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THE PRINCESSES.

manner

of surrendering my money, as if a pis- | and she continued to live under her tol was held at my breast. Every idea Majesty's paternal care in Windsor of charity is equally out of the question Castle, until her royal mother's death. in both cases ; though in the one the In this castle she had private apartmoney is demanded for God's sake, ments fitted up in a style of simplicity and in the other for my own sake.” and elegance, suited to her refined

I remember,” rejoined the second, taste, with an extensive collection of being once singled out by a terrible books and paintings. Her visits to woman with an enormous belly, who the surrounding nobility, accompanied stuck to me brawling for charity the by her sisters, were frequent; and whole length of Piccadilly. Being whenever the weather was fair, she determined not to give her a farthing, rarely omitted either to walk or ride I was quickening my pace to shake daily through the beautifully romantic ber off, when unluckily I met a lady of scenery of the parks. She had a horse my acquaintance, and stopped to speak that was a particular favourite, which, to her. The hideous wretch with the always on her return from riding, parbelly, taking advantage of the inci- took from her Royal Highness' own dent, renewed her clamour so loud hand, a dainty repast which was ready and so woefully, that I was afraid she at the palace door. would have been delivered in the open She usually made presents of select street; and so I was obliged to throw publications on particular occasions, her a shilling, which put an end to her and generally on New Year's Day, to labour, and my pangs, at once.” most of her domestic servants. Her

purchases were frequently made by herself, visiting the shops personally; on which occasions her conduct and

were so humble and unas(With Portraits.)

suming, as to endear her to all to whom

she was known. AMONG the illustrious females who The charitable institutions in Windhave adorned the Royal Families of sor are very extensive, all of which this country, few have been more be- received an annual bounty from her loved than the present, and few we royal hand; and every object of dispresume have ever rendered them-tress coming to her knowledge, reselves more deserving of national af-ceived immediate relief. fection.

Her Royal Highness, with her sisCharlotte Augusta Matilda, Princess ters, added much to the beautiful Royal of England, was born on the walks round the palace. On the north 29th of September, 1766. Her amiable side of the castle formerly was a piece character endeared her to the nation of land forming a plantation, called during the earlier part of her life, and the Slopes, running in extent from her illustrious virtues, rather more than west to east, about three-quarters of her exalted rank, rendered her an ob- a mile, which had continued a rough ject of affection to a foreign prince, to uncultivated spot for many years. It whom she was afterwards united. She is now tastefully laid out, affording was married to Frederick Charles Wil- one of the most interesting walks liam, Elector of Wirtemberg, May 18, about the Royal residence, and has a 1797, who was created king in 1805. private communication with Frogmore This prince died in 1816, leaving her Lodge. Both inhabitants and stranin a state of disconsolate widowhood, gers have free access to this delightto mourn over those common cala- ful spot, where every accommodation mities from which no station can is offered to the weary, by several exempt the human race. Springing alcoves, or thatched cottages, with from a monarch remarkable for his seats. benevolence, we have no doubt, that Her Royal Highness is displaying she still continues to display thosc great taste in the present alterations charitable virtues in a foreign land, now making at Frogmore. The whole which, in her native country, from her mansion has undergone considerable earliest infancy she was taught to alterations; and the improvements venerate and cherish.

made through the gardens, are well Her Hoyal Highness the Princess worthy the admiration of every stranAugusta, was born November 8, 1768, ger; to whom Her Royal Highness

scene.

frequently grants permission to take a Many of the rural spots in Frogmore a few hours' walk.

gardens, and other parts, were laid We would recommend none to take out after her own taste. An exquithis rural rambie without joining the sitely beautiful grotto, forming an inretired hermit, who will always be teresting feature in the Frogmore garfound with his cross and bible in de- dens, is said to have been completed vout meditation. The following lines, by her. said to be the composition of Her The Princess Mary was born April Royal Highness Princess Elizabeth, 25, 1776. She was married to William are descriptive of this interesting Frederick, Duke of Gloucester. This

princess has always supported an Ye whom variety delights,

amiable character, following the steps, Descend awhile from Windsor's heights ; and copying the example, of her elder And in this hovel deign to tread,

sisters. She was the constant attendant Quitting the castle for the shed.

upon the late Princess Amelia, during Such were the Muse's favourite haunts, From care secluded, and from wants.

the long period she was confined to What nature needs, this hat can give,

her bed, watching her with unceasing Could we as nature dictates live;

affection till she expired. To assist For see, on this plain board at noon, in works of charity she was ever ready; Are placed a platter and a spoon,

and many benevolent institutions enWhích, tho' they mark no gorgeous treat, Suggest 'tis reasonable to eat.

joyed the benefits of her influence and What though the sun's meridian light

liberality. Beams not thro' our hovel bright;

The Princess Sophia, was born Tho others need, we need him not,

Nov. 3, 1777. The same amiable spirit Coolness and gloom befit a cot;

by which her sisters have been disOur bours we count without the sun, These sands proclaim him as they run.

tinguished, breathes in her conduct, Sands within a glass confin'd,

and dignifies her actions. Charmed Glass which ribs of iron bind :

with domestic life, her retiring virtues For Time, still partial to this glass,

shun the eye of popular observation, Made it dnrable as brass,

on which account her real worth is less That, placed secure upon a shelf, None might crush it but himself.

generally known ; but it is rendered Let us hence the day prolong,

the more valuable, because it is not With loyal and with nuptial song.

obtrusive. Affable, charitable, and Such as with duteous strains address’d, sympathetic, she has an ear that atMay gratify each royal guest;

tentively listens to tales of woe, and Thrice happy should our rural toils

a hand that is always ready, by Be requited by their smiles.

administering to the necessities of the Her Royal Highness the Princess indigent, to soothe them in afilietion, Elizabeth, was born May 22, 1770. and alleviate their distress. Literature,-the Arts,–Charitable In

In 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 på. laudable undertaking: Her bountiful tronage ; and among the poor who hand, which on all occasions was bave 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.

their fathers. She wrote many pieces of poetry, Destitute of that haughtiness which and prose; acquired a considerable too frequently associates with royalty

, knowledge of drawing; and is said to they have rendered themselves beloved have engraved or etched a series of in their respective neighbourhoods, by plates. Most of her own productions their dignified condescenson, without she had printed at Frogmore, at the compromising their illustrious descent, press of Her Majesty, who had fitted or degrading themselves in the up a small private printing-office. She the most exalted, who cannot but adestablished a school of girls, at Old mire in their conduct, a lovely feature Windsor Cottage, the country resi- which pride will not permit them to dence of Her Royal Highness, which imitate. They have taught ambition a was supported entirely by herself. lesson by their own humility; and in

of

eyes

structed avarice by their genuine li- | stances, assumed the garb of disinberality. The nation has long ex- terested charity. perienced the effects of their benevolence; and the consequences of their influcnce among those who have been

Literary Notices. honoured with titles, wealth, and station, will in all probability extend

A new Edition of Walton and Cotton's Complete Angler,

is preparing for the Press, by Mr. Bagster. It will be print beyond the present generation of ed in a size for the pocket; with entirely new Embellishmankind. Their example has ope- In the course of November will be published, in One rated among many, who have been

Vol. 8vo.“ Augustus; or, The Ambitious Student."

On the 1st November was published, the whole of the strangers to their motives; so that Rev. G. C. Smith's Works, (Author of the Boatswain's

Mate,) in 24 Parts, illustrated with a Portrait of the Author, even ostentation has, in several in- half-bound, 5s.

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.

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Solution to the 4th Question of No. 15, col. 387. The equation for determining (a), found by Des Cartes' method, is a + 2 pa +p?m4r. a* - q* = 0

for a substitute y .'. + 2 p y + p - 4r. y - = Let m', n’ and t 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.

.', 2 p =- m2 n - t? and - 0.=(-mo) x (1) x ( - )=- mo na t whence q = mnt. Now let us use the first value of y (mo) and the value of a, (m) then, if the component quadratics are a2 + ax + b = 0 and 2 ax + c = 0, we have

ma + n + ť A 응

m - (n + the a = m, b = 2

2

+ ma

nt

p+ a

2

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2

2

4

p + a' tu

m - (+) and c = 2

mm - (n + the ... the component quadratics become 20 + mou +

4

(m + 1)
and
- mx +

zacz + mx +

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and x

2

m

n

t m
or

nt

or

2
n+t

n+t
+
or

+
2
2

2
2
2
2
2

2 which are the four values of x, and may be written in the following form, if m +n +t

=s, then x = -sors - m, or s — n or s-t, which values 2 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.

I am, Sir, your's obediently,

WM. White. Let s = the sum of n terms, Then will 2 x + 3 x” + 4x + 5 x* + &c. (n + 1) X" =s And x + 2 x2 + 3.2.2 + 4x* + 5x + &c. (n + 1)2 + 1 = 8x + x By Subtraction x + x2 + x3 + x + &c. x" —s— sx — x + (n + 1) x** ! Divided by x gives 1 + x + x2 + x* + &c. --5-1+(n+1) x" Again by Sub. x"

x + 1 + (n + 1) 2*+1-(n+1)x"

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S

1=2s

-SX

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Or, sxo — 2 sx + s = (n + 1) -(n + 2)x" +? cca + 2x
Sos =
(n + 1) *** 2 -(n + 2)x" + 1. x? + 2 x

sum of n terms.
x2 2 x + 1
Again, let s = their sum, when continued ad infinitum.
Then 2 x + 3x + 4x + 5 x* + &c. =s

And x + 2 x? + 3 x + 4x* + &c. = sx + x
By Subtraction x + x + x + x* + &c. = 8 — sx —
Divided by x gives 1 + x + 2m + x + &c. =

-1

S

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2 x

ra So s =

=sum, when continued ad infinitum. x-2x+1

MR. EDITOR, 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.

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