Á sin. b. sin. C = a sin c. . sin. B; Å . sin. B = a cosect. c; A. sin. c = a cosect. C; A. cos. C. sin. b B. sin. a; À sin. c. cos B = C. sin. a; Ċ. sin. aa. cot B; B sin. a = a . cot. C; B. cot. B = C. cot. C; B. tan C = Ċ. tan. B.

If B and C be constant, the following equations exhibit the principal relations among the fluxions of the other parts.

a. sin. b = À cosect. C; & sin. C = A cosect. b; a. sin. C. cos. b = c. sin. A; b`sin. A A cot. c; b. cot. b = c. cot. C.

=

Examples of the Application of the fluxional Analogies of Spherical Triangles.

Example 1.-When is that part of the equation of time which depends on the obliquity of the ecliptic a maximum?

=

Here if b denote the sun's longitude, and c his right ascension, we have b:c:: sin. 2 b: sin. 2 c, and when ¿ = c, sin. 2 b = sin. 2 c, hence 2 b and 2 c must be supplements of each other, or in the first quadrant of the ecliptic that part of the equation of time which depends on the obliquity is a maximum, when b+c= = 90°. Example 2.-The error in altitude being given to find the corresponding error in the hour angle.

In the last figure let A be the zenith, B the pole, and C the object observed. Then B. sin. A. sin. c = b, or Bb cosect. A. cosect. C; whence B is a minimum, when cosect. A is a minimum, or when A is 90°, that is, when the object is on the prime vertical.

We shall terminate this article with a few miscellaneous problems to illustrate the method of applying the fluxionary calculus in the different departments of science.

Problem 1.-The force of attraction above the earth being inversely as the square of the distance from the centre, it is proposed to determine the time, velocity, and other circumstances of a heavy body falling from any given height,

the descent in the first second at the earth's surface being 193 inches.

When a is very great with respect to r, vis

nearly = (1-2). √gr, or nearly =

√ 4 g r, and it is accurately equal to this latter quantity when a is infinite. Making r = 3965 miles, and the distance of the sun 12,000 times that quantity, this expression gives v = 6·9405 miles per second, the velocity acquired in falling from the distance of the sun; and that acquired in falling from the distance of the moon would be found to be 6.8927 miles per second.

Problem 2.-To determine the resistance of a

fluid to any body moving in it with a curved end, as a sphere, or a cylinder with a hemispherical end.

Let BEAD be a section through the axis CA, of the solid moving in the direction of its axis.

To any point in the curve draw the tangent EG, meeting the axis produced in G, draw the ordinates EF, e f, indefinitely near to each other, and draw a e parallel to

CG.

D

B

E

a e

Ff

xr becomes

p n v2 p2 8 g

the resistance for the be substituted for r, we

=

r 2

have p n v2 ď2 32g

2

any instant. Then sin. x2: r2 :: 1:ƒ the force at the distance r, that of gravity at the earth's surface being considered as 1; and tv = 2 gr2 i - i, also v i —— 2 gƒ x= and by tak" x2 ing the correct fluent of this last equation, we get v = 4 gr.a-x

is 4gr.a

a

strikes the earth.

; which when x = r,

the velocity when the body

FLY, v. a. & v. n.

FLY BOAT, n. s.
FLYER, OF FLIER, n. s.
FLIGHT', n. s.
FLIGHT'Y, adj.

Pret. flew, or fled; part. fled, or flown, Sax. Fleogan.

V. n.

(To fly is properly to use wings, and gives FLIGHT INESS, n. s. flew and flown. To flee is to escape, or to go away. Sax. plean, and makes fled they are now confounded. To move through the air with wings; to pass through the air; to pass away with the idea of swiftness or escape; to move with rapidity; applied to a violent and sudden separation of adhering parts: to shiver; to burst asunder with a sudden explosion. Sax. rlean; Germ. flichen. To run away; to attempt escape. In this sense the verb is properly to flee, when fled is formed : the verb active is used in the sense of to strain; to decline; to avoid; to refuse association with; to quit by flight; to attack by a bird of prey. It is probable that flew was originally the preterite of fly, when it signified volation, and fled when it signified escape: flown should be confined likewise to volation; but these distinctions are now confounded. We know not any book except the Scriptures in which fly and flee are carefully kept separate. The substantives are more restricted, and somewhat different in their application. Flyer, or flier, is one that flies, or runs away; one that uses wings: it is used in mechanics and in architecture: in the one to the wheel in a machine of a particular use and construction; Dr. Johnson says it is that part of a machine which, by being put into a more rapid motion than the other parts, equalises and regulates the motion of the rest, as in a jack; in the other it is the technical name for a certain kind

of stairs. Stairs made of an oblong square figure, whose fore, and back sides are parallel to each other, and so are their ends: the second of these flyers stand parallel behind the first, the third behind the second, and so are said to fly off from one another.-Moxon's Mech. Exer. Flight is the act of flying, or escaping from danger; the act of using wings; removal from place to place, by means of wings, or impelled by fear: a flock of birds flying together; the birds produced in the same season; a volley; a shower: the space rassed by flying; heat of the imagination; sally of the soul; excursion on the wing; the power of flying; a shower of arrows. Flightiness is applied to wildness and irregularity of mind, or conduct: the adjective signifies fleeting; swift; wild full of imagination. Flyboat is a kind of vessel, nimble and light for sailing.

Fowl that may fly above the earth in the open firGen. i. 20.

mament of heaven.

Abiathar escaped and fled after David.

1 Sam. xxii. Man is born unto trouble as the sparks fly upward. Job v. Ye shall flee, as ye fled from before the earthquake. Zech. xiv. 5.

As birde flyeth up in the aire,
And liveth by birdes that ben meke,
So these ben flowe up in despair
And shenden sely soules eke.

Chaucer. The Plowman's Tale.

Flie, fro the prese and dwell with sothfastnesse,
Suffise unto thy gode though it be small,

For horde hath hate, and climbyng tikilaesse, Prece hath envie, and wele it brent oer all

Chaucer

So that I mighten lyven and nat faile
To morowe for to taken my bataile,
I ne wolde never fro this place flye,
Tyl that ye shulde the very profe ye
For, now if that the soth I shall you say,
I have loved you ful many adaie.

Id. Legend Ariadne. They hit one another with darts, as the others do with their hands, which they never throw counter, but at the back of the flier. Sandy's Journal,

For he so swift and nimble was of flight, That from this lower tract he dared to stie Up to the clowdes, and thence with pineons light To mount aloft unto the crystall skie, To view the workmanship of heaven's height: Whence down descending, he along would flie Upon the streaming rivers, sport to finde; And oft would dare to tempt the troublous winde. Spenser. Muiopotmos. Which when the valient elf perceived, he leapt, As lion fierce, upon the flying prey. Spenter.

These men's hastiness the warier sort of you do not commend: ye wish they had held themselves longer in, and not flown so dangerously abroad before the feathers of the cause had been grown. Hooker.

At the first flight of arrows sent
Full threescore Scots they slew.

Chery Chase.

Denham.

He grieves so many Britons should be lost; Taking more pains, when he beheld them yield, To save the fliers than to win the field. Waller. Sleep flies the wretch; for when with cares opprest, And his tossed limbs are wearied into rest, Then dreams invade. Dryden's Juvenal.

Dedalus, to fly the Cretan shore, His heavy limbs on jointed pinions bore, The first who sailed in air. Id. Æneid. He thinks by flight his mistress must be won, And claims the prize because he best did run.

Dryden.

As striplings whip the top for sport, On the smooth pavement of an empty court, The wooden engine flies and whirls about. Id. Fowls, by Winter forced, forsake the floods, And wing their hasty flight to happier lands. Id. The scouts with flying speed Return, and through the city spread the news.

In half-whipt muslin needles useless lie, And shuttle-cocks across the counter fly.

Id.

Gay.

You now a more delusive art must try, And tempt their hunger with the curious fly. Id. Earth rolls back beneath the flying steed. Pope. I'll fly from shepherds, flocks, and flow'ry plains; From shepherds, flocks, and plains I may remove, Forsake mankind, and all the world but love.

Id.

Strange graces still, and stranger flights she had; Was just not ugly and was just not mad.

Id.

It is not only the utmost pitch of impiety, but the highest flight of folly, to deride these things.

Tillotson.

If there were any certain height where the flights of ambition end, one might imagine that the interest of France were but to conserve its present greatness. Temple.

Above an hundred arrows, discharged on my left hand, pricked me like so many needles; and besides they shot another flight into the air as we do bombs.

meaning. The following phrases are instanced by Dr. Johnson:

To fly at. To spring with violence upon; to fall on suddenly.

A servant that he bred, thrilled with remorse, Opposed against the act, bending his sword To his great master; who, thereat enraged, Flew on him, and amongst them felled him dead. Shakspeare. Though the dogs have never seen the dog-killer, yet they will come forth, and fly at him.

Bacon's Natural History. No honour, no fortune, can keep a man from being miserable, when an enraged conscience shall fly at him, and take him by the throat. South.

This is an age that flies at all learning, and enquires especially into faults.

To fly in the face. To insult.

Id.

This would discourage any man from doing you good, when you will either neglect him, or fly in his face; and he must expect only danger to himself. Swift's Drapier's Letters. To fly in the face. To act in defiance. Fly in nature's face!

-But how, if nature fly in my face first?
-Then nature's the aggressor.

To fly off. To revolt.

Dryden.

Deny to speak to me! They're sick, they're weary, They have travelled all the night! mean fetches; The images of revolt and flying off.

The traytor Syphax

Shakspeare.

Flew off at once with his Numidian horse.
Addison's Cato.

To fly out. To burst into passion.
How easy is a noble spirit discerned,
From harsh and sulphurous matter that flies out
In contumelies, makes a noise, and stinks.

Ben Jonson's Catiline. Passion is apt to ruffle, and pride will fly out into contumely and neglect. Collier of Friendship.

To fly out. To break out into license. You use me like a courser spurred and reined; If I fly out, my fierceness you command. Dryden.

Papists, when unopposed, fly out into all the pageantries of worship; but, when they are hard pressed by arguments, lie close intrenched behind the council of Trent. Id.

To fly out. To start violently from any direction. All bodies, moved circularly, have a perpetual endeavour to recede from the centre, and every moment would fly out in right lines, if they were not restrained. Bentley's Sermons.

To let fly. To discharge.

The noisy culverin, o'ercharged, lets fly, And bursts, unaiming, in the rended sky.

fish, is to angle with a hook baited with a fly, either natural or artificial: a fly-flap is a fly or flapper to keep flies off.

For lo the gentil kinde of the lion;
For whan a flie offendeth him or biteth,
He with his taile awaie the flie ysmiteth
Al easily, for of his genterie

Him deineth nat to wrecke him on a flie,
As doth a curre or els another best.

Chaucer. Leg. of Good Women, Prologue.
Like as the fly that seeth the flame,
And thinks to play her in the fire,
That found her woe and sought her game
Where grief did grow by her desire.

Earl of Surrey.
The fresh young Flie, in whom the kindly fire
Of lustfull youth began to kindle fast,
Did much disdaine to subject his desire
To loathsom sloth, or houses in ease to wast,
But ioyd to range abroad in fresh attire
Through the wide compas of the ayrie coast.

Spenser. Muiopotmos. As flies to wanton boys, are we to the gods; They kill us for their sport.

Shakspeare.

To sacrifice the honour of sacred things to our vain pleasure, being like the ridiculous fondness of that people, which, as Elian reporteth, worshipping a fly,

did offer up an ox thereto.

Barrow.

I am unwilling to believe that he designs to play tricks, and to flyblow my words, to make others distaste Stillingfleet.

them.

There was more need of Brutus in Domitian's days, to mend, than of Horace, to laugh at a flycatcher.

Dryden.

The swallow was a flycatcher as well as the spider. L'Estrange.

My country neighbours begin to think of being in general, before they come to think of the fly in their Locke. sheep, or the tares in their corn. To prevent the fly, some propose to sow ashes with the seed. Mortimer's Husbandry.

Pope.

Swift.

So morning insects, that in muck begun, Shine, buz, and flyblow in the setting sun. Like a fly-blown cake of tallow; Or, on parchment, ink turned yellow. To heedless flies the window proves A constant death. Thomson's Summer. None save the Spanish Fly and Attic Bee As yet are strongly stinging to be free. Byron. FLY, or Musca, in entomology, a large order of insects, the distinguishing characteristic of which is, that their wings are transparent. By this they are distinguished from beetles, butterflies, grasshoppers, &c. See ENTOMOLOGY.

FLY, in mechanics, a cross, with leaden weights at its ends; or rather, a heavy wheel at right angles, to the axis of a windlass, jack, or the like; by means of which the force of the power, whatever it is, is not only preserved, but equally distributed in all parts of the revolution of the machine. See MECHANICS.

FLY, ELECTRIC. See ELECTRICITY. FLY ISLAND, an island in the South Pacific Ocean, discovered by Le Maire and Schouten, in the year 1616, and so named from the number of flies seen there. It is covered with trees, and a lagoon seems to be formed in the interior by the flowing of the tide. The navigators observed a few naked inhabitants. Long. 150° 20′ W., lat. 15° S.

FLY, VEGETABLE, a very curious natural production, chiefly found in the West Indies. It

resembles the drone both in size and color, excepting that it has no wings, more than any other British insect. In the month of May it buries itself in the earth, and begins to vegetate. By the end of July the tree has arrived at its full growth, and resembles a coral branch: it is about three inches in height, and bears several little pods, which dropping off, become worms, and then flies, like the British caterpillar. Such was the account originally given of this extraordinary production. But several boxes of these flies having been sent to Dr. Hill, for his examination, his report was as follows: "There is in Martinique a fungus of the clavaria kind, different in species from those hitherto known. It produces soboles from its sides; I call it, therefore, clavaria sobolifera. It grows on putrid animal bodies, as our fungus ex pede equino, from the dead horse's hoof. The cicada is common in Martinique, and in its nymph state, in which the old authors call it tettigometra, it buries itself under the dead leaves to await its change; and, when the season is unfaVorable, many perish. The seeds of the clavaria find a proper bed in this dead insect, and grow. This is the solution of the mystery; though the untaught inhabitants suppose a fly to vegetate, and though there is a Spanish drawing of the plants growing into a trifoliate tree, and it has been figured with the creature flying with this tree upon its back. Mr. Edwards treats of this extraordinary production in his Gleanings of Natural History.

FLY, HONEYSUCKLE. See LONICERA.

FLY, HONEYSUCKLE, AFRICAN. See HALLERIA. FLYING, the progressive motion of a bird, or other winged animal in the air. The parts of birds chiefly concerned in flying are the wings and the tail; by the former, the bird sustains and wafts himself along; and, by the latter, he is assisted in ascending and descending, to keep his body poised and upright, and to obviate the vacillations thereof. It is by the largeness and strength of the pectoral muscles, that birds are so well disposed for quick, strong, and continued flying. These, muscles, which, in men, are scarcely a seventieth part of the muscles of the body, in birds exceed and outweigh all the other muscles taken together. The tail, Messrs. Willoughby, Ray, and many others, imagined to be principally employed in steering and turning the body, as a rudder; but Borelli has shown that this is the least use of it. Its chief use is to assist the bird in its ascent and descent in the air, and to obviate the vacillations of the body and wings; for, as to turning the body to this or to that side, it is performed by the wings and inclination of the body, and but very little by the help of the tail. The flying of a bird, in fact, is a very different thing from the rowing of a vessel. Birds do not vibrate their wings towards the tail, as oars are struck towards the stern, but waft them downwards; nor does the tail of the bird cut the air at right angles, as the rudder does the water; but is disposed borizontally, and preserves the same situation what way soever the bird turns. In effect, as a vessel is turned about on its centre of gravity to the right, by a brisk application of the oars to the