Querela geometrica, or, Geometry's complaint of the injuries lately received from Mr. Thomas VVhite in his late tract entituled, 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.

The same in English.

OF The Center of Gravity. Lib. 2. Cap. 3. Pag. 58.

THus far (Courteous Reader) had this new, and no less specious then 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 appa∣rent danger, or unexpected surprize. For indeed I rather knew, then ima∣gin••d, that I travelled in the High Road; and looking stedfastly on my main design, I was wholly attentive to the means whereby to tread out a commodious Path for the Discovery of the Center of Gravity in the Spiral. True it is, while I was plodding there∣on, a serious reflection came often into my minde, which yet, for certain rea∣sons, was not sufficient to retard my course, viz. That if this Dimension of the Spiral Line were so obvious, easie, and commodious a thing, as I found it

to be; as also strengthned with such firm Demonstrations, how came it to pass that the great Archimedes, who (to speak with Pappus) did with admi∣rable Dexterity demonstrate those Proprieties of the Spiral Line, which other men had onely 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 my self; to wit, that Archimedes had ei∣ther 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 Center of Gravity in Plains; for he omitted the Tract of the Center of Lines. And as that objection deter∣red me not from enquiring the Center of Gravity in Lines, especially Circular ones, so here I encouraged my self in hope to discover something new con∣cerning 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 then 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 pro∣ceeded: 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 ex∣act proportion I had already discover∣ed, 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 Demon∣strations, both Geometrical and Arith∣metical, which I thought to be irrepre∣hensible. 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 mis∣chief.

I took in hand therefore to examine

that Examen, promising my self to discover a thousand errours in his Cal∣culation, rather then one in my own Inventions. But it happened far otherwise; for I found that the Spiral Line of the first Revolution was great∣er then the Semi-circumference of the first Circle: so that I saw I was fairly to submit. 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 above-mentioned, obstinately defend my own Assertion, though I knew it to be false? Should I dare to question, accuse, yea and condemn Archimedes and Euclide 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 ob∣served. It is in the Appendix to his Cyclometriques; where he thus speaks.

I grant, saith he, that in all other Arts and Sciences, errour may be tollerably committed oftentimes: but in the Mathematiques it ought not so much as once. For as an old Wri∣ter saith, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, &c. All things digested

by Art ought to have an unreprove∣able evidence. So that a Mathe∣matician observing his own errour, ought 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 Cor∣rectour, he is ill-deserving towards the man; but if he do not presently amend his errour, he wrongs the Science it self.

Yet this honest Sca∣liger was so far from correcting his own over-sights, published both in black and red, that with greater igno∣rance and animosity he still obstinately defended them. I concluded therefore with my self, that something was amiss in my Deductions. But where this Snake (the errour) lay, I could neither so easily nor so presently perceive. Wherefore I gave these my Writings to a Friend well skilled in the Mathe∣maticks 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 errour my self, and was able (as the Proverb saith) to point it out with my finger. Onely I could not well resolve, whe∣ther 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 concering the measu∣ring of the Spiral Line; or other∣wise should publish whatsoever I had written on that Subject, toge∣ther with this Examen. Many and various Reasons occurred to me Pro and Con: but at last those prevailed, which for the good of others incli∣ned me to think that some part at least even of those my Labours, was not to be denied to the publick. Especially seeing there wanted not the examples of Authours, 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 †business, for the avoiding of those rocks of in∣conveniences and errour, which other∣wise they will most easily run upon. And lastly, that Geometry it self, even according to the judgement of the same Joseph Scaliger, (cap. 1. prop. 4. num. 7.) should rejoyce, 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 shew where the errour lyes: and lastly to examine all the severall 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 Rea∣der. But finding them loose and slip∣pery, (to avoid offence) I have justly laid them aside. However, it might (perhaps) be matter not unworthy our consideration, to think what man∣ner 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 ra∣ther leave to other persons to per∣form.

The First Proposition. More particularly to declare the occasion of this following Exa∣men, and of Doubting.

THat there was one, who before me had attempted the measu∣ring 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 professour 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 minde, whether I did either then, or any other time know it in all my life. Whilest therefore I was writing the precedent Chapter, I hapned to think on the afore mentioned Examen; yet remembring it but confusedly, and in gross. Whereupon being at Gratz in Stiria, I wrote to Vienna where the said Mathematician then was, and en∣treated 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 my self, or did in the least measure doubt of my own writings and discoveries; but onely that I might communicate what I had

written, with those of my own pro∣fession. He forthwith yielded to my request, and sent me this following Ex∣amen, or rather refutation of the things I had asserted; which having my self 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 shew, that the Spiral Line is greater then the Semi circumference to wit, the Semi-circumference of the first Circle that contains it.

Notes

• See how unjustly Mr. White charges this Au∣thor of obstinacy, for not confessing his errour.
• See if this Au∣thors not expunging his errour proceeded out of pride, as he is accu∣sed.
• †

  Where is now that great vani∣ty Master White would pin upon this Authour?