merit the greatest encouragement from a nation that owes to it both its riches and security. And not only does the common art of navigation depend on mathematics, but whatever improvements are to be made in naval architecture, whether the vessels are designed for merchantmen or ships of war; whether for swift running, bearing a great sail, or lying near the wind.

Since, then, it has been shown how much mathematics improve the mind, how subservient they are to other arts, and how immediately useful to the common-wealth, there needs no other argument to recommend them to the study of mankind, or any motives to a government to encourage them. PHILO.

Sheffield, April 17th, 1818.

SOLUTIONS TO THE QUERIES.

QUESTION 1. By Lysis.

A person by his natural strength is just able to raise up a weight of 300lbs. what weight would he be able to sustain on an inclined plane, the elevation of which is 60°. by means of a rope going round the weight, (which is circular,) one end whereof is attached to the top of the plane, the other held in his hand?

Solution, by Messrs. Aaron Arch and J. C. Wallace, Sheffield. The power requisite to sustain any weight on an inclined plane the weight sine of the plane's elevation radius, (EM. Mech. Prop. 14, cor. 1), whence 86603: 1:300: 364.4lbs. the weight which he could sustain, provided the rope was attached to the weight instead of the plane; but under the given conditions the weight will be doubled, or = 692·8lbs. the answer required,

By Mr. T. S. Davies, Sheffield.

As by the conditions of the question the given power is doubled, put al= 600, and x the weight required, and (per GREGORY's Mechanics, art. sin 60 154,) a = x whence 692.82lbs. cos 0

It was also answered by Mr. Moses Avadith and the proposer.

QUESTION 2. By the same.

If a square and an equilateral triangle be inscribed in a circle, a circle inscribed in both the square and triangle, and again a square and an equilateral triangle be inscribed in these circles, and so on ad infinitum: What will be the diameter of the circle, when the sum of the areas of the triangles minus the rectangle of the sum of the areas of the squares and triangles is a maximum?

x2

Putting

x2 x2

Solution, by Mr. T. S. Davies, Sheffield.

diameter of the circle, the series of squares is as follows:

++ &c. ad inf. = x2 then let the side of the equilateral tri

4 8

and consequently 2-4 is a maximum; whence a, diameter required.

The same results were found by Messrs. Aaron Arch, Moses Avadith, J. C. Wallace, and the proposer.

QUESTION 3. By Philo.

In what order must I plant 14 trees to make 21 rows, each row to contain 3 trees?

Find an arc, such that the nth power of its sine multiplied into its cosine, shall be the greatest possible.

QUESTION 8. By Mr. J. C. Wallace, Sheffield.

A pole, six feet long, casts a shadow from a lamp on an horizontal plane : the height of the lamp is twelve feet, and the distance of its base from that of the pole is six feet; what angle does the pole make with the plane, its shadow being a maximum?

QUESTION 9. By the same.

In a plane inclined to the horizon at a given angle, to determine geome. trically, a point such that two straight lines being drawn from this point to two other points situated above the plane, and given in position, the times of descent down these lines may be equal.

Original Poetry.

MARY, THE MANIAC.

Scene, a Village near the Banks of the Clyde.

TREAD softly, Sir, nor with too rude a step
Approach this cot-it is Misfortune's dwelling.
The man bowed down with age and gray with years,
Whose all of hope is now beyond the grave,

If chance he pass this melancholy home,

Here stays his step, recals his thoughts from heav'n,
Leans on his staff, and gives a tear to Mary:
The village-boys, freed and let loose from school,
Noisy and boisterous and rude with play,
Suspend their gambols and forego their mirth
As they approach the spot where Mary dwells.

Behold that beauteous maiden! how unlike
That lovely form to any thing that's earthly!
Her eye so wild is fixed on vacancy;
And she appears so void of consciousness,
So strangely lost, and dead to all around,
That thought and life seem to have passed away,

And left the lovely temple they had hallowed

Lorn and desolate. Poor, hopeless sufferer!

How changed from what she was! Some few years past
She was the pride and idol of the village;
Blithsome and joyous as the voice of Spring,
Gay as the mountain-kid, as guileless too;
And in that cheek, now cold and colourless,
Love's roses blossomed. Nay, the finest form
By happiest pencil traced, or waked to life,
And to perfection wrought, by that diviner art
That breathes a soul through marble, never yet
Imaged a being half so fair and lovely.

Then every day of life, each hour that passed,

But added to the happiness of Mary;

And duly as the morn and evening came

She knelt before her God, and thanked him for the blessing.

Her hand was plighted to a neighbouring youth,

Aud Connell prized the treasure he had won

A virtuous woman's love-dearer than life.
One fair antumnal morn, when not a cloud
Hung on the face of heaven; when the air,
Balmy and fresh, o'er the charmed senses came,
Breathing tranquillity;

He crossed the still, clear waters of the Clyde,
To ask a parent's blessing on his love.

At eve returning, what a change was there!
Loud blew the angry winds, the rain beat fast;
And it is said, on that disastrous night,
Voices, not human, mingled with the blast,
Foreboding misery;-and shadowy forms,
Unseen by all but visionary seers,
Rode on the storm, directing all its fury.

O, 'twas a night of horror! Darkness all,
Save where the livid lightning's frequent flash
Gleamed wildly o'er the troubled stream, unfolding
All its terrors. Awhile poor Connell's bark,
Borne down the raging river's rapid course,

Strove with the storm; then sunk— to rise no more!

The next day came: the storm, which all night long Beat at the heart of Mary, was at rest :

Anxious she sought the margin of the Clyde
To hail her love's return. O'er the wide stream
She cast a fearful look-O, God of Heaven!
Borne on the wave that still rolled rapidly,
The lifeless Connell floated at her feet!-

I would not hear again the shriek she utter'd,
To add another year to my existence-
Sleeping or waking it is ever with me.

I marked the agony that shook her frame,
The deep convulsive throb of hopeless anguish ;
And when her eye unclosed to misery,
She clasp'd her breathless lover in her arms,
And press'd him closely to her beating heart,

As she would warm him into life again;

And then looked wildly round, and loudly laughed;

But so unlike the joyous laugh of mirth-
'Twas full of suffering, and betokened mad ness.

E'er since that fatal morn has Mary been
The piteous thing you see her. There she sits,
Spending the day in tedious wretchedness.
And oft with her dark hair her fingers play,
Which thus she plaits, as if with studious care,
And then disparts again.

Her wayward faney

Th' imperfect image sometimes dimly sees
Of him she loved; and as his parting words
Fall heavy on her heart, sudden she starts-
With hasty steps, and strange disordered mien

She seeks the church-yard where her Connell lies;
And as she sits unconscious on his grave,

With stedfast gaze she eyes the village-clock,
And marks the lingering hands: as they approach
The promised moment of her love's return,
Then the fixed wildness of her eye subsides,
And a faint roseate hue warms her pale cheek,
Something like hope, yet more allied to fear.
Illusion all! The bell that tolls the hour
Dissolves the vision of her wildered brain,
And gives her back to self-consuming sorrow.
Again she shrieks-as she beheld once more
The breathless Connell floating at her feet;
And folding close her arms across her bosom,
She hurries to her dwelling.-Poor Mary!
The grave alone can cure thee.

Sheffield, March 18, 1818.

R.

SONNET TO THE NORTHERN STAR

HAIL to thee, Northern Star, for thou dost beam
From that fair land where first my infant eye
Saw life, and felt it as a lovely dream,

A floating vision betwixt earth and sky,
Seen on the bosom of the deep blue stream

When the sun travels in his glory high!
Far other sense of life, since then, hath press'd
Upon my heart; yet have I ever found

The land that gave me birth was hallow'd ground,
Enshrined by fancy, by affection bless'd;

And when the varying landscape round me spread, In all the pomp of glowing beauty rife,

Still would my mind those dearer scenes retread Known in the sunny morn-the morn of early life!

CHARADE.

My first, in animated nature's round,

B. H.

Alive or dead, of use to man is found.

Should the wide stream the traveller's course impede,
My next is hailed with joy, a friend in need.
My whole, a twin, to sacred truth allied,
Gives birth to science, and is Britain's pride.

JUVENIS.