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 and 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 Transac tions 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 Grange (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.

This misrepresentation (as Mr. W. acknowledges in p. 273) is on the authority of M. La Croix; who in the 2d Volume of his Traité du Calcul Différential et du Calcul Intégral, art. 422, speaks of the aforesaid substitution as the device of M. La Grange; and in the table of contents refers to the Memoires de l'Acad. des Sciences de Turin for the year 1785, for the origin of it. Yet the fact is,

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 Croix refers, acknowledges that he had seen Landen's Paper on the Ellipsis and Hyperbola (in which the substitution 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= 4) 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 Author 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.

A brief account of Art. VI. of the Philosophical Transactions for 1811, will conclude the present discussion.

This Paper, as its title indicates, consists of two principal parts; the Demonstration, and the Observations.

The Rectification of the Hyperbola by means of two Elipses, 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 Mathematical 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) is 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. 460 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 M'Laurin, 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 comparison 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 his 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 differences the geometrical progression which has place in it, decreasing somewhat swifter than the powers of the frac tion, even in the most disadvantageous 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 quickened by numeral co-efficients,) will

be

be sufficient for all common uses. Having obtained this series, he gives (in Art. 24,) a new and very convenient formula, for computing the difference 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 law 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 Sir 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 ob tained 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. Hellius 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 ad

vantage

vantage of descending series appear: more examples of their utility might be given: and it might easily be shown, that there are cases in which such series have the advantage, even when the ascending series have a good rate of convergency. I trust, how ever, that enough has been done in this Paper, to satisfy all candid and competent judges of the matter, that the rectification of the Hyperbola by means of two Ellipses is more curious than useful; that the advantage of computing by descending series, is, in many cases, very great; and that such series will often answer the end of a transformation without the trouble of making it.”

and integrity, the writer is convinced he would not intentionally bave inserted any thing that would not have borne the test of the strictest investigation. In one instance, however, this is not the case: and truth being the sole object in view, more especially that the character of Milton may not be liable to a charge of inconsistency, the writer of this may easily be pardoned for attempting to clear up a point relative to the Poet's first marriage into the family of Powell; in which, according to Mr. Todd's account, there is most certainly a considerable inaccuracy.

The first Life of Milton was written by Phillips, his sister's son, who may reasonably be supposed to know the circumstances connected with his uncle's first marriage. His words are :

With respect to the two Authors, the distinction is very obvious. The one borrows largely from books; the other takes from his own store: the one delights in Gallicisms, and is often obscure; the other is plain and perspicuous.

More might be said respecting the different tempers of the writers, but nothing that would not be self-evident to every reader of the two Papers.

Mr. URBAN,

THE

Inner Temple. HE life of our great Poet Milton has occupied the attention of many able pens. Every minute occurrence of his memorable career, which industry, joined to the spirit of modern inquiry, could at this distance of time recover, has been laid before the publick, and points out the high estimation in which his memory is now held. Indeed no genuine admirer of the Poet will regard any circumstance connected with the family of Milton, or which serves to throw light on the transactions of those times, as trivial. Much curious information, and many valuable notices, collected by the late Mr. Thomas Warton, are prefixed to his edition of Milton's Juvenile Poems.

It is owing to the commendable zeal and assiduity of a late writer of 'his life, the Rev. H. J. Todd, that even an additional harvest has been gleaned to adorn what the Author modestly terms an" unadorned narration:" and from his acknowledged talents

a little after, he (Milton) took a journey into the country, nobody about bim certainly knowing the reason:-after about a month's stay, home he returned a married man that went out a batchelor; his wife being Mary, the eldest daughter of Mr. Richard Powell, then a Justice of the Peace, of Forrest-hill, near Shotover in Oxfordshire.” (Phillips's Life of Milton, p. 22.)

Mr. Todd (on the authority of the late Mr. T. B. Richards) asserts, that "Milton married a daughter of Justice Powell of Sandford, in the vici nity of Oxford, and lived at a house at Forrest-hill, about three miles from Sandford.” (Todd's Life of Milton, p. 25, 2d edition.)

The late Mr. Richards had certainly great opportunities of making inqui ries concerning the family into which Milton married, having resided many years in the early part of his life, at Bensington, within ten miles of Oxford. But, if indeed he ever did make inquiries, he has in this instance been most strangely mistaken; baying confounded the family of Richard Powell, Justice of the Peace, of Forrest-hill, with an antient Roman Catholic family, the Powells of Saudford.

To prove this point satisfactorily, it will be necessary to pursue the following plan :

1. Give a genealogical sketch of the Powells of Saudford.

2. State the result of an accurate examination of the parish register at Sandford.

3. Col

.

.3. Collect the incidental notices scattered through the works of those eminent Antiquaries Hearne and Anthony Wood, relating to this family. The first part proposed, the writer is the better enabled to accomplish, being possessed of a curious Pedigree of the family, commencing with Maurice Ap Howel of Guernan, co. Cardigan, to the death of the late John Powell, esq. of Sandford, A. D. 1730, without issue male.

I. The manor of Sandford belonged in antient times to Sir Thomas de Saundford, who, in the reign of King Stephen, or thereabouts, gave it to the Knights Templars. At the suppression of religious houses, it was granted by King Henry VIII. to Edward [Edmund] Powell. (Tanner's Not. Mon. ed. 1744, p. 414.)

Arms: Arg, a lion ramp. Sab. debruised by a fess engrailed Gules.

1. Edmond P. (to whom the manor was granted) settled at Sandford 33 Henry VIII. A. D. 1542.

2. Edmond P. his son ob. 1592, sepult. ap. Sandford. He left two sons, 1. Edmond, his successor; 2. Sir William P. of Tutbury and Rolleston Park, co. Stafford. ob. s. p. 1656.

3. Edmond P. married two wives, 1. Frances, daughter of - Gifford, of Chillington, co. Stafford, by whom he had three daughters, who died young and unmarried. Secondly, Cicely, daughter of Richard Fogge, of Dane-court, co. Kent, by whom he had two sons: 1. Edmond, born 1804, his successor; and 2. William: also six daughters. 1. Thomazine, born 1603, married Richard Spicer, of London, Doctor of Physick. 2. Philippa. 3. Anne, born 1607, married Richard Betham. 4. Mary, died young. 5. Mary, born 1609. 6. Cecilia, born 1611, buried at Sandford 1641. This Edmond P. dying in 1632, was succeeded by his eldest son.

born 1632, succeeded his father; married Catharine, daughter of William Petre, of Stanford Rivers, co. Essex, and died 1678. He had two sons: 1. Edmund, who married Anne, sister to Rowland fourth Lord Dormer, and died v. p. without issue. 2. John, who succeeded his father.

6. John P. married Anne, daughter of Thomas Wyndham, and dying Aug. 1730, without issue male, was sueceeded in his estates by his two daughters and coheiresses. 1. Winifred, born 1705, married to Sir Francis Curzon, of Waterbury, co. Oxon. bart. whom she survived, and died 1764, s. p. 2. Catharine, born 1709, married in 1732 to Henry Roper, 10th Lord Teynham, and died 1765.

From this short, though comprehensive extract, comprising the names of every individual of the Powell family from the above-mentioned pedigree, it appears then, that from 1542, when the manor of Sandford was granted to Edmond P. till 1730, during a space of nearly 200 years, there never was any one of this family named Richard. That at the time Milton is said to have married (in 1643) Mary, the eldest daughter of Mr. Richard P. of Forrest-hill, Edmond Powell born 1604, fourth in descent from the original grantee, both in name and possession, was Lord of the Manor of Sandford; that he was then 39 years of age; and of his daughters, the third, named Mary, died unmarried in 1703.

Besides, the family were rigid Roman Catholics, and connected by marriage with several antient families of that persuasion; as Gifford of Chillington, Napier of Halywell, co. Oxon. Dormer, Petre, Throgmorton, &c. as set forth in the Pedigree. That they continued in this profession is evi dent. The two daughters and coheiresses of the last John Powel mar

man Catholic families in England, Curzon and Roper, although they have both since that time conformed to the Establishment. While Milton was a rigid Presbyterian, whose principles must have ill accorded with

* Henry Curzon, esq. of Waterbury, Colonel of the Oxford Volunteers, was a Candidate to represent the city of Oxford, in Parliament at the last general Elec, tion.

those