarum Disciplinarum ambitio me semper traxit & defixit: Geometricorum hunc fructum & speravi & tuli, ut eorum rigorem ad Metaphysica traducerem. Caeteroqui oblectamento mihi erant, cum deforet potiorum commoditas. Talis cum sim, non à me haec habes, sed ab eo, qui ex legibus providentiae suae ea Gubernationi Ecclesiae suae in hoc rerum articulo opportuna & fecit & vidit: Illi accepta refer. Mihi si grataris, injuriarum te postulo, quod plus in me oneris aggeras, quam cui sim ferendo: & in Deum, à quo avertis quale quale à te debetur benignitatis praemium. Quod superest, tibi consule, & ostentum a coelo ad te delapsum ne contemnito. In English thus. A GOLDEN SHIELD: OR, A Geometrical Defence As an APOLOGY for all his other Writings in the obscurer Sciences. By Mr. Thomas White. As for the Preface, the first part thereof concerns not our present purpose, as speaking only of the profound research made into nature by that worthy Gentleman, Sir Kenelm Digby, (whose learning and respects to Mr. White would never certainly have suffered this Geometrical Treatise to pass abroad, had he seen it before it was published) I shall only English the second, which Mr. White falls upon by occasion of an objection that some things touched in Sir Kenelmes Philosophy seemed not fully agreeable to Sciences of higher consequence. The refutation of which objection Mr. White undertook, and so gives a brief account of such writings as he had published to that effect: and comes at last to this present Tutela which he is about to publish, intending it as a defence of all he hath hitherto written, and whereof he speaks to his Reader in these words, Tu modo apud temet, etc. as we saw just now. In English thus. TAke gentle Reader this into thy serious consideration, either the Author (meaning himself) hath performed the things he here presents thee with, by his own Industry, or by God's peculiar Providence. If they be the fruits of his own industry, and proceed from that vigour of wit, by which he is also ready to perform many the like: certainly he is a person whose other writings formerly published for the common good, ought not to be contemned. Yea, for this very reason, they deserve a profound consideration, as proceeding from such a Father. But in case thou consider them as coming from the Almighty, then think with thyself; how much greater thy obligation is to examine the rest of his Works. Lookest thou upon me? Behold the man, whom no man will call a Geometrician if he be one himself. Neither did I ever desire to complete myself in that faculty; I was never taught it, nor did I ever profess to study it. I never read over any chief Author in that Science, no not so much as Euclid; but was always delighted with other studies. From Geometry I both expected and attained sufficient for transferring its vigorous proceed to the Metaphysic. Otherwise it was only my recreation when I wanted better employment. Since therefore I am such, thou hast not these things from me, but from him, who in his Divine providence, both saw and fitted them for the Government of his Church in this present conjuncture. Give him the thanks; for if thou apply them to me, I shall expostulate the injury done me in laying a greater weight upon me than I am able to bear; and thou wilt lose the reward thou shouldest receive from God. It remains that thou neglect not thy own good, nor contemn a wonder come down to thee from Heaven. Thus Mr. White in his Preface. By all which, and by the Title of his Book, he plainly shows, that according to the clearness and solidity of the following Demonstrations in Geometry, men must take the just measure of the solidity and strength of his other Demonstrations in Philosophy and Divinity. And thus much for those places of the Preface mentioned in the precedent Letter. I will now set down those that concern the dispute between Mr. White and Gulden, which are only two; the former is in Nota secunda before the tenth Proposition: the latter in Nota Quarta at the end of the twelfth. Ex Tutela Geometrica ante Prop. 10. Nota Secunda. Finieram, & regulam cum circino consecraturiebam; cum ab amicis monitus sum, quam in Exercitatione Geometrica exhibueram spiralis ad Peripheriam Circuli aequationem, à magni nomin●… Mathematico, & prius excogitatam, & eàdem demonstratione confirmatam, & posterioribus consiliis repudiatam fuisse, & oppositâ demonstratione reprobatam. Conscius eram, non indiligentèr apodixi meae invigilavisse. Terruit tamen hominem (cui omnia alia prae Mathess praehabita fuerant) tot notis veritatis impressus rumor. Tollo de Tabula manum, & cum typis mandavissem, quae sunt praemissa, caeter a usque ad examinationem hujus improperii sustinenda decrevi. Author oppositionis erat quidam Paulus Guldenus, ex Societate Jesus, editor justi voluminis, quod pro Geometrico suppositum Centrobaryca appellavit. Quid agerem? ubi degebam, opus illud non apparebat, & negotium quod illic gerebam, ad umbilicum perductum erat, & jam egelidum ver monebat aestivam sedem Ciconiarum monitu●… vestigare. Contuli me itaque Lugdunum Batavorum, & gratiâ clariff. Mathematum ibidem Professoris Examinationem Problematis mei aggredior. Primò ipsam revisi; apparuit constantissima: summam tibi sic accenseo, &. In English thus. I Had even now ended, and began to lay my Rule and Compass aside, when I was admonished by my friends, that the equation of a Spiral to the circumference of a Circle, which I had demonstrated in my Exercitatione Geometrica, had been found out before and confirmed with the same Demonstration, by a famous Mathematician: who afterwards retracted it, and by a contrary Demonstration, shown it to be false. I was conscious to myself that I had not slightly examined my Demonstration: nevertheless a rumour with so many marks of truth somewhat frighted me, being a man that esteemed all other learning before the Mathematical. I presently made a stop, and having printed the precedent part de Quadratura, I deferred the rest, till I had examined this reproach. The retracting Author was one Paul Gulden of the Society of Jesus, who had printed a complete Volume, which he pretending to be a Geometrical Work, called it Centrobaryca, What should I do? Where I then lived. this Book was not to be had, and my work in hand was even now finished. Besides the Spring drawing on, did invite me by the crying of the Storks, to seek my Summer habitation. I went therefore to Leyden in Holland, and with the favour of the most famous Professor of Mathematics there, I begin to examine my Problem; I revised it, and found it most solid, Take here the sum thereof, etc. This is Mr. Whites relation of the beginning of the Dispute between him and Gulden, handsomely, as you see, contrived, that he may not seem to have stolen the said Demonstration out of Gulden, but to have fallen upon it himself, or had it from Heaven. Yet all this will not serve his turn, but still the Reader will imagine those words in his Preface, where (speaking of this Demonstration together with the rest) he says, Non haec à me habes, sed ab eo qui ex legibus Providentiae suae, etc. are to be changed thus, (applying them to this particular) Non haec à me habes, sed à Paulo Guldeno, qui ante trigintaferè annos ea primò invenit, & postea retractavit, etc. After this relation Mr. White proceeds to confirm the said Demonstration and infringe the Retractation of Gulden, showing his computation to be false, as he imagines: which done, he makes his Nota Quarta, or invective against Gulden in these words. Nota Quarta. Calculus itaque Guldenianus imperitas est, & qualem ab ipso acceptari (neque enim vel talem ipse instruxit) decebat, homine prorsus Amathematico, ut legenti ipsius scripta pronum est patere. Nam dum proportionem Spiralis ad circulum ad struere conaretur, assumpsit sine probatione propositionem prorsus improbabilem, nempe, lineas intra aliam ductas esse minores illâ. Et si enim videatur de Inscriptis velle loqui, tamen quas ipse scribit nihil minus sunt quans Inscriptae, cum circum scriptam non accedant nisi altero duntaxat termino. Rursus aequali temeritate vult Arcus circuli esse proportionaliter medios inter arcus Spiralis aequalium angulorum. Sed (quod foedissimum est) tantae vanitatis est, ut cum erravisse sese putaverat, neque delendo tegere, neque candidè confiteri sustinuerit, sed excusationes texere quasi in ipso errore egregiè se gesserit, ostentare pergat, Quae (utpote de sumptae ex locis Logicis vel Rhetoricis) clarè docent hominem officij Geometrici (quod hac respust) esse prorsus ignarum, & ex eo Semidoctorum genere, qui cum ex magnorum virorum scriptis egregia multa depeculati fuerint, ut sua faeciant, additis quibusdam levibus, justi voluminis oftentatione se vulgò discentium ostentant: & (quod perniciosissimum est) mixtis incertis, sacrum Scientiae nomen denigrant, ut abundè egit noster Guldenus; Saltitationem telluris circa centrum, & consistentiam Centri in puncto imaginario, in Geometricum tractatum inferciens. H●…c coactus sum de homine caeteroqui ignoto prodere quia umbra Tomi illustris, per opinionem consequam, officiebat veritati, quam ejusdem studiosis offerebam. Quantumvis operae pretium erat lectorem monitum redàere de exitiali hâc Sciolorum Sectâ, quae sub professione facultatis garriendi omnem certitudinem, tum è Scientiis, tum ex Fide Christianâ tollere molitur. In English thus. WHerefore Guldens calculation is unskilful, and such as was fit to be received by him (for he made it not himself * But he there made another more accurate. ) a man no way versed in Mathematic, as his Reader will easily perceive. For endeavouring to give the proportion of a Spiral to a Circle, he assumes without proof a proposition wholly improbable, viz. that lines drawn within another line, are less than it. For though he seem to intent to speak of lines inscribed; yet those he describes are nothing less than such, since they touch the circumscribed but with one end only. In like manner with equal temerity he will have the Arches of a Circle to be mean proportionals between the Arches of a Spiral of equal * Both these are retracted by Gulden, prop. 4. Angles. But that which above all is the most detestable, his vanity is so great, that when he thought he had erred, he could neither endure to suppress it, nor candidly to acknowledge it, but proceeds framing excuses, and brags, as if therein he had carried himself very gallantly. All which, being but flourishes of Rhetoric, clearly show him to be a man wholly ignorant of what belongs to Geometry; which uses no such Arts: and that he is one of those petty Scholars, who having stolen divers excellent things out of other men's Writings, that they may make them seem their own, add some few trivial matters, and then boast themselves amongst their Scholars as the Authors of a great Volume: and (which is most pernicious) by mingling many uncertainties, defile the Sacred Name of Science; as this Gulden hath done to the full, thrusting into a Geometrical Treatise the Dancing of the Earth about its Centre, and the Consistency of the same Centre in an Imaginary Point. These things I was forced to publish of a man otherwise unknown to me, because the shadow of so fair a Tome, through the opinion it had gained, hindered the light of that truth, which I proposed to those that sought it. Nevertheless it was worth the labour, to admonish the Reader of this pernicious Sect of Sciolists, who under profession of the Faculty of Prating, endeavour to remove all certainty, as well from Sciences, as from Christian Faith. Hitherto we have heard, or rather seen Mr. Whites gall against this Author, through whose sides he unworthily seems to endeavour the wounding of his whole Order, under the title of Exitialis Sciolorum Secta. Truly a man would wonder to see, how studiously and completely Mr. White here acts the part of a malicious Detractor, seeking every way to defame his Adversary: for besides what we have already heard in the precedent Letter, he intimates him here to be so little versed in the Mathematics, as not to be able to make even the computation of a Polygone of twelve sides: for speaking (as we have heard) of such a Calculation which he terms unkilful, and calls it Guldens, (though he tells us withal, that Gulden made it not himself, but received it from another) he maliciously inserts in a Parenthesis, Neque enim vel talem ipse instruxit, thereby to insinuate, that Gulden was unable to frame so trivial a Computation, as he (Mr. White) esteemed this. Wherein yet he could not be ignorant of the great injury he did this Author; who having set down the said Calculation of twelve sides, acknowledging it to be none of his own, but sent him from another, he examines and approves it in prop. I. c. 3. lib. 2. Which done, he presently in prop. 3. exhibites another of his own far more accurate, as consisting of a thousand sides, and performed by a different way of computation, with laborious Tables expressing the quantity of each particular side: thence also probably inferring that the proportion of the said Spira●… to the half Circumference is as 1961 to 1818 proximè. Yet Mr. White would take no notice of this Computation, although it were so near the place from whence he took that other of twelve sides. But having attained his end in this he proceeds, finding a means to asperse him with a censure of absolute ignorance in the Mathematics: where that the Reader may see Mr. Whites malicious way of proceeding, he must know that Gulden (as we shall presently hear in his own words) having (before the printing or publishing the often mentioned Demonstration of the Spiral) discovered by help of a Friend an error therein, thought good notwithstanding to print it, not as a true one, but as erroneus. This he performs in c. 2. lib. 2. and presently c. 3. shows and refutes all the particular errors committed in c. 2. to the end that others advertised of the errors committed in cap. 2. might beware of committing the like; sciant (saith he) sibi cavere à scopulis. Now here Mr. White plays his game, and taking no notice at all of this, plays upon Gulden, as if he had affirmed the contents of cap. 2. to be really true; which, as I said, Gulden did publish as absolutely false: than which a more unworthy proceeding can hardly be imagined, as will now appear. For in the above named Nota Quarta, Mr. White terms him, Hominem prorsus Amathematicum, a man utterly ignorant of the Mathematics, in assuming (as he says) without proof a Proposition wholly improbable, viz. that Lines drawn within another are less than it. But in this Mr. White extremely wrongs the Author, who only sets down cap. 2. Prop. 3. the said Proposion as erroneous, which he took at first sight to be true; and afterwards cap. 3. Prop. 1. at large declares the error. Which had he not done, I dare boldly affirm, that all Mr. Whites Geometry would never have been able to discover. But by this proceeding Mr. White thinks to attain what he aimed at; viz. the depressing of Gulden, and the exalting of himself. For by concealing Guldens refutation of the said Proposition, on the one side he would give the world to understand, that Gulden was not able to see and rectify the error: on the other he would gain to himself the opinion of a sharp wit, and deep insight into the Principles of Geometry, in being able to detect what so great a Mathematician (as he reports in his Nota secunda that Gulden was esteemed) could not perceive: whereas indeed if Gulden had not put it into Mr. Whites head, it had never been there. But to seek the glory of a great Wit and profound Mathematician, as also to purchase an esteem to all his other Writings by such Arts as these, is a thing most unworthy of a Gentleman. And though such Artifices may for a time, by some more affected to him then learned to discover them, be received with applause; yet at last they will be discovered, as here they are, and instead of the hoped glory, bring nothing but shame to such as use them. Lastly, he accuses Gulden of rashness for affirming (as he says) the circular Arches to be mean proportionals between the Spiral Arches of equal Angles. Wherein he uses the like artifice as before, by making Gulden assert even what he absolutely denies. For this Assertion corresponds only to what Gulden says in the said cap. 2. Prop. 8. where he delivers it as false, and after proves the falsity, cap. 3. prop. 4, n. 6. Thus much for Mr. Whites invective against Gulden; let us now, if you please, hear Gulden speak himself, and see whether there appear in his words that great vanity, stubbornness, want of candour, and such ostentation as Mr. White reproaches him with: and thereupon frame a judgement of them both, accordingly. Gulden therefore in the Preface to the cap. 3. gives a full account of his retracting the Demonstration of the spiral: which is as followeth. De Centro Gravitatis, Lib. 2. Cap. 3. Pag. 58. Examen eorum quae proxime praecedenti capite tradita sunt. HVcusque ergo, amice Lector, novae haec, & non minùs jucunda quàm pulchra, de lineis Spiralibus speculatio non tam traxit, quàm tum equis quàm velis suavissimè nos provexit, nullum sive in aequore sive in montibus nobis timentes vel periculum apertum, vel insidias latentes. Nam regiâ nos incedere viâ scivimus potiùs, quàm arbitrati sumus; & solùm intenti fuimus quâ ratione, Scopum videlicet nostrum intuendo, nobis paulatim semitam ad indagandum Lineae Spiralis Gravitatis Centrum commodè praepararemus. Occurrer at saepius quidem, dum haec tractaremus, non levis cogitatio; quae tamen ob certas rationes nos in caepto retardare minimè visa est: nimirum si haec Spiralis Lineae dimensio tam obvia, tam commoda, tamque ordinata est, ut nos eam invenimus; tam facilè etiam progreditur, támque firmis potest roborari Demonstrationibus; cur cam non Magnus Archimedes, qui ea, quae alii de Lineâ Spirali proposuerant, ipse admirabili (ut cum Pappo loquamur) quâdam aggressione demonstravit; cur eam (inquam) Dimensionem libro suo De Spiralibus non inseruit? Sed facilè nobis ipsis responsum dedimus: ipsum videlicet Archimedem plura alia, quae tamen alii post ipsum tractârunt, velneglexisse, vel studio ac volens praeteriisse; vel etiam jam ab ipso tractata, injuriâ temporum, intercidisse. Quemadmodum etiam fecit, cum ae Planorum centro ageret gravitatis: omisit enim Tractatum de centro Linearum. Et sicut nos non absterruit illa objectio ab indagatione Centri Gravitatis Linearum, praesertim Circularium; sic & hìc nobis ipsis fecimus animos novi aliquid circa Spirales inveniendi, quod vel ipsum Archimedem fugere, aut posteros nos latere potuerit. Et nostro quidem judicio rem tunc felicitèr & incepimus, & in eâdem magnis etiam itineribus perreximus. Sed ecce dum hoc loco in medio quasi essemus cursu, portumque etsi valdè procul adhuc positum, jam à longè tamen jam jam conspiceremus, vela repentè & erant contrahenda, & securitatis causâ ad legenda littora prora convertenda. Incidi enim tunc primùm in Examen quoddam, per numeros institutum, dimensionis Lineae Spiralis; de quo antè quidem inaudiveram, sed qualitèr ant quâ ratione dimensio illa progrederetur, cum rectâne, an cum curuâ, aut purè circulari, Spiralis illa linea conferretur, omninò nesciebam: tantum abest, ut ips●… justam proportionem, à nobis inventam, inde redargui posse suspicarer. Quare ex primo illius aspectu nequaquam sum territus; quippe qui meis fidebam Demonstrationibus, tam Geometricis quam Arithmeticis, quas irreprehensibiles esse judicavi. Hostem tamen qualem qualem saepe audivi spernendum esse minimè: quippe qui, quando minimè putamus, vel obesse vel nocere possit. Examinavi igitur examen illud; quo in labore mille potius sperabam me inventurum in calculo hujus examinis errores, quàm vel unicum in meis inventis. Sed contra quasi accidit. Reperi enim Lineam Spiralem primae Circulationis majorem esse Semiperipheriâ primi Circuli; & tacitè victusque manus dare debui. Quid facerem? Dissimularemne? Tacerem? Mordicus mea, instar Circuli Quadratorum suprà nominatorum, defenderem? Et Archimedem ipsum Euclidemque in jus vocarem, accusarem, condemnarem? Nequaquam. Sed servandum mihi duxi id, quod ipse Jos. Scaliger sibimet quidem servandum praescripsit; at minime servavit. Sic enim in Appendice ad sua Cyclometrica habet. Primum, aio, in omnibus Scientiis & Artibus posse saepe tolerabilitèr peccari: in Mathematicis ne semel quidem debere. Nam ut ait quidam vetus Scriptor, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Itaque cum Mathematicus errorem suum deprehenderit, primus occupare debet Me, Me, adsum, qui feci. Postquam autem per alium id cognoverit, si non statim Castigatori gratias agit, malè de homine; si non corrigit, malè de Mathemetica meritus est. Verum bonus ille Scaliger praecipuos errores suos, atro & rubro colore jam editos, non solum non correxit; sed & majori inscitiâ animique tumore pertinacitèr defendit. Conclusi ergo, in meis inventis alicubi peccatum esse. At ubi lateret anguis in herbâ, non it a facilè neque statim videre aut judicare potui. Quare Scripta mea amico in Mathematicis benè docto dedi perlegenda, ut an ea sibi constarent animum diligentèr adverteret, suumque mihi de iis judicium candidè aperiret, eundem rogavi. Verùm enim verò dum ille differret lectitandi operam, ipse errorem meum reperi, & digito (quod aiunt) demonstrare potui. In eo solùm haesi, an totum illud Caput praecedens, cum principi meo intento ac fini nihil tolleret; parum, etiamsi omnia vera essent, adferret, omittendum esset, totumque negotium mensurationis Lineae Spiralis dissimulandum: an vero, prout jam scripta essent omnia, unà cum hoc Examine edenda. Occurrebant rationes plures & variae, prò & contrà: Vicerunt tamen illae, quae in bonum aliorum laborem horum, partem saltem aliquam, ipsis communicandum esse, caeteris praelatae sunt, suasêrunt & persuaserunt. Praesertim cum non defint exempla Auctorum, qui & cum laude suâ & cum utilitate Lectorum idem factitârunt: qui si nullum alium hinc auferrent fructum, is saltem satis esse posset, quod inveniant cautionem, ut si curiosiùs sive de Spiralibus, sive de alijs inquirere velint, sibi sciant cavere & à scopulis, & ab alijs incommodis, in quae facillimè incurrere possent. Geometria profectò ipsa, secundum judicium Josephi Scaligeri suprà cap. primo Propos. 4. num. 7. adductum, sibi inprimis gratulabitur, quòd accessione saltem novorum aliquot Epichirematum locupletata sit. Primum igitur indicandum est, qualiter in cognitionem venimus inventa nostra lubricae esse fidei; deinde, ubinam haereat error, ostendendum: tum singulae propositiones examinandae, & pro meritis approbandae vel reprobandae, & siquidem id fieri commodè poterit, erroneae corrigendae. Omisimus autem plurima jam conscripta, bonoque ordine ac methodo digesta; quae si rationibus suis solidè nixa fuissent, Lectori plurimùm oblectationi esse potuissent: cum verò vacillantia ea inventa fuerint, ne fastidio potius essent, jure meritò ea praeterivimus. Non esset autem abs re cogitationem suscipere, qualisnam aut quae sit illa flexuosa linea, & quâ arte illa compendiose describi possit, quae illas haberet proprietates, quas frustrà Spirali attribuimus. Id quod in aliud tempus, vel potius aliis faciendum reservamus. Propositio prima. Occasionem Examinis hujus ac Dubitationis pressius declarare. FVisse quendam, qui Dimensionem Lineae Spiralis ante nos instituerit, memini me aliquando audivisse ex P. Hieronymo Kinig Societatis nostrae Mathematico, & olim in Ingolstadianâ, Dilinganâ, ac Pragensi Academiis earundem Mathematicarum Disciplinarum Professore accuratissimo, mihi & Romae & alibi notissimo; imò eundem Lineam illam alicui alteri aequalem, sine tamen ullâ Demonstratione, pronunciasse; quod assertum dictus ille Professor examinaverit: cui vero Lineae illam adaequaverit, non solùm tunc scire, verum an aliquando id sciverim planè meminisse non potni. Cum ergo versarer in scriptione superioris Capitis, venit in mentem Examen illud quod diximus, recordatione tamen satis confusâ. Existens ergo Graecii in Stiriâ scribo Viennam, ubi Mathematicus ille manebat; & siquid haberet his de rebus in Scripto, ad me mitteret rogavi: non quòd incertus essem de meis jam inventis & scriptis, aut ullo modo de iisdem dubitarem, sed ut illud ipsum cum meis conferrem. Annuit ipse, & sequens ad me misit, sive Examen, sive contra Asserta instantiam & reprobationem: quam cum examinassem, ut suprà diximus, eam veritati consonam esse, manifestè deprehendi. Sic autem se habebant illa. 2. Propositum sit demonstrare, Lineam Spiralem majorem esse Semicircumferentiâ; Intellige Circuli primieam comprehendentis. The same in English. OF The Centre of Gravity. Lib. 2. Cap. 3. Pag. 58. THus far (Courteous Reader) had this new, and no less specious than delightful speculation on the subject of Spiral Lines rather swiftly advanced me, by Sea and Land as I may say, then slowly drawn me, who apprehended nothing either of apparent danger, or unexpected surprise. For indeed I rather knew, then imagined, that I traveled in the High Road; and looking steadfastly on my main design, I was wholly attentive to the means whereby to tread out a commodious Path for the Discovery of the Centre of Gravity in the Spiral. True it is, while I was plodding thereon, a serious reflection came often into my mind, which yet, for certain reasons, was not sufficient to retard my course, viz. That if this Dimension of the Spiral Line were so obvious, easy, and commodious a thing, as I found it to be; as also strengthened with such firm Demonstrations, how came it to pass that the great Archimedes, who (to speak with Pappus) did with admirable Dexterity demonstrate those Proprieties of the Spiral Line, which other men had only hinted at; how came it, I say, to pass that he did not insert this Dimension into his Book of Spirals? But I easily answered myself; to wit, that Archimedes had either neglected or purposely omitted many other things, which have since his time been treated by others; or else it must be, that what he wrote thereof, hath perished by the injury of time. The like he did when he treated of the Centre of Gravity in Plains; for he omitted the Tract of the Centre of Lines. And as that objection deterred me not from enquiring the Centre of Gravity in Lines, especially Circular ones, so here I encouraged myself in hope to discover something new concerning Spirals, which hitherto had escaped both Archimedes and all that had come after him. And truly in my opinion I began the business happily enough, and had made a great progress therein; but on a sudden when I was half way on my Voyage, and came within Kenning of the Port, I was fain to strike Sail, and for security, to make directly towards the shore: for I than first lighted on a certain Examen of the Dimension of the Spiral Line, performed by numbers: whereof indeed, I had heard before, but could not tell how or which way that Dimension proceeded: or whether that Spiral were to be compared with a right Line or crooked, or a pure Circular Line. So far was I from suspecting, that the exact proportion I had already discovered, could be disproved thereby. So that I was nothing at all troubled at the first sight of this Examen, as being very confident of my own Demonstrations, both Geometrical and Arithmetical, which I thought to be irreprehensible. Yet I had often heard, that an enemy, how mean soever he seems, ought not to be contemned; for when we lest think of him, he may hap to stand in our way, if not do us a mischief. I took in hand therefore to examine that Examen, promising myself to discover a thousand errors in his Calculation, rather than one in my own Inventions. But it happened far otherwise; for I found that the Spiral Line of the first Revolution was greater than the Semicircumference of the first Circle: so that I saw I was fairly to submit. * See how unjustly Mr. White charges this Author of obstinacy, for not confessing his error. For what should I do? Should I dissemble the matter? should I hold my peace and conceal it? Or should I, with those Squarers of the Circle , obstinately defend my own Assertion, though I knew it to be false? Should I dare to question, accuse, yea and condemn Archimedes and Euclid himself, to maintain my own opinion? By no means. I resolved therefore to observe the Rule, which Joseph Scaliger once prescribed to himself, but never observed. It is in the Appendix to his Cyclometriques; where he thus speaks. I grant, saith he, that in all other Arts and Sciences, error may be tolerably committed oftentimes: but in the Mathematics it ought not so much as once. For as an old Writer saith, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, etc. All things digested by Art ought to have an unreprovable evidence. So that a Mathematician observing his own error, aught before all others to cry out, 'Tis I, 'tis I, here I am that did it. But if he comes to know it by means of some other person, unless he presently gives thanks to his Corrector, he is ill-deserving towards the man; but if he do not presently amend his error, he wrongs the Science itself. Yet this honest Scaliger was so far from correcting his own oversights, published both in black and red, that with greater ignorance and animosity he still obstinately defended them. I concluded therefore with myself, that something was amiss in my Deductions. But where this Snake (the error) lay, I could neither so easily nor so presently perceive. Wherefore I gave these my Writings to a Friend well skilled in the Mathematics to read them over; entreating him to consider as attentively as he could, whether they were consistent or not; and that he would impart his judgement of them to me with all candour and clearness. But to tell you the plain truth, while my Friend deferred somewhat the pains of reading my Writings, I in the mean time discovered the error myself, and was able (as the Proverb saith) to point it out with my finger. Only I could not well resolve, whether I should now wholly lay aside the foregoing Chapter, (seeing that to do so would not be any prejudice to my principal intent, as on the other side it would not have added much to it, though every thing therein had proved true) and so dissemble the whole matter concerning the measuring of the Spiral Line; or otherwise should publish whatsoever I had written on that Subject, together with this Examen. Many and various Reasons occurred to me Pro and Con: but at last those prevailed, which for the * See if this Authors not expunging his error proceeded out of pride, as he is accused. good of others inclined me to think that some part at least even of those my Labours, was not to be denied to the public. Especially seeing there wanted not the examples of Authors, who to their own praise, and the benefit of their Readers have done the like; who though they should happen to reap no other profit by it, yet were this alone sufficient, that they have here a Caveat given them, that in case they should themselves desire to search more curiously into this Subject of Spiral Lines, or any other of like nature, they should proceed warily and advisedly in the † Where is now that great vanity Master White would pin upon this Author? business, for the avoiding of those rocks of inconveniences and error, which otherwise they will most easily run upon. And lastly, that Geometry itself, even according to the judgement of the same Joseph Scaliger, (cap. 1. prop. 4. num. 7.) should rejoice, being enriched thereby with the Addition of some New Endeavours. The first thing therefore here to be done is to declare, how I came to know that my Inventions were but doubtful and uncertain; next to show where the error lies: and lastly to examine all the several Propositions, with approbation or rejection of them according to their merits; yea (where it may conveniently be done) by rectifying and correcting those which are erroneous. Yet many things I have omitted, though already written, and digested by me into due order and method; which had they been built upon good and solid grounds, would have given great delight to the Reader. But finding them lose and slippery, (to avoid offence) I have justly laid them aside. However, it might (perhaps) be matter not unworthy our consideration, to think what manner of Bending Line that is, (and also how it may be compendiously drawn, and described) which might be found to have all those properties, which in the Spiral Line we have hitherto but vainly sought. But that's a thing I must defer to some other time, or rather leave to other persons to perform. The First Proposition. More particularly to declare the occasion of this following Examen, and of Doubting. THat there was one, who before me had attempted the measuring of the Spiral Line, I remember well to have heard long since from the mouth of Father Hierome Kinig, a Mathematician of our Society, and formerly a most accurate professor of those Sciences in the Universities of Ingolstad, Dilingen, and Prague; whom I knew very well both at Rome and elsewhere; yea, that the abovesaid person had affirmed (but without any Demonstration given) that the said Spiral Line was exactly equal to some other Line: which Assertion of his the said professor did also examine. But to what Line he made the Spiral to be equal, I cannot possibly call to mind, whether I did either then, or any other time know it in all my life. Whilst therefore I was writing the precedent Chapter, I happened to think on the afore mentioned Examen; yet remembering it but confusedly, and in gross. Whereupon being at Gratz in Stiria, I wrote to Vienna where the said Mathematician than was, and entreated him, that if he had any thing of this Subject, he would send it in writing to me; not that I was then any way jealous of myself, or did in the least measure doubt of my own writings and discoveries; but only that I might communicate what I had written, with those of my own profession. He forthwith yielded to my request, and sent me this following Examen, or rather refutation of the things I had asserted; which having myself examined, (as I said before) I manifestly found it agreeable to truth. Now that which he wrote was as followeth. 2. The second Proposition may be to show, that the Spiral Line is greater than the Semi circumference to wit, the Semicircumference of the first Circle that contains it. Though I had here ended my additional Vindication of a worthy Mathematician by the name of Gulden, because I found him so written by the Author of the precedent Letter, as following Mr. White, who names him Paulus Guldenus; yet I thought it not amiss to advertise the Reader before I took Pen from Paper, that his Adversary could not be ignorant that he calls himself Paulus Guldinus in his own printed works, not Guldenus. Whereby one may probably conjecture, that this was a mere affected mistake in Mr. White, to confirm his Reader in the belief of that incredible story he delivers, that he had never either seen, nor heard of this Author's Work, till his own was ready for the Press. FINIS.