frequently grants permission to take a a few hours' walk.

We would recommend none to take this rural ramble without joining the retired hermit, who will always be found with his cross and bible in devout meditation. The following lines, said to be the composition of Her Royal Highness Princess Elizabeth, are descriptive of this interesting

scene.

YE whom variety delights,
Descend awhile from Windsor's heights;
And in this hovel deign to tread,
Quitting the castle for the shed.
Such were the Muse's favourite haunts,
From care secluded, and from wants.
What nature needs, this hut can give,
Could we as nature dictates live;
For see, on this plain board at noon,
Are placed a platter and a spoon,
Which, tho' they mark no gorgeous treat,
Suggest 'tis reasonable to eat.

What though the sun's meridian light
Beams not thro' our hovel bright;
Tho' others need, we need him not,
Coolness and gloom befit a cot;
Our hours we count without the sun,
These sands proclaim him as they run.
Sands within a glass confin'd,
Glass which ribs of iron bind :
For Time, still partial to this glass,
Made it durable as brass,
That, placed secure upon a shelf,
None might crush it but himself.
Let us hence the day prolong,
With loyal and with nuptial song.
Such as with duteous strains address'd,
May gratify each royal guest;
Thrice happy should our rural toils
Be requited by their smiles.

Many of the rural spots in Frogmore gardens, and other parts, were laid out after her own taste. An exqui sitely beautiful grotto, forming an interesting feature in the Frogmore gardens, is said to have been completed by her.

The Princess Mary was born April 25, 1776. She was married to William Frederick, Duke of Gloucester. This princess has always supported an amiable character, following the steps, and copying the example, of her elder sisters. She was the constant attendant upon the late Princess Amelia, during the long period she was confined to her bed, watching her with unceasing affection till she expired. To assist in works of charity she was ever ready; and many benevolent institutions enjoyed the benefits of her influence and liberality.

The Princess Sophia, was born Nov. 3, 1777. The same amiable spirit by which her sisters have been distinguished, breathes in her conduct, and dignifies her actions. Charmed with domestic life, her retiring virtues shun the eye of popular observation, on which account her real worth is less generally known; but it is rendered the more valuable, because it is not obtrusive. Affable, charitable, and sympathetic, she has an ear that attentively listens to tales of woe, and a hand that is always ready, by administering to the necessities of the indigent, to soothe them in affliction, and alleviate their distress.

Her Royal Highness the Princess Elizabeth, was born May 22, 1770. Literature, the Arts,-Charitable InIn these respects, the Princesses of stitutions, the Widow, the Orphan, England appear to be actuated by one and the Friendless, have reason to common spirit. Our benevolent inhail with unspeakable joy this happy stitutions have acquired respectability event. She was the patroness of every and vigour under their fostering palaudable undertaking. Her bountiful tronage; and among the poor who hand, which on all occasions was have been relieved by their bounty, united with her sympathizing heart, their munificence will be long rememrendered her departure an irreconcile-bered with grateful affection, when able loss to the needful, who had so they shall be gathered to the sleep of long partaken of her charity.

She wrote many pieces of poetry, and prose; acquired a considerable knowledge of drawing; and is said to have engraved or etched a series of plates. Most of her own productions she had printed at Frogmore, at the press of Her Majesty, who had fitted up a small private printing-office. She established a school of girls, at Old Windsor Cottage, the country residence of Her Royal Highness, which was supported entirely by herself.

their fathers.

Destitute of that haughtiness which too frequently associates with royalty, they have rendered themselves beloved in their respective neighbourhoods, by their dignified condescenson, without compromising their illustrious descent, or degrading themselves in the eyes of the most exalted, who cannot but admire in their conduct, a lovely feature which pride will not permit them to imitate. They have taught ambition a lesson by their own humility; and in

structed avarice by their genuine li- | stances, assumed the garb of disinberality. The nation has long ex-terested charity.

Literary Notices.

A new Edition of Walton and Cotton's Complete Angler, is preparing for the Press, by Mr. Bagster. It will be print

perienced the effects of their benevo-
Îence; and the consequences of their
influence among those who have been
honoured with titles, wealth, and
station, will in all probability extend
beyond the present generation of
mankind. Their example has ope-
rated among many, who have been
strangers to their motives; so that
even ostentation has, in several in-half-bound, 58.

ments.

ed in a size for the pocket; with entirely new EmbellishIn the course of November will be published, in One Vol. 8vo. "Augustus; or, The Ambitious Student."

On the 1st November was published, the whole of the Rev. G. C. Smith's Works, (Author of the Boatswain's Mate,) in 24 Parts, illustrated with a Portrait of the Author,

SOLUTIONS OF MATHEMATICAL QUERIES.

MR. EDITOR,-Your correspondent P. in col. 734, of your valuable Magazine, proposes three Queries, and requests a Solution to the 4th Mathematical Query in No. 15, col 387, which I have given below. The first of P.'s Queries, if I rightly understand its meaning, is included in that Solution: for it is there proved, that we shall obtain the same value of x in the Biquadratic, whichever root of y we make use of. With respect to the second Query, in the instances alluded to, by your correspondent, Bonnycastle must certainly have intended the cubic to be solved by the method of divisors, or by a table of Co.sines or by the converging series, (though he has neglected to mention this point) in which cases there is no necessity that any of the roots of the Biquad. should be impossible: but, if the cubic is to be solved by Cardan's method, that of Des Cartes will certainly not apply, except two of the roots are impossible or imaginary.

Your correspondent P. says, that "three of the roots, frequently all the four, are rational;" he should recollect, that there cannot be three roots rational, and one irrational, or impossible, because the latter always enter equations by pairs, (when the coefficients thereof are rational.) Upon the whole, Des Cartes' method is exceedingly laborious, and must yield, as well as every other, to that of approximation, both with respect to accuracy and expedition. By inserting this communication, you will very much oblige, your's &c. Lynn Regis, Sept. 11, 1820. PETRARCH.

Solution to the 4th Question of No. 15, col. 387. The equation for determining (a), found by Des Cartes' method, is a + 2 pa' + p2-4r. a2 - q3 o

for a substitute y y3 +2 p y2 + p2

4r. y-q=0

Let m2, n' and 'be the values of y or a; then, since the coefficient of the second term is equal to the sum of all the roots, with their signs changed, &c.

...2p=

m2 — n2 — t2

and q.= (— m2) × (— n2) × (— t2) = mn't whence q = mnt. Now let us use the first value of y (m2) and the value of a, (m) then, if the component quadratics are x2 + ax + b = 0 and x2

ax +co, we have

2

4

which are the four values of x, and may be written in the following form, if m+n+t—s, then x = s or s m, or s -n or s t, which values being involved in the same manner as n and t, we shall plainly arrive at the same values of n by using n or t in the operation, instead of m: and it may easily be shewn, that if we use the negative values of m, n, and t, that the first component quadratic will be the same as the second of the former two, and the second the same as the first of the former; so, that it is immaterial which root of a is used, as each of them will obtain the same value of x.

Portsea, October, 1820.

MR. EDITOR, SIR,-If the following Solution to the Question, appearing in col. 865, of your valuable Miscellany, under the name of Thomas Dixon of Broughton, be consistent, will thank you for its insertion.

Let s

Then will 2 x + 3 x2

I am, Sir, your's obediently,

WM. WHITE.

the sum of n terms,
+ 4 x3 + 5 x1 + &c. (n + 1) x" = s

1

And x+2x2+ 3 x3 + 4 x2 + 5 x5 + &c. (n + 1) x" + 1 = sx + x

By Subtraction x + x2 + x3 + xa + &c. x" = 8
Divided by x gives 1 + x + x2 + x3 +&c. x3-- =

-X8

S

x

Again by Sub. x” — 1 — 2 s

x

Or, sx2 - 2 sxs = (n + 1) x " + 2 −

(n + 2) x* * 1 — x2 + 2 x

Sos=

(n+1)*_(n+2) +} x2+2x
x2 2x+1

sum of n terms.

Again, let s their sum, when continued ad infinitum.
Then 2x+3x2 + 4x3 + 5 x1+ &c. = s

And x+2x2+ 3 x3 + 4 x1 + &c. = sx+ x

By Subtraction x + x2+x3 + xa + &c. = s — sx — x
Divided by a gives 1 + x + x2+ x3+&c. =

SIR, I shall consider myself greatly obliged, by your inserting the following remarks. I am, Sir, your's respectfully,

Liverpool, 8th Nov. 1820.

H. PERKINS.

1st. Lest the question proposed by me in col. 960, of your Magazine, should be found fault with, in consequence of there being given four equations, and but three unknown quantities, I beg leave to observe, that it is proposed in the Key to Vyse's Tutor, page 356, where the answers are given, but no steps of the solution exhibited.

2d. However ingenious the construction of Polygons, as exhibited in col. 864 of your number for October last may appear, it is certainly not a geometrical construction, for I believe, geometry no where furnishes us with the method of trisecting an arc of a circle, which must however be done, before the arc CB can possibly be divided into six parts, otherwise than by trial. 3d. Your correspondent George Dunn, in his question relative to the sailing of a ship, should say whether or not she is supposed to sail on the arc of a great circle. If not, I think the question cannot be answered by pure

Algebra.

4th. In order to throw some additional light on the sum of the nth powers of two quantities expressed in general terms, I would refer your Mathematical readers to Walker's Elements of Algebra, Section 299, commencing at page 167.

COMMERCIAL REPORT, NOVEMBER 21, 1820.

Much languor continues to prevail in business, and the changes which have taken place during the month, have all tended to a decline: a material amelioration can hardly be expected during this dull season of the year, as purchases generally are circumscribed, and dealers unwilling to increase their stocks. Yet as the public mind will now be more tranquillized respecting domestic concerns, we may anticipate a considerable renovation in many branches of commerce. The manufacturers in our neighbourhood are in full employ, and the weekly consumption of Cotton Wool was scarcely ever known to be so great.

Of Cotton Wool the supplies continue abundant; and as the quantities offering for sale, exceed the immediate demand, prices continue to give way. Speculators evince no desire of touching this article, and a reduction of price has been the natural result. The total import of this article from the 1st of January up to this day, is 433,861 bags and bales.

Sugars have receded in price, and the decline since our last may be estimated at 2s. to 3s. per cwt. and sales are heavy. We have not experienced any demand for Foreign or East India Sugars. West India Molasses are selling at 25s. 6d. to 27s. per cwt.

Coffee. Nothing of moment has taken place in this article; indeed our stock will scarcely suffice to supply the demand for internal consumption, unless we receive some fresh imports.

Brazil Cocoa continues in demand; about 600 bags have been taken for shipment last week, at 50s. to 52s. per cwt.

Rums.-250 puncheons of Leewards have been lately sold for export, common quality at 1s. 9d. 21 O. P. at 2s. 6d. to 28 O. P. 2s. 6d. to 2s. 9d. per gallon.

American produce. Tobacco.-good qualities of Virginia Leaf command a ready sale, and are wanted for the Irish market; all other descriptions of Leaf and Stemmed remain dull of sale at low prices.

Pot and Pearl Ashes.—The sales are extremely limited; the market is steady, but without change or briskness.

Rice. The grocers buy but sparingly.

-Tar and Turpentine.—During the last week a quantity of American Tar was disposed of at 18s. per barrel, and nearly the same quantity of Turpentine, of inferior quality, at 9s. 6d. per cut. Buenos Ayres Hides are taken out of the market with avidity when offered.

B. A. Hides at Sid. to 10 d. the latter of which were of prime quality.

The last sales were 4500

Dry Salteries.-Rough Brimstone has been sold at £23. 5s. per ton, but is now held at £24 per ton. Sicily Shumac, scarce and in good demand.--Valonia is dull, prime quality sells at £17 per ton.Campeachy Logwood sells freely at £8 per ton. Other kinds of Dyewoods are neglected.—Quercitron Bark of fine quality is becoming scarce. Philadelphia Bark has been sold at 21s. per cut. New York of low quality has not brought more than 13s. per cwt.

At a public sale of East India Goods on the 17th instant, the following are the rates at which the undermentioned articles sold. Indigo, at 4s. 10d. to 6s. per lb.-Turmeric, 12s. per cwt.-Munjeet, 38s. per cwt.-Shell Lac, 45s. per cwt.-Gum Arabic, 67s. to 69s. per cwt.-Nux Vomica, 10s. per cwt.-Ginger, 18s. per cwt.-Rice, 9s. per cwt. exclusive of the duties of the customs. For 242 bags of Galanga Root, and 100 boxes of Star Aniseeds, there were no buyers.

The market for Tallow has given way in price, and the demand continues limited.

Seed Oils are low, and Fish Oils are drooping in price. The last sales of Whale Oil were at £25 per Pale Seal at £30 per ton. Cod Oil ut £23. 10s. per ton.

tun.

The arrivals of Fruit this year have been unusually early, and our market possesses at present both great abundance and variety. Prices rule as under:-Figs, Turkey, in drums, 80s. to 84s. per cwt.Ditto flat half chests, 75s. per cwt.-Ditto, quarter ditto, 80s. per cwt.-Currants, old 105s. new 110s. = per cwt.-Raisins, Lexia, 50s. per cwt.-Black Smyrna, 56s. per cwt.-Red ditto, 66s. per cwt.Blooms, 95s. per cwt.-Muscatells, 100s. to 120s. per cwt.-Jordan Almonds, £11 to £12 per cwt.French Plums, 78s. to 84s. per cut.-Prunes, 45s. to 47s.

We purpose to present to our readers next month, an epitome of the Imports for the year, with a compendious view of the variations of price which the leading articles of trade have undergone. Corn Market. The arrivals this week from Ireland, have been very extensive, far exceeding the demand. This day's market is as well attended by country dealers, who at the commencement, did not seem anxious to purchase. The importers were obliged to submit to a reduction of 2d. per bushel on Wheat, and 1d. per bushel on Oats; the latter were heavy sale throughout; but towards the clos Wheats were in request, and considerable sales were made. In other articles, there was no alteration

Bourdeaux, 26 0. fort on the Main, 1544. Ex. M. Madrid, 364. effect. Cadiz, 364 effect. Barcelona, 36. Gibraltar, 304. Leghorn, 47. Genoa, 44. Venice, Italian Liv. 27. 60. Malta, 45. Naples, 39. Palermo, 115. Lisbon, 51. Oporto, 51. Rio Janeiro, 54. Dublin, 6. 12: 8. Ex. M. Hamburg, 37 Paris, 3 days' sight, 25: 7.

PRINTED BY H. FISHER, LIVERPOOL, PRINTER IN ORDINARY TO HIS MAJESTY;