JOUR Mathematical Readers (and

Y doubtless you have such) will infallibly be pleased with the following discussion of two different methods suggested for the Rectification of the Hyperbola. The one proposes to effect it by means of two Ellipses; the other shews that it may be better done by an appropriate Theorem. The former is Mr. Woodhouse, then Tutor at Caius College, Cambridge; the latter, Mr. Hellins, Vicar of Potters-Pury in Northamptonshire. The discussion refers to two Papers, pubfished at different times in the Philosophical Transactions; that of Mr. Woodhouse, at Art. X. p. 219, in the Volume for 1804: that of Mr. Hellins, at Art. VI. p. 110, in the Volume for 1811. It was written originally for a respectable Journal, but by accident deferred, and finally prevented from appearing there. But considering it as a question interesting and important to Mathematicians, I am induced to forward it to you.

R.

N. B. It is written by a very emineut Mathematician and Professor. PHILOSOPHICAL TRANSACTIONS FOR

THE YEAR 1811. PART I. Art. VI. On the Rectification of the Hyperbola by means of two Ellipses, proving that Method to be circuitous, and such as requires much more Calculation than is requisite by an appropriate Theorem; in which Process a new Theorem for the Rectification of that

Curve is discovered.

To which are added, some Observations on the Rectification of the Hyperbola: among which the great Advantage of descending Series over ascending Series, in many cases, is clearly shown, and several Methods are given for computing the constant Quantity by which those Series differ from each other. By the Rev. John Hellins, B. D. F.R. S. and Vicar of Potters-Pury, in

Northamptonshire. Being an Appen dix to his former Paper on the Rectification of the Hyperbola, inserted in the Philosophical Transactions for the year 1802. Communicated by Nevil Maskelyne, D.D. F. R.S. Astronomer,

The Rectification of the Ellipsis, and of the Hyperbola, are problems of the same class; and, by a judicious application of appropriate theorems, may be solved with equal facility. Yet, since the discovery that the latter of these problems might be solved by means of the former, that method of solving it has been considered as the best by several eminent mathematicians. The Rectification of the Ellipsis is the main subject of Art. X. in the Philosophical Transactions for 1804; in which Paper Mr. Woodhouse, the writer of it, has applied the Rectification of the Ellipsis to the Rectification of the Hyperbola, and to the solution of a problem in Physical Astronomy.

It must be evident to every intelligent Reader, and appears also from Mr. Woodhouse's own references, that the greater part of the matter contained in his Paper was taken from other books, and no small part of it from French books, some of which were by no means easy to be procured, especially in time of War.

So scarce were the foreign books required, that two years elapsed before they could be procured. In that interval, however, and even to the present time, I have not heard of any correction which this Paper has received from its Author. But several of its errors have been pointed out in different periodical publications; and a few of them are noticed by the second writer, Mr. Hellins. A few brief remarks on Art. X. of the Philosophical Transactions for 1804 shall therefore suffice, with respect to that part of the subject.

The differential notation of Leibnitz, which is used throughout this Paper, instead of the fluxional notation of Newton, displays such a partiality for foreigners, and so much disrespect to the great inventor of Fluxions, as could not be expected

from any Englishman, and particularly from a Member of the University of Cambridge. The new notation also of the co-efficients of a binomial quantity raised to a given power, proposed by Mr. Woodhouse (p. 227) to be used “for the sake of conciseness," is rather surprizing; since the Newtonian method of denoting such co-efficients by the letters A, B, C, &c. is both more simple and

more concise.

The

The writer also falls into blemishes of style, which might easily have been avoided. Such, for instance, as the following phrases, borrowed from the French: The whole integral.""Integral from x=0 to x=1." "Integral" (of a quantity) "between z=0 and z=1." This is not the mathematical language of England; and it is a pity if the Author, in studying French mathematicians, has forgotten his English Masters.

Of his Algebraic processes, some are very obscure, and some are erroneous; so that to a person not otherwise acquainted with the subjects, they could hardly be intelligible. The following processes aud results may be noted as erroneous. The process in p. 231, and the series derived from it in p. 232; also the process in p. 233, and that in p. 260, and the theorem derived from it in p. 261, for rectifying the Hyperbola by means of two Ellipses. The form of the fluent which Mr. W. assumes in p. 276, shews such a want of skill in series as is very inconsistent with the high tone in which he speaks on the subject.

Mr. Woodhouse is erroneous also when he speaks (p. 236 and 237) of Fagnani's Theorem as necessary in the investigation of Euler's Series (given in p. 235) for computing a quadrantal arch of an excentric Ellipsis. Had Mr. W. been acquainted with a Paper on Series, written by the Rev. J. Hellins, and published by the Royal Society in their Trausactions for 1798, he might have perceived that Fagnani's Theorem is not at all necessary in that investigation.

Mr. W. is erroneous again when he speaks of M. La Grauge (which he does more than once) as the discoverer of a substitution, by which the fluxions of Elliptic and Hyperbolic arches are transformed into others of which the fluents are attainable in swiftly converging series.

that a similar substitution was used, and a like result obtained, by our countryman, Mr. John Landen, at least ten years before M. La Grange's Paper appeared, as may be seen in the Philosophical Transactions for 1775. And the same device may be found in his Mathematical Memoires, vol. I. p. 32. Nay, M. La Grange himself, in the very Paper in the Turin Memoires to which M. La Crois refers, acknowledges that he had seen Landen's Paper on the Ellipsis and Hyperbola (in which the substi tution is used) by the mention which. he there makes of that Paper! It is no great commendation of a tutor in an English University, to be better acquainted with French books than with those that are valuable in English; and still less can he be excused, if, through carelessness, or partiality, he gives to one Author that praise which is due to another.

The grossly erroneous assertion in p. 273, respecting series of the swiftest convergency for computing the values of A and B. (which the Author affirms to be when the index is = 1⁄2) is borrowed, with the exception of the peremptory mode of expression, from M. La Grange! Nullius in verba, the judicious motto of the Royal Society of London, might have warned the Autbor against this fault.

Notwithstanding these faults of the Paper (No. X. for 1804), and others which may be found in it, still it is not without its value, as a synopsis of the ingenious devices of several eminent mathematicians of this island, and of more on the Continent, for rectifying the Ellipsis, and by that means solving a difficult problem in Physical Astronomy. It is valuable also for showing that several methods of computation, very different in Algebraic characters, are founded on the same principle, and are in fact the same. It is impossible therefore not to regret that the Author did not draw it up in a manner more conducive to his own credit.

inonstration, and the Observations.

The Rectification of the Hyperbola by means of two Ellipses, is an invention of the late Mr. John Landen, F.R.S. which was first published in

the

the Philosophical Transactions for the year 1775, and afterwards in Vol. I. of his Mathematicul Memoirs, in 1780.

In the beginning of this Paper, Mr. Hellins speaks of this method as a display of great ingenuity, and observes that it has "justly obtained the notice, and called forth the praises of eminent mathematicians both in this Island and on the Continent." He next adverts to Landen's representation of himself, as the first who solved the Problem of computing the difference between the length of the infinite arch of an Hyperbola and its asymptote, (a problem of great importance in the rectification of that curve,) although it had been solved before both by M'Laurin and Simpson, in their Treatises of Fluxions; but candidly, and, as we believe, justly, attributes this misrepresentation of fact to the failure of Landen's memory, who was old, and much encumbered with other business. He then proceeds to demonstrate, That the Rectification of the Hyperbola by means of two Ellipses (the mode recommended in the former Paper) ís circuitous, and such as requires much more Calculation than is requisite by an appropriate Theorem. This proposition is fairly and fully proved. Indeed, no one who deserves the name of a Mathematician, can cast his eye on the new Theorem given in Art. 9 of this Paper, and withhold his assent from the proposition.

Among the observations which make up the second part of this Paper, the first is: that, when the convergency of the ascending series (which is a new series given in Art. 11 of this Paper) ceases to be swift, then a good rate of convergency will take place in some of the descending series to be found in his former paper on the Rectification of the Hyperbola, published by the Royal Society in their Transactions for 1802. This naturally introduces the consideration of the constant difference which subsists between the ascending and the descending series given in that Paper. It is a curious fact, that this constant difference is no other than the difference between the length of the infinite arch of the Hyperbola and its asymptote, as is easily perceived by what is done from p. 400 to p.

465 of the volume last mentioned, where also methods are given for computing it. But as methods of computing this difference have been proposed by MLaurin, Simpson, and Landen, he gives a brief statement of their methods, and compares them with such of his own as he has offered to the publick. The first comparison is of a series in Art. 808 of M⭑Laurin's Fluxions, with another in Art. 435 of Simpson's Fluxions, and with a third given in the former part of this Paper, by which it appears that each of these series has, in this case, the same rate of convergency, and the three may be said to coincide. The next compa rison is of Landen's method of computing the said difference by means of two Elliptic arches, with the series before mentioned; which affords a striking instance of the inutility of rectifying the Hyperbola by means of two Ellipses. The third comparison is of a series derived from Landen's Theorem in his second Memoir, Art. 5, (for Landen cannot be said to have finished his work,) with those of M'Laurin and Simpson, before mentioned; by which it appears, that when the transverse axis of an Hyperbola is much greater than the conjugate axis, the series thus obtained converges much faster than the old series: and consequently that Landen had some reason for setting a value on that Theorem. It appears also, by this comparison, that, when the transverse axis of the Hyperbola is less than the conjugate, Landen's method of computing the difference in question is not wanted, since the old series (which is simpler in its form than that which is derived from hi Theorem,) converges swiftly enough to answer the purpose. He then proceeds to show, that, by a combination of Landen's Theorem with the new one given in the former part of this Paper, a series of more rapid convergency is obtained for computing the aforesaid difference; the geometrical progression which has place in it, decreasing somewhat swifter than the powers of the fraction or even in the most dis advantageous case, viz. when the ratio of the axes of the Hyperbola is as 1000 to 786, or as 4 to 3 nearly so that twelve terms of this series (its convergency being quickned by numeral co-efficiente,) will

be

be sufficient for all common uses. Having obtained this series, he gives (in Art. 24,) a new and very conve nient formula, for computing the dif ference before mentioned.

Mr. Hellins next (in Art. 25) adverts to p. 466 and 467 of the Philosophical Transactions for 1802, and shows that the difference between the ascending series and the descending series, there inserted, is the very expression which Mr. Landen obtained, by a very different method, in Art. 5 of his second Memoir, and on which he set a considerable value. This difference, as was before observed, (and is proved in this Article,) is the difference between the infinite arch of the Hyperbola and its asymptote; which difference he denotes by the letter d, the character by which we also, for the sake of brevity, shall denote that difference in the remaining part of our account of this Paper. It clearly appears, by the process in Art. 25 of this Paper, that, when the same geometrical progression obtains both in the ascending and in the descending series, the latter will be most eligible for arithmetical computation, on account of the absence of a column of quantities in that series which enters into the other. So that the formula for computing the value of d, by the descending series, will be more convenient in practice, than the formula for computing it by the ascending series. The first of these formulæ (which may be called Landen's Theorem) is given in Art. 25, the second is given in Art. 27 of this Paper. In Art. 28, Mr. H. refers again to his Paper in the Philosophical Transactions for 1798, for a method of transforming the series given in Art. 25, for computing the value of d, into others which converge twice as fast: And, in the next Article, he transforms one of his own descending series for the rectification of the Hyperbola (inserted in the Philosophical Transactions for 1802,) into a pair of series for computing the value of d, each converging by the powers of the fraction where a, which denotes the transverse semi-axis, is supposed to be much greater than 1, which denotes the conjugate semiaxis; so that this series will converge very swiftly. In Art. 30, the last mentioned pair of swiftly converging series is transformed into another

pair of a simpler form, but having the same rate of convergency; the operations being similar to those which he had described in his former papers inserted in the Philosophical Transactions for 1798 and 1800.

Art. 31 and 32 contain the investigation of the low which the co-efficients of the new pair of series observe ad infinitum; which law is discovered by a fluxionary process, for which we must refer our mathematical readers to the Paper itself, as it cannot be abridged, nor will the nature of our plan admit of its insertion here. Mr. H. then says, with great truth, "Thus, by the common application of Șir Isaac Newton's doctrine of Fluxions and infinite series, without any assistance from, or regard to, Landen's Theorem, we have obtained a pair of series for computing the value of d, which converge by the powers of, and of which we can find as many terms as we please. And by a similar process, may Euler's series for computing the quadrantal arch of an Ellipsis be obtained without any use of Fagnani's Theorem, or the tentative methods, and strange artifices as Mr. Woodhouse calls them, which appear in Euler's Paper."

In Art. 34, that ratio of the axes of an Hyperbola is pointed out, which serves as a limit of the use of the single series, and of the pair of series, before spoken of, for computing the value of d. And in the next Article, the pair of series is accommodated to the Hyperbola of which the semi-axes are 1 and b.

Mr. Hellins had shown in Art. 24, that, even in the most disadvantageous case, the value of d might be computed by a series converging somewhat swifter than the powers of; he remarks in Art. 36, that series of much swifter convergency may be obtained for that purpose, by means of a transformation of the fluxion of the arch of the Hyperbola; but that such transformations were foreign from his present design.

By the examples which are given in the remaining pages of this Paper, the great advantage of descending series over ascending series, in the rectification of the Hyperbola, is very obvious; and Mr. H. concludes his Paper with this just remark : "In these examples the use and advantage