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Oeuvres complètes. Tome VI. Correspondance 1666-1669 (1895)

Oeuvres complètes. Tome VI. Correspondance 1666-1669

(1895)–Christiaan Huygens–rechtenstatus Auteursrecht onbekend

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No 1709.
[J. Collins] à [R. Moray].
Appendice au No. 1708.
[février 1669].

La copie se trouve à Leiden, coll. Huygens.

The State of ye Controversy between Mr. Hugenius and Mr. James Gregory.

 

Mister Gregory being in Italy, A. 1667. publisheth a litle book Intitul'd vera Circuli et Hyperbolae Quadratura in propria sua Proportionis specie inventa et demonstrata1).

He sends divers of these Books abroad to ye Mathematicians of Italy, and to Monsieur Hugenius also, desiring their judgement of it, before they wrote against it, yt might by private letters satisfy their doubts, before they should require any such thing in print2). He likewise sent some of ym into England, where it met with a good Character in ye Philosophical Transactions3). From others he receiues thanks and their approbation; from Monsieur Hugenius he heard nothing, till he met with ye French Journal des Scavans4), wherein Hugenius publisht diverse Exceptions against ye Book, as.

I. That ye Author had not prov'd ye maine matter in question, his Assertion

according to ye Title being, That ye dimension of ye Circle, Hyperbola, and their portions are quantities non-Analytick, viz to this sense, That it is not possible by any Analytical Operations whatsoeuer to obtaine, or by any Aequation whatsoeuer to expresse ye true quantity or measure of any Portion of a Circle, Ellipsis or Hyperbola, and consequently these figures can neuer be Geometrically squar'd. Doctor Wallis doth not deny this doctrine; nor is ye Author displeas'd, but rather desirous, yt it should be tested to ye uttermost: and though ye Author asserts he hath demonstrated it, yet if in ye Iudgment of ye Learned he falls short thereof, he is willing publikely in print to acknowledge his failure.

II. But Hugenius tells ye world, yt himself in Anno 1654 had publisht better Approaches for the Circle, than Gregory. To this Gregory answers5), yt he was not conversant in ye writings of Hugenius (being a Traveller without Books;) which is likely to be true, since upon diligent search there hath not been any of Hugenius his books to be found in any Stationers shop in London for 12. years past, and probably they are as scarce in Italy; which perchance might induce Mister Gregory to write a Treatise of this kind. But as to ye matter, Mister Gregory answers, yt indeed ye methods by him used in yt Book (he beginning his Polygons with a whole Quadrant) are somewhat more tedious than those of Hugenius; but at ye end of his 25. Proposition he hath such Approaches, as are more exact and lesse laborious than any of those of Monsieur Hugens, but made not use of ym, by reason yt at yt time he could not demonstrate ye same Geometrically; but saith, he hath now demonstrated ym in his Exercitationes Geometricae6), ye said Approaches being mis-vnderstood and mis-applied by Monsieur Hugens, Mister Gregory in his Answer7) not yet printed, vndertakes to show him his Error.

And ye Authors sense concerning his owne method is this.

Approximatio mea in Circuli mensura videtur Hugenianae anteponenda ob has rationes;

1. Mea est generalis, Circulo, Ellipsi et Hyperbolae applicabilis, quod mirandum, Hugeniana est particularis, soli Circulo propria; et elegantes illae Analogiae Geometricae et Harmonicae, Circulo, Ellipsi et Hyperbolae communes, à me (ni fallor) primo erant inventae.

2. Mea est simplex et vniformis, quippe in illa solummodò examino Polygona; in Hugeniana autem opus est non solum Polygona, sed etiam adscriptas, et Centra gravitatis computare.

And ye Author in his Exercitationes Geometricae has made his former methods much more easy and expedit than they were, and seem now very much to transcend any thing of Hugenius.

Secondly, variety of methods doth much advance Inventions. We give an instance in division; ye turning of ye Divisor into a Binomial, and ye Quote into an Infinite series, was ye medium, by which Monsieur Mercator8) so happily squared ye Hyperbola9); and this was no other, than to make yt laborious and difficult, which in itself was easy. In like manner ye Perpendicular of an Oblique Plaine Triangle may be found, from ye 3. sides giuen, easily in ye common road, but by ye help of a more troublesom proportion, you shall not only find ye thing required, but likewise solue most of those hard cases about Oblique Plaine Triangles in Oughtred's10) Clavis11), and divers others in Billy's12) Diophanty Geometra13);

ye like in Spherick Trigonometry, where ye performing of some common cases by unwonted more laborious proportions hath enlargd ye Doctrine.

Though we could give many instances, yet we will insist vpon no other, but these excellent things, to which ye Author has often asserted, he hath further apply'd and advanced his own method, by Approaches to any exactnesse required.

1. The finding ye Square root, Cube root, or ye root of any Power of any Number proposed.

2. The finding of some root of any possible Aequation proposed, however affected, and thereby depressing ye Aequation to find another root, and consequently all ye roots.

3. The finding of any Mean in a rank of continual proportionalls between ye Extreams given.

4. The interpolation of such ranks, whose roots are not an Arithmetical progression, and yet their 2d, 3d, 4th etc. differences are equal: Such ranks may be derived from any Arithmetical progressions by making facts of each successive part, and solids from each ternary of Numbers etc.

III. Monsieur Hugenius grants Mister Gregory's Quadrature of ye Hyperbola to be very good, but saith, he belieues, it cannot appear to be new to ye Royal Society, in regard he communicated ye same manner of measuring ye Hyperbolae to ym, when he shew'd how to weigh ye Air in severall distances from ye Earth. To which, tis answer'd, yt Hugenius did by a letter to Sir R. Moray14) (as it now appears, but hath been hitherto vnknowne to Mister Gregory and most of ye Society) shew, How to compute ye Weight of ye Air by Logarithmes, but gaue no ground, reason or demonstration thereof; and it seems (as Hugenius saith since) it is built vpon computing ye Area of ye Hyperbola by ye difference of Logarithmes, which in ye Hyperbola are represented by Spaces contain'd between the Hyperbolick Curve, and Asymptote, and two Lines parallel to ye other, which are Proportional to ye differences of Logarithmes, but this is not so much as mentioned, in ye letter.

Moreouer, Monsieur Hugens his Rule for measuring those Area's was invented here by Mister Barrow, before he euer heard of it from Monsieur Hugens, as Doctor Wallis hath signify'd to him15); and neither of ym did very well like it. Therefore Mister Barrow set himself to ye finding out of other Rules for finding ye Area of an Hyperbola without Logarithmes, which he was pleas'd to

impart; wherein again it seems very mysterious, yt ye same Rules are likewise applicable to ye Circles, mutatis signis.

Besides, all Quadratures of the Hyperbola will make ye Logarithmes, but ye Rule of Hugenius supposes ym made already; and is therefore no proper quadrature of ye Hyperbola, nor to be compared with Mister Gregory's methods.

IV. Monsieur Hugens saith, yt ye same matter and ye very same rule for making ye Logarithmes giuen by ye Author, was a thing knowne, and vpon record in ye Parisian Library, and theresore not New.

Though it were so, yet we are beholding to Mister Gregory as ye first, yt reduced the Calculation of Logarithmes to Geometry. We may take an answer from what Doctor Wallis saith for himself page 76. de Cycloide16).

Dummodo enim ipsi sua apud se premunt Inventa, nec juris publici faciunt, iniquum plané esset, nì et alios patiant ea quae ipsi celant, itidem invenire, atque interim inveniendi, si qua sit, gloriam reportare.

And we may say, as Doctor Wallis in ye like case, Nos saltem Gregorio plus debemus, qui demonstrationes suas palam factas vulgavit, quam illis, qui suas adhuc supprimunt.

Moreouer, may not their Rule sor Logarithmes differ as much from Mister Gregory's, as Hugenius his Rule for ye Area of an Hyperbola differs from Mister Gregory's quadrature; and so, how well their Rule performs it, remains a just scruple, seing yt what Hugenius has found out, since this controversy arose, he does acknowledge to be inconsiderable, in comparison to what Monsieur Mercator has done; but Mister Gregory asserts, his method to be wholy different from yt of Mercator, and more general as performing ye Converse, to wit, a Logarithme giuen to find its Number, whether as easy or not, he leaues to Experience; and even in Mercators method he hath shew'd how to abridge ye work, by leaving out all ye even powers, which Mercator takes in: And those methods Mister Gregory has demonstrated Geometrically.

Vpon ye whole, Monsieur Hugens seems blameable for beginning these comparisons, quasi ex animo vilipendendi, as appears from his reason rendred, why Gregory's quadrature of ye Hyperbola should not seem new to ye Royal Society; on the other side it were to be wisht, that Mister Gregory had been more mild with yt generous person, who hath deserv'd well of ye republick of Learning.


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