by cross pieces and stanchions. On each side of the —v—"■ •>
ship are six or seven ports, H, about 18 inches broad
and 15 inches high j and having their lids to open
downward, contrary to the usual method.
Against every port is placed an iron chamber (a),
which, at the time of firing the ship, blows out the
port-lid, and opens a passage for the flame. Immedi¬
ately under the main and fore-shrouds is fixed a wood¬
en funnel M; whose lower end communicates with a
fire-barrel (b), by which the flame passing through
the funnel is conducted to the shrouds. Between the
funnels, which are likewise called jire-trunks, are twoFr^/rnwr’s
scuttles, or small holes, in the upper deck, serving also ^J/ur’pt‘
to let out the flames. Both funnels must be stopped
with plugs, and have sailcloth or canvas nailed close
over them, to prevent arty accident happening from
above to the combustibles laid below.
The ports, funnels, and scuttles, not only commu¬
nicate the flames to the outside and upper works of
the ship and her rigging ■, but likewise open a passage
for the inward air, confined in the fire-room, which is
thereby expanded so as to force impetuously through
these outlets, and prevent the blowing up of the decks,
which must of necessity happen from such a sudden
and violent rarefaction of the air as will then be produ¬
ced.
On each side of the bulk-head behind is cut a hole,
L, of sufficient size to admit a trough of the same di¬
mensions as the others. A leading trough, LI, whose
foremost end communicates with another trough with¬
in the fire-room, is laid close to this opening, from
whence it extends obliquely to a sally-port, I, cut
through the ship’s side. The decks and troughs are
well covered with melted rosin. At the time of the
firing
(a) 1 he iron chambers are 10 inches long and 3.3 in diameter. They are breeched against a piece of
wood fixed across the ports, and let into another a little higher. When loaded, they are almost filled with
Corn-powder, and have a wooden tompion well driven into their muzzles. They are primed with a small piece
of quick-match thrust through their vents into the powder, with a part of it hanging out. When the ports
are blown open by means ol the iron chambers, the port-lids either fall downwards or are carried away by the
explosion.
(b) The fire-barrels ought to be of a cylindrical form, as most suitable to contain the reeds with which they are
filled, and more convenient for stowing them between the troughs in the fire-room. Their inside chambers should
not be less than 21 inches, and 30 inches is sufficient for their length. The bottom parts are first well stored
with short double-dipped reeds placed upright; and the remaining vacancy is filled with fire-barrel composition
well mixed and melted, and then poured over them. The composition used for this purpose is a mass of sulphur,
pitch, tar, and tallow.
There are five holes, of three-fourths of an inch in diameter and three inches deep, formed in the top of the
composition while it is yet warm ; one being in the centre, and the other four at equal distances round the sides
of the barrel. When the composition is cold and hard, the barrel is primed by filling these holes with fuse
composition, which is firmly driven into them, so as to leave a little vacancy at the top to admit a strand of quick-
match twice doubled. The centre hole contains two strands at their whole length, and every strand must be
driven home with mealed powder. The loose ends of the quick-match being then laid within the barrel, the
whole is covered with a dipped curtain, fastened on with a hoop that slips over the head of the barrel, to which
it is nailed.
The barrels should be made very strong, not only to support the weight of the composition before firing, when
they are moved or carried from place to place, but to keep them together whilst burning : for if the staves are
too light and thin, so as to burn very soon, the remaining composition will tumble out and be dissipated, and the
intention of the barrels, to carry the flame aloft, will accordingly be frustrated.
The curtain is a piece of coarse canvas, nearly a yard in breadth and length, thickened with melted composi¬
tion, and covered with saw-dust on both sides.
4 M 2
FIR [ 644 ] FIR
firing either of the leading troughs, the flame is imme¬
diately conveyed to the opposite side of the ship, wheie-
by both sides burn together.
’ The spaces N, 0, behind the fire-room, represent the
cabins of the lieutenant and master, one of which is on
tiie starboard, and the other on the larboard side. 1 he
captain’s cabin, which is separated from these by a bull;-
head, is exhibited also by P.
Four of the eight fire-barrels are placed under the
four fire-trunks ; and the other four between them,
two on each side the fire-scuttles, where they are se¬
curely cleated to the deck. The longest reeds (c) are
put into the fore and aft trough, and tied down : the
shortest reeds are laid in the troughs athwart, and tied
down also. The bavins (d), dipped at one end, are
tied fast to the troughs over the reeds, and the curtains
are nailed up to the beams, in equal quantities, on each
side of the fire-room.
The remainder of the reeds are placed in a position
nearly upright, at all the angles of every square in the
fire-room, and there tied down. If any reeds are left,
they are to be put round the fire-barrels, and other
vacant places, and there tied fast.
Instructions to Prime.
Take up all your reeds, one after another, and
strewr a little composition at the bottom of all the
troughs under the reeds, and then tie them gently dovTn
again : next strew composition upon the upper part of
the reeds throughout the fire-room j and upon the said
composition lay double quick-match upon all the
reeds, in all the troughs : the remainder of the com¬
position strew over all the fire-room, and then lay your
bavins loose.
Cast oil’ all the covers of the fire-barrels, and hang
the quick-match loose over their sides, and place lead¬
ers of quick-match from the reeds into the barrels, and
from thence into the vent of the chambers, in such a
manner as to be certain of their blowing open the
pots, and setting fire to the barrels. Two troughs of
communication from each door of the fire-room to the
sally ports, must be laid with a strong leader of quick-
match, four or five times double : also a cross-piece to
go from the sally-port, when the ship is fired, to the
communication trough, laid with leaders of quick-
match, that the fire may be communicated in both sides
at once.
What quick-match is left place so that the fire may
be communicated to all parts of the room at once,
especially about the ports and fire-barrels, and see that
the chambers are well and fresh primed. [N. B. The
port-fire used for firing the ship, burns about 12 minutes. Fiie.
Great care must be taken to have no powder on board —v—
when the ship is fired.]
The sheer hooks (represented by A) are fitted so
as to fasten on the yard-arms of the fire-ship, where
they hook the enemy’s rigging. The fire-grapplings
(I?) are either fixed on the yard-arms, or thrown by
hand, having a chain to confine the ships together, or
fasten those instruments wherever necessary.
When the commanding officer of a fleet displays the
signal to prepare for action, the fire-ships fix their sheer
hooks, and dispose their grapplings in readiness. The
battle being begun, they proceed immediately to prime,
and prepare their fire-works. When they are ready
for grappling, they inform the admiral thereof by a
particular signal.
To avoid being disabled by the enemy’s cannon dur¬
ing a general engagement, the hre-ships continue suf¬
ficiently distant from their line of battle, either to wind¬
ward or to leeward.
They cautiously shun the openings or intervals of
the line, where they would be directly exposed to the
enemy’s fire, from which they are covered by lying on
the opposite side of their own ships. They are atten¬
tively to observe the signals of the admiral or his se¬
conds, in order to put their designs immediately in
execution.
Although no ship of the line should be previously
appointed to protect any fire-ship, except a few of the
smallest particularly destined to this service, yet the ship
before whom she passes in order to approach the ene¬
my, should escort her thither, and assist her with an
armed boat, or whatever succour may be necessary in
her situation.
The captain of the fire-ship should himself be parti¬
cularly attentive that the above instructions are punc¬
tually executed, and that the yards may be so braced
when he falls alongside of the ship intended to be de¬
stroyed, that the sheer-hooks and grapplings fastened
to the yard-arms, &c. may eflectually hook the enemy.
He is expected to be the last person who quits the ves¬
sel j and being furnished with every necessary assist¬
ance and support, his reputation will greatly depend on
the success of his enterprise.
Lambent Fires, as the shining of meat at certain
seasons, the luminousness of the sea, of insects, vapours,
&c. See Light, Chemistry Index; Fire-FUcs, En¬
tomology ImAv; GLOW-Worm, &c.
Port-FiRE. See PoRT-Firc.
Spur-Fire. See Spur-Fire.
FiRE-TForks, are preparations made of gunpowder,
sulphur,
(c) The reeds are made up in small bundles of about a loot in circumference, cut even at both ends, and
tied together in two places. They are distinguished into two kinds, viz. the long and short j the iormer 01
which are four feel, and the latter two feet five inches in length. One part of them are singly dipped, 1. e. at
one end : the rest are dipped at both ends in a kettle of melted composition. After being immersed about
seven or eight inches in this preparation, and then drained, they are sprinkled over with pulverized sulphur upon
(d] The bavins are made of bii’ch, heath, or other brush-wood, which is tough and readily kindled.. ^)€y
are usually two or three feet in length, and have all their brush-ends lying one way, the other ends being tied
too-ether with small cords. They are dipped in composition at the bush-ends, whose branches are afterwards
confined by the hand, to prevent them from breaking off by moving about: and. also to make them burn more
fiercely. After being dipped in the same manner as the reeds, they also are sprinkled with sulphur.
F I 'R [ 645 ] FIR
Fire,
sulphur, ami other Inflammable ami combustible ingre¬
dients, used on occasion of public rejoicings and other
solemnities.
The invention of fire-works is by M. Mahudel at¬
tributed to the Florentines and people of Sienna *, who
found out likewise the method of adding decorations
to them of statues, with fire issuing from their eyes and
mouths.
The art of preparing and managing these is called
pyrotechtu/. See Pyrotechny.
FIRING, in the military art, denotes the discharge
of the fire-arms \ and its object is to do the utmost ex¬
ecution to the enemy.
The method of firing by platoons is said to have
been invented by Gustavos Adolphus, and first used
about the year 1618 : the reason commonly given for
this method is, that a constant fire may be always
kept up. There are three difl’erent ways of platoon
firing •, viz. standing, advancing, and retreating. Rut
previous to every kind of firing, each regiment or bat¬
talion must be told off in grand divisions, subdivisions,
and platoons, exclusively of the grenadiers, which form
two subdivisions or four platoons of themselves. In
firing standing, either by divisions or platoons, the first
fire is from the division or platoon on the right j the
second fire from the left', the third from the right
again $ and so on alternately, till the firing comes to
the centre platoon, which is generally called the colour
platoon, and does not fire, remaining as a reserve for
the colours. Firing advancing is performed in the same
manner, with this addition, that before either division
or platoon fires, it advances three paces forward. Fir¬
ing retreating varies from either of the former me¬
thods ; for before either division or platoon fires, if
they are marching from the enemy, it must go to
the right about, and after firing, to the left about
again, and continue the retreat as slow and orderly as
possible.
In hedge firing the men are drawn up two deep, and
in that order both ranks are to fire standing. Oblique
firing is either to the right and left, or from the right
and left to the centre, according to the situation of
the object. The Prussians have a particular contri¬
vance for this purpose ; if they are to level to the
right, the rear ranks of every platoon make two quick>
hut small paces to the left, and the body of each sol¬
dier turns one-eighth of a circle, and vice versa. Pa¬
rapet firing depends on the nature of the parapet over
which the men are to fire, and also upon that of the
attack made to possess it. This method of firing is
sometimes performed by single ranks stepping on the
banquette and firing ; each man instantly handing his '
arms to the centre rank of the same file, and taking
his back in the room of it *, and the centre rank giv¬
ing it to the rear to load, and forwarding the arms
of the rear to the front rank ; by which means the
front rank men can fire six or seven rounds in a minute-
with exactness. Parapet firing may also be executed
two deep, when the banquette is three feet broad, or in
field works where no banquettes are made. Square
firing is performed by a regiment or body of men
drawn up in a hollow square, in which case each front,
is generally divided into four divisions or firings, and
the flanks of the square, being the weakest part, are
covered by four platoons of grenadiers. The first fire
is from the right division of each fitce; the second nrin <
from the left division of each face, &c, and the grena- (j f’
diers make the last fire. Street firing is practised in FirnuVus.
two ways ; either by making the division or platoon v~ '
that has fired to wheel by half-rank to the right and
lelt outwards from the centre, and to march in that
order by half divisions down the flanks on each side of
the column, and to draw up in the rear, and go on
with their priming and loading j or, to make the divi¬
sion or platoon, after firing, to face to the right and
left outwards from the centre, and one half rank to
fellow the other $ and in that order to march in one
centre file down on each side of the column into the
rear, and there draw up as before.
Fiking Iron, in Farriery, an instrument not unlike
the blade of a knife j which being made red hot is ap¬
plied to a horse’s hams, or other places standing in
need of it, as in preternatural swellings, farcy, knots,
&c. in order to discuss them.
FIRKIN, an English measure of capacity for things
liquid, being the fourth part of the barrel: it contains
eight gallons of ale, soap, or herrings 5 and nine gal-'
Ions of’beer.
FIRLOT, a dry measure used in Scotland. The -
oat firlot contains 2l^th pints of that country ; the
wheat firlot ccntains about 2211 cubical Indies ; and
the barley firlot, 31 standard pints. Hence it appears
that the Scotch wheat firlot exceeds the English bushel
by 33 cubical inches.
FIRMAMENT, in the ancient astronomy, the
eighth heaven or sphere ; being that wherein the fixed
stars were supposed to be placed. It is called the-
eighth, with respect to the seven heavens or spheres of
the planets which it surrounds.
It is supposed to have two motions j a diurnal mo¬
tion, given it by the primum mobile, from east to west,
about the poles of the ecliptic ; and another opposite
motion from west to east ; which last it finishes, accord¬
ing to Tycho, in 25,412 years j according to Ptolemy,
in 36,000; and according to Copernicus, in 258,000;
in which time the fixed stars return to the same precise
points wherein they were at the beginning. This
period is commonly called Plato’s year, or the great
year.
In various places of Scripture the word firmament is
used for the middle region of the air. Many of the
ancients allowed, with the moderns, that the firma¬
ment is a fluid matter ; though they, who give it the
denomination of firmament, must have taken it for a
solid one.
FIRMAN, is a passport or permit granted by the
Great Mogul to foreign vessels, to trade within the
territories of his jurisdiction.
FIRMICUS Maternus, Julius, an ecclesiastical
writer, who lived about the middle of the fourth centu¬
ry. Nothing is known with certainty respecting his
country, profession, or character, as we find no mention
made of him in the writings of ancient authors. Some
say that he was by birth a Sicilian, and practised in the
forum as a barrister for some time, becoming a convert
to Christianity when far advanced in years ; which ap¬
pears to derive considerable support from different pas¬
sages in his writings. He was author of a treatise F)e
errore profianarum religionum, which was dedicated to
the emperors Constantius and Constans. This work
must i
F I R
t 646 ]
F I S
Firmiout must have been written between 340 anti 350, in which
!! Constans was slain by Magnentius. It is allowed to be
First Fruits. a )earned, able, and well written performance, in which
' v the reasonableness of the Christian religion is strongly
contrasted with the absurdity and immorality of the gen¬
tile creed. It must not be dissembled, however, that
he sometimes betrays such a spirit ot intolerance .as is
wholly incompatible with the genius ol the Christian
religion, which breathes nothing but benevolence to¬
wards the whole human race. The arguments em¬
ployed by him in its defence are disgraced by an ex¬
hortation to the civil power to propagate it by force of
arms, and to crush the advocates of error by severe
edicts. This work was first published at Strasburg in
1362, at Heidelberg in I559> an<^ 16io.
The greater part of critics ascribe to him a work en¬
titled Astronomicorum, scu de Mnt/iesi, lib. viii. In it
he treats of the power and influence of the stars, agree¬
ably to the doctrine of the Egyptians and Babylonians,
blending a considerable degree ol mathematical know¬
ledge with the unmeaning jargon of judicial astrology-
Those who imagine that so good a man as Firmicus
could not have been the author of such an absurd per¬
formance, should remember that it was probably com¬
posed prior to his conversion, when such absurdities
would constitute a part of his creed.
FIRMNESS, denotes the consistence of a body, or
that state wherein its sensible parts cohere in such a
manner, that the motion of one part induces a motion
in the rest.
FIRST-born. See Primogeniture, for the li¬
teral meaning of the term.
In Scripture it is also used often in a figurative sense
for that which is first, most excellent, most distinguish¬
ed in any thing. Thus it is sam of Christ (Col. i. 5,)>
that he is “ the first-born of every creature $” and in
Revelation (i. 5.) he is called “ The first-begotten of
the dead that is, according to the commentators,
begotten of the Father before any creature was pro¬
duced ; and the first who rose from the dead by his own
power. 44 rl he first-born of the poor, (Isa. xiv. 3®*)
signifies, The most miserable of all the poor ; and in
Job (xviii. 13.) “ The first-born of death that is,
The most terrible of all deaths.
First Fruits {primitice), among the Hebrews, were
oblations of part of the fruits of the harvest, offered
to God as an acknowledgment of his sovereign domi¬
nion. The first of these fruits was offered in the name
of the whole nation, being either two loaves of bread,
or a sheaf of barley which was thrashed in the court
of the temple. Every private person was obliged to
bring his first fruits to the temple and these consisted
of wheat, barley, grapes, figs, apricots, olives, and
dates.
There was another sort of first fruits which were
paid to God. When bread was kneaded in a family,
a portion of it was set apart and given to the priest
or Levite who dwelt in the place ; if there was no priest
or Levite there, it was cast into the oven, and consumed
by the fire. These offerings made a considerable part
of the revenues of the Hebrew priesthood.
First Fruits are frequently mentioned in ancient Chri¬
stian writers as one part of the church revenue. One
of the councils of Carthage enjoins, that they should
Fish.
consist only of grapes and corn $ which shows, that this p;m
was the practice of the African church. n
First Fruits in the church of England, are the
profits of every spiritual benefice for the first year,
according to the valuation thereof in the king’s
books.
FISC, {Fisells'), in the Civil Law, the treasury of a
prince or state j or that to which all things due to the
public do fall. The word is derived from the Greek
an<^ edu¬
cated in the collegiate church of that place. In 1484,
he removed to Michael house in Cambridge, of which
college he was elected master in the year .1495. Hav¬
ing applied himself to the study of divinity, he took
orders $ and, becoming eminent as a divine, attracted
the notice of Margaret countess of Richmond, mother
of Henry VII. who made him her chaplain and con¬
fessor. In 1501, he took the degree of doctor of di¬
vinity', and the same year was elected chancellor of the
university. In the year following, he was appointed
Lady Margaret’s first divinity professor ; and in 1504,
consecrated bishop of Rochester; which small bishopric
he would never resign, though he was offered both Ely
and Lincoln. It is generally allowed, that the foun¬
dation of the two colleges of Christ church and St
John’s, in Cambridge, was entirely owing to Bishop
Fisher’s persuasion and influence with the countess of
3
Richmond: he not only formed the design, but super-
intended the execution. On the promulgation of Mar- Fisherj’,
tin Luther’s doctrine, our bishop was the first to enter v“—v—-
the lists against him. On this occasion he exerted all
his influence, and is generally supposed to have written
the famous book by which Henry VIII. obtained the
title of Defender of the Faith. Hitherto he continued
in favour with the~kingbut in 1527, opposing his di¬
vorce, and denying his supremacy, the implacable
Harry determined, and finally effected, his destruction.
In 1543, the parliament found him guilty of mispri¬
sion of treason, for concealing certain prophetic speech¬
es of a fanatical impostor, called the Holy Maid of
Kent, relative to the king’s death ; and condemned
him, with five others, in loss of goods and imprisonment
during his majesty’s pleasure ; but he was released on
paying 300I. for the king’s use.
King Henry being now married to Anne Boleyn,
his obsequious parliament took an oath of allegiance
proper for the occasion. Bins oath the bishop of Ro¬
chester steadily refused 5 alleging, that his conscience
could not be convinced that the king’s first .marriage
was against the law of God. For refusing this oath of
succession, he was attainted by the parliament of 1534 5
and committed to the Tower, where he was cruelly
treated, and where he would probably have died a na¬
tural death, had not the pope created him a cardinal.
The king, now positively determined on his destruc¬
tion, sent Rich the solicitor general, under the pre¬
tence of consulting the bishop on a case of conscience,
but really with a design to draw him into a conver¬
sation concerning the supremacy. The honest old bi¬
shop spoke his mind without suspicion or reserve, and
an indictment and conviction of high treason was the
consequence. He was beheaded at Tower Hill on the
22d of June'1535, in the 77th year of his age.. Thus
died this good old prelate; who, notwithstanding his
inflexible enmity to the Reformation, was undoubtedly
a learned, pious, and honest man. He wrote several
treatises against Luther, and other works, which were
printed at Wurtzburg in I597> 0116 vo^ume folio.
FISHERY, a place where great numbers of fish are
caught.
The principal fisheries for salmon, herrings, mackrel,
pilchards, &c. are along the coasts of Scotland, Eng¬
land, and Ireland : for cod, on the banks of Newfound¬
land : for whales, about Greenland ; and for pearls, in
the East and West Indies.
Free-FmiERY, in Law, or an exclusive right of fish¬
ing in a public river, is a royal franchise ', and is con¬
sidered as such in all countries where the feoda! po¬
lity has prevailed: though the making such grants, and
by that means appropriating, what it seems unnatural to
restrain, the use of running water, was prohibited for
the future, by King John’s Great Charter } and the ri
vers that were fenced in his time were directed to be
laid open, as well as the lorests to be disforested. This
opening was extended by the second and third charters
of Henry HI. to those" also that were fenced under
Richard I. ; so that a franchise of free fishery ought now
to be as old at least as the reign of Henry II. This
differs from a several of piscary, because he that has a
several fishery must also be the owner of the soil, which
in a free fishery is not requisite. It differs also from a
common fishery in that the free fishery is an exclusive
*
FIS [ 649 ] FIS
,hety. r*gty> th® common fishery is not so : and therefore, in a
-y ■ n> free fishery, a man has property in the fish before they
are caught j in a common piscary, not till afterwards.
Some indeed have considered a free fishery not as a royal
franchise j but merely as a private grant of a liberty to
fish in the several fishery of the grant. But the consi¬
dering such right as originally a flower of the preroga¬
tive, till restrained by Magna Charta, and derived by
royal grant (previous to the reign of Richard I.) to
such as now claim it by prescription, may remove some
difficulties in respect to this matter with which our law
books are embarrassed.
Fishery, denotes also the commerce of fish, more
particularly the catching them for sale.
Were we to enter into a very minute and particular
consideration of fisheries, as at present established in
this kingdom, this article would swell beyond its pro¬
per bounds j because, to do justice to a subject of such
concernment to the British nation, requires a very am¬
ple and distinct discussion. We shall, however, ob¬
serve, that since the Divine Providence hath so emi¬
nently stored the coasts of Great Britain and Ireland
with the most valuable fish j and since fisheries, if suc¬
cessful, become permanent nurseries for breeding ex¬
pert seamen ; it is not only a duty we owe to the Su¬
preme Being, not to despise the wonderful plenty he
hath afforded us, by neglecting to extend this branch
of commerce to the utmost; but it is a duty we owe
to our country, for its natural security, which depends
upon the strength of our royal navy. No nation
can have a navy where there is not a fund of business
to breed and employ seamen without any expence to
the public, and no trade is so well calculated for
training up these useful members of society as fish¬
eries.
The situation of the British coasts is the most advan¬
tageous in the world for catching fish : the Scottish
islands, particularly those to the north and west, lie
most commodious for carrying on the fishing trade to
perfection ; for no country in Europe can pretend to
come up to Scotland in the abundance of the finest fish,
with which its various creeks, bays, rivers, lakes, and
coasts, are replenished. Of these advantages the Scots
seem indeed to have been abundantly sensible ; and their
traffic in herrings, the most valuable of all the fisheries,
is noticed in history so early as the ninth century.
The frequent laws which were enacted in the reigns of
James III. IV. and V. discover a steady determined
zeal for the benefit of the native subjects, and the full
restoration of the fisheries, which the Dutch had lat¬
terly found means to engross $ and do honour to the
memory of those patriots whom modern times affect to
call barbarians.
The expedition of James V. to the Hebrides and
western parts of the Highlands, and his assiduity in
exploring and sounding the harbours, discovered a fix¬
ed resolution in that active prince, to civilize the in¬
habitants, to promote the valuable fisheries at their
doors, «nd to introduce general industry. His death,
at an early period, and the subsequent religious and
civil commotions in the kingdom, frustrated those
wise designs, and the western fisheries remained in their
original state of neglect. At length, 1602, James VI.
resumed the national purposes which had been thus
chalked out by his grandfather. “ Three towns
VOL. VIII. Part II. i
(says Dr Robertson) which might serve as a retreat Fishery,
for the industrious, and a nursery for arts and com-
merce, were appointed to be built in different parts of
the Highlands j one in Cantire, another in Lochaber,
and a third in the isle of Lewis $ and in order to draw
the inhabitants thither, all the privileges of the royal
boroughs were to be conferred upon them. Finding
it, however, to be no easy matter to inspire the inhabi¬
tants of those countries with the love of industry, a
resolution was taken to plant amongst them colonies of
people from the more industrious counties. The first
experiment was made in the isle of Lewis ; and as it
was advantageously situated for the fishing trade (a
source from which Scotland ought naturally to derive
great wealth), the colony transported thither was drawn
out of Fife, the inhabitants of which were well skilled
in that branch of commerce. But before they had
remained there long enough to manifest the good ef¬
fects of this institution, the islanders, enraged at see¬
ing their country occupied by those intruders, took
arms, and surprising them in the night time, murdered
some of them, and compelled the rest to abandon the
settlement. The king’s attention being soon turned
to other objects, particularly to his succession to the
English crown, we hear no more of his salutary pro¬
ject.”
The Scottish fisheries were, however, resumed by
Charles I. who “ ordained an association of the three
kingdoms, for a general fishing within the hail seas
and coasts of his majesty’s said kingdoms ; and for
the government of the said association, ordained, that
there should be a standing committee chosen and no¬
minated by his majesty, and his successors from time
to time,” &c. &c. Several persons of distinction em¬
barked in the design, which the king honoured with
his patronage, and encouraged by his bounty. He also
ordered Lent to be more strictly observed ; prohibited
the importation of fish taken by foreigners; and agreed
to purchase from the company his naval stores and the
fish for his fleets. Thus the scheme of establishing a
fishery in the Hebrides began to assume a favourable
aspect; but all the hopes of the adventurers were frus¬
trated by the breaking out of the civil wars, and the
very tragical death of their benefactor.
In 1661, Charles II. the duke of York, Lord Claren¬
don, and other persons of rank and fortune, resumed the
business of the fisheries with greater vigour than any of
their predecessors. For this purpose the most salutary
laws were enacted by the parliaments of England and
Scotland ; in virtue of which, all materials used in, or
depending upon the fisheries, were exempted from all
duties, excises, or imposts whatever. In England, the
company were authorised to set up a lottery, and to
have a voluntary collection in all parish churches; houses
of entertainment, as taverns, inns, ale-houses, were
to take one or more bai’rels of herrings, at the stated
price of 30s. per barrel ; and 2s. 6d. per barrel was to
be paid to the stock of this company on all imported fish
taken by foreigners. Some Dutch families were also
invited, or permitted to settle in Stornaway. The her¬
rings cured by the Royal English company gave ge¬
neral satisfaction, and, as mentioned above, brought
a high price for those days. Every circumstance at¬
tending this new establishment seemed to be the result
of a judicious plan and thorough knowledge of the
4 N business,
FIS [ 65° ] FIS
Eisber^. business, when the necessities of the king obliged him
y-~~' to withdraw his subscription or bounty •, which gave
such umbrage to the parties concerned, that they soon
after dissolved.
In 167*/, a new royal company was established in
England, at the head of which was the duke oi \ork,
the earl of Derby, &c. Besides all the privileges which
former companies had enjoyed, the king granted this
new company a perpetuity, with power to purchase
lands 5 and also 20I. to be paid them annually, out
of the customs of the port of London, for every dog¬
ger or buss they should build and send out for seven
years to come. A stock of 10,980!. was immediately
advanced, and afterwards 1600I. more. This small
capital was soon exhausted in purchasing and fitting
out busses, with other incidental expences. rlhe com¬
pany made, however, a successful beginning ; and one
of their busses or doggers actually took and brought
home 32,000 cod fish } other vessels had also a favour¬
able fishery. Such favourable beginnings might have
excited fresh subscriptions, when an unforeseen event
ruined the whole design beyond the possibility of reco¬
very. Most of the busses had been built in Holland,
and manned with Dutchmen; on which pretence the
French, who were then at war with Holland, seized
six out of seven vessels, with their cargoes and fishing
tackle: and the company being now in debt, sold, in
1680, the remaining stores,-&c. A number of gen¬
tlemen and merchants raised a new subscription of
6o,oool. under the privileges and immunities of the
former charter. This attempt also came to nothing,
owing to the death of the king, and the troubles of the
subsequent reign.
Soon after the Revolution this business was again re¬
sumed, and upon a more extensive scale ; the proposed
capital being 300,000!. of which ioo,oool. was to
have been raised by the surviving patentees or their
successors, and 200,0001. by new subscribers. Copies
of the letters patent, the constitution of the company,
and terms of subscription, were lodged at sundry
places in London and Westminster, for the perusal of
the public, while the subscription was filling. It is
probable, that King William’s partiality to the Dutch
fisheries, the succeeding war, or both of these circum¬
stances, frustrated this new attempt ; of which we
have no farther account in the annals of that reign or
since.
The Scottish parliament had also, during the three
last reigns, passed sundry acts for erecting companies
and promoting the fisheries; but the intestine commo¬
tions of that country, and the great exertions which
were made for the Darien establishment, enfeebled all
other attempts, whether collectively or by individuals,
within that kingdom.
In 1749, his late majesty having, at the opening
of the parliament, warmly recommended the improve¬
ment of the fisheries, the house of commons appointed
a committee to inquire into the state of the herring
and white fisheries, and to consider of the most proba¬
ble means of extending the same. All ranks of men
were elevated with an idea of the boundless riches
that would flow into the kingdom from this source. A
subscription of 500,000!. was immediately filled in the
city, by a body of men who were incorporated for 21
years by the name of The Society of the Free British
Fishery. Every encouragement was held out by go¬
vernment, both to the society, and to individuals who ^
might embark in this national business. A bounty of
36s. per ton was to he paid annually out of the customs,
for 14 years, to the owners of all decked vessels or bus¬
ses, from 20 to 80 tons burden, which should be built
after the commencement of the act, for the use of, and
fitted out and employed in, the said fisheries, whether
by the society or any other persons. At the same time
numerous pamphlets and newspaper essays came forth ;
all pretending to elucidate the subject, and to convince
the public with what facility the herring fisheries might
he transferred from Dutch to British hands. This
proved, however, a more arduous task than had been
foreseen by superficial speculators. The Dutch were
frugal in their expenditures and living ; perfect masters
of the arts of fishing and curing, which they had carried
to the greatest height and perfection. They were in
full possession of the European markets ; and their fish,
whether deserving or otherwise, had the reputation of
superior qualities to all others taken in our seas. With
such advantages, the Dutch not only maintained their
ground against this formidable company, but had also
the pleasure of seeing the capital gradually sinking,
without having procured an adequate return to the ad¬
venturers ; notwithstanding various aids and efforts of
government from time to time in their favour, particu¬
larly in 1757, "’hen an advance of 20s. per ton was
added to the bounty.
In 1786 the public attention was again called to the
state of the British fisheries, by the suggestions of Mr
Dempster in the house of commons, and by different
publications that appeared upon the subject: in conse¬
quence of which the minister suffered a committee to
be named, to inquire into this great source of national
wealth. To that committee it appeared, that the best
way of improving the fisheries was to encourage the in¬
habitants living nearest to the seat ol them to become
fishers : And it being found that the north-western
coast of the kingdom, though abounding with fish and
with fine harbours, was utterly destitute of towns, an act
was passed for incorporating certain persons therein
named, by the style of “ The British Society for extend¬
ing of the fisheries and improving the sea coasts of the
kingdom ;” and to enable them to subscribe a joint
stock, and therewith to purchase lands, and build there¬
on free towns, villages, and fishing stations, in the
Jlighlands and islands in that part of Great Britain
called Scotland, and for other purposes. The isle of
Mull, Loch Broom, the isles of Sky and of Cannay,
have already been pitched upon as proper situations for
some of these towns. The progress of such an underta¬
king from its nature must be slow, but still slower when
carried on with a limited capital arising from the sub¬
scriptions of a few public-spirited individuals. But it is
not to be doubted hut that it will ultimately tend to
the increase of our fisheries, and to the improvement of
the Highland part of this kingdom. Its tendency is
also to lessen the emigration of a brave and industrious
race of inhabitants, too many of whom have already
removed with their families to America.
1. Anchovy Fishery. The anchovy is caught in the
months of May, June, and July, on the coasts of Cata¬
lonia, Provence, &.c. at which season it constantly re¬
pairs up the straits of Gibraltar, into the Mediterranean.
^ 1 Collins
F 1 s [ 651 ] FIS
Collins says they are also found in plenty on the west¬
ern coasts of England and Wales.
The fishing for them is chiefly in the night time j
when a light being put on the stern of their little fishing
vessels, the anchovies flock round, and are caught in the
nets. But then it is asserted to have been found by
experience, that anchovies taken thus by fire,,are nei¬
ther so good, so firm, nor so proper for keeping, as those
which are taken without fire.
When the fishery is over, they cut off the heads,
take out their gall and guts, and then lay them in bar¬
rels and salt them. JThe common way of eating an¬
chovies is with oil, vinegar, &c. in order to which they
are first boned, and the tails, fins, &c. slipped ofl‘.—
Being put on the fire, they dissolve almost in any liquor.
Or they are made into a sauce by mincing them with
pepper, &c. Some also pickle anchovies in small delft
or earthen pots, made on purpose, of two or three
pounds weight, more or less, which they cover with
plaster to keep them the better. Anchovies should be
chosen small, fresh pickled, white on the outside and
red within. They must have a round back ; for those
which are fat or large are often nothing but sardines.
Besides these qualities, the pickle, on opening the pots
or barrels, must be of a good taste, and not have lost
its flavour.
2. Cod Fishery. There are two kinds of cod fish j
the one green or white cod, and the other dried or cur¬
ed cod j though it is all the same fish, differently pre¬
pared 5 the former being sometimes salted and barrel¬
led, then taken out for use ; and the latter having lain
some competent time in salt, dried in the sun or smoke.
We shall therefore speak of each of these apaft ; and
first of the
Green. The chief fisheries for green cod are in
the bay of Canada, on the great bank of Newfound¬
land, and on the isle of St" Peter, and the isle of
Sable ; to which places vessels resort from divers parts
both of Europe and America. They are from 100
to 150 tons burden, and will catch between 30,000 and
40,000 cod each. The most essential part of the
fishery is, to have a master who knows how to cut up
the cod, one who is skilled to take off the head pro¬
perly, and above all a good salter, on which the pre¬
serving of them, and consequently the success of the
voyage depends. The best season is from the begin¬
ning of February to the end of April ; the fish, which
in the winter retire to the deepest water, coming then
on the banks and fattening extremely. What is
caught from March to June keeps well j but those
taken in July, August, and September, when it is warm
on the banks, are apt to spoil soon. Every fisher takes
but one at a time: the most expert will take from 330
to 400 in a day $ but that is the most 5 the weight of
the fish and the great coldness on the bank fatiguing
very much. As soon as the cod is caught, the head
is taken off; they are opened, gutted, and salted ; and
the salter stows them in the bottom of the hold, head
and tail, in beds a fathom or two square; laying layers
of salt and fish alternately, but never mixing fish caught
on different days. When they have lain thus three or
four days to drain off the water, they are replaced in
another part of the ship, and salted again ; where they
remain till the vessel is loaded. Sometimes they are cut
in thick pieces, and put in barrels for the conveniency
of carriage. - ‘ y
Fri/. Ihe principal fishery for this article is, from
Cape Rose to the Bay des Exports, along the coast of
Placentia, in which compass there are divers com¬
modious ports for the fish to be dried in. These,
though of the same kind with the fresli cod, are
much smaller, and therefore fitter to keep, as the salt
penetrates more easily into them. The fishery of both
is much alike ; only this latter is most expensive, as it
takes up more time and employs more hands, and yet
scarce half so much salt is spent in this as in the other.
I he bait is herrings, of which great quantities are
taken on the coast of Placentia. When several vessels
meet and intend to fish in the same part, he whose shal¬
lop first touches ground becomes entitled to the qua¬
lity and privileges of admiral: he has the choice of his
station, and the refusal of all the wood on the coast at
1,1S ariival. As fast as the masters arrive, they unrigg
a:l their vessels, leaving nothing but the shrouds to su¬
stain the mast ; and in the mean time the mates pro¬
vide a tent on shore, covered with branches of trees,
and sails over them, with a scaffold of great trunks of
pines, 12, 13, 16, and often 20 feet high, commonly
from 40 to 60 feet long, and about one-third as much,
in breadth. While the scaffold is preparing, the crew
are a-fishmg ; and as fast as they catch, they bring their
fish ashore, and open and salt them upon moveable
benches ; but the main salting is performed on the
scaffold. When the fish have taken salt, they wash and
hang them to drain on rails ; when drained, they are
laid on kinds of stages, which are small pieces of wood
laid across, and covered with branches of trees, having
the leaves stripped off for the passage of the air. On
these stages, they are disposed, a fish thick, head against
tail, with the back uppermost, and are turned carefully
four times every 24 hours. When they begin to dry,
they are laid in heaps 10 or 12 thick, in order to re¬
tain their warmth ; and every day the heaps are en¬
larged till they become double their first bulk ; then
two heaps are joined together, which they turn every
day as before : lastly, they are salted again, beginning
with those first salted ; and being laid in huge piles,
they remain in that situation till they are carried on
hoard the ships, where they are laid on the branches
of trees disposed for that purpose, upon the ballast, and
round the ship, with mats to prevent their contracting
any moisture.
There are four sorts of commodities drawn from cod,
viz. the sounds, the tongues, the roes, and the oil ex¬
tracted from the liver. The first is salted at the fishery
together with the fish, and put in barrels from 600 to’
700 pounds. The tongues are done in like manner, and
brought in barrels from 400 to 500 pounds. The roes
are also salted in barrels, and serve to cast into the sea
to draw fish together, and particularly pilchards. The ■■
oil comes in barrels, from 400 to 520 pounds, and is
used in dressing leather. In Scotland they catch a
small kind of cod on the coasts of Buchan and all along-
the Murray frith on both sides; as also in the friths of
lorth, Clyde, &c. which is much esteemed. Thev
salt and dry them in the sun upon rocks, and some¬
times in the chimney.
3. Coral Fishery. See Coral.
4 N 3
4. Herring
Fishery.
* Hisi. of
Commerce.
FIS [ 652 ] FIS
4. Herring Fishery. Our great stations for this
fishery are oft' the Shetland and Western isles, and off
the coast of Norfolk, in which the Dutch also share.
There are two seasons for fishing herring ; the first from
June to the end of August) and the second in autumn,
when the fogs become very favourable lor this kind of
fishing. The Dutch begin their herring fishing on the
24th of June, and employ a vast number of vessels
therein, called busses, being between 45 and 60 tons
burden each, and carrying three or four small cannon.
They never stir out of port, without a convoy, unless
there be enough together to make about 18 or 20 can¬
non among them, in which case they are allowed to go
in company. Before they go out they make a verbal
agreement, which has the same force as if it were in
writing. The regulations of the admiralty of Holland
are partly followed by the French and other nations,
and partly improved and augmented with new ones ;
as, that no fisher shall cast his net within 100 fathoms
of another boat : that while the nets are cast, a light
shall be kept on the hind part of the vessel : that when
a boat is by any accident obliged to leave oft fishing,
the light shall be cast into the sea) that when the
greater part of a fleet leaves oft fishing, and casts an*
chor, the rest shall do the same, &c.
Mr Anderson * gives to the Scots a knowledge of
great antiquity in the herring fishery. He says that
the Netherlanders resorted to these coasts as early as
A. D» 836, to purchases alted fish of the natives ; but,
imposing on the strangers, they learned the art, and
took up the trade, in after times of such immense emo¬
lument to the Dutch.
Sir Walter Raleigh’s observations on that head, ex¬
tracted from the same author, are extremely worthy
the attention of the curious, and excite reflections on
the vast strength resulting from the wisdom ot well ap¬
plied industry.
In 1603, he remarks, the Dutch sold to different
nations as many herrings as amounted to 1,759,000!.
sterling. In the year 1615, they at once sent out
2000 busses, and employed in them 37,000 fishermen.
In the year 1618, they sent out 3000 ships, with
50,000 men to take the herrings, and 9000 more ships
to transport and sell the fish *, which by sea and land
employed 150,000 men, besides those first mentioned.
All this wealth was gotten on our coasts, while our at¬
tention was taken up in a distant whale fishery.
The Scottish monarchs seemed for a long time to di¬
rect all their attention to the preservation of the salmon
fishery, probably because their subjects were such no¬
vices in sea affairs. At length James III. endeavoured
to stimulate his great men to these patriotic undertak¬
ings : for by an act of his third parliament, he com¬
pelled certain lords spiritual and temporal, and bur¬
rows, to make ships, busses and boats, with nets and
other pertinents, for fishing : that the same should be
made in each burgh ) in number according to the
substance of each burgh, and the least of them to
be of twenty tons : and that all idle men be com¬
pelled by the sheriffs in the country to go on board the
same.”
Numerous indeed have been the attempts made at
different periods to secure this treasure to ourselves, but
without success. In the late reign,-a very strong effort
was made, and bounties allowed, for the encouragement
of British adventurers : the first was of 30s. per ton to FUtwry,
every buss of 70 tons; and upwards. This bounty wasy*-
afterwards raised to 50s. per ton, to be paid to such
adventurers as were entitled to it by claiming it at the
places of rendezvous. The busses are from 20 to 90 tons
burden, but the best size is 8o. A vessel of 80 tons ought
to take ten lasts, or 120 barrels of herrings, to clear
expences, the price of the fish to he admitted to be a
guinea a barrel. A ship of this size ought to have 18
men, and three boats : one of 20 tons should have six
men, and every five tons above require an additional
hand. To every ton are 280 yards of nets ; so a vessel
of 80 tons carries 20,000 square yards : each net is
12 yards long, and 10 deep, and every boat takes out
from 20 to 30 nets, and puts them together so as to
form a long train ) they are sunk at each end of the
train by a stone, which weighs it down to the full ex¬
tent : the top is supported by buoys, made of sheeps-
skin, with a hollow stick at the mouth fastened tight:
through this the skin is blown up, and then stopped
with a peg, to prevent the escape of the air. Sometimes
these buoys are placed at the top of the nets : at other
times the nets are suffered to sink deeper, by the length¬
ening the cords fastened to them, every cord being
for that purpose 10 or 12 fathoms long. But the best
fisheries are generally in more shallow water.
Of the Scots fishery in the Western isles, the follow¬
ing account is given by Mr Pennant t. “ The fishing is t
always performed in the night, unless by accident. The^“*
busses remain at anchor, and send out their boats a little
before sunset: which continue out, in winter and sum¬
mer, till day-light; often taking up and emptying their
nets, which they do 10 or 12 times in a night, in case
of good success. During winter it is a most dangerous
and fatiguing employ, by reason of the greatness and
frequency of the gales in these seas, and in such gales
are the most successful captures : but by the Providence
of heaven the fishers are seldom lost; and, what is won¬
derful, few are visited with illness. They go out well
prepared, with a warm great coat, boots, and skin
aprons, and a good provision of beef and spirits. The
same good fortune attends the busses, which in the tem¬
pestuous season, and in the darkest nights, are conti¬
nually shifting in these narrow seas, from harbour to
harbour. Sometimes 80 barrels of herrings are taken
in a night by the boats of a single vessel. It once hap¬
pened in Loch Slappan, in Sky, that a buss of 80 tons
might have taken 200 barrels in one night, with
10,000 square yards of net; but the master was obliged
to desist, for want of a sufficient number of hands to
preserve the capture. The herrings are preserved by
salting, after the entrails are taken out. This last is
an operation performed by the country people, who get
three-halfpence per barrel for their trouble ) and some¬
times, even in the winter, can gain fifteen pence a-day.
This employs both women and children ) but the salt¬
ing is only intrusted to the crew of the busses. The
fish are laid on their backs in the barrels, and layers of
salt between them. The entrails are not lost, for they
are boiled into an oil 8000 fish will yield ten gal¬
lons, valued at one shilling the gallon. A vessel of
80 tons, takes out 144 barrels of salt ; a drawback of
2s. 8d. is allowed for each barrel used by the foreign
or Irish exportation of the fish ; but there is a duty of
is. per barrel for the home consumption, and the same
for
FIS [ 653 ] FIS
pislery. for those sent to Ireland. The barrels are made of oak
i staves chiefly from Virginia 5 the hoops from several
parts of our own island, and are either of oak, birch,
hazel or willow •, the last from Holland, liable to a du¬
ty. The barrels cost about 3s. each, they hold, from
500 to 800 fish, according to the size of the fish ; and
are made to contain 22 gallons. The barrels are in¬
spected by proper officers j a cooper examines if they
are statutable and good ; if faulty, he destroys them,
and obliges the maker to stand to the loss.
“ Loch Broom has been celebrated for three or four
centuries as the resort of herrings. They generally
appear here in July ; those that turn into this bay are
part of the brigade that detaches itself from the west-.
ern column of that great army which annually deserts
the vast depths of the arctic circle, and comes, heaven-
directed, to the seats of population, offered as a cheap
food to millions, whom wasteful luxury or iron-hearted
avarice bath deprived, by enhancing the price, of the
wonted supports of the poor. The migration of the
fish from their northern retreat is regular j their visits
to the Western isles and coasts certain j but their at¬
tachment to one particular loch extremely precari¬
ous. All have their turns $ that which swarmed with
fish one year, is totally deserted the following j yet
the next loch to it may be crowded with the shoals.
These changes of place give often full employ to
the busses, who are continually shifting their harbour
in quest of news respecting these important wanderers.
They commonly appear here in July ; the latter end
of August they go into deep water, and continue
there for some time, without any apparent cause :
in November, they return to the shallows, when a new
fishery commences, which continues till January : at
that time the herrings become full of roe, and are use¬
less as articles of commerce. Some doubt, whether
those herrings that appear in November are not part
of a new migration j for they are as fat, and make
the same appearance, as those that composed the first.
The signs of the arrival of the herrings are flocks of
gulls, who catch up the fish while they skim on the
surface, and of gannets who plunge and bring them,
up from considerable depths. Both these birds are
closely attended to by the fishers. Cod fish, haddocks,,
and dog fish, follow the herrings in vast multitudes
these voracious fish keep on the outsides of the co¬
lumns, and may be a concurrent reason of driving the
shoals into bays and creeks. In summer, they come
into the bays generally with the warmest weather,
and with easy gales. During winter, the hard gales
from north-west are supposed to assist in forcing them
into shelter. East winds are very unfavourable to the
fishery.”
Herrings are cured either white or pickled, or red.
Of thej^/'i^, those done by the Dutch are the most
esteemed, being distinguished, into four sorts, accord¬
ing to their sizes j and the best are those that are fat,
fleshy, firm, and white, salted the same day they are
taken, with good salt, and well barrelled. The Bri¬
tish cured herrings are little inferior, if not equal, to
the Dutch : for in spite of all their endeavours to con¬
ceal the secret, their method of curing,, lasting,, or
casking the herrings, has been discovered, and is as
follows. After they have hauled in their nets, which
they drag in the stern of their vessels backwards and
forwards in traversing the coast, they throw them upon Fishery,
the ship’s deck, which is cleared of every thing for ' ■' 1 v1""
that purpose : the crew is separated into sundry divi¬
sions, and each division has a peculiar task j one part
opens and guts the herrings, leaving the milts and
roes ; another cures and salts them, by lining or rub¬
bing their inside with salt; the next packs them, and
between each row and division they sprinkle handfuls
of salt; lastly, the cooper puts the finishing hand to all,
by heading the casks very tight, and stowing them in
the hold.
Red herrings must lie 24 hours in the brine, in¬
asmuch as they are to take all their salt there $ and
when they are taken out, they are spitted, that is, strung
by the head on little wooden spits, and then hung in a
chimney made for that purpose. After which, a fire
of brushwood, which yields a deal of smoke, but no
flame, being made under them, they remain there till
sufficiently smoked and dried, and are afterwards bar¬
relled up for keeping.
5. Lobster Fishery. Lobsters are taken along the
British channel, and on the coast of Norway, whence
they are brought to London for sale } and also in the
frith of Edinburgh, and on the coast of Northumber- -
land. By 10 and II W. III. cap. 24. no lobster is to
be taken under eight inches in length, from the peak
of the nose to the end of the middle fin of the tail $
and by 9 Geo. II. cap. 33. no lobsters are to be taken
on the coast of Scotland from the 1st of June to the
1st of September.
6. Mackerel Fishery. The mackerel is a summer fish
of passage, found in large shoals, in divers parts of the
ocean, not far north ; but especially on the French and
English coasts. The fishing is usually in the months
of April, May, and June, and even July, according to
the place. They enter the English channel in April,
and proceed up the straits of Dover as the summer
advances ; so that by June they are on the coasts of
Cornwall, Sussex, Normandy, Picardy, &c. where the
fishery is most considerable. They are an excellent
food .fresh ; and not to be despised, when well prepar¬
ed, pickled, and put up in barrels j a method of pre¬
serving them chiefly, used in Cornwall.
The fish is taken two ways; either with a line or
nets : the latter is the more considerable, and is usually
performed in. the night-time. The rules observed in,
the fishing for mackerel are much the same as those al- .
ready mentioned in the fishery of herrings.
There are two ways of pickling them : the first is,
by opening and gutting them, and filling the belly
with salt, crammed in as hard as possible, with a stick;,
which done, they range them in strata or rows, at the
bottom of the vessel, strewing salt between the layers.
In the second way, they put them immediately into
tubs full of brine, made of fresh water and salt; and
leave them.to steep, till they have imbibed salt enough
to make them keep; after which, they are taken,
out, and barrelled up, taking care to press them close
down.
Mackerel are not cured or exported as merchandise,
except a few by the Yarpiouth and Leostoff merchants, ,
but are generally consumed at home ; especially in
the city , of London, and the sea-ports between the
Thames and Yarmouth, east, and the Land’s End of
Cornwall, west,.. > dJir
7. Oyster r
Ftsbery.
IS [ 654. ]
This fishery Is principally car- bittern
| See Os-
trm. Con-
ehology
Index.
7. Oyster FisueryX „ ...
rled on at Colchester In Essex Feversham ami Milton
in Kent j the Isle of Wight; the Swales of the Med¬
way ; and Tenbv on the coast of Wales. From Fever-
sham, and adjacent parts, the Dutch have sometimes
loaded a hundred large hoys with oysters in a year.
They are also taken in great quantities near Portsmouth,
and in all the creeks and rivers between Southampton
and Chichester : many of which are carried about by
sea to London and to Colchester, to be fed in the pits
about Wavenhoe and other places.
8. Pearl Fishery. See Pearl, Conchology In¬
dex, and Ceylon.
9. Pilchard Fishery. The chief pilchard fisheries are
along the coasts of Dalmatia, on the coast of Bretagne,
and along the coasts of Cornwall and Devonshire. That
of Dalmatia is very plentiful: that on the coasts of
Bretagne employs annually about 300 ships. Of the
pilchard fishery on the coast of Cornwall the following
account is given by Dr Borlase : “ It employs a great
number of men on the sea, training them thereby to
naval affairs 5 employs men, women, and children, on
land, in salting, pressing, washing and cleaning j in
making boats, nets, ropes, casks, and all the trades
depending on their construction and sale. Ihe poor
are fed with the offals of the captures, the land with
the refuse of the fish and salt •, the merchant finds the
gains of commission and honest commerce, the fisher¬
man the gains of the fish. Ships are often freighted
hither with salt, and into foreign countries with the
fish, carrying off at the same time part of our tin. Of
the usual produce of the great number of hogsheads ex¬
ported each year for ten years, from 1747 to 1756 in¬
clusive, from the four ports of Fowey, lalmouth, Pen¬
zance, and St Ives, it appears that Fowey has exported
yearly 1732 hogsheads 5 Falmouth, 14,631 hogsheads
and two thirds; Penzance and Mounts-Bay 12,149
hogsheads and one-third; St Ives, 1280 hogsheads:
in all amounting to 29,795 hogsheads. Every hogshead
for ten years last past, together with the bounty allow¬
ed for each hogshead exported, and the oil made out of
each hogshead, has amounted, one year with another
at an average, to the price of ll. 13s. 3^* 5 80 that the
cash paid for pilchards exported has, at a medium, an¬
nually amounted to the sum of 49’532^*
numbers that are taken at one shooting out of the nets
are amazingly great. Mr Pennant says, that Dr Bor¬
lase assured him, that on the 5th of October I7^7»
there were at one time enclosed in St Ives’s Bay yooo
hogsheads, each hogshead containing 35,000 fish; in all
245 millions.
The pilchards naturally follow the light, which con¬
tributes much to the facility of the fishery ; the season
is from June to September. On the coasts of I ranee
they make use of the roes of the cod fish as a bait ;
which, thrown into the sea, makes them rise from the
bottom, and run into the nets. On our coasts there
are persons posted ashore, who, spying by the colour
of the water where the shoals are, make signs to the
boats to go among them to cast their nets. When
taken, they are brought on shore to a warehouse,
where they are laid up in broad piles, supported with
hacks and sides ; and as they are piled, they salt them
with bay salt; in which lying to soak for 30 or 40 days,
they run out a deal of blood with dirty pickle and
F I S
then they wrash them clean in sea water; and, Fi»hm.
when dry, barrel and press them hard down to squeeze '*1—* -J
out the oil, which issues out at a hole in the bottom of
tllG Ccislv#
10. Salmon Fishery \. The chief salmon fisheries
in Europe are in England, Scotland, and Ireland, in the0^o^ ^
rivers, and sea coasts adjoining to the river mouths.4^.
The most distinguished for salmon in Scotland are, the
river Tweed, the Clyde, the Tay, the Dee, the Don,
the Spey, the Ness, the Bewly, &c. in most of which
it is very common, about the height of summer, espe¬
cially if the weather happens to be very hot, to catch
four or five score salmon at a draught. Jbe chief ri¬
vers in England for salmon are, the Tyne, the Trent,
the Severn, and the Thames. rlhe fishing is performed
with nets, and sometimes with a kind of locks or wears
made on purpose, which in certain places have iron or
wooden grates so disposed, in an angle, that being im¬
pelled by any force in a contrary direction to the course
of the river, they may give way and open a little at the
point of contact, and immediately shut again, closing
the angle. The salmon, therefore, coming up into the
rivers, are admitted into these grates, which open, and
suffer them to pass through, but shut again, and pre¬
vent their return. The salmon is also caught with a
spear, which they dart into him when they see him
swimming near the surface of the water. It is custom¬
ary likewise to catch them with a candle and lanthorn,
or wisp of straw set on fire ; for the fish naturally fol¬
lowing the light, are struck with the spear, or taken in
a net spread for that purpose, and lifted with a sudden
jerk from the bottom.
“ The capture of salmon in the Tweed, about the
month of July (says Mr Pennant J) is prodigious. In
a good fishery, often a boatload, and sometimes near^0®
two, are taken in a tide : some few years ago there were
above 700 fish taken at one haul, but from 50 to 100
is very frequent. The coopers in Berwick then begin
to salt both salmon and grilses in pipes and other large
vessels, and afterwards barrel them to send abroad, ha¬
ving then far more than the London markets can take
off thei” hands.
“ Most of the salmon taken before April, or to the
settino' in of the warm weather, is sent Iresh to Lon¬
don in baskets: unless now and then the vessel is dis¬
appointed by contrary winds of sailing immediately ; in
which case the fish is brought ashore again to the coopers
offices, and boiled, pickled, and kitted, and sent to the
London markets by the same ship, and fresh salmon
put in the baskets in lieu of the stale ones. At the be¬
ginning of the season, when a ship is on the point of
sailing, a fresh clean salmon will sell from a shilling to
eighteen pence a pound ; and most of the time that this
part of the trade is carried on, the prices are from five
to nine shillings per stone ; the value rising and falling
according to the plenty of fish, or the prospect of a fair
or foul wind. Some fish are sent in this matter to Lon¬
don the latter end of September, when the weather
grows cool ; but then the fish are full of large roes,
grow very thin bellied, and are not esteemed either pa¬
latable or wholesome. _
“ The season for fishing in the Tweed begins No¬
vember 30th, but the fishermen work very little till af¬
ter Christmas : it ends on Michaelmas day ; yet the
corporation of Berwick (who are conservators of thd
river)
FIS [ 655 ] FIS
ithery. river) indulge the fisherwYen with a fortniglit past that
■-v—— * time, on account of the change of the style.
“ There are on the river 41 considerable fisheries,
extending upwards about 14 miles from the mouth
(the others above being of no great value), which are
rented for near 5400I. per annum : the expence at-
, tending the servants wages, nets, boats, &c. amount
to 5000I. more j which together makes up the sum
10,400!. Now, in consequence, the produce must
defray all, and no less than 20 times that sum of fish
will effect it 5 so that 208,000 salmon must be caught
there one year with another.
“ Scotland possesses great numbers of fine fisheries
on both sides of that kingdom. The Scotch in early
times had most severe laws against the killing of this
fish *, for the third offence was made capital, by a law
of Janies IV. Before that, the offender had power to
redeem his life. They were thought in the time of
Henry VI. a present worthy of a crowned head : for
in that reign the queen of Scotland sent to the duchess
of Clarence 10 casks of salted salmon; which Henry
directed to pass duty free. The salmon are cured in
the same manner as at Berwick, and a great quantity
is sent to London in the spring; but after that time,
the adventurers begin to barrel and export them to fo¬
reign countries ; but we believe that commerce is far
less lucrative than it was in former times, partly owing
to the great increase of the Newfoundland fishery, and
partly to the general relaxation of the discipline of ab¬
stinence in the Romish church.
“ Ireland (particularly the north) abounds with this
fish : the most considerable fishery is at Cranna, on the
river Ban, about a mile and a half from Coleraine.
When I made the tour of that hospitable kingdom in
1754, it was rented by a neighbouring gentleman for
6201. a-year ; who assured me, that the tenant, his
predecessor, gave 1600I. per annum, and was a much
greater gainer by the bargain, for the reasons before
mentioned, and on account of the number of poachers
who destroy the fish in the fence months.
“ The mouth of this river faces the north ; and is
finely situated to receive, the fish that roam along the
coast in search of an inlet into some fresh water, as
they do along that end of the kingdom which op¬
poses itself to the northern ocean. We have seen near
Ballicastle, nets placed in the sea at the foot of the
promontories that jut into it, which the salmon strike
into as they are wandering close to shore ; and numbers
are taken by that method.
“ In the Ban they fish with nets 18 score yards
long, and are continually drawing night and day the
whole season, which we think lasts about four months,
two sets of 16 men each alternately relieving one ano¬
ther. The best drawing is when the tide is coming
in : we were told, that at a single draught there were
once 840 fish taken.
“ A few miles higher up the river is a wear where
a considerable number of fish that escape the nets are
taken. We were lately informed, that, in the year
1760, about 320 tons were taken in the Cranna fish¬
ery.”
Curing Salmon. When the salmon are taken, they
open them along the back, take out the guts and gills,
and cut out the greatest part of the bones, endeavour¬
ing to make the inside as smooth as possible : they then
salt the fish in large tubs for the purpose, where they irj^cjr,
lie a considerable time soaking in brine ; and about w—
October they are packed close up in barrels, and sent
to London, or exported up the Mediterranean. They
have also in Scotland a great deal of salmon salted in
the common way, which after soaking in brine a com¬
petent time, is well pressed, and then dried in smoke :
this is called kipper, and is chiefly made for home con¬
sumption ; and if properly cured and prepared, is reck¬
oned very delicious.
Sturgeon f Fishery. The greatest sturgeon fishery f See Acci-
is in the mouth of the Volga, on the Caspian sea : VenS£r>
where the Muscovites employ a great number of hands,
and catch them in a kind of enclosure, formed by huge
stakes representing the letter Z repeated several times.
These fisheries are open on the side next the sea, and
close on the other ; by which means the fish ascending
in its season up the river, is embarrassed in these nar¬
row angular retreats, and so is easily killed with a
harping iron. Sturgeons, when fresh, eat deliciously ;
and in order to make them keep, they are salted or
pickled in large pieces, and put up in cags from 30 to
50 pounds. But the great object of this fishery is the
roe, of which the Muscovites are extremely fond, and
of which is made the cavear, or kavia, so much esteem¬
ed by the Italians. See Caveak.
Tunny Fishery. The tunny (a species of Scom¬
ber), was a fish well known to the ancients, and
made a great article of commerce : And there are still
very considerable tunny fisheries on the coasts of Sici¬
ly, as well as several other parts of the Mediterrrmean.
The nets are spread over a large space of sea by
means of cables fastened to anchors, and are divided
into several compartments. The entrance is always
directed, according to the season, towards that part of
the sea from which the fish are known to come. A
man placed upon the summit of a rock high above the
water, gives a signal of the fish being arrived ; for he
can discern from that elevation what passes under the
waters infinitely better than any person nearer the sur¬
face. As soon as notice is given that the shoal of fish
has penetrated as far as the inner compartment, or the
chamber of death, the passage is drawn close, and the
slaughter begins.
The undertakers of these fisheries pay an acknow¬
ledgment to the king, or the lord upon whose land
they fix the main stay or foot of the tonnara ; they
make the best bargain they can : and, till success has
crowned their endeavours, obtain this leave for a small
consideration ; but the rent is afterwards raised in pro¬
portion to their capture.
The tunny enters the Mediterranean about the vernal
equinox, travelling in a triangular phalanx, so as to cut
the waters with its point, and to present an extensive
base for the tides and currents to act against, and impel
forwards. These fish repair to the warm seas of Greece
to spawn, steering their course thither along the Euro¬
pean shores, but as they return, approach the African
coast; the young fry is placed in the van of the squadron
as they travel. They come back from the east in
May, and abound on the coast of Sicily and Calabria
about that time. In autumn they steer northward, and
frequent the neighbourhood of Amalfi and Naples ;
but during the whole season stragglers are occasionally
caught.
When
FIS
t 655 ]
F I S
Fishery. When taken in May, the usual time of their ap-
>■■■—y——> pearance in the Calabrian bays, they are full of spawn,
and their flesh is then esteemed unwholesome, apt to
occasion headachs and vapours } the milts and roes are
particularly so at that season. To prevent these bad
eft’ects, the natives fry them in oil, and afterwards salt
them. The quantity of this fish consumed annually
in the Two Sicilies almost exceeds the bounds of calcu¬
lation. From the beginning of May to the end of Oc¬
tober it is eaten fresh, and all the rest of the year it is
in use salted. The most delicate part is the muzzle.
The belly salted was called tarantallum^ and accounted
a great delicacy by the Romans 5 its present name is
Surra. The rest of the body is cut into slices, and
put into tubs.
Turbot Fishery, Turbot grows to a large size,
some of them weighing from 23 3° pounds. They
are taken chiefly off the north coast of England, and
others off the Dutch coast. The large turbot (as well
as several other kinds of flat fish) are taken by the hook
and line, for they lie in deep water j the method of
taking them in wears or staked nets being very preca¬
rious. When the fishermen go out to fish, each person
is provided with three lines, which are coiled on a flat
oblong piece of wicker work } the hooks being baited,
and placed regularly in the centre of the coil. Each
line is furnished with 14 score of hooks, at the distance
of six feet two inches from each other. The hooks are
fastened to the lines upon sneads of twisted horse hair
27 inches in length. Wben fishing, there are always
three men in each coble, and consequently nine, of
these lines are fastened together, and used in one.line,
extending in length near three miles, and furnished
with 2520 hooks. An anchor and a buoy are fixed at
the first end of the line, and one more of each at the
end of each man’s linesj in all four anchors, which
are common perforated stones, and four buoys made
of leather or cork. The line is always laid across the
current. The tides of flood and ebb continue an equal
time upon our coast, and, when undisturbed by winds,
run each way about six hours $ they are so rapid that
the fishermen can only shoot and haul their lines at the
turn of tide, and therefore the lines always, remain
upon the ground about six hours j during which time
the myxine glutinosa of Linnseus will frequently pene¬
trate the fish that are on the hooks, and entirely devour
them, leaving only the skin and bones. The same
rapidity of tides prevents their using hand lines $ and
therefore two of the people commonly wrap themselves
in the sail, and sleep, while the other keeps a strict
look-out, for fear of being run down by ships, and to
observe the weather. For storms often rise so suddenly,
that it is with extreme difficulty they can sometimes
escape to the shore, leaving their lines behind.
Besides the coble, the fishermen have also a five-men
!boat, which is 40 feet long and 15 broad, and 25 tons
burden $ it is so called, though navigated by six men
and a boy, because one of the men is commonly hired
to cook, *&c. and does not share in the profits with
the other five. This boat is decked at each end, but
open in the middle, and has two large lug sails. All our
able fishermen go in these boats to the herring fishery
at Yarmouth in the latter end of September, and re¬
turn about the middle of November. The boats are
^then laid up till the beginning of Lent, at which time
3
they go off in them to the edge of the Dogger, and Fishery,
other places to fish for turbot, cod, ling, skates, &c. Fishgaid
They always take two cobles on board ; and when they
come upon their ground, anchor the boat, throw out
the cobles, and fish in the same manner as those do
who go from the shore in a coble: with this difference
only, that here each man is provided with double the
quantity of lines, and instead of waiting the return of
the tide in the coble, return to their boat and bait their
other lines j thus hauling one set and shooting another
every turn of tide. They commonly run into harbour
twice a-week to deliver their fish.
The best bait is fresh herring cut in pieces of a pro¬
per size ; the five-men boats are always furnished with
nets for taking them. Next to herrings are the lesser
lampreys. The next baits in esteem are small had¬
docks cut in pieces, sand worms, and limpets, here called
flidders ; and when none of these can be had, they use
‘bullock’s liver. The hooks are two inches and a half
long in the shank, and near an inch wide between the
shank and the point. The line is made of small cord¬
ing, and is always tanned before it is used.
Turbots are extremely delicate in their choice of
baits j for if a piece of herring or haddock has been 12
hours out of the sea, and then used as bait, they will
not touch it.
Whale Fishery. See Balden a, Cetology Index.
'Whales are chiefly caught in the north seas j the
largest sort are found about Greenland or Spitzbergen.
At the first discovery of this country, whales not being
used to be disturbed, frequently came into the very bays,
and were accordingly killed almost close to the shore ,
so that the blubber being cut off was immediately boiled
into oil on the spot. The ships in those times took in
nothing but the pure oil and the whalebone, and all the
business was executed in the country j by which means
a ship could bring home the product of many more
whales than she can, according to the present method
of conducting this trade. The fishery also was then so
plentiful, that they were obliged sometimes to send
other ships to fetch off the oil they had made, the
quantity being more than the fishing ships could bring
away. But time and change of circumstances have
shifted the situation of this trade. The ships coming in
such numbers from Holland, Denmark, Hamburgh,
and other northern countries, all intruders upon the
English, who were the first discoverers of Greenland,
the whales were disturbed, and gradually, as other fish
often do, forsaking the place, were not to be killed so
near the shore as before: but are now found, and have
been soever since, in the openings and space among the
ice, where they have deep water, and where they go
sometimes a great many leagues from the shore.
The whale fishery begins in May, and continues all
June and July ; but whether the ships have good or
bad success, they must come away, and get clear of
the ice by the end of August j so that in the month
of September at farthest they may be expected home,;
but a ship that meets with a fortunate and early fishery
in May may return in June or July.
A particular account of the recent history and pre¬
sent state of the British fisheries will be found in the
article Fisheries, in the Supplement.
FISHGARD, or Fisgard, a town of Pembroke¬
shire, situated on a steep cliff on the sea shore, 254 miles
from
FIS [ 657 ] FIS
"ishery, from London, at the influx of the river Gwaine into
'ishing. the sea, which here forms a spacious bay. It is govern*
1 - v ed by a mayor, a bailiff, and other officers 5 and here
vessels may lie safely in five or six fathoms water. The
inhabitants have a good trade in herrings, and annually
cure, between Fisgard and Newport, above 1000 bar¬
rels of them. The town sends one member to parlia¬
ment.
FISHING, in general, the art of catching fish,
whether by means of nets, of spears, or of the line and
hook.
Fishing in the great, performed by the net, spear,
or harpoon, for fish that go in shoals, has been explain¬
ed in the preceding article. That performed by the
rod, line, and hook, for solitary fish, is usually termed
Angling : See that article ; and for the particular
manner of angling for the different kinds of fish, see
their respective names, as Dace, Eel, Perch, under
Ichthyology.
Here we shall give an account of the following:
See Cv- I* ^ie (so called on account of the barb
nm.lch-W beard that is under his chops), though a coarse fish,
iology gives considerable exercise to the angler’s ingenuity.
They swim together in great shoals, and are at their
worst in April, at which time they spawn, but come
soon in season j the places whither they chiefly resort,
are such as are weedy and gravelly rising grounds, in
which this fish is said to dig and root with his nose like
a swine. In the summer he frequents the strongest,
swiftest, currents of water j as deep bridges, wears, &c.
and is apt to settle himself among the piles, hollow
places, and moss or weeds; and will remain there im¬
moveable ; but in the winter he retires into deep wa¬
ters, and helps the female to make a hole in the sands
to hide her spawn in, to hinder its being devoured by
other fish. He is a very curious and cunning fish j for
if his baits be not sweet, clean, well scoured, and kept
in sweet moss, he will not bite ; but well ordered and
curiously kept, he will bite with great eagerness. The
best bait for him is the spawn of a salmon, trout, or
any other fish ; and if you would have good sport with
him, bait the places where you intend to fish with it a
night or two before, or with large worms cut in pieces ;
and the earlier in the morning or the later in the
evening that you fish, the better it will be. Your rod
and line must be both strong and long, with a running
plummet on the line ; and let a little bit of lead be
placed a foot or more above the hook, to keep the bul¬
let from falling on it; so the worm will be at the bot¬
tom, where they always bite ; and when the fish takes
the bait, your plummet will lie and not choke him.
By the bending of your rod you may know when he
bites, as also with your hand you will feel him make a
strong snatch *, then strike, and you will rarely fail, if
vou play him well; but if you manage him not dex¬
terously, he will break your line. The best time for
fishing is about nine in the morning, and the most pro¬
per season is the latter end of May, June, July, and the
beginning of August.
leeCj/- The BleakX, is an eager fish, caught with all
fc/t-sorts of worms bred on trees or plants 5 as also with
flies, paste, sheep’s blood, &c. They may be angled
for with half a score of hooks at once, if they can be
all fastened on; he will also in the evening take a na¬
tural or artificial fly. If the day be warm and clear,
VOL. VIII. Part II. f
there is no fly so good for him as the small fly at the Fishing,
top of the water, which he will take at any time of the v——'
day, especially in the evening $ but if the day is cold
and cloudy, gentles and caddis are the best $ about two
feet under water. No fish yields better sport to a young
angler than the bleak. It is so eager, that it will leap
out of the water for a bait.
There is another way of taking bleak, w hich is by
whipping them in a boat, or on a bank side in fresh
water in a summer’s evening, with a hazel top about
five or six feet long and a line twice the length of the
rod. But the best method is with a drabble, thus :
Tie eight or ten small hooks across a line two inches
above one another; the biggest hook the lowermost,
(whereby you may sometimes take a better fish), and
bait them with gentles, flies, or some small red worms,
by which means you may take half a dozen or more at
a time.
3. For the Bream ||, observe the following directions, || See Cy-
W'hich will also be of use in carp fishing.—Procure aboUtf™”^
a quart of large red worms ; put them into fresh moss,
well washed and dried every three or four days, feeding
them with fat mould and chopped fennel, and they will
be thoroughly scoured in about three weeks.
Let your lines be silk and hair, but all silk is the
best; let the floats be either swan-quills or goose-quills.
Let your plumb be a piece of lead in the shape of a
pear, with a small ring at the little end of it; fasten the
lead to the line, and the line hook to the lead, about
ten or twelve inches space between lead and hook will
be enough ; and take care the lead be heavy enough to
sink the float. Having baited your hook well with a
strong worm, the worm will draw the hook up and
down in the bottom, which will provoke the bream to
bite the more eagerly. It will be best to fit up three
or four rods or lines in this manner, and set them as
will be directed, and this will afford you much the
better sport. Find the exact depth of the water if
possible, that your float may swim on its surface directly
over the lead; then provide the following ground bait.
Take about a peck of sweet gross-ground malt; and
having boiled it a very little, strain it hard through a
bag, and carry it to the water side where you have
sounded ; and in the place where you suppose the fish
frequent, there throw in the malt by handfuls squeezed
hard together, that the stream may not separate it before
it comes to the bottom ; and be sure to throw it in at
least a yard above the place where you intend the hook
shall lie, otherwise the stream will carry it down too far.
Do this about nine o’clock at night, keeping some of
the malt in the bag, and go to the place about three
the next morning; but approach very warily, lest you
should be seen by the fish ; for it is certain that they
have their centinels watching on the top of the water,
while the rest are feeding below. Having baited your
hook so that the worm may crawl to and fro, the better
to allure the fish to bite, cast it in at the place where
you find the fish to stay most, which is generally in the
broadest and deepest part of the river, and so that it
may rest about the midst of your bait that is on the
ground. Cast in your second line so that it may resta g^0,.fc>
yard above that, and a third about a yard below it -Diet.
Let your rods lie on the bank, with some stones to keep
them down at the great ends ; and then withdraw your¬
self, yet not so far but that you cgn have your eye
4 O upon
FIS
[ 658 ]
F I S
r,.],;*, upon all the floats *, ami when you ece one bitten and
1 V^L. carried away, do not be too hasty to run in, but give
time to the fish to tire himself, and then touch him
gently. When you perceive the float sink, creep to
the waterside, and give it as much line as you can.
If it is a bream or carp, they will run to the other side •,
which strike gently, and hold your rod at a bent a lithe
while 5 but do not pull, for then you will spoil all ;
but you must first tire them before they can be landed,
for they are very shy. If there are any carps in the
river, it is an even wager that you take one or more of
them ; but if there are any pike or perch, they will be
sure to visit the ground bait, though they will not
touch it, being drawn thither by the great resort of the
small fish ; and until you remove them, it is in vain to
think of taking the bream or carp. In this case, bait
one of your hooks with a small bleak, roach, or gud¬
geon, about two feet deep from your float, with a little
red worm at the point of your hook ; and it a pike he
there, he will be sure to snap at it. This sport is good
till nine o’clock in the morning *, and in a gloomy day,
till night •, but do not frequent the place too much, lest
the fish grow shy.
f See Carp, 4. The Carp ■f'. A person who angles for carp
and Oypri- arm himself with abundance of patience, because ol its
nut‘ extraordinary subtility and policy ; they always choose
to lie in the deepest places, either of ponds or rivers,
•where there is but a small running stream.
Further, observe, that they will seldom bite in cold
weather } and you cannot be too early or too late at
the spot in hot weather ; and if he bite, you need not
fear bis hold •, for he is one of those leather-mouthed
fish that have their teeth in their throat.
Neither must you forget, in angling for him, to have
a strong rod and line j and since he is so very wary, it
will be proper to entice him, by baiting the ground with
a coarse paste.
He seldom refuses the red worm in March, the
caddis in June, or the grashopper in June, April, and
September.
This fish does not only delight in worms, but also in
sweet paste ; of which there is great variety j the best
is made of honey and sugar, and ought to be thrown
into the water some hours before you begin to angle ;
neither will small pellets thrown into the water two or
three days before be worse for this purpose, especially
if chickens guts, garbage, or blood mixed with bran
and cow dung, be also thrown in.
But more particularly, as to a paste very proper for
this use, you may make it in the manner following :
Take a sufficient quantity of flour, and mingle it with
veal, cut small, making it up with a compound of
honey j then pound all together in a mortar till they
are so tough as to hang upon the hook without washing
off. In order to effect which the better, mingle whitish
wool with it; and if you keep it all the year round,
add some virgin wax and clarified honey.
Again, If you fish with gentles, anoint them with
honey, and put them on your hook, with a deep scarlet
dipped in the like, which is a good way to deceive the
fish.
Honey and crumbs of wheat bread, mixed together,
make also a very good paste.
In taking a carp either in pond or river, if the angler
intends to add profit to his pleasure, he must take a
peck of ale-grains, and a good quantity of any blood
to mix with the grains, baiting the ground with it y—,
where he intends to angle. This food will wonder¬
fully attract the scale-fish, as carp, tench, roach, dace,
and bream.
Let him angle in a morning, plumbing his ground,
and angling for carp with a strong line ; the bait must
be either paste or a knotted red worm ; and by this
means he will have sport enough.
Description of proper Baits for the several sorts of Fim
referred to in the annexed Table.
Flies.~\ I. Stone fly, found under hollow stones at
the sides of rivers, is of a brown colour, with yellow
streaks on the back and belly, has large wings, and 1$
in season from April to July. 2. Green drake, lound
among stones by river sides, has a yellow body striped
with green, is long and slender, with wings like a but¬
terfly, his tail turns on his hack, and from May to mid¬
summer is very good. 3* Oak-fly, found in the body of
an oak or ash, with its head downwards, is of a brown
colour, and excellent from May to September. 4. Palmer
flv or worm, found on leaves of plants, is commonly
called a caterpillar, and when it conies to a fly is excel¬
lent for trout. 5. Ant fly, found in ant hills from June
to September. 6. The May fly is to be found playing
at the river side, especially against rain. 7. The black
fly is to be found upon every hawthorn after the buds
are come off.
Pastes.] I. Take the blood of sheep’s hearts, and
mix it with honey and flour worked to a proper con¬
sistence. 2. Take old cheese grated, a little butter
sufficient to work it, and colour it with saffron : in
winter use rusty bacon instead of butter. 3. Crumbs
of bread chewed or worked with honey or sugar, moist¬
ened with gum ivy water. 4. Bread chewed, and work¬
ed in the hand till stiff.
Worms.] 1. The earth hob, found in sandy ground
after ploughing ; it is white, with a red head, and big¬
ger than a gentle : another is found in heathy ground,
with a blue head. Keep them in an earthen vessel well
covered, and a sufficient quantity ol the mould they
harbour in. They are excellent from April to Novem¬
ber. 2. Gentles to he had from putrid flesh : let them
lie in wheat bran a few days before used. 3. Hag
worms, found in the roots of flags ; they are of a pale
yellow colour, are longer and thinner than a gentle, and
must be scowered like them. 4. Cow-turd bob, or clap
bait, found under a cow turd from May to Michael¬
mas ; it is like a gentle, hut larger. Keep it in its na¬
tive earth like the earth hob. 5. Caddis worm, or cod
bait, found under loose stones in shallow rivers ; they
are yellow, bigger than a gentle, with a black or blue
head, and are in season from April to July. Keep
them in flannel bags. 6. Lob worm, found in gardens;
it is very large, and has a red head ; a streak down the
hack, and a flat broad tail. 7. Marsh-worms, found in
marshy ground ; keep them in moss ten days before you
use them : their colour is a bluish red, and are a good
bait from March to Michaelmas. 8. Brandling red
worms, or blood worms, found in rotten dunghills and
tanners bark; they are small i'ed worms, very good for
all small fish, have sometimes a yellow tail, and are
called tag-tail.
Fish
uiiiog.
FIS [ 659 3 FIS
Fish and lmects.~\ 1. Minnow. 2. Gudgeon. 3.
Koacli. 4. Dace. 5. Smelt. 6. Yellow frog. 7.
Snail slit. 8. Grasliopper.
Fishing F/y, a bait used in angling for divers kinds
of fish. See Fishing.
The fly is either natural or artificial.
I. Natural flies are innumerable. The more usual
for this purpose are mentioned in the preceding page.
The^e are two ways to fish with natural flies j ei¬
ther on the surface of the water, or a little under¬
neath it.
In angling for chevin, roach, or dace, move not
your natural fly swiftly when you see the fish make at
it : but rather let it glide freely towards him with the
stream : but if it be in a still and slow water, draw the
fly slowly sidewise by him, which will make him eager¬
ly pursue.
II. The artificial fly is seldom used but in bluster¬
ing weather, when the waters are so troubled by the
winds, that the natural fly cannot be seen, nor rest upon
them. Of this artificial fly there are reckoned no less
than 12 sorts, of which the following are the principal.
I. For March, the dun fly j made of dun wool, and
the feathers of the partridge’s wing \ or the body made
of black wool, and the feathers of a black drake. 2. For
April, the stone fly ; the body made of black wool,
dyed yellow under the wings and tail. 3. For the
beginning of May, the ruddy fly ; made of x'ed wool,
and bound about with black silk, with the feathers of
a black capon hanging dangling on his sides next his
tail. 4. For June, the greenish fly ; the body made of
black wool, with a yellow list on either side, the wings
taken off the wings of a buzzard, bound with black
broken hemp. 5. The moorish fly, the body made of
duskish wool, and the wings of the blackish mail of a
drake. 6. The tawney fly, good till the middle of
June j the body made of tawney wool, the wings made
contrary one against the other of the whitish mail of
a white drake. 7. For July, the wasp fly ; the body
made of black wool, cast about with yellow silk, and
the wings of drakes feathers. 8. The steel fly ; good
in the middle of July ; the body made with greenish
wool, cast about with the feathers of a peacock’s tail,
and the wings made of those of the buzzard. 9. For
August, the drake fly ; the body made with black wool
cast about with black silk ; his wings of the mail of a
black drake, with a black head.
The best rules for artificial fly fishing are,
X. To fish in a river somewhat disturbed with rain :
or in a cloudy day, w'hen the waters are moved by a
gentle breeze : the south wind is best •, and if the wind
blow high, yet not so but that you may conveniently
guard your tackle, the fish will rise in plain deeps; but
if the wind be small, the best angling is in swift streams.
2. Keep as far from the water side as may be ; fish
down the stream with the sun at your back, and touch
not the water with your line. 3. Ever angle in clear
rivers, with a small fly and slender wings; but in mud¬
dy places, use a larger. 4. W hen, after rain, the wa¬
ter becomes, brownish, use an orange fly ; in a clear
day, a light-coloured fly ; a dark fly for dark waters,
&c. 5. Let the line be twice as long as the rod,
unless the river be encumbered with wrood. 6. For
every sort of fly, have several of the same differing in
colour, to suit with the different complexions of seve¬
ral waters and weathers. 7. Have a nimble eye, and
active hand, to strike presently with the rising of the
fish ; or else he will be apt to spue out the hook. 8.
Let the fly fall first into the water, and not the line,
which will scare the fish. 9. In slow rivers, or still
places, cast the fly across the river, and let it sink a
little in the water, and draw it gently back with the
current.
Salmon flies should be made with their wings stand¬
ing one behind the other, whether two or four. This
fish delights in the gaudiest colours that can be; chief¬
ly in the wings, which must be long, as well as the tail.
I ishing by means of birds, a method peculiar to the
Chinese, who train certain birds for the purpose in the
same manner as falcons are taught to pursue game.
For this purpose they have trained a species of pelican,
resembling the common corvorant, which they call the
Lcu-txe, or fishing bird. Sir George Staunton, who,
when the embassy was proceeding on the southern
branch of the great canal, saw those birds employed,
tells us, that on a large lake, close to the east side of
the canal, are thousands of small boats and rafts, built
entirely for this species of fishery. On each boat or
raft are ten or a dozen birds, which, at a signal from
the owner, plunge into the water; and it is astonishing
to see the enormous size offish with which they return,
grasped within their bills. They appeared to be so well
trained, that it did not require either ring or cord about
their throats to prevent them from swallowing any por¬
tion ol their prey, except what their master was pleased
to return to them for encouragement and food. The
boat used by these fishermen is of a remarkable light
make, and is often carried to the lake, together with
the fishing birds, by the men who are there to be sup¬
ported by it.
The same author saw the fishermen busy on the great
lake Wee-chaung-hee ; and he gives the following ac¬
count of a very singular method practised by them for
catching the fish of the lake without the aid of birds,
of net, or of hooks. To the one side of a boat a flat
board, painted white, is fixed, at an angle of about 41;
degrees, the edge inclining towards the water. On
moonlight nights the boat is so placed that the painted
board is turned to the moon, from whence the ravs of
light striking on the whitened surface, give to it the
appearance of moving water ; on which the fish be¬
ing tempted to leap on their element, the boatmen
raising with a string the board, turn the fish into the
boat.
Water-fowl are much sought after by the Chinese,
and are taken upon the same lake by the following in¬
genious device. Empty jars or gourds are suftered to
float about upon the water, that such objects may be¬
come familiar to the birds. The fisherman then wades
into the lake with one of these empty vessels upon his
head, and walks gently towards a bird ; and lifting up
his arm, draws it down below the surface of the water
without any disturbance or giving alarm to the rest,
several of whom he treats in the same manner, until he
fills the bag he had brought to hold his prey. The
contrivance itself is not so singular, as it is that the
same exactly should have occurred in the new continent,
as Ulloa asserts, to the natives ofLarthagena, upon the
lake Cienega de Tesias.
Fishing Floats, are little appendages to the line,
4 O 2 serving
FIS [
serving to keep the hook ami bait suspended at the
proper depth, to discover when the fish has hold of
them &c. Of these there are divers kinds j some
made5 of Muscovy duck quills, which are the best for
slow waters ; but for strong streams, sound cork, with¬
out flaws or holes, bored through with a hot iron, in¬
to which is put a quill of a fit proportion, is prefer-
660 ] FIS
able: pare the cork to a pyramidal form, and make it
smooth.
Fishing Hook, a small instrument made of steel wire,
of a proper form to catch and retain fish.
The fishing hook in general ought to be long in the
shank, somewhat thick in the circumference, the point
even and straight j let the bending be in the shank.
Fishing.
FIS r 661 ] FIT
lUbiflg, setting the hook on, use strong but small silk,
issnre*,- laying the hair on the inside of your hook $ for if it be
'■’Y*—on the outside,, the silk will fret and cut it asunder.
There are several sizes of these fishing hooks, some
big, some little: and of these, some have peculiar
names j as, I. Single hooks. 2. Double hooks j which
have two bendings, one contrary to the other. 3. Snap¬
pers, or gorgers, which are the hooks to whip the artifi¬
cial fly upon, or bait with the natural fly. 4. Springers,
or spring hooks ; a kind of double hooks, with a spring,
which flies open upon being struck into any fish, and
bo keeps its mouth open.
FismxG-Line, is either made of hair twisted ; or silk ;
or the Indian grass. The best colours are the sorrel,
white, and gray j the two last for clear waters, the first
for muddy ones. Nor is the pale watery green de-
spisable $ this colour is given artificially, by steeping
the hair in a liquor made of alum, soot, and the juice
of walnut leaves, boiled together.
Fishing Rod, a long slender rod or wand, to which
the line is fastened, for angling.—Of these there are
several sorts $ as, 1. A troller, or trolling rod, which
has a ring at the end of the rod, for the line to go
through when it runs off a reel. 2. A whipper, or
whipping rod *, a top rod, that is weak in the middle,
and top heavy, but all slender and fine. 3. A dropper $
which is a strong rod and very light. 4. A snapper,
or snap rod j which is a strong pole, peculiarly used
for the pike. 5. A bottom rod j being the same as
the dropper, but somewhat more pliable. 6. A snig¬
gling or procking stick j a forked stick, having a short
strong line, with a needle, baited with a lob worm :
this is only for eels in their holes.
Fishing Frog, or Angler. See Lophius.
Right of Fishing, and property of fish. It has been
held, that where the lord of the manor hath the soil
on both sides of the river, it is good evidence that he
hath a right of fishing 5 and it puts the proof upon him
who claims liberum piscariam : but where a river ebbs
and flows, and is an arm of the sea, there it is common
to all, and he who claims a privilege to himself must
l,i Diet, prove it; for if the trespass is brought for fishing there,
the defendant may justify, that the place where is bra-
chium mar is, in quo unusquisque subditus domini regis
habet et habere debet liberam piscariam. In the Severn
the soil belongs to the owners of the land on each side j
and the soil of the river Thames is in the king, but the
fishing is common to all. He who is owner of the soil
of a private river, hath separalis piscaria ; and he that
hath libera piscaria, hath a property in the fish, and may
bring a possessory action for them $ but communis pis¬
caria is like the case of all other commons. One that
has a close pond in which there are fish, may call them
pisces sms, in an indictment, &c. but he cannot call
them bona et catalla, if they be not in trunks. There
needs no privilege to make a fish pond, as there doth
in the case of a warren. See Franchise.
FISSURES, in Geology, certain interruptions, that
in a horizontal or parallel manner divide the several
strata of which the body of our globe is composed.
See Geology /We#.
Fissure of the Bones, in Surgery, is when they are
divided either transversely or longitudinally, not quite
through, but cracked after the manner of glass, by any
external force. See Surgery.
FISTULA, in the ancient music, an instrument of
the wind kind, resembling our flute or flageolet.
The principal wind instruments of the ancients, were
the tibia and the fistula. But how they were consti¬
tuted, wherein they differed, or how they were played
upon, does not appear. All we know is, that the
fistula was at first made of reeds, and afterwards of other
matters. Some had holes, some none ; some again were
single pipes ; others a combination of several j witness
the syringa of Pan.
I istula, in Suigery, a deep, narrow, and callous
ulcer, generally arising from abscesses.
It differs from sinus, in its being callous, the latter
not. See Surgery Index.
Fistula, in Farriery. See Farriery Index.
FISTULARIA, or Tobacco-Pipe Fish ; a genu**
of fishes, belonging to the order of abdominales. See
Ichthyology Index.
FIT. See Paroxysm.
Dr Cheyne is of opinion, that fits of all kinds, whe-*
ther epileptic, hysteric, or apoplectic, may be cured
solely by milk diet, of about two quarts of cows milk
a day, without any other medicine.
i I1CHES, in Husbandry, a sort of pulse, more ge¬
nerally known by the name of chick-pea. See Cicer,
Botany and Agriculture Index.
Fitches are cultivated either for feeding cattle, or
improving the land. They make a wholesome and nou¬
rishing food, whether given in the straw or thrashed
out. When sown only to improve the soil, they are
ploughed in just as they begin to blossom, by which
means a tough stiff clay soil is much enriched.
FITCH ET, a name used in some places for the
weasel, called also the foumart. See Mustela, Mam¬
malia Index.
Fistula
Fitz-Ste-
pben.
FITCHY, in Heraldry, (from the French fishl, i.e. -
fixed) ; a term applied to a cross when the lower branch
ends in a sharp point: and the reason of it Mackenzie
supposes to be, that the primitive Christians were wont
to carry crosses with them wherever they went $ and
when they stopped on their journey at any place, they
fixed those portable crosses in the ground for devotion’s
sake.
FITZ, makes part of the surname of some of the na¬
tural sons of the kings of England, as Fit%-roy; which
is purely French, and signifies the “ king’s son.”
FITZHERBERT, Sir Anthony, a very learned
lawyer in the reign of King Henry VIII. was descended
from an ancient family, and born at Norbury in Der¬
byshire. He was made one of the judges of the court
of common pleas in 1523 j and distinguished himself
by many valuable works, as well as by such an honour¬
able discharge of the duties of his office, as made him
esteemed an oracle of the law. His writings are, The
Grand Abridgement ; The Office and Authority of
Justices of Peace; the Office of Sheriffs, Bailifts of
Liberties, Escheators, Constables, Coroners, &c.; Of
the Diversity of Courts ; The New Natura Brevium ;
Of the Surveying of Lands, and The Book of Hus¬
bandry. He died in 1538.
FITZ-STEPHEN, William, a learned monk bf
Canterbury, of Norman extraction, but born of respect¬
able parents in the city of London. He lived in the
12th century ; and beingattached to the service of Arch¬
bishop Becket, was present at the time of his murder*
In .
F I X
Fi^s In the year 1174, he wrote in Latin, The Life of St
fi Thomas, archbishop and martyr; in which, as Becket
Fixlmi'lner. was a native of the metropolis, he introduces a clescrip-
' tion of the city of London, with a miscellaneous detail
of the manners and usages of the citizens: this is de¬
servedly considered as a great curiosity, being the ear¬
liest professed account of London extant. 1* itz-htephen
died in 1191.
FIVES, or Vives. See Farriery.
FIXATION, in Chemistry, the rendering any vo¬
latile substance fixed, so as not to fly ofl upon being ex¬
posed to a great heat: hence,
FIXED BODIES, are those which bear a considerable
degree of heat without evaporating, or losing any of
their weight. Some of the most fixed bodies are dia¬
monds, gold, &c.
Fixed or Fixable Air, an invisible and permanently,
elastic fluid, superior in gravity to common atmosphe¬
ric air and most other aerial fluids, exceedingly destruc¬
tive to animal life j produced in great quantities, natu¬
rally from combustible bodies, and artificially by many
chemical processes. From its acid properties it has
obtained the name of aerial acid, cretaceous acid, and
carbonic acid; from its noxious qualities, it has been
called mephitic air, or mephitic gas; and, from the cir¬
cumstance of being produced in vast quantities dming
the combustion of charcoal, it first obtained from A an
Helmont the name of gas sylvestre. The term fixed air
has been given from its property of readily losing its
elasticity, and fixing itself in many bodies, particularly
those of the calcareous kind } and though some objected
to the propriety of the term, the fluid in question is so
well known by the name of fixed air, that we choose still
to retain it. See CHEMISTRY Index. For an account
of the apparatus for impregnating water with fixed air
or carbonic acid, see Materia Medica Index.
Fixed Stars, are such as constantly retain the same
position and distance with respect to each other } by
which they are distinguished from erratic or wandering
stars, which are continually shifting their situation and
distance. The fixed stars are properly called stars;
the rest have the peculiar denomination of planet and
comet. See Astronomy Index.
FIXITY, or Fixedness, in Chemistry, is in a pe¬
culiar manner used for the affection opposite to volati¬
lity j i. e. the property whereby bodies bear the action
of the fire, without being dissipated in fumes.
F1XLMILLNER, Placidus, an eminent astro¬
nomer, was born at Achleiten near Linz, in Austria,
on the 28th of May, 1720. He received the rudiments
of his education in the monastery of Kremsmunster, of
which his uncle Alexander was abbot. Here he stu¬
died during six years, and delighted so much in drawing-
straight and curve-line figures, that his mother called
him the almanack-maker. He went afterward to Salz¬
burg, where he studied a regular course of philosophy,
and particularly turned his attention to mathematics
under a professor Stuard, whose method of teaching that
science was truly extraordinary, as he never made use
of any figures, and yet conveyed such a clear idea of
every proposition as made it perfectly easy. He was
admitted as a novice into Kremsmunster in 1737, and
the next year he took the solemn vow in presence of
his uncle. After being two years in this monastery,
during which time he devoted every leisure hour to the
F I X
study of mathematics and philosophy, he wrent to Fixlmilliuj
Salzburg to finish his studies in divinity and jurispru- —v—
dence, acquiring at that time a competent knowledge
of oriental and modern languages, history and antiqui¬
ties. In the year I 745, he obtained the degree of D. D.
after which he received priest’s orders in his own monas¬
tery, and was created professor of ecclesiastical law,
which office he held for 40 years, discharging the du¬
ties belonging to it till within a few days of his death,
He was also chosen dean of the higher schools, and re¬
gent of the young nobility, which he retained during
life.
He wrote a commentary on the Jus Canonicum, not-
withstanding his extensive epistolary correspondence,
and the management of the whole business of the mo¬
nastery 5 but this work was never published. He was,
by the intreaties of his friends, induced to publish his
Reipublicce sacra origines divincc, seu Ecclesice Christi
exteriorjunctura, imperium, ethierarchia, exprimigenia
ejus institutione eruta et demonstrata. His commendable
diligence procured him universal esteem, but it was his
knowledge of astronomy which rendered him illustri¬
ous. H is uncle Alexander fitted up an apartment for
containing the instruments necessary for the dissemina¬
tion of mathematical knowledge, and he also erected an
observatory, which was begun in 1748, and completed
in 1758, under the direction of Anselm Dering of Ems-
dorf, a celebrated architect. While the observatory
was building, Fixlmillner led a life of retirement and
severe study, his favourite subject during these ten
years being astronomy. When it was finished, one
Dobler, a celebrated mathematician, was appointed first
astronomer-, but the successors of Fixlmillner’s uncle
having discovered his extensive mathematical know¬
ledge, made him an offer of the astronomical depart¬
ment, and the sole direction of the observatory. This
place he accepted in the year 1762, still retaining
his chair as professor of ecclesiastical law. He was not
yet master of the learning which practical astronomy
requires, to remedy which defect he attentively perused
Lalande’s Exposition duCalculAstronomique, soon after
which he obtained the large astronomical Work of the
same great man, and in 1766 he published his Meridi-
anus speculce Astronomicce Cremifanensis, by which he
acquired considerable reputation. Ten years after this
period he gave the world his Decennium Asttonomicum,
containing many curious and important particulars re¬
specting the theory and practice of astronomy. His
Acta Astronomica Cremifanensis, which did not appear
till after his decease, still farther increased his astro¬
nomical reputation j and he was a large contributor
to many periodical publications in different coun¬
tries.
He made and collected a number of observations of
the planet Mercmy, which were at that period both
scarce and difficult, the importance of which was pub¬
licly acknowledged by Lalande, as they greatly assisted
him in constructing his tables ot that planet. Fixlmill¬
ner was one of the first astronomers who calculated the
orbit of the new planet Uranus (Georgium Sidus), and
his tables respecting it may be seen in the Berlin al¬
manack for 1789. He also proved the truth of what
was formerly conjectured, that the 34th star of Taurus,
which Flamstead observed in 1690, was the new planet.
It may be said of most philosophers, that they observe %
great
[ 662 ]
FLA [ 663 ] FLA
imiiincr prent deal, and calculate little, but tbe conduct of
jl Fixlmillner was exactly the reverse. He turned his
| attention to the observation of the solar spots more than
t"'~Y' any of his predecessors, which he noticed in the years
1767, 1776, 1777, I77^» and 1782, from which he
deduced important inferences respecting the revolution
of the sun on his axis.
He had » genius uncommonly adapted to the study
of mechanics, by which he was enabled to invent a new
micrometer, and a machine for grinding concentric
circles. As an additional proof of Ins profound inven¬
tive genius, he resided in the country, by which means
he was in a great measure deprived of literary assistance,
yet to the very close of life he was a singular instance
of the most indefatigable zeal, diligence, and persever¬
ance. He was little subject to the influence of the tur¬
bulent passions ;—perhaps less so than most other men.
Like the laws of nature, which it was his chief delight
to study, he was simple, uniform, and constant; and
such were the mildness and integrity of his character,
that he could not fail to acquire the love and esteem of
mankind. His high reputation never inspired him
with vanity, and he rather wished to conceal than to
propagate what was written in his praise. It gave gene¬
ral joy to his monastic brethren to celebrate the anni¬
versary of the fiftieth year of his residence in it, which
he did not long survive. His health was very much
impaired by his intense application, and he finished his
career on the 27th of August 1791, in the 71st year
of his age.
FLACCUS, Caius Valerius, an ancient Latin
poet, of whom we have very imperfect accounts remain¬
ing. He wrote a poem on the Argonautic expedition ;
of which, however, he did not live to finish the eighth
book, dying at about 30 years of age. John Baptisto
Pius, an Italian poet, completed the eighth book of
the Argonautics ; and added two more from the fourth
of Apollonius j which supplement was first added to
Aldus’s edition in 1523.
FLAGS, in the army, are small banners of distinc¬
tion stuck in the baggage waggons to distinguish the
baggage of one brigade from another ; and of one bat¬
talion from another; that they may be marshalled by
the waggon-master general according to the rank of
their brigades, to avoid the confusion that might other¬
wise arise.
Flag, in the marine, a certain banner or standard,
by which an admiral is distinguished at sea from the
inferior ships of his squadron ; also the colours by which
one nation is distinguished from another. See Plate
CCXVIII.
In the British navy, flags are either red, white, or
blue ; and are displayed from the top of the main-mast,
fore-mast, or mizen-mast, according to the rank of the
admiral. When a flag is displayed from the flag-staff
on the main-mast, the officer distinguished thereby is
known to be an admiral ; when from the foremast, a
vice-admiral; and when from the mizen-mast, a rear-
admiral.
The first flag in Great Britain is the royal standard,
which is only to he hoisted when the king or queen are
on board the vessel : the second is that of the anchor
of hope, which characterizes the lord high admiral, or
lords commissioners of the admiralty: and the third is
the union flag, in which the crosses of St George and
2.
St Andrew are blended. This last is appropriated to Flag,
the admiral of the fleet, who is the first military officer ^BSe^aH*
under the lord high admiral. * . tl~>’ r
The next flag after the union is that of the white
squadron at the main-mast head ; and the last, which
characterizes an admiral, is the blue, at the same mast
head.
For a vice-admiral, the first flag is the red, the se¬
cond the white, the third the blue, at the flag staff on
the fore-mast.
'I he same order proceeds with regard to the rear-ad¬
mirals, whose flags are hoisted on the top of the mizen-
mast : the lowest flag in our navy is accordingly the
blue on the mizen-mast.
To Lower or Strike the Flag^ in the marine, is to
pull it down upon the cap, or to take it in, out of
the respect, or submission, due from all ships or fleets
inferior to those any way justly their superiors. To
lower or strike the flag in an engagement is a sign of
yielding.
The way of leading a ship in triumph is to tie the
flags to the shrouds, or the gallery, in the hind part
of the ship, and let them hang down towards the wa¬
ter, and to tow the vessels by the stern. Livy relates,
that this was the way the Homans used those of Car¬
thage.
To Heave out the Flag., is to put out or put abroad
the flag.
To Hang out the White Flag, is to ask quarter; or
it shows when a vessel is arrived on a coast, that it has
no hostile intention, but comes to trade or the like.
The red flag is a sign of defiance and battle.
Flag is also used for a sedge, a kind of rush.
Coj'ti-Flag. See Gladiolus, Botany Index.
Sweet-scented Flag. See Acorus, Botany Index.
FLAG-Ojficers, those who command the several squa¬
drons of a fleet; such are the admirals, vice-admirals,
and rear-admirals.
The flag officers in our pay, are the admiral, vice-
admiral, and rear-admiral, of the white, red, and blue.
See Admiral, Flag, and Fleet.
FLAG-Ship, a ship commanded by a general or flag-
officer, who lias a right to cany a flag, in contradi¬
stinction to the secondary vessels under the command
thereof.
FLAG-Stone, a kind of sand-stone of a slaty structure,
on account of which it is much employed for the pur¬
pose of paving foot-paths or the floors of apartments in
which wood is unsuitable.
FLAGELLANTES, a set of wild fanatics who
chastised and disciplined themselves with whips in pub¬
lic.
The sect of the Flagellantes had its rise in Italy in
the year 1260 ; its author was one Rainier a hermit y
and it was propagated from hence through almost all
the countries of Europe. It was in all probability no
more than the effect of an indiscreet zeal. A great
number of persons of all ages and sexes made proces¬
sions, walking two by two with their shoulders bare,
which they whipped till the blood ran down, in order
to obtain mercy from God, and appease his indigna¬
tion against the wickedness of the age. They were
then called the devout j and having established a supe¬
rior, be was called x\\e general of the devotion. Though
the primitive Flagellantes were exemplary in point of
morals.
FLA [ 664 ] FLA
Fia 'ellan. nioraii, yet they were joined by X turbulent rabble
tes who were infected with the most ridiculous and impi-
li ous opinions j so that the emperors and pontiffs thought
Flake. Droper t0 pUt an end to this religious frenzy, by de-
^ ^ f claring all devout whipping contrary to the divine law,
and prejudicial to the soul’s eternal rest.
This sect revived in Germany towards the middle
of the next century, and rambling through many
provinces, occasioned great disturbances. They held
among other things, that flagellation was of equal
virtue with baptism and the other sacraments 5 that
the forgiveness of all sins was to be obtained by it
from God without the merits of Jesus Christ; that the
old law of Christ was soon to be abolished, and that a
new law enjoining the baptism of blood to be admini¬
stered by whipping was to be substituted in its place ;
upon which Clement VII. by an injudicious as well as
unrighteous policy, thundered out anathemas against
the Flagellantes, who were burnt by the inquisitors in
several places j but they were not easily extirpated.
They appeared again in Thuringia and Lower Saxony
in the 15th century, and rejected not only the sacra¬
ments, but every branch of external worship, and pla¬
ced their only hopes of salvation in faith and flagella¬
tion, to which they added other strange doctrines con¬
cerning evil spirits. Their leader Conrad Schmidt and
many others were committed to the flames by German
inquisitors in and after the year I4I4*
FLAGEOLET, or Flajeolet, a little flute, used
chiefly by shepherds and country people. It is made of
box or other hard wood, and sometimes of ivory j and
has six holes besides that at the bottom, the mouth¬
piece, and that behind the neck.
FLAIL, an instrument for thrashing corn. It con¬
sists of the following parts. I. The hand-stafl, or piece
held in the thrasher’s hand. 2. The swiple, or that part
which strikes out the corn. 3. The caplins, or strong
double leathers, made fast to the tops of the hand-staff
and swiple. 4. The middle band, being the leather
thong or fish skin that ties the caplins together.
FLAIR, in sea language. The seamen say that
the work doth flair over, when a ship is housed in
near the water, so that the work hangs over a little too
much, and this is let out broader aloft than the due
proportion will allow.
FLAKE, in the cod fishery, a sort of scaffold or
platform, made of hurdles, and supported by stanchions,
used for drying cod fish in Newfoundland. These
flakes are usually placed near the shores of fishing har¬
bours.
Flake, in Gaj'denivg, a name given by the florists
to a sort of carnations which are of two colours only,
and have very large stripes, all of them going quite
through the leaves.
VVhite Flake, in Faulting, is lead corroded by means
of the pressing of grapes, or a ceruse prepared by
the acid of grapes. It is brought from Italy, and
far surpasses, both with regard to the purity of its
whiteness and the certainty of its standing, all the ce¬
ruse or white lead made with us in common. It is
used in oil or varnish painting for all purposes where
a very clean white is required. The white flake should
be procured in lumps as it is brought over and levi-
gated by those who use it j because that which the co-
1 iourmen sell in a prepared state is levigated and mixed
3
up with starch, and often with white lead, and worse £]8j((!
sophistications. Q
FLAMBEAU, or Flamboy, a luminary made of t Flame,
several thick wicks, covered over with wax, serving to 'r^
burn at nights in the streets 5 as also at funeral proces¬
sions, illuminations, &c.
Flambeaux differ from links, torches, and tapers.-—
They are made square, sometimes of white wax and
sometimes of yellow. They usually consist of four
wicks or branches near an inch thick, and about three
feet long, made of a sort of coarse hempen yarn half
twisted. They are made with the ladle much as torches
or tapers are; viz. by first pouring the melted wax on
the top of the several suspended wicks, and letting it
run down to the bottom. This they repeat twice.
After each wick has thus got its proper cover of wax,
then lay them to dry; then roll them on a table, and
so join four of them together by means of a red hot
iron. When joined, they pour on more wax till the
flambeau is brought to the size required, which is
usually from a pound and a half to three pounds. The
last thing is to finish their form or outside, which they
do with a kind of polishing instrument of wood by
running it along all the angles formed by the union of
the branches.
The flambeaux of the ancients were different from
ours. They were made of woods dried in furnaces or
otherwise. They used divers kinds of wood for this
purpose 5 the wood most usually was pine. Pliny says,
that in his time they frequently also burnt oak, elm,
and hazel. In the seventh book of the iEneid, men¬
tion is made of a flambeau of pine; and Servius on
that passage remarks, that they also made them of the
cornel-tree. -'I
FLAMBOROUGH head, in Geography, a cape
or promontory on the eastern coast of Yorkshire, five
miles east of Burlington, and 2x5 from London.-—
E. Long. 20°. N. Lat. 54. 15.—This was the Fleam-
burg of the Saxons $ so called, as some think, from the
lights made on it to direct the landing of Ina, who in
547 joined his countrymen in these parts with a large
reinforcement from Germany, and founded the king¬
dom of Northumberland. In the time of Edward the
Confessor, Flamborough was one of the manors of
Harold, earl of the West Saxons, afterwards king of
England. On his death, the Conqueror gave it to
Hugh Lupus ; who, in perpetual alms, bestowed it
on the monastery of Whitby.—The town is on the
north side, and consists of about 150 small houses, en¬
tirely inhabited by fishermen 5 few of whom, as is said,
die in their beds, but meet their fate in the element
they are so conversant in. The cliffs of the Head are
of a tremendous height and amazing grandeur. Be¬
neath are several vast caverns ; some closed at the end,
others pervious, formed with a natural arch. In some
places the rocks are insulated, and of a pyramidal
figure, soaring up to a vast height. The bases of most
are solid, but in some pierced through and arched.
The colour of all these rocks is white, from the dung
of the innumerable flocks of migratory birds, which
quite cover the face of them, filling every little projec¬
tion, every hole that will give them leave to rest.
FLAME, is a general name for every kind of lu¬
minous vapour, provided the light it emits hath any
considerable degree of intensity. The xi^mejiame, how¬
ever,
FLA [ 665 ] FLA
iflame. ever* i3 mos^ generally applied to such as are of a coni-
-n/S cal figure, like those arising from our common fires j
without this they are commonly called luminous vapours,
or simple lights.
According to Sir Isaac Newton, flame is only red-
hot smoke, or the vapour of any substance raised from
it by fire, and heated to such a degree as to emit
light copiously. This definition seems to be the most
accurate and expressive of any. It is certain, that
bodies are capable of emitting flame only in proportion
to the quantity of vapour that rises from them. Thus
wood, coals, &c. which emit a great quantity of
vapour, flame violently 5 while lead, tin, &c. which
emit but a small fume, can scarce be perceived to flame
at all.
This rule, however, is by no means to be depended
upon in all cases. Some vapours seem to be in their
own nature uninflammable, and capable of extinguish¬
ing flame j as those of water, the mineral acids, sal-am¬
moniac, arsenic, &c.: while others take fire on the
slightest approach of a flaming substance j such as ether,
spirit of wine, &c. These last-mentioned substances
also exhibit a remarkable phenomenon j namely, that
they cannot be made to flame without the approach of
some substance actually in flames beforehand. Thus,
spirit of wine poured on a red-hot iron, though instant¬
ly dissipated in vapour, will not flame j but if a burn¬
ing candle touch its surface, the whole is set in a flame
at once. The case is otherwise with oils, especially
those of the grosser kind $ for the vapours will readily
be changed into flame by the mere increase of heat,
without the approach of any flaming substance.
There is, however, no kind of vapour, perhaps, that
is incapable of being converted into flame, provided it
is exposed to a sufficient degree of heat. Thus the va¬
pour of water made to pass through burning coals
produces an exceedingly strong and bright flame.—
It is remarkable, that this kind of vapour seems to be
more powerful than almost any other in absorbing heat,
and detaining it in a latent state. When any quantity
of aqueous vapour is condensed, more heat will be sepa¬
rated from it than would have been sufficient to heat an
equal bulk of iron red hot.—It is most probably to this
property which all vapours have of absorbing heat, and
detaining it in a latent state, that we are to attribute
the phenomena of flame, and also the exceeding great
elasticity of steam. It is certain, that vapours, of wa¬
ter at least, have a much greater power of absorbing
and retaining heat, than the water from which they are
raised. In open vessels, water cannot be heated more
than to 212 degrees of Fahrenheit’s thermometer j but
in Papin’s digester, where the vapour is forcibly con¬
fined, it has been heated to 400 of the same degrees ;
and, no doubt, might have been heated a great deal
more, had the vessels been strong enough to bear the
expansile force of the steam. On opening the vessels,
however, the excess of heat was found to have resided
entirely in the vapour; for the water in the vessel very
soon sunk down to 212, while the steam issued forth
with great violence.
From these experiments it appears, that the steam of
water, after it has absorbed as much heat in the latent
state as it can contain, continues to absorb or detain
among its particles, an unlimited quantity of sensible
heat; and if the steam could be confined till this quan-
Vol. VIII. Part II. t
tity became great enough to be visible by emission of Flame,
light, there cannot be the least doubt that the vapour Flamen.
would then be converted into flame. >r~~
In what manner the heat is detained among the par¬
ticles of steam, is perhaps impossible to be explained;
but to this heat we must undoubtedly ascribe the vio¬
lent expansive force of steam of every kind. It seems
probable, that when smoke is converted into flame, the
latent heat with which the vapour had combined, or
rather that which made an essential part of it, breaks
forth, and adds to the quantity of sensible heat which
is already present. This seems probable, from the
sudden explosion with which all flames break out. If
a vessel full of oil is set over the fire, a smoke or vapour
begins to arise from it; which grows gradually thicker
and thicker; and at last begins to shine in some places
very near the surface of the oil, like an electric light,
or sulphur just kindled. At this time the oil is very
hot, as well as the steam which issues from it. But this
last is continually giving off its sensible heat into the at¬
mosphere ; so that at the distance of an inch or two
from the surface of the oil, the heat of the steam will
not exceed 400 degrees of Fahrenheit, or perhaps may
not be so much; but if a burning candle is held in the
steam for a moment, the whole is immediately con-
vei’ted into flame, with something like an explosion ;
after which the oil burns quietly until it is all con¬
sumed. The flame, as soon as it appears, is not only
much hotter than the steam from whence it was pro¬
duced, but even than the oil which lies below it.
Whence, then, has this sudden and great increase of
heat arisen ? It could not be the heat of the va¬
pour, for that was greatly inferior; nor could it be
communicated from the oil, for that could communi¬
cate no more than it had to itself. The candle, in¬
deed, would communicate a quantity of heat to the va¬
pour which touched its flame ; but it is impossible that
this quantity should extend permanently over a surface
perhaps 100 times larger than the flame of the candle,
in such a manner as to make every part of that surface
equally hot with the flame of the candle itself; for this
would be to suppose it to communicate 100 times more
heat than really was in it. The heat therefore must
have originally resided in the vapour itself; and as, in
the freezing of wTater, its latent heat is extricated and
becomes sensible, and the water thereupon loses its flui¬
dity ; so, in the ascension of vapour, the latent heat
breaks forth with a bright flash, and the vapour is then
totally decomposed, and converted into soot, ashes, or
water, according to the different nature of the sub¬
stances which produce it, or according to the intensity
of the heat.—Several other hypotheses have been in¬
vented to solve the phenomena of burning and flaming-
bodies; for an account of which, see Ignition and
Heat, Chemistry Index.
Flames are of different colours, according to the sub¬
stances from which they are produced. Thus, the
flame of sulphur and spirit of wine is blue ; the flame
of nitre and zinc, of a bright white ; that of copper, of
a greenish blue, &c.—These varieties aft'ord *n oppor¬
tunity of making a number of agreeable representations
in fireworks, which could not be done if the flame
produced from every difl’erent substance was of the same
colour. See Pyrotechnics.
FLAMEN, in Roman antiquity, the name of an
4 P order
FLA I 6(
ri«*en order of priests, instituted by Romulus or Numa : au-
y thors not being agreed on this head. .
Jlaminhu. They were originally only three, viz. the JUcimen
' * Diulis, Flamen Martialis, and Flamen Quirinalis. The
Flamen Dialis was sacred to Jupiter, and a person of
the highest consequence and authority in the state.
He discharged several religious duties which properly
belonged to the kings, and was honoured with many
eminent privileges beyond all other officers, hut was
obliged to observe several superstitious restraints. The
Flamen Martialis tvas sacred to Mars, and was ordain¬
ed to inspect the rites of that god. The ITamen
Quirinalis was sacred to, and superintended the rites
of, Quirinus Romulus. The Flamines last mentioned,
though of high authority, were much inferior to the
.Flamen Dialis. All three were chosen by the people,
and consecrated by the Fontifex Maximus. In latter
times several priests of the same order and name were
added to them, but inferior in power. The whole
number at last amounted to 15; the three first of
whom were senators, and called Flcimines majores y the
other 12 taken from among the people, being deno¬
minated Flamincs wifnorw.—-Some authors tell us the
Romans had a Flamen for every deity they worship¬
ped. The greater Flamines wore the robe edged with
purple, like the great magistrates; had an ivory chair,
and a seat in the senate. They wore a little band ot
thread about their heads, whence their name is said to
be derived, quasi Filamines.—F^he wife of the Flamen
Dialis was called Flammica, and wore a flame-colour¬
ed habit, on which was painted a thunderbolt, and
above her head-dress she had green oak boughs, to
indicate that she belonged to Jupiter the thunderer, to
whom the oak was sacred. The Flamines wore each
of them a hat or cap called Flammeum or Apex.
FLAMINGO, in Ornithology. See Phoenicop-
terus, Ornithology Index.
FLAMINIUS, or Flamininus, T. Q. a cele¬
brated Roman, raised to the consulship in the year of
Rome 554, though under the age of 30. He was trained *
in the art of war agai nst Hannibal; and he showed him¬
self capable in every respect to discharge with honour
the great office with which he was intrusted. He was
sent at the head of the Roman troops against Philip king
of Macedon, and in his expedition he met with uncom¬
mon success. The Greeks gradually declared themselves
his firmest supporters; and he totally defeated Philip on
the confines of Epirus, and made all Locris, Phocis,
and Thessaly, tributary to the Roman power. He
granted peace to the conquered monarch, and proclaim¬
ed all Greece free and independent at the Isthmian
games. This celebrated action procured the name of
Patrons of Greece to the Romans, and insensibly paved
their way to universal dominion. Flaminius behaved
among them with the greatest policy; by his ready
compliance to their national customs and prejudices, he
gained uncommon popularity, and received the name of
father and deliverer of Greece. He was afterwards
sent ambassador to King Prusias, who had given refuge
to HantUbal; and there his prudence and artifice hasten¬
ed out of the world a man who had long been jhe ter¬
ror of the Romans. Flaminius was found dead in his
bed, after a life spent in the greatest glory, in which he
had imitated with success the virtues of his model
Scipio.
56 3 FLA
Flaminius, or Flaminio, Mark Antony, one of FlamuH;
the best Latin poets in the 16th century, of Imola in g
Italy, son and grandson of very learned men. The FlamsUi
pope had chosen him secretary to the council in 1545 ; ~
but he refused that employment, because, favouring
the new opinions, he would not employ his pen in
an assembly where he knew these opinions were to he
condemned.— He paraphrased 38 of the psalms in Latin
verse, and also wrote notes on the Psalms ; and some let¬
ters and poems which are esteemed. He died at Rome
in 1550.
FLAMSTED, a town of Hertfordshire in England,
five miles from St Albans and Dunstable, stands on the
river Verlam, and was of old called Verlamstede. The
land in the vicinity is a clay so thickly mixed with
flints, that, after a shower, nothing appears hut a heap
of stones; and yet it bears good corn even in dry sum¬
mers. This fertility is imputed to a warmth in the flint, .
which preserves it from cold in the winter; and to its
closeness, which keeps it from the scorching rays of the
sun in the summer. Edward VI. when an infant, was
brought hither for his health ; and, it is said, the bed¬
stead he lay on, which is curiously wrought, is still pre¬
served in the manor bouse near the town.
FLAMSTEED, John, an eminent English astrono¬
mer, and the first who obtained the appointment of as¬
tronomer-royal, was born at Derby in the year 1646.
He was educated at the free school of Derby, where he
was head scholar at 14 years of age, at which period his
constitution, naturally tender and delicate, was much
tried by a severe illness. When some of his companions
went to the university, the state of his health prevented
him from accompanying them. He afterwards met with
a book T)e Sphcera, written by John Sacrobosco, which
was perfectly suited to the natural turn of his genius,
and therefore he perused it with uncommon satisfaction,
translating as much of it into English as he thought
could be necessary for him; and from the Astromma
Carolina; of Strut he learned the method of calculating
eclipses, and ascertaining the places of the planets.^ Mr
Hatton, a mathematician, sent him Kepler’s labulce
Rudolphince, and RicciolVs Almagestum Novum, together
with some other astronomical works to which he was
as yet a stranger. In 1669 he calculated an eclipse of
the sun, which had been omitted in the Ephemerides
for the following year, together with five appulses of the
moort to fixed stars, and sent them to Lord Brouncker,
president of the Royal Society, who submitted them to
theexamination of that learned body, by which theywere
greatly applauded, and he received a letter of thanks
from Mr Oldenburg the secretary. He likewise receiv¬
ed a letter of thanks from Mr Collins, one of the mem¬
bers. In 1670 he was invited to come up to London
by his father, that he might become personally acquaint¬
ed with his learned correspondents, ot which he gladly
accepted, and had an interview with Mr Oldenburg
and Mr Collins, by the latter of whom he was intro¬
duced to Sir Jonas Moore, who became the warm friend
and patron of Mr Flamsteed. In consequence of this
journey he became acquainted with many astronomical
instruments, and was presented by Sir Jonas Moore with
Tovvnley’s micrometer, who also assisted him in pro¬
curing glasses at a moderate rate for the construction of
telescopes. On his way home again he returned by
Cambridge, where he paid a visit to the celebrated Dr
JBarrow
FLA t 667 ] FLA
wsteed, Barrow anti Sir Isaac, then Mr Newton, and entered
mders- a student of Jesus college.
In the year 1672, he made large extracts from the
letters of Gascoigne and Crabtree, by which his know¬
ledge of dioptrics was very much improved ; and during
the same year he made a number of celestial observa¬
tions when the weather would permit, which were af¬
terwards published in the Philosophical Transactions.
In 1673 he composed a treatise on the true and ap¬
parent diameters of the planets, when at their greatest
and least distance from the earth, which even the great
Newton did not scruple to borrow, and made some use
of it in his Principia in 168 5'. He published an Ephe-
meris in 1674, in which he exposed the folly and
absurdity of astrology, and the same year he drew up a
table of the tides for the use of the king, with,an astro¬
nomical account of their ebbing and flowing, which Sir
Jonas Moore assured him would be w'ell received by his
majesty. Sir Jonas received from Mr Flamsteed a pair
of barometers, with directions how to use them, which
he presented to the king and the duke of York, to whose
notice he embraced every opportunity of introducing
Mr Flamsteed.
Having taken the degree of M. A. at Cambridge,
he formed the resolution of entering into holy orders,
when Sir Jonas wrote to him to come to London,
where he had an appointment for him very different
from that of the church. Betas he found that nothing
could make him abandon the resolution he bad formed,
he obtained a situation for him which was perfectly con¬
sistent with the character of a clergyman. This was
the new office of astronomer to the king, with a salary
of lool.perannum. Pie received ordination atEly-house
by Bishop Gunning, in Easter 1675 ; and on the 10th
of August in the same year the foundation stone of the
royal observatory at Greenwich was laid, which receiv¬
ed the designation of Flamsteed house, in honour of the
first astronomer royal. Till this edifice was erected, he
made his observations in the queen’s house at Green¬
wich, and in 1681 his Doctrine of the Sphere was pub¬
lished by Sir Jonas Moore in his System of the Mathe¬
matics. Notwithstanding his extraordinary merit, he
never rose higher in the church than to the living of
Burslow in Surrey, although he was deservedly esteem¬
ed by the greatest men in the nation. He correspond¬
ed with the great Newton, Dr Plalley, Mr W. Moly-
neux. Dr Wallis, and many others 5 and M. Cassini
and he imparted their discoveries to each other with
the utmost confidence and cordiality. But none of his
works contributed so much to render his fame immortal
as his “ Historia Caelestis Britannica,” in three vols.
folio. Mr Flamsteed was suddenly carried off by a
strangury on the 31st of December 1719 j and not¬
withstanding the extreme delicacy of his constitution and
incessant labours, he reached the 73d year of his age.
FLANDERS, a province of the Netherlands, bound¬
ed by the German sea and the United Provinces on the
north, by the province of Brabant on the east, by Hain-
ault and Artois on the south, and by another part of
Artois and the German sea on the west 5 being about
60 miles long and 50 broad.
Flanders is a perfectly champaign country, with not Flanders
a rising ground or hill in it, and watered with many Flanel.'
fine rivers and canals. Its chief commodities are fine v—v——^
lace, linen, and tapestry.
In this country some important arts were invented
and improved. Weaving in general was greatly im¬
proved, and that of figures of all sorts in linen was in¬
vented $ also the art of dyeing cloths and stuffs, and
of oil colours; the curing of herrings, &c. The ma¬
nufactures of this country are not now in the flourishing
state they were formerly ; yet silk, cotton, and woollen
stufts, &c. are still manufactured here in great quanti¬
ties. This province had counts of its own from the
ninth century to the year 1369, when it went by mar¬
riage to the dukes of Burgundy j and afterwards from
them, by marriage also, to the house of Austria. France,
in 1667, seized the southern part ; and the States Ge¬
neral obtained the northern. It was overrun by the
French in 1794, but was united to the new kingdom of
the Netherlands in 1814.
For a more particular history of Flanders, see the
article Netherlands.
FLANFL, or Flannel, a kind of slight, loose,
woollen stuff, composed of a woof and warp, and wove
on a loom with two treddles, after the manner of
baize.
Dr Black assigns as a reason why flanel and other
substances of the kind keep the body warm, that they
compose a rare and spongy mass, the fibres of which
touch each other so lightly, that the heat moves slowly
through the interstices, which being filled only with air,
and that in a stagnant state, give little assistance in con¬
ducting the heat. From the experiments of Count
Rumford, it appears, that there is no relation betwixt
the power which the substances usually worn as clothing
have of absorbing moisture, and that of keeping the
body warm. Having provided a quantity of each of
these substances mentioned below, he exposed them,
spread out upon clean china plates, for the space of 24
hours to the warm and dry air of a room which had
been heated by a German stove for several months, and
during the last six hours had raised the thermometer to
85° of Fahrenheit ; after which he weighed equal quan¬
tities of the different substances with a very accurate
balance. They were then spread out upon a china
plate, and removed into a very large uninhabited room
upon the second floor, where they were exposed 48
hours upon a table placed in the middle of the room,
the air of which was at 450 of Fahrenheit. At the end
of this space they were weighed, and then removed in¬
to a damp cellar, and placed on a table in the middle
of the vault, where the air was at the temperature of
450, and which by the hygrometer seemed to be fully
saturated with moisture. In this situation they were
suffered to remain three days and three nights $ the
vault being all the time hung round with wet linen
cloths, to render the air as completely damp as possible.
At the end of three days they were weighed, and the
weights at the different times were found as in the fol¬
lowing table.
4 P 2 Sheep’s
FLA
Flauel.
Weight af
ter being
diied in the
hot room.
Parts
1000
Weight af-
tercoming
out of the
cold room.
1084
IO72
1065
1067
ioj7
i°54
1046
1044
Weight af¬
ter reuiain-
ng 72 b. in
the vault.
1163
1125
mi
1112
1107
1103
1102
1082
Sheep’s wool
Beaver’s fur
The fur of a Russian hare
Eider down
f Raw single thread j
Silk < Ravellingsofwhite 7 1
[ taffety. V
r Fine lint
Linen-J Ravellingsoffine 1
linen j
Cotton wool io43 1089
Ravellings of silver lace j 1000 1000
On these experiments our author observes, that though
linen, from the apparent ease with which it receives
dampness from the atmosphere, seems to have a much
greater attraction for water than any other j yet it
would appear from what is related above, that those
Bodies which receive water in its unelastic form with
the greatest ease, or are most easily wet, are not those
which in all cases attract the moisture of the atmo¬
sphere with the greatest avidity. “ Perhaps (says he),
the apparent dampness of linen to the touch, arises
more from the ease with which that substance parts
with the water it contains, than from the quantity of
water it actually holds : in the same manner as a body
appears hot to the touch, in consequence of its parting
freely with its heat j while another body which is really
at the same temperature, but which withholds its heat
with great obstinacy, affects the sense of feeling much
less violently. It is well known that woollen clothes,
such as flanels, &c. worn next the skin, greatly promote
insensible perspiration. May not this arise principally
from the strong attraction which subsists between wool
and the watery vapour which is continually issuing from
the human body ? That it does not depend entirely on
the warmth of that covering, is clear j for the same de¬
gree of warmth produced by wearing more clothing of
a different kind, does not produce the same effect. The
perspiration of the human body being absorbed by a
covering of flanel, it is immediately distributed through
the whole thickness of that substance, and by that
means exposed, by a very large surface, to be carried
off by the atmosphere j and the loss of this watery va¬
pour which the flanel sustains on the one side by evapo¬
ration, being immediately restored from the other, in
consequence of the strong attraction between the flanel
and this vapour, the pores of the skin are disencumber¬
ed, and they are continually surrounded by a dry and
salubrious atmosphere.”
Our author expresses his surprise, that the custom of
wearing flanel next the skin should not have prevailed
more universally. He is confident it would ’prevent a
number of diseases ; and he thinks there is no greater
luxury than the comfortable sensation which arises from
wearing it, especially after one is a little accustomed to
it. “ It is a mistaken notion (says he), that it is too
warm a clothing for summer. I have worn it in
the hottest climates, and at all seasons of the year ;
and never found the least inconvenience from it. It
is the warm bath of perspiration confined by a linen
shirt, wet with sweat, whiclf renders the summer heats
668 ] FLA
of southern climates so unsupportable; but flanel pro- p]anel
motes perspiration, and favours its evaporation ; and j)
evaporation, as is well known, produces positive cold. Flats.
It has been observed that new flanel, after some time
wearing, acquires the property of shining in the dark,
but loses it on being washed. Philosophical Transact
tions, N° 483. § 7.
FLANK, or Flanc, in the manege, is applied to
the sides of a horse’s buttock, &c. In a strict
sense, the flanks of a horse are the extremes of the
belly, where the ribs are wanting, and are below the
loins.
The flanks of a horse should he full, and at the top
of each a feather. The distance between the last rib
and haunch-bone, which is properly the flank, should
be short, which they term ivell coupled, such horses be¬
ing most hardy, and fit to endure labour.
A horse is said to have no flank if the last of the
short ribs be at a considerable distance from the haunch-
bone ; as also when the ribs are too much straitened in
their compass.
Flank, in War, is used by way of analogy for the
side of a battalion, army, &c. in contradistinction to
the front and rear.
To attack the eneiyiy inflank, is to discover and fire
upon them on the side. See File.
Flank, in Fortification, is a line drawn from the
extremity of the face towards the inside of the work.
Or, flank is that part of a bastion which readies from
the curtain to the face, and defends the opposite face,
the flank, and the curtain. See Fortification.
Oblique or Second Flank, or Flank oflthe Curtain,
is that part of the curtain from whence the face of the
opposite bastion can be seen, being contained between
the lines rasant and fichant, or the greater and less lines
of defence ; or the part of the curtain between the
flank and the point where the fichant line of defence
terminates.
Covered, Low, or Retired Flank, is the platform of
the casement which lies hid in the bastion ; and is other¬
wise called the Orillon.
Fichant Flank, is that from whence a cannon play¬
ing, fires directly on the face of the opposite bas¬
tion.
Rasant or Raxant Flank, is the point from whence
the line of defence begins, from the conjunction of
which with the curtain, the shot only raseth the face
of the next bastion, which happens when the face can¬
not be discovered but from the flank alone.
FLAT, in sea-language, denotes a level ground ly¬
ing at a small depth under the surface of the sea, and
is also called a shoal or shallow.
Flat-Bottomed Boats are such as are made to swim in
shallow water, and to carry a great number of troops,
artillery, ammunition, &c. They are constructed with
a 12-pounder, bow-chase, and an 18 pounder, stern-
chase ; their keel is from 90 to 100 feet, and from 12
to 24 feet beam : they have one mast, a large square
main-sail, and a jib-sail j are rowed by 18 or 20 oars,
and can carry 400 men eacli. The gun takes up one
bow, and a bridge the other, over which the troops are,
to march. Those that carry horses have the fore-part
of the boat made to open when the men are to mount
and ride over a bridge.
FLATS, in Music. See Interval.
FLATUS,
FLA [ 669 ] FLA
iFlaius FLATUS, X LATULENCE, in Medicine ; vapours ge-
B nerated in the stomach and intestines, chiefly occasion-
FJax. ed by a weakness of these parts. They occasion disten-
sions, uneasy sensation, and sickness, and often a con¬
siderable degree of pain. See MEDICINE Index.
FLAVEL, John, an eminent non-conformist mini¬
ster, was educated at University-college, in Oxford
and became minister first of Deptford, and afterwards of
Dartmouth in Devonshire, where he resided the great¬
est part of his life, much respected and admired for his
preaching ; although he was persecuted on account of
his principles, when in 1685, several of the aldermen of
the town, attended by the rabble, carried about a ridi¬
culous effigy of him, to which were affixed the Bill of
Exclusion and the Covenant. Upon this occasion, he
thought it prudent to withdraw from the town j not
knowing what treatment he might meet with from a
riotous mob, headed by magistrates who were them¬
selves among the lowest of mankind. Part of his Diary,
printed with his Remains, must give the reader a high
idea of his piety. He died in 1691, aged 61 ; and af¬
ter his death, his works, which consisted of many pieces
of practical divinity, were printed in two volumes folio.
Among these, the most famous are his “ Navigation
Spiritualized, or a New Compass for seamen, consisting
of 32 points of pleasant observations and serious reflec¬
tions,” of which there have been several editions in
8vo ; and his “ Husbandry Spiritualized, &c. with oc¬
casional meditations upon beasts, birds, trees, flowers,
rivers and several other objects,” of which also there
have been many editions in 8vo.
FLAX, in Botany. See Linum, Botany Index.
The following particulars with regard to the manner
of raising flax, have been some years past warmly re¬
commended by the trustees for fisheries, manufactures,
and improvements in Scotland.
Of the choice of the Soil, and preparing the ground for
Flax. A skilful flax-raiser always prefers a free open
deep loam j and all grounds that produced the preced¬
ing year a good crop of turnip, cabbage, potatoes,
barley, or broad clover, or have been formerly laid
down rich, and kept for some years in pasture.
A clay soil, the second or third crop after being
limed, will answer well for flax ; provided, if the
ground be still stiff, that it be brought to a proper
mould, by tilling after harvest to expose it to the win¬
ter frosts.
All new grounds produce a strong crop of flax, and
pretty free of weeds. W hen a great many mole heaps
appear upon new ground, it answers the better for flax,
after one tilling.
Flax seed ought never to be sown on grounds that
are either too wet or dry; but on such as retain a na¬
tural moisture : and such grounds as are inclined to
weeds ought to be avoided, unless prepared by a care¬
ful summer fallow.
If the linseed be sown early, and the flax not al¬
lowed to stand for seed, a crop of turnip mav be got
after the flax that very year} the second year a crop
of bear or barley may be taken j and the third year,
grass seeds arc sometimes sown along with the linseed.
This is the method mostly practised in and about the
counties of Lincoln and Somerset, where great quanti¬
ties of flax and hemp are every year raised, and where
these crops have long been capital articles. There, old
ploughed grounds are never sown with linseed, unless Flax,
the soil be very rich and clean. A certain worm, called —v—■
in Scotland the coup worm, abounds in grounds newly
broken up, and greatly hurts every crop but flax. In
small enclosures surrounded with trees or high hedges,
the flax, for want of free air, is subject to fall before
it be ripe j and the droppings of rain and dew from
the trees prevent the flax, within the reach of the trees,
from growing to any perfection.
Of preceding crops, potatoes and hemp are the best
preparation for flax. In the fens of Lincoln, upon
proper ground of old tillage, they sow hemp, dunging
well the first year ; the second year, hemp without
dung; the third year, flax without dung; and that same
year, a crop of turnip eaten on the ground by sheep ;
the fourth year, hemp with a large coat of dung ; and
so on for ever.
If the ground be free and open, it should be but once
ploughed ; and that as shallow as possible, not deeper
than 2J inches. It should be laid flat, reduced to a
fine garden mould by much harrowing, and all stones
and sods should be carried off.
Except a little pigeons dung for cold or sour
ground, no other dung should be used preparatory for
flax ; because it produces too many weeds, and throws
up the flax thin and poor upon the stalk.
Before sowing, the bulky clods should be broken, or
carried off the ground ; and stones, quickenings, and
every other thing that may hinder the growth of the
flax should be removed.
Of the choice of Linseed. The brighter in colour,
and heavier the seed is, so much the better ; that
which when bruised appears of a light or yellowish
green, and fresh in the heart, oily and not dry, and
smells and tastes sweet, and not fusty, may be depend¬
ed upon.
Dutch seed of the preceding year’s growth, for the
most part, answers best; but it seldom succeeds if kept
another year. It ripens sooner than any other foreign
seed. Philadelphia seed produces fine lint and few
bolls, because sown thick, and answers best in wet cold
soils. Riga seed produces coarser lint, and the greatest
quantity of seed. Scots seed, when well winned and
kept, and changed from one kind of soil to another,
sometimes answers pretty well ; but should be sown
thick, as many of its grains are bad, and fail. It springs
well, and its flax is sooner ripe than any other; but
its produce afterwards is generally inferior to that from
foreign seed.
A kind has been lately imported called Memmelseed;
which looks well, is short and plump, but seldom grows
above eight inches, and on that account ought not to
be sown.
Of sowing Linseed. The quantity of linseed sown
should be proportioned to the condition of the soil ;
for if the ground be in good heart, and the seed sown
thick, the crop will be in danger of falling before it is
ready for pulling. From 11 to 12 pecks Linlithgow
measure of Dutch or Riga seed, is generally sufficient
for one Scots acre; and about 10 pecks of Philadelphia
seed, which, being the smallest grained, goes farthest.
Riga linseed, and the next year’s produce of it, is pre¬
ferred in Lincolnshire.
The time for sowing linseed is from the middle of
March to the end of April, as the ground and season
answer 5. ;
t 670 ]
Fias.
the less the
FLA
answer j but the earlier the seed is sown
' crop interferes with the corn harvest.
Late sown linseed may grow long, but the flax upon
the stalk will be thin and poor.
After sowing, the ground ought to be harrowed till
the seed is well covered, and then, (supposing the soil,
as before mentioned, to be free and reduced to a fine
mould) it ought to be rolled.
When a farmer sows a large quantity of linseed, he
may find it proper to sow a part earlier and part later,
that in the future operations of weeding, pulling, wa¬
tering, and grassing, the work may be the easier and
more conveniently gone about.
It ought always to be sown on a dry bed.
Of Weeding Flax. It ought to be weeded when the
crop is about four inches long. If longer deferred, the
weeders will so much break and crook the stalks, that
they will never perhaps recover their straightness again
and when the flax grows crooked, it is more liable to
be hurt in the rippling and swingling.
Quicken grass should not be taken up j for, being
strongly rooted, the pulling of it always loosens a deal
of the lint.
If there is an appearance of a settled drought, it
is better to defer the weeding, than by that opera¬
tion to expose the tender roots of the flax to the
drought.
How soon the weeds are got out, they ought to be
carried off the field, instead of being laid in the fur¬
rows, where they often take root again, and at any rate
obstruct the growth of the flax in the furrows.
Of Pulling Flax. When the crop grows so short
and branchy, as to appear more valuable for seed than
flax, it ought not to be pulled before it be thoroughly
ripe; but if it grows long and not branchy, the seed
should be disregarded, and all the attention given to
the flax. In the last case it ought to be pulled after
the bloom has fallen, when the stalk begins to turn
yellow, and before the leaves fall, and the bolls turn
hard and sharp-pointed.
When the stalk is small, and carries few bolls, the
flax is fine : but the stalk of coarse flax is gross, rank,
branchy, and carries many bolls.
When the flax has fallen, and lies, such as lies ought
to be immediately pulled, whether it has grown enough
or not, as otherwise it will rot altogether.
When parts of the same field grow unequally, so
that some parts are ready for pulling before other parts ;
only what is ready should be pulled, and the rest should
be suffered to stand till ready.
The flax-raiser ought to be at pains to pull and keep
by itself, each different kind of lint which he finds
in his field j what is both long and fine, by itself j
what is both long and coarse, by itself 5 what is both
short and fine, by itself $ what is both short and coarse,
by itself*, and in like manner every other kind by it¬
self that is of the same size and quality. If the differ¬
ent kinds be not thus kept separate, the flax must be
much damaged in the watering and the other succeed¬
ing operations.
"Wrhat is commonly called under-growth may be ne¬
glected as useless.
Few persons that have seen pulled flax, are ignorant
of the method of laying it in handfuls across each
FLA
other ; which gives the flax sufficient air, and keeps the
handfuls separate and ready for the rippler. u.
Of Stacking up Flax during the Winter, and Winning
the Seed. If the flax be more valuable than the seed,
it ought by no means to be stacked up j for its own na¬
tural juice assists it greatly in the watering ; whereas,
if kept long unwatered, it loses that juice, and the
harle adheres so much to the boon, that it requires
longer time to water, and even the quality of the flax
becomes thereby harsher and coarser. Besides, the flax
stacked up over year, is in great danger from vermine
and other accidents; the water in spring is not so soft
and warm as in harvest; and near a year is thereby
lost of the use of the lint : but if the flax be so short
and branchy as to appear most valuable for seed, it
ought, after pulling, to be stocked and dried upon the
field, as is done with corn ; then stacked up for win¬
ter, rippled in spring; and after sheeling, the seed
should be well cleaned from bad seeds, &c.
Of Rippling Flax. After pulling, if the flax is to
he regarded more than the seed, it should be allowed
to lie some hours upon the ground to dry a little, and
so gain some firmness, to prevent the skin or harle,
which is the flax, from rubbing off in the rippling; an
operation which ought by no means to be neglected, as
the bolls, if put into the water along with the flax,
breed vermine there, and otherwise spoil the water. The
bolls also prove very inconvenient in the grassing and
breaking.
In Lincolnshire and Ireland, they think that rip¬
pling hurts the flax ; and therefore, in place of rip¬
pling, they strike the bolls against a stone.
The handfuls for rippling should not be great, as
that endangers the lint in the rippling comb.
After rippling, the flax-raiser will perceive, that he
is able to assort each size and quality of the flax by it¬
self more exactly than he could before.
Of Watering Flax. A running stream wastes the
lint, makes it white, and frequently carries it away.
Lochs, by the great quantity and motion of the wa¬
ter, also waste and whiten the flax, though not so much
as running streams. Both rivers and lochs water the
flax quicker than canals.
But all flax ought to be watered in canals, which
should be digged in clay ground if possible, as that
soil retains the water best: but if a firm retentive soil
cannot be got, the bottom or sides of the canal, or
both the bottom and sides, may be lined with clay ; or
instead of lining the sides with clay, which might
fall down, a ditch may be dug without the canal, and
filled with clay, which will prevent both extraneous
water from entering, and the water within from run¬
ning off.
A canal of 40 feet long, six broad, and four deep,
will generally water the growth of an acre of flax.
It ought to be filled with fresh soft water from a
river or brook, if possible, two or three weeks before
the flax is put in, and exposed all that time to the heat
of the sun. The greater way the river or brook has
run, the softer, and therefore the better, will the water
be. Springs, or short runs from hills, are too cold,
unless the water is allowed to stand long in the canal.
Water from coal or iron is very bad for flax. A little
of the powder of galls thrown into a glass of water,
will
Flax.
Flax.
FLA [67
will immediately discover if it comes from minerals of
o' that kind, by turning it into a dark colour, more or
less tinged in proportion to the quantity of vitriol it
contains.
The canal ought not to be under shade : which, be¬
sides keeping the sun from softening the water, might
make part ot the canal cooler than other parts, and so
water the flax unequally.
The flax-raiser will observe, when the water is
brought to a proper heat, that small plants will be
rising quickly in it, numbers of small insects and rep¬
tiles will be generating there, and hubbies of air ris¬
ing on the surface. If no such signs appear, the wa¬
ter must not be warm enough, or is otherwise unfit for
flax.
Moss holes, when neither too deep nor too shallow,
frequently answer well for watering flax, when the wa¬
ter is proper, as before described.
The proper season for watering flax is from the end
of July to the end of August.
The advantage of watering flax as soon as possible
after pulling, has been already mentioned.
I he flax being sorted after rippling, as before men¬
tioned, should next be put in beets, never larger than
a man can grasp with both his hands, and tied very
slack with a band of a few stalks. Dried rushes an¬
swer exceedingly well for binding flax, as they do not
rot in the water, and may be dried and kept for use
again.
The beets should he put into the canals slopewise,
or half standing upon end, the root end uppermost.
Upon the crop ends, when uppermost, there frequent¬
ly breeds a deal of vermine, destructive of the flax,
which is effectually prevented by putting the crop end
downmost.
The whole flax in the canal ought to be carefully
covered from the sun with divots ; the grassy side of
which should be next the flax, to keep it clean. If it
is not thus covered, the sun will discolour the flax,
though quite covered with water. If the divots are
not weighty enough to keep the flax entirely under wa¬
ter, a few stones may be laid above them. But the flax
should not be pressed to the bottom.
When the flax is sufficiently watered, it feels soft to
the gripe, and the harle parts easily with the boon or
show, which last is then become brittle, and looks whit¬
ish. When these signs are found, the flax should be
taken out of the water, beet after beet; each gently
rinsed in the water, to cleanse it of the nastiness which
has gathered about it in the canal ; and as the lint is
then very tender, and the beet slackly tied, it must be
carefully and gently bandied.
Great care ought to be taken that no part be over
done ; and as the coarsest rvaters soonest, if different
kinds be mixed together, a part will be rotted, when
the rest is not sufficiently watered.
When lint taken out of the canal is not found suffi¬
ciently watered, it may be laid in a heap for 12, 18,
or 24 hours, which will have an effect like more water-
ing ; but this operation is nice, and may prove danger¬
ous in unskilful hands.
After the flax is taken out of the canal, fresh lint
should not be put a second time into it, until the former
water be run off, and the canal cleaned, and supplied
with fresh water.
1 ] FLA
Of Grassing Flax. Short heath is the best field for
grassing flax ; as, when wet, it fastens to the heath,
and is thereby prevented from being blown away by
the wind. The heath also keeps it a little above the
earth, and so exposes it the more equally to the wea¬
ther. When such heath is not to be got, links or
clean old lea ground is the next best. Bong grass
grounds should be avoided, as the grass growing
through the lint frequently spots, tenders, or rots it;
and grounds exposed to violent winds should also be
avoided.
1 he flax, when taken out of the water, must be
spread very thin upon the ground : and being then
very tender, it must be gently handled. The thinner
it is spread the better, as it is then the more equally
exposed to the weather. But it ought never to be
spread during a heavy shower, as that would wash and
waste the harle too much, which is then excessively
tender, but soon after becomes firm enough to bear the
rains, which, with the open air and sunshine, cleans,
softens, and purifies the harle to the degree wanted,
and makes it blister from the boon. In short, after the
flax has got a little firmness by being a few hours
spread in dry weather, the more rain and sunshine it
gets the better.
Jf there be little danger of high winds carrying off
the flax, it will be much the better of being turned
about once a-week: If it is not to be turned, it ought
to be very thin spread. The spreading of flax and
hemp requires a deal of ground, and enriches it
greatly.
The skilful flax-raiser spreads his first row of flax
at the end of the field opposite to the point from
whence the most violent wind commonly comes, pla¬
cing the root-ends foremost; he makes the root-ends
of every other row overlap the crop ends of the former
row three or four inches, and binds down the last row
with a rope ; by which means the wind does not easily
get below the lint to blow it away ; and as the crop
ends are seldom so fully watered as the root ends, the
aforesaid overlapping has an effect like giving the crop
ends mere watering. Experience only can fully teach
a person the signs of flax being sufficiently grassed: then
it is of a clearer colour than formerly ; the harle is blis¬
tered up, and easily parts with the boon, which is then
become very brittle. The whole should be sufficiently
grassed before any of it is lifted ; for if a part be lifted
sooner than the rest, that which remains is in great
danger from the winds.
A dry day ought to be chosen for taking up the
flax ; and if there is no appearance of high wind, it
should be loosed from the heath or grass, and left loose
for some hours, to make it thoroughly dry.
As a great quantity of flax can scarcely be all equal¬
ly watered and grassed, and as the different qualities
will best appear at lifting the flax off the grass ; there¬
fore at that time each different kind should he gathered
together, and kept by itself; that is, all of the same co¬
lour, length, and quality.
The smaller the beets lint is made up in, the better
for drying, and the more convenient for stacking, hous¬
ing, &c. and in making up these beets, as in every
other operation upon flax, it is of great consequence
that the lint be laid together as it grew, the root ends
together, and the crop ends together.
Fsliows
Flax.
FLA
t 672 ]
FLA
Fkx. Follow an estimate of the Expence, Produce, and Profit of a Scots Acre of Flax.—-supposing the season favour-
"—y——^ that no accidental losses happen, and that the farmer is neither unskilful nor negligent.
Ground rent, labouring the ground, and leading the flax
Linseed from 2I. to 4^* Per hogshead, the medium
3s. 4d. per peck — ““
Clodding and sowing — —
Weeding — . T“ ,
Pulling, rippling, putting in, and covering in the
water — —
Taking out of the water, grassing, and stacking
Breaking and scutching, at 2s. per stone —
Total expence
Produce at ios. per stone
Linseed sold for oil at is. per peck —
The chaff of the bolls is well worth the expence of
drying the seed as it is good food, when boiled
and mixed with bear, for horses.
Total produce
A medium crop.
L. 2 10 o
1168
for 11 pecks.
020
012 o
o 14 o
080
300
for 30 stones.
L. 9
8
L. 15 o o
for 30 stones.
0160
Balance for profit
L. 15 16
L. 6 14
A great crop.
L. 3 10 o
1100
for 9 pecks.
020
080
o 15 o
0 12 0
4 0 0
for 40 stones.
L. 10 17
An extra crop.
L. 5 o o
168
for 8 pecks.
020
nothing.
100
0180
600
for 60 stones.
L. 20 o o
for 40 stones.
0180
L. 20 18
L. 10
L. 14
8
L. 30 o o
for 60 stones.
100
L. 31
L. 16 13
The above estimate being made several years ago,
the expence and profit are now different j but the pro¬
portions of each are probably the same. There is
nothing stated here as expence of the canal in which
the flax is watered ; because that varies much according
to circumstances.
It is a certain fact, that the greater the crop is, the
better is the quality of the same kind of flax.
The advantage of having both a crop of flax and a
crop of turnip the same year—or of sowing grass seeds
along with the linseed—and of reducing the ground to
a fine garden mould, free of weeds, ought to be attend¬
ed to.
For Cambric and fine Lawn. The ground must be
a rich light soil, rather sandy, but cannot be too rich.
It ought to be ploughed in September, or the be¬
ginning of October, first putting a little hot x’otten
dung upon it. In January it ought to have a second
ploughing, after a hard frost j and when you intend
to s
seem worthy of particular attention.
Of the watering of Flax by a new method, so as to
shorten labour, to add probably to the strength of
the fax, and to give it a much finer colour, which
would render the operation of bleaching safer and
less tedious.
“ Though the following reflections have for their
5 ] FLA
object an improvement in the very essential article of Flax,
watering of flax, yet I must advertise the reader, that
they are only theory, and must depend entirely for
their truth and justification upon future experiments,
skilfully and judiciously made. Should repeated trials
prove the advantage of the new method proposed, we
may venture to affirm, that it would be an improve¬
ment that would increase the national income in the
agricultural branch many thousand pounds annually,
would add greatly to the perfection of the linen manu¬
facture, and over and above would suppress a very dis¬
agreeable nuisance, which the present method of water¬
ing flax occasions during some part of the summer in
every flax-growing country.
“ The intention of watering flax, is, in my opinion,
to make the boon more brittle or friable, and, by soak¬
ing, to dissolve that gluey kind of sap that makes
the bark of plants and trees adhere in a small degree
to the woody part. The bark of flax is called the
harle ; and when separated from the useless woody part,
the boon, this harle itself is called flax. To effect this
separation easily, the practice has long prevailed, of
soaking the flax in water to a certain degree of fer¬
mentation, and afterwards drying it. For this soak¬
ing some prefer rivulets that have a small current, and
others stagnant water in ponds and lakes. In both
methods the water acts as in all other cases of infusion
and maceration 5 after two or three weeks it extracts
a great many juices of a very strong quality, which in
ponds give the water an inky tinge and offensive smell }
and in rivulets mix in the stream and kill the fish.
Nay, if this maceration be too long continued, the
extracted and fermented sap will completely kill the
flax itself. For if, instead of two or three weeks, the
new flax were to lie soaking in the water four or five
months, I presume it would be good for nothing but to
be thrown upon the dunghill -, both harle and boon
would in time be completely rotted 5 yet the harle or
flax, when entirely freed from this sap, and manufac¬
tured into linen, or into ropes, might lie many months
under water without being much damaged j as linen,
it may be washed and steeped in scalding water twenty
times without losing much of its strength and as pa¬
per, it acquires a kind of incorruptibility.
“ It appears then essential to the right management
of new flax, to get rid of this pernicious vegetable
sap, and to macerate the boon j but from the com¬
plaints made against both the methods of watering now
in use, there is reason to think that there is still great
room for improvement in that article. In rivulets, the
vegetable sap, as it is dissolved, is carried off by the
current to the destruction of the fish. I his prevents
the flax from being stained j but the operation is te¬
dious, and not complete, from the uncertainty of
knowing when it is just enough, and not too much,
or perhaps from neglect. In ponds, the inky tinge of
the water often serves as a kind of dye to the flax,
which imbibes it so strongly, that double the labour in
bleaching will hardly bring the linen made of such
flax to an equality in whiteness with linen made of
flax untinged. This seems to be equally unwise, as
though we were to dye cotton black first, in order to
whiten it afterwards, These ponds, besides, become
a great nuisance to the neighbourhood; the impreg¬
nated water is often of such a pernicious quality, that
cattle,
2
FLAGS .
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r LAX .
EtK«]»ia [ Mixli-na
J"/ ~ < S'rrfioji ofSreaTc
AIo^-ul !J '!
rLOATIXG BODIES
Fiff.l.
B A
f
FLA. [ 675 ] FLA
Fla*- cattle, however thirsty, will not drink of it; and the
effluvia of it may perhaps be nearly as infectious as it
is offensive. If this effluvia is really attended with
any contagious effects in our cold climate, a thing
worth the inquiring into, how much more pernicious
must its effects have been in the hot climate of Egypt,
a country early noted for its great cultivation of
flax ?
“ I have often thought that the process of watering
might be greatly improved and shortened by plunging
the new flax, after it is rippled, into scalding water j
which, in regard to extracting the vegetative sap, would
do in five minutes more than cold water would do in
a fortnight, or perhaps more than cold water could do
at all, in respect to the clearing the plant of sap.
Rough almonds, when thrown into scalding water, are
blanched in an instant *, but perhaps a fortnight’s ma¬
cerating those almonds in cold water would not make
them part so easily with their skins, which are the same
to them as the harle is to the flax. Were tea leaves
to be infused in cold water a fortnight, perhaps the tea
produced by that infusion would not be so good to the
taste, or so strongly tinged to the eye, as what is ef¬
fected by scalding water in five minutes. By the same
analogy, I think, flax or any small twig would be made
to part with its bark much easier and quicker by be¬
ing dipped in boiling water, than by being steeped in
cold water.
“ This reflection opens a door for a great variety
of new experiments in regard to flax. I would there¬
fore recommend to gentlemen cultivators and farmers,
to make repeated trials upon this new system, which
would soon ascertain whether it ought to be adopted
in practice or rejected. One thing, I think, we may
be certain of, that if the Egyptians watered their flax
in our common manner, they undoubtedly watered it
in very warm water, from the great heat of their cli¬
mate, which would probably make them neglect to
think of water heated by any other means than that
of the sun. A good general practice can only be esta¬
blished upon repeated trials. Though one experiment
may fail, another with a little variation may succeed $
and the importance of the object desired to be obtain¬
ed will justify a good degree of perseverance in the
prosecution of the means. In this view, as the Chi¬
nese thread is said to be very strong, it would be worth
while to be acquainted with the practice of that distant
nation, in regard to the rearing and manufacturing of
flax, as well as with the methods used by the Flemings
and the Dutch.
“ Boiling water perhaps might at once clear the
new flax from many impurities, which when not re¬
moved till it be spun into yarn, are then removed with
difficulty, and with loss of substance to the yarn. Why
should not the longitudinal fibres of the flax, before
they be spun into yarn, be made not only as fine but
as chan as possible ? Upon the new system proposed,
the act of bleaching would begin immediately after the
rippling of the flax ; and a little done then might per¬
haps save much of what is generally done after the
spinning and weaving. To spin dirty flax with a view
of cleaning it afterwards, appears to be the same impro¬
priety as though we were to reserve part of the dressing
given to leather till after it is made into a glove.
“ Should the plunging of the flax into the boiling
water not suffice to make the boon brittle enough, as I
am inclined to think it would not, then the common
watering might be added ; but in that case probably
hall the time usually given to this watering would suf¬
fice, and the flax might then be laid in clear rivulets,
without any apprehension of its infecting the water
and poisoning the fish, or of being discoloured itself;
lor the boiling water into which it had been previously
put, woultl have extracted all the poisonous vegetative
sap, which I presume is what chiefly discolours the flax
or kills the fish.
“ On the supposition that the use of boiling water
in the preparation of flax may be found to be advan¬
tageous and profitable, I can recollect at present but
one objection against its being generally adopted.
Every flax-grower, it may be said, could not be ex¬
pected to have conveniences for boiling water suffi¬
cient for the purpose •, the consumption of water
would be great ; and some additional expence would
be incurred. In answer to this I shall observe, that
I presume any additional expence would be more than
reimbursed by the better marketable price of the flax ;
for otherwise any new improvement, if it will not quit
cost, must be dropt, were it even the searching after
gold. In a large caldron a great deal of flax might
be dipt in the same water, and the consumption per¬
haps would not be more than a quart to each sheaf.
Even a large household pot would be capable of con¬
taining one sheaf after another j and I believe the
whole objection would be obviated, were the practice
to prevail with us, as in Flanders and Holland, that
the flax-grower and the flax-dresser should be two di¬
stinct professions.
“ 1 shall conclude with recommending to those who
are inclined to make experiments, not to be discou¬
raged by the failure of one or two trials.—Perhaps the
flax, instead of being just plunged into the scalding
water, ought to be kept in it five minutes, perhaps a
quarter of an hour, perhaps a whole hour. Should five
minutes or a quarter of an hour, or an hour, not be
sufficient to make the boon and harle easily separate,
it might perhaps be found expedient to boil the flax
for more than an hour j and such boiling when in this
state might in return save several hours boiling in the
article of bleaching. It is not, I think, at all probable
that the boiling of the flax with the boon in it would
prejudice the harle j for in the course of its future ex¬
istence, it is made to be exposed 20 or 40 times to
this boiling trial ; and if not detrimental in the one
case, it is to be presumed it would not be detrimental
in the other. Perhaps after boiling, it would be
proper to pile up the flax in one heap for a whole day,
or for half a^day, to occasion some fermentation $ or
perhaps immediately after the boiling, it might be
proper to wash it with cold water. The great object,
when the flax is pulled, is to get the harle from the
boon with as little loss and damage as possible $ and if
this is accomplished in a more complete manner than
usual, considerable labour and expence will be saved in
the future manufacturing of the flax. On this ac¬
count I think much more would be gained than lost,
were the two or three last inches of the roots of the
stems to be chopped off, or clipped oft’, previous to the
flax being either watered or boiled. When the flax is
watered, care should be taken not to spread it out to
^ 4 Q 2 dry,
FLA
[ 676 ]
FLA
Flax.
dry, when there is a hazard of its being exposed in its
wet state to frost.”
To what we have now salt! we shall add the follow-
ing short account of the flax husbandry of Ireland, in a
letter which appeared in the Farmers Magazine, vol.
vii. page 35. .
“ Having for several years (says the writer) been enga¬
ged in the culture of flax, I devoted a part of last summer
to a tour through the manufacturing districts ot Ireland.
Here that branch of husbandry has long been established
over a large extent of the country, and conducted with
very considerble success. As some of the processes in
this culture, which are followed with advantage, are
either unknown to the Scots farmers, or are periormed
in a very awkward and inefficient manner, it might, I
conceive, prove of no small benefit, were some of your
intelligent corespondents induced to lay belore them a
plain sketch of the peculiar management observed by
the Irish peasantry in this important article. I am the
more desirous it should appear in your pages, because a
periodical work on husbandry, conducted by a practical
farmer, appears before the public with manifest advan¬
tage, and is received with that sort of deference which
is due to experience and authority.. The discussions of
actual cultivators regarding the objects of their own
profession, however new they may as yet be in the an¬
nals of agriculture, are far more likely to prove useful,
that the writings of those volunteers in this favourite
science, who are merely speculative and theoretical. I
freely confess to you, Sir, that I found with pleasure
your work widely circulated in the sister kingdom j and
that the cause uniformly given for its popularity, was a
degree of confidence placed in the practical skill of its
conductor.
“ During my progress through Ireland, the several
processes of stee/u'ttg, drying, and sktitching,viP,xeix\ hand,
and I think I found a peculiarity of management in
these sufficient to affect the success of the whole business,
and to confer a decided superiority on the produce of
an acre of flax in Ireland over that in Scotland, both
in quantity and value. It is no uncommon thing for a
farmer in this country, who wishes to make up a sum
for his rent, to sell a part of his lint on the foot, as it is
termed ; and for this he will commonly receive from
30 to 40 guineas per acre.
“ 1. The method of Steeping.—As soon as the crop has
attained the proper degree of ripeness, (which is some¬
what below your standard of maturity), the flax is pul¬
led, and carried to a stagnant pool, dug lor this pur¬
pose, moderately deep. It is allowed to remain there
only irom five to seven days, according to the tempera¬
ture of the weather. After the fermentation in the
steeping process has been carried to a degree sufficient
to produce the requisite laxity of fibre, the flax is taken
out of the pool, and spread very thinly on the stubble
of the hay meadow. There, instead of remaining till
it is merely dried, it is continued for three or four
weeks, till the grower conceives it ready for skutching.
This blenching process, if I am allowed to call it so,
which, in Scotland, is either unknown, or continued
merely till the crop is dried, has many advantages*, the
most obvious one is, that it enables the farmer, every
time he examines it, to ascertain exactly (by rubbing
on his band) the precise point at which the fermentation
has arrived,and thus to perceive the tenacity.andstrength
of his flax ‘7 while the adhesion of the fibre has been suf¬
ficiently weakened, to admit of the skutcher cleansing it
completely of the woody parts. It is, I am apprehen¬
sive, only the practical flax farmer who is able to judge
of the importance and delicacy of this part of the hus¬
bandry. It is so remarkable, that of two acres of flax,
under precisely the same seed and culture, and of equal
fertility, it frequently happens that the one shall yield
a produce thrice the value of the other, merely trom
superior accuracy in ascertaining the proper time of con¬
tinuing the steeping and blenching processes. In Scot¬
land, therefore, I suspect the practice is faulty and de¬
fective ; because there the whole process of fermentation
is completed by steeping alone j whereas, in Ireland, it
is begun only in the steep, and completed by blenching
on the meadow, to that precise point which the safety
of the produce requires.
« 2. Smoking and Drying.—The Irish peasant seems
to possess another advantage, almost equally decisive, in
his mode of drying the flax, before he submits it to the
skutcher or beater. After the lint has remained a suf¬
ficient length of time on the blenching green, it is ga¬
thered up a second time into sheafs, (beats, provincial-
ly), and seems tolerably dry. In this state it is deemed
by the Scots growers fully prepared for the flax-mill ;
but far otherwise by the Irish farmer, who never sub¬
mits it to the hands of the beaters till h has undergone
a thorough smoking over a peat fire. For this purpose,
he raises, at the back of a ditch, a small hurdle thinly
wrought with osiers, and places it on four posts of wood,
at the height of four feet above the level of the ground.
A pretty strong fire of peats being kindled below, the
heat and smoke pervade every part ot the flax, which
is placed perpendicularly above the hurdle. Ihis pro¬
cess is continued, and fresh quantities of flax regularly
added, till the whole crop is brought to a state of dry¬
ness, which, in this moist climate, can never be eflect-
ed by the sun and the weather alone : by this operation
a degree of brittleness and friability is produced on the
straw, which greatly facilitates the ensuing work, and
admits of an easy separation of the fibre from the wood.
It is evident, that the less friction required in skutching,
the less waste or diminution must be occasioned in clear¬
ing the flax; and consequently, the greater must be
the grower’s produce from the mill. Ibis part of the
process is equally delicate with that described above,
and requires, if possible, still greater attention on the
part of the workmen, since it is clear that, by a care¬
less management of the fire, the whole crop may be de¬
stroyed.
“ 3, Cleansing and Dressmg.—The flax husbandry or
Ireland derives no small benefit from the application of
hand-labour in the beating and skutching of lint, thus
superseding the use of the mill. The most careful and
expert workmen are not always able to temper the ve¬
locity of machinery so exactly, as to preserve flax that
has been oversteeped or blenched to excess: while the
steady and regulated impetus of the hand skutch can
easily be modified, as the circumstances of each case
may require ; a matter of obvious advantage, because
the best flax-mills seldom produce an equal quantity of
lint, nor equally clean, with that which is obtained by
the hand. Besides this, the price of labour in this
part of the united kingdom, still continues so moderate,.
as to preclude any considerable degree of saving in ex¬
pence
Flat.
FLA [ 677 ] F L E
Wa*. pence by the use of machinery. In proof of this, the
«—V 1 flax millers in Scotland, I find, are charging this sea¬
son from three to four shillings for dressing a stone of
flax j while, at the place I am now writing, the same
quantity is dressed by the hand for thirteenpence, or one
British shilling. In Scotland, where hands are scarce,
and the price of labour consequently high, I certainly
would not recommend the disuse of the flax-mill ; on
the contrary, I am persuaded that it is chiefly owing to
our superior machinery, and excellent implements of
husbandry, that wre are at all enabled to maintain a
competition with our neighbours in the present state of
our skill in flax husbandry, and subjected to the dis¬
advantage of paying double price for labour.
“ 4. Preservation of Flax-seed.—The last peculiarity
of management, which I shall at present notice as advan¬
tageous to the flax husbandry of Ireland, is the inven¬
tion of a flax barn for the preservation of seed. Enjoy¬
ing a climate perhaps still more moist and unsteady than
that of Great Britain, the farmers here were, for a long
series of years, unable to supply themselves with this
article, and were obliged to commission seed annually
from America and the Baltic, to supply the increased
demands of an extended culture, to the large amount
of 200,0001. sterling. The annual expenditure of cash
long continued to operate as a drain on the stock of the
laborious farmer, and prevented the accumulation of
his capital; an evil of the most serious magnitude, un¬
der which the Irish peasantry still labour, and from
which, till very lately, they had not even a prospect of
relief. By the practice in universal use, if the farmer
stored up his lint in the barn-yard with the rest of his
crop in harvest, he might, it is true, preserve his seed }
but in doing so, he uniformly lost his flax to a far
greater value from overdryness, when wrought in the
spring.
“ If, on the other hand, he attempted to separate his
seed during the lint harvest by means of the rippling-
comb, he had no means of preventing it from being al¬
most invariably destroyed by the wetness of the climate.
Various methods had been attempted to overcome this
difficulty, but without success j till llobert Tennant,
Esq. of Strangmore, linen-inspector, near Dungannon,
contrived the plan of a flax-barn, which seems perfect¬
ly competent to the preservation of seed. It has al¬
ready been erected, and has proved successful on a small
scale j the seed cured in it remained during the winter
perfectly fresh, and nothing seems wanting to complete
this improvement in our flax husbandry, but a larger
capital in the hands of a few of our farmers. This flax
barn is constructed on wooden posts, roofed on the top,
but left perfectly open at each side y it is supplied with
various stages of floors of basket-work, placed regu¬
larly at two feet distance above each other. Thus,
the air, having free access to the seed on all sides,
preserves it fresh and well-coloured for any length of
time.
“ Thi&contrivance was suggested to Mr Tennant, it is
said, almost casually, by noticing the great effect pro¬
duced on cloth, by drying-houses in bleachfields. He
had in fact been employed by the Linen Board of Ire¬
land, in teaching the new process of bleaching to the
manufacturers, by means of the oxymuriate of lime;
and, in the course of seven or eight years, this method
of whitening linen has been established over the whole
kingdom, with the exception of hardly a single field. Fla*,
Lord Northland and Mr Foster, who invited this Flea,
gentleman from Scotland, and patronized him in this '-“■“■v—
part of the kingdom, have enjoyed the satisfaction of
beholding a more essential improvement effected in
the linen manufacture, in the short space already
mentioned, than had ever taken place in a century
before.
“ It was my intention, when I began this letter, to
have presented you a more minute description of a flax
farm, and to have laid before your readers, a more de¬
tailed account of the flax husbandry of Ireland in gene¬
ral. I find, however, that I have already exceeded
the ordinary bounds prescribed to the contributors to
your useful work; therefore conclude, with expressing
a hope, that the few hints already offered, will incline
some of your correspondents to treat of a subject certain¬
ly of sufficient importance to merit attention. For a
branch of husbandry cannot be deemed contemptible,
which affords sustenance to upwards of two millions of
people ; and which, at the same time, adds to the ge¬
neral resources of the empire, no less a sum than seven
millions sterling annually. These circumstances, too,
I trust, will plead my excuse for holding up a portion
of Irish husbandry to the imitation of your numerous
readers among the cultivators of Scotland, who are at
present justly celebrated for their agricultural know¬
ledge in every part of the world.”
Flax mads to resemble Cotton. In the Swedish Trans¬
actions for the year 1747, a method is given of pre¬
paring flax in such a manner as to resemble cotton in
whiteness and softness, as well as in coherence. For
this purpose a little sea water is to be put into an
iron pot or an untinned copper kettle, and a mixture
of equal parts of birch ashes and quicklime strewed
upon it: A small bundle of flax is to be opened and
spread upon the surface, and covered with more of the
mixture, and the stratification continued till the vessel
is sufficiently filled. The whole is then to be boiled;
with sea water for ten hours, fresh quantities of water
being occasionally supplied in proportion to the eva¬
poration, that the water may never become dry. The
boiled flax is to be immediately washed in the sea
by a little at a time, in a basket, with a smooth stick
at first while hot; and when grown cold enough to
be borne by the hands, it must be well rubbed,
washed with soap, laid to bleach, and turned and
watered every day. Repetitions of the washing with
soap expedite the bleaching; after which the flax is
to be beaten, and again well-washed; when dry it
is to be worked and carded in the same manner as com¬
mon cotton, and pressed betwixt two boards for 48
hours. It is now fully prepared and fit for use. It
loses in this process near one half its weight, which is
abundantly compensated by the improvement made in
its quality.
The filamentous parts of different vegetables have
been employed in difl’erent countries for the same me¬
chanic uses as hemp and flax among us. See Fila¬
ment.
Earth-Flax. See Amianthus, Botany Index.
New Zealand Flax Plant. See Phormium, Bo¬
tany Index.
Toad-FLAX. See Linaria, Botany Index.
FLEA. See Pulex, Entomology Index.,
FlBAt
F L E
FlEA-Bane. See CoNYZA, BOTANY Lw'ex.
FiEA-Bitten, that colour of a horse which is white
or fray, spotted all over with dark reddish spots.
FLEAM, in Surgery and Farriery, an instrument
for letting blood of a man or horse. A case of fleams,
as it is called by farriers, comprehends six sorts of in¬
struments; two hooked ones, called and used
for cleansing wounds ; a pen knile ; a sharp-pointed
lancet for making incisions ; and two fleams, one sharp
and the other broad pointed. 1 hese last are somewhat
like the point of a lancet, fixed in a flat handle, and no
longer than is iust necessary to open the vein.
FLECHIER, Esprit, bishop of Nismes, one of the
most celebrated preachers of his age, and the publisher
of many panegyrics and funeral orations, was boin at
Perne in Avignon in 1632. He was nominated to the
bishopric of Lavaur in 1685, and translated to Nismes
in 1687. At this latter place he founded an academy,
and took the presidentship upon himself: his own pa¬
lace was indeed a kind of academy, where he applied
himself to train up orators and writers, who might serve
the church, and do honour to the nation. He published,
besides his panegyrics and funeral orations, 1. A His¬
tory of the Emperor Theodosius, that ot Cardinal Xi-
menes, and that of Cardinal Commendon. 2. Several
Sermons. 3. Miscellaneous Works. 4. Letters, &c.
He died in 1710.
ELECKNOE, Richard, an English poet in the
reign of Charles II. more remarkable for Mr Dryden’s
satire on him than for any works of his own. He is
said to have been originally a Jesuit, and to have had
good English connexions in the Catholic interest.
When Dryden lost the place of poet laureat on the Re¬
volution, its being conferred on Flecknoe, for whom
he had a settled aversion, gave occasion to his poem
entitled Mac Flecknoe; one of the best written satires in
our language, and from which Pope seems to have ta¬
ken tlm hint for his Dunciad. Flecknoe wrote some
pi a vs; but could never get more than one of them act¬
ed, and that was damned.
FLEECE, the covering of wool shorn off the bodies
of sheep. See Wool.
Golden Fleece. See Argonauts, and Golden
Fleece
FLEET, commonly implies a company of ships of
war, belonging to any prince or state : but sometimes
it denotes any number of trading ships employed in a
particular branch of commerce.
The admirals of his Britannic majesty’s fleet are di¬
vided into three squadrons, viz,, the red, the white, and
the blue. When any of these officers are invested with
the command of a squadron or detachment of men of
war, the particular ships are distinguished by the colours
of their respective squadron: that is to say, the ships
of the red squadron wear an ensign whose union is dis¬
played on a red field; the ensigns of the white squa¬
dron have a white field ; and those of the blue squa¬
dron a blue field ; the union being common to all three.
The ships of war, therefore, are occasionally annexed
to anv of the three squadrons, or shifted from one to
another.
Of whatsoever number a fleet of ships of war is com¬
posed, it is usually divided into three squadrons; and
these, if numerous, are again separated into divisions.
The admiral, or principal officer, commands the centre ;
FEE
the vice admiral, or second in command, superintends p|eet
the van guard ; and the operations of the rear are di- |j
rected by the rear admiral, or the officer next in rank. Flemim
See the article Division. , gm^3‘
The disposition of a fleet, while proceeding on * 'r^“
voyage, will in some measure depend on particular cir-
cumstances ; as the difficulty of the navigation, the
necessity of dispatch, according to the urgency or im¬
portance of the expedition, or the expectation of an
enemy in the passage. The most convenient order is
probably to range it into three lines or columns, each
of which is parallel to a line close hauled according to
the tack on which the line of battle is designed to be
formed. This arrangement is more useful than any ;
because it contains the advantages of every other form,
without their inconveniences. Die fleet being thus
more enclosed will more readily observe the signals, and
with greater facility form itself into the line ot battle,
a circumstance which should be kept in view in every
order of sailing. See Naval Tactics.
Fleet, is also a noted prison in London, where
persons are committed for contempt of the king and his
laws, particularly of his courts of justice; or for debt,
where any person will not or is unable to pay his cre¬
ditors.
There are large rules and a warden belonging to
the Fleet prison ; which had its name from the float
or fleet of the river or ditch, on the side whereof it
stands.
FLEETWOOD, William, a very learned English
bishop in the beginning of the 18th century, of an an¬
cient family in Lancashire. He distinguished himself
during King William’s reign, by his Inscriptionvm An-
tiquarum Sylloge, by several sermons he preached on
public occasions, and by his Essay on Miracles. He
was designed by King WAliam to a canonry of Wlnd-
sor. The grant did not pass the seals before the king’s
death ; but the queen gave it him, and he was installed
in 1702. In 1703, he took a resolution to retire ; and
in 1707, published, without his name, his Chromcon
Pretiosum. In 1708, he was nominated by the queen
to the see of St Asaph. The change of the queen’s
ministry gave him much regret. In he publish¬
ed a pamphlet entitled, “ The 13th chapter of the Ro¬
mans vindicated from the abusive senses put upon it.
In 1714, he was translated to the bishopric of Ely;
and died in 1723, aged 67. He published several other
sermons and tracts, and was a man of great learning
and exemplary piety.
FLEMINGIANS, or Flandrians, in ecclesiasti¬
cal history, a sect ot rigid Anabaptists, who acquired
this name in the i6lh century, because most ot them
were natives of Flanders, by way of distinction from
the Waterlandians. In consequence of some dis¬
sensions among the Flemingians relating to the treat¬
ment of excommunicated persons, they were divided
into two sects, distinguished by the appellations of
Flandrians and Frieslanders, who differed from each
other in their manners and discipline. Many of these
in process of time came over to the moderate commu¬
nity of the Waterlandians, and those who remained se¬
parate are still known by the name of the old Flemin¬
gians or Flandrians; but they are comparatively few
in number. These maintain the opinion of Menno with
respect to the incarnation of Christ; alleging, that his
[ 678 ]
f
I FI
'
F L E
Fltmin-
gialis
II
'Fletcher.
body was produced by tbe creating power of the Holy
Ghost, and not derived from his mother Mary.
FLEMISH, or the Flemish tongue, is that which
we otherwise call Low Dutch, to distinguish it from the
German, whereof it is a corruption and a kind of dia¬
lect. See German.
It differs from the Walloon, which is a corruption
of the French language. The Flemish is used through
all the provinces of the Netherlands.
Flemish Bricks, a neat, strong, yellow kind of bricks,
brought from Flanders, and commonly used in paving
yards, stables, &c. being preferable for such purposes
to the common bricks. See the article Bricks.
FLESH, in Anatomy, a compound substance, con¬
sisting of the various softer solids of the animal body,
and so denominated in contradistinction to bones. See
Anatomy, passim.
Flesh is also used, in Theology, in speaking of the
mysteries of the incarnation and eucharist. “ The word
was made Verhum caro factum est.
The Romanists hold, that the bread in the sacrament
of the supper is turned into the real flesh of Jesus
Christ. See Transubstantiation.
Flesh is sometimes also used by botanists for the
soft pulpy substance of any fruit, enclosed between the
outer rind or skin and the seeds or stone j or for that
part of a root, fruit, &c. fit to be eaten.
FhEsu-Colour. See Carnation.
FLETA, the name given to an unknown writer
who lived about the end of the reign of Edward II.
and beginning of Edward III. and who being a pri¬
soner in the Fleet, wrote there an excellent treatise on
the common law of England.
FLETCHER. See Beaumont and Fletcher.
Fletcher, Andrew, of Salton, a celebrated Scots
patriot and political writer, was descended from an an¬
cient family who trace their origin to one of the fol¬
lowers of William the Conqueror. He was the son of
Sir Robert Fletcher of Salton and Innerpeffer, and
born in the year 1650. The tuition of our author was
committed by his father, on his deathbed, to Mr (af¬
terwards Bishop) Burnet, then his parish minister j by
whose care he received a pious, learned, and polite
education. Endowed with uncommon genius, and pos¬
sessed of virtues and abilities peculiarly suited to the
times in which he lived, Mr Fletcher quickly shone
forth the ornament of his country, and the champion
of its freedom. Having in the course of his classical
studies and historical reading been impressed with an
enthusiastic admiration both of ancient and modern
republics, he had early contracted an ardent love of
liberty, and an aversion to arbitrary rule. Hence his
spirit the more readily took alarm at certain measures in
the reign of Charles II. Being knight of the shire for
Lothian to that parliament where the duke of York
was commissioner, he openly opposed the designs of
that prince and the bill of accession. Pie had a share
with Lord Viscount Stair in framing the test act, by
which the duke of York complained that he lost Scot¬
land. On these accounts he became peculiarly ob¬
noxious to the duke j and was at last obliged to flee to
Holland, to avoid the fatal consequences of prosecutions
which on various pretence were commenced against
him. Being cited before the privy council andjusti-
[ 679 ]
F L E
ciary courts, and not appearing, he was declared trai- Fletcher,
tor, and his estate confiscated. 1—»-v—»
In Holland, he and Mr Baillie of Jerviswood were
the only persons whom the earl of Argyle consulted
concerning the designs which were then in agitation.
In 1681 they came over to England, in order to con¬
cert matters with their party in that country j and
were the only two who were intrusted so far as to be
admitted to the secrets of Lord Russel’s council of six.
Mr Fletcher managed his part of the negotiation with
so much address, that administration could find no
pretext for seizing him j nor could they fix upon him
those articles on account of which Mr Baillie was con¬
demned $ to whose honour let it be remembered, that
although offered a pardon on condition of his accusing
his friend, he persisted in rejecting the proposal with
indignation.
Mr Fletcher having joined the duke of Monmouth
upon his landing, received a principal command under
him ; but the duke was deprived of his services on
the following occasion, as related by Sir John DA-Memoirs of
rymple. Being sent upon an expedition, and not Great Bri-
esteeming “ times of danger to be times of ceremony,**™ and
he had seized for his own riding the horse of a country n an< ’
gentleman [the mayor of Lynne] which stood ready
equipped for its master. The master, hearing this, ran
in a passion to Fletcher, gave him opprobrious language,
shook his cane, and attempted to strike. Fletcher,
though rigid in the duties of morality, having been
accustomed to foreign service both by sea and land, in
which he had acquired high ideas of the honour of a
soldier and a gentleman, and of the affront of a cane,
pulled out his pistol, and shot him dead on the spot.
The action was unpopular in countries where such re¬
finements were not understood. A clamour was raised
against it among the people of the country : in a body
they waited upon the duke with their complaints ; and
he was forced to desire the only soldier, and almost the
only man of parts, in his army, to abandon him. With
Fletcher all Monmouth’s chance of success in war left
him.” But, in a manuscript memoir belonging to the
family, we have the following notice concerning Mr
Fletcher’s connection with Monmouth, in which his
separation from that prince is very differently accounted
lor: “ To Lord Marischal Mr Fletcher explained the
motives which first induced him to join, and afterwards
abandon, the duke of Monmouth. The former he
ascribed to the duke’s manifesto in Scotland relating to
religion, and in England to liberty. For the latter he
accounted by the disgust produced in his own mind and
that of his associates, when the duke declared himself
king, and broke faith with all who embarked with him
on his principles. He complained heavily of the ac¬
count commonly given of the death of the mayor of
Lynne : and mentioned to Lord Marischal, in proof of
the contrary, that he did not leave the duke till he came
to Taunton, where he was proclaimed king, several
weeks after the death of the mayor of Lynne.”
Seeing all the efforts of himself and his friends in
favour of liberty frustrated at Taunton, he endeavour¬
ed to secure his own personal freedom by taking his
passage in the first ship bound to a foreign country. It
was his misfortune to land in Spain j where he was
immediately arrested, cast into prison, and guarded by
three
Ifletclier.
Memoirs
of the fa¬
mily of Sal
ton, MS.
F L E [ 680 ] F L E
three different bande of soldiers, till a vessel should be ryas given him in t-eeotnpense of all his sufferings On Fkycte
orenared to carry him a victim in chains to the court the contrary, he, together noth the duke of Hamilton,^
R„t on the morning before the ship could was distinguished by marks of royal and ministerial
he looked penTve Jough the bars that se- dislike. Still, whatever private resentment he might
cured the window of his room, he was hailed by a ve- entertain ,t appeared that his ruling principle was the
nerable personage who made signs to speak with him. good of his country j and that to tins grand object ot
"ihe prison doors he found open ; and whilst his friend- his heart he was willing to sacrifice all persona cons.-
Iv conductor waved to him to follow him, he passed derations. For when, m 169a the abdicated king
brough three different guards of soldiers all fast asleep, meditated an invasion, Mr Fletcher addressed a letter
\V°thont being permitted to offer his thanks to his de- (preserved in Sir John Dalrymple s collection) to the
liverer he found himself obliged to prosecute with all duke of Hamilton, ... which every argument is em-
speed the journey, in which he was directed by a per
son concerning whom he could never collect any infor¬
mation ; and in disguise he proceeded in safety through
Spain. He felt a peculiar pleasure in relating to his
friends instances of the care of Providence which he had
experienced during his exile ; and entertained them
often with narratives of this kind, which he always
mingled with religious reflections. Of these, another
may he here mentioned. Happening in the evening to
pass the skirt of a wood at a few miles distance from a
city where he intended to lodge, he came to a place
where two roads met. After he had entered upon the
road on the right, he was accosted by a female ot a
respectable figure, who warned him to turn back, and
take the road to the left; for that in the other there
was danger which he could not escape if he continued
to proceed. His friendly monitor suddenly retired into
the wood, out of which she had issued no less unexpect¬
edly. Having arrived at the city, the inhabitants were
soon after alarmed by an account of the robbery and
murder of several travellers who that evening had fallen
into the hands of a banditti upon the very way in
which he had intended to travel. From these and other
instances of preservation from dangers, the devotion of
his mind, habituated from his infancy to an intercourse
with heaven, led him to conclude that he was in a pe¬
culiar manner the care of Providence, and that in cri¬
tical cases his understanding received its direction from
a supernatural impulse*
During bis exile, he maintained a frequent and ex¬
tensive correspondence with the friends of liberty at
home j and he partly employed himself in making a
curious collection of books, which compose the best
private library in Scotland. But his genius also
prompted him to engage in more active employments.
He repaired to Hungary, and served Several campaigns
as a volunteer under the duke of Lorrain with great
reputation. At length, understanding that the great
design then projected in Holland, and upon the issue
of which he considered the liberties ot Britain to be
suspended, had attained a considerable degree of ma¬
turity, he hastened thither j where his counsels and ad¬
dress were of eminent service. He came over with
King William j and in zeal, activity, penetration, and
political skill, proved inferior to none of the leaders in
the Revolution.
Such, however, was his magnanimity, that from a
survey of King William’s papers, it appears that while
others laboured to turn this grand event to the emo¬
lument of themselves and the aggrandisement of their
family, Mr Fletcher asked nothing. His estate had
been’forfeited, and his house abandoned to military
discretion j his fortune was greatly shattered, and his
family reduced to circumstances of distress. Nothing
3
_ in which
ployed with skill and energy to engage his grace to
forget his injuries, and in the present crisis to employ
the extensive influence and authority he then possessed
in the cause of freedom and of his country. This
letter produced its full effect j and the duke returned
to his duty, from which he had in part begun to de¬
viate.
To follow our author through all the mazes of his
political life subsequent to the Revolution, is beyond
our purpose, and would exceed our limits. One or
two circumstances more shall therefore suffice. Being
elected a member for the parliament 1683, he showed
an uniform zeal for the interest of his country. The
thought of England’s domineering over Scotland was
what his generous soul could not endure. The indig¬
nities and oppression which Scotland lay under galled
him to the heart; so that in his learned and elaborate
discourses, he exposed them with undaunted courage
and pathetical eloquence. In that great event, the
Union, he performed essential service. He got the
act of security passed, which declared that the two
crowns should not pass to the same head till Scotland
was secured in her liberties civil and religious. There¬
fore Lord Godolphin was forced into the Union, to
avoid a civil w'ar after the queen’s demise. Although
Mr Fletcher disapproved of some of the articles, and in¬
deed of the whole frame of the Union j yet, as the act
of security was his own work, he had all the merit of
that important transaction.
We must not omit mentioning, that in the ardour of
his political career Mr Fletcher forgot not the interests
of the place that gave him birth. He esteemed the
education of youth one of the noblest objects of go¬
vernment. On this subject he wrote a treatise, still
extant, most characteristic of himself j and he establish¬
ed at Salton a foundation for the same purpose, of
great utility while it lasted.
This great man died at London 1716, aged 66. His
remains were conveyed to Scotland, and deposited in
the family vault at Salton.
That Mr Fletcher received neither honours nor emo¬
luments from King William, may perhaps be in part
attributed to himself j a circumstance, however, which
must add greatly to the lustre of his character. His
uncomplying virtue, and the sternness of his principles,
were ill calculated to conciliate courtly favour. He
was so zealous an assertor of the liberties of the people,
that he was too jealous of the growing power of all
princes; in whom he thought ambition so natural, that
he was not for trusting the best of kings with the
power which ill ones might make use of against their
subjects j he was of opinion that all princes were made
by, and for the benefit of, the people ; and that they
should have no power but that of doing good. This,
which
F L E [ 681 ] ELI
ietcher. which made him oppose King Charles, and invade King
•' James, led him also to oppose the giving so much power
to King William, whom he would never serve after his
establishment. So we are told by the author of Short
Political Characters, a MS. in the library of the late
T. fiawlinson, Esq—Mr Lockhart, in his Memoirs,
p. 72. expresses a belief that his aversion to the Eng¬
lish and to the Union was so great, that, in revenge to
them, he was inclined to side with the abdicated fa¬
mily: “ But (adds he) as that was a subject not fit to
be entered upon with him, this is only a conjecture
from some inuendos I have heard him make j but so
far is certain, he liked, commended, and conversed
with high-flying Tories, more than any other set of
men j acknowledging them to be the best country¬
men, and of most honour, integrity, and ingenuity.”
It seems difficult to reconcile this with Mr Fletcher’s
avowed principles and the general tenor of his con¬
duct. May we suppose, that chagrin, if not at the
neglect or the ill treatment which he had himself re¬
ceived from government since the Revolution, yet at
the public measures relating to his native country,
might have occasioned him to relent in his sentiments
with regard to the exiled family?—In the family me¬
moirs already quoted, we are informed, That, “ Lord
Mariscbal held Mr Fletcher’s character in high admi¬
ration and that, “ when governor of Neufchatel,
where Rousseau resided about the year 1766, he pre¬
vailed with this very extraordinary genius to write the
life of a man whose character’and actions he wished to
have transmitted to posterity with advantage. For
this purpose his lordship applied to an honourable re¬
lation of Mr Fletcher’s for materials: which by him
were transmitted to Lord Marischal: but the design
failed through Rousseau’s desultory and capricious dis¬
position.” This anecdote must appear incompatible
with the known loyalty and attachments of the Earl
Marischal, unless we suppose him to have been privy
to some such sentiments of Mr Fletcher as those al¬
luded to by Mr Lockhart for how could we suppose
him anxious to promote a composition, in which the
task would be to celebrate principles diametrically op¬
posite to his own, and to applaud actions subversive of
that royal family in whose cause he had ventured his
life, and forfeited his fortune, and foregone his coun¬
try !—But however these circumstances may be recon¬
ciled, as the integrity, disinterestedness, and public
spirit of Mr Fletcher, have been universally acknow¬
ledged, there is reason to believe, that all his sentiments
and actions were founded in honour, and that he never
once pursued a measure further than he judged it to be
for the interest of his country.
Mr Fletcher was master of the English, Latin,
Greek, French, and Italian languages ; and well versed
in history, the civil law, and all kinds of learning. In
his travels, he had not only acquired considerable
knowledge in the art of war, but also became versant
in the respective interests cf the several princes and
states of Europe. In private life, he was affable to his
friends, and free from all manner of vice. He had a
penetrating, clear, and lively apprehension j but is said
to have been too much wedded to opinions, and im¬
patient of contradiction.—He possessed an uncommon
elevation of mind, accompanied with a warmth of
temper, which would suffer him to brook from no rank
Vol. VIII. Part H. f
among men, nor in any place, an indignity. Of this Fletcher *
he exhibited a singular proof in the Scots parliament. 1!
I he earl of Stair, secretary of state and minister for , f light,
Scotland, having in the heat of debate used an im- ’
proper expression against Mr Fletcher, he seized him
by his robe, and insisted upon public and immediate
satisfaction. His lordship was obliged instantly to beg
his pardon in presence of parliament.
Mr I letcher was by far the finest speaker in the
parliament of Scotland j the earl of Stair alone rivalled
him. The latter was famed for a splendid, the former
for a close and nervous, eloquence. He formed his
style on the models of antiquity; and the small volume
of his works, Sir John Dalrymple observes, though
imperfectly collected, is one of the very few classical
compositions in the English language.
FLETEWOOD, William, an eminent English
lawyer and recorder of London, in the reign of Queen
Elizabeth. He was very zealous in suppressing mass-
houses, and committing popish priests; but once rush¬
ing in upon mass at the Portuguese ambassador’s house,
he was committed to the Fleet for breach of privilege,
but soon released. Mr Wood says, “ He was a learned
man, and a good antiquary, but of a marvellous merry
and pleasant conceit.” He was a good popular speaker,
and wrote well upon subjects of government. His prin¬
cipal works are, 1. Annalium tam regum Edwardi V.
Richardi III. et Henrici VII. quam Henrici VIII.
2. A Table of the Reports of Edmund Plowden. 3. The
Office of a Justice of Peace. He died about the vear
I LEVILLEA, a genus of plants belonging to the
dioecia class. See Botany Index.
ILEURI, Claude, an able French critic and histo¬
rian, was born at Paris in 1640. He applied himself
to the law, was made advocate for the parliament of
Paris, and attended the bar nine years j he then entered
into orders, and was made preceptor to the princes of
Conti. In 1689, the king made him sub-preceptor to
the dukes of Burgundy, Anjou, and Berry j and in
1706, when the education of these young princes was
completed, the king gave him the priory of Argente-
ville belonging to the Benedictines in the diocese of
Paris. In 1716, he was chosen counsellor to Louis
XV. and died in I723’ H6 was the author of a great
number of esteemed French works j the principal of
which are, I. An ecclesiastical history, in 20 volumes,
the last of which ends with the year 1414. 2. The
manners of the Israelites and Christians. 3. Institutions
of ecclesiastical law. 4. An historical catechism. 5.
On the choice and method of study. 6. The duties of
masters and servants, &c.
FLEXIBLE, in Physics, a term applied to bodies
capable of being bent or diverted from their natural
figure or direction.
FLEXOR, in Anatomy, a name applied to several
muscles, which are so called from their office, which is
to bend the parts to which they belong $ in opposition
to the extensors, which open or stretch them. See
Anatomy, Table of the Muscles.
FLIGHT, the act of a bird in flying 5 or the man¬
ner, duration, &c. thereof.
Almost every kind of bird has its particular flight j
the eagle’s flight is the highest j the flight of the spar¬
row-hawk and vulture is noble, and fit for high enter-
4 R . prise
F L I
Flight, prise and combat. The flight of some birds is low,
Flint. Weak, and transient *, the flight of the partridge and
V"—' pheasant is but of short continuance •, that ot the dove
is laboured ; that of the sparrow undulatory, &c.
The augurs pretend to foretel future events from the
flight of birds. See Augury.
Flight. In melting the lead ore in the works at
Mend ip, there is a substance which flies away in the
smoke which is called the flight. The workmen find it
sweetish upon their lips, if their faces happen to be in the
way of the smoke, which they avoid as much as possible.
This, falling on the grass, kills cattle that feed thereon ;
and being gathered and carried home, kills rats and mice
in their houses 5 that which falls on the sand, they gather
and melt upon a flag hearth into shot and sheet lead.
FLINT, a species of simple stones, chiefly composed
of siliceous earth. See MINERALOGY Index.
Breaking of Flints. The art of cutting, or rather
breaking flini stones into uniform figures, is by some
supposed to be one of the arts now lost. That it was
known formerly, appears from the ancient Bridewell
at Norwich, from the gate of the Augustin friars at
Canterbury, that of St John’s Abbey at Colchester, and
the (rate near Whitehall, Westminster. But that the
art fs not lost, and that the French know it, appears
from the platform on the top of the royal observatory
at Paris ; which, instead of being leaded, is paved with
flint cut or broken into regular figures.
Gun Flints. For the method of manufacturing,
see Mineralogy Index. ^
Flints, in the glass trade. _ The way of preparing
flints for the nicest operations in the glass trade is this.
Choose the hardest flints, such as are black and will
resist the file, and will grow white when calcined in
the fire. Cleanse these of the white crust that adheres
to them, then calcine them in a strong fire, and throw
them while red hot into cold water j wash off the ashes
that may adhere to them, and powder them in an non
mortar, and sift them through a very fine sieve j pour
upon this powder some weak aquafortis, or the phlegm
of aquafortis, to dissolve and take up any particles of
iron it may have got from the mortar ; stir this mix¬
ture several times, then let it rest, and in the morning
pour off the liquor, and wash the powder several times
with hot water and afterwards dry it for use. You
will thus have a powder for making the purest glass as
perfectly fine and faultless as if you had used rock-
crystal itself.
The washing off the ferruginous particles with aqua¬
fortis is not necessary when the glass intended to be
made is to be tinged with iron afterwards j but when
meant to be a pure white, this is the method that will
secure success.
Flint, the chief town of Flintshire, in North Wales.
It is commodipusly seated on the river Dee j and is but
a small place, though it sends one member to parlia¬
ment. It was formerly noted for its castle, where
Richard II. took shelter on his arrival from Ireland ;
but having quitted it, he was taken prisoner by the duke
of Lancaster. The castle now is in a ruinous condition.
This castle stands close to the sea on a rock, which in
various parts forms sevex*al feet of its foundation. It
covers about three quarters of an acre. The assizes
are still held in the town. It is 195 miles north-west
of London. Population 1433 1811.
FLO
FLINTSHIRE, a county of Wales, bounded on Flintshire j,!
the north-east and east by an arm of the sea, which is |j
properly the month of the river Dee ; on the north- Floath'5
west by the Irish sea; and on the south-south-west.
and west by Denbighshire. It is the least ot all the |j
counties in Wales, being but 33 miles in length and
9 in breadth. It is divided into five hundreds; in
which are two market towns and 28 parishes ; and
the population in 1811 was 46,518. The greatest
part of this county lies in the diocese of St Asaph, and
the rest belongs to that of Chester. It sends two mem¬
bers to parliament, one for the county and one for Flint;,
and pays one part of the land tax. I he air is cold,
but healthful. It is full of hills, intermixed with a few
valleys, which are very fruitful, producing some wheat
and plenty of rye. The cows, though small, yield a
great quantity of milk in proportion to their size, and
are excellent beef. The mountains are well stored
with lead, coal, and millstones. This county also pro¬
duces good butter, cheese, and honey. See Flint¬
shire, Supplement.
FLIP, a sort of sailors drink, made of malt liquor,
brandy, and sugar mixed.
FLOAT, a certain quantity of timber bound to¬
gether with rafters athwart, and put into a river to be
conveyed down the stream ; and even sometimes to
carry burdens down a river with the stream.
FLOAT-Boards, those boards fixed to water wheels of
under-shot mills, serving to receive the impulse of the
stream, whereby the wheel is carried round. See the
articles Wheel and Mill.
It is no advantage to have too great a number of
float-boards ; because, when they are struck by the
water in the best manner that it can be brought to
come against them, the sum of all the impulses will be
but equal to the impulse made against one float-board
at right-angles, by all the water coming out of the pen¬
stock through the opening, so as to take place on the
float-board. The best rule in this case is to have just
so many, that each of them may come out of the water
as soon as possible, after it has received and acted with
its full impulse. As to the length of the float-board, it
may be regulated according to the breadth.of the mill.
See Mill.
Floats for Fishing. See Fishing Flouts.
FLOATAGES, all things floating on the surface of
the sea or any water ; a word much used in the com¬
missions of water bailiffs.
FLOATING BODIES are those which swim on the
surface of a fluid, the most interesting of which are
ships and vessels employed in war and commerce. It is
known to every seaman, of what vast moment it Is to
ascertain the stability of such vessels, and the positions
they assume when they float freely on the surface of the
water. To be able to accomplish this, it is necessary to
understand the principles on which that stability and
these positions depend. This has been done with great
ingenuity by Mr Atwood, of whose reasoning the fol¬
lowing is a summary account, taken from the Philoso¬
phical Transactions for 179^*
A floating body is pressed downwards by its own
weight in a vertical line passing through its centre of
gravity; and it is supported by the upward pressure of
a fluid, which acts in a veitical line that passes through
the centre of gravity of the part which is under the
water;
[ 682 ]
FLO, [ 683 ] FLO
ifloaiing water ; and without a coincidence between these two
Bodies, lines, in such a manner as that both centres of gravity
•‘—r—’ may be in the same vertical line, the solid will turn on
an axis, till it gains a position in which the equilibrium
of floating will he permanent. From this it is obvious¬
ly necessary to find what proportion the part immersed
bears to the whole, to do which the specific gravity of
the floating body mu^t be known, after which it must
be found by geometrical methods, in which positions
the solid can be placed on the surface of the fluid, so
that both centres of gravity may be in the same verti¬
cal line, when any given part of the solid is immersed un¬
der the surface. These things being determined, some¬
thing is still wanting, for positions may be assumed in
which the circumstances now mentioned concur, and
yet the solid will assume some other position wherein it
will permanently float. If the specific gravity of a cylin¬
der be to that of the fluid on which it floats as 3 to 4, and
its axis to the diameter of the base as 2 to X : if it be
placed on the fluid with its axis vertical, it will sink to
& depth equal to a diameter and a half of the base j and
while its axis is preserved in a vertical position by out¬
ward force, the centres of gravity of the whole solid
and immersed part will remain in the same vertical line $
but when the external force is removed, it will deviate
from its upright position, and will permanently float
with its axis horizontal. If we suppose the axis to be
half the diameter of the base, and placed vertically, the
solid will sink to the depth of three-eighths of its dia¬
meter, and in that position it will float permanently.
If the axis he made to incline to the vertical line, the
solid will change its position till it permanently settles
with its axis perpendicular to the horizon.
Whether a solid floats permanently, or oversets when
placed 011 the surface of a fluid, provided the centre of
gravity of the solid and that of the immersed part be
in the same vertical line, it is said to be in a position of
equilibrium, of which there are three kinds j the equi¬
librium of stability, in which the solid permanently floats
in a given position ; the equilibrium of instability, in
which the solid spontaneously oversets, if not supported
by external force j and the equilibrium of indifl’erence,
or the insensible equilibrium, in which the solid rests on
the fluid indifferent to motion, without tendency to right
itself when inclined, or to incline farther.
If a solid body floats permanently on the surface of
a fluid, and external force be applied to turn it from its
position, the resistance opposed to this inclination is
termed the stability offloating. Some ships at sea yield
to a given impulse of the wind, and suffer a greater in¬
clination from the perpendicular than others. As this
resistance to heeling, duly regulated, has been consider¬
ed of importance in the construction of vessels, many
eminent mathematicians have laid down rules for ascer¬
taining the stability of ships from their known dimen¬
sions and weight, without recurring to actual experiment.
Bouguer, Euler, Chapman, and others, have laid down
theorems for this purpose, founded on the supposition
that the inclinations of ships from their quiescent posi¬
tions are evanescent, or very small in a practical point
of view. But ships at sea have been found to heel io°,
' 20°, or 30°, and therefore it may be doubted how far
such rules are applicable in practice. If statics can be
applied to naval architecture, it seems necessary that
the rules should be extended to those cases in^which the
angles of inclination are of any magnitude, likely to
occur in the practice of navigation. A solid body
placed on the surface of a lighter fluid, at such a depth
as corresponds to the relative gravities, cannot alter its
position by the joint action of its own weight and the
pressure of the fluid, except by turning on some horizon¬
tal axis passing through the centre of gravity j hut, as
many axes may be drawn through this point of the float¬
ing body, in a direction parallel to the horizon, and the
motion ot the solid regards only one axis, this must be
determined by the figure of the body and the particu¬
lar nature ot the case. When this axis of motion is as¬
certained, and the specific gravity of the solid found,
the positions of permanent floating will be determined,
by finding the several positions of equilibrium through
winch the solid may be conceived to pass, while it turns
round the axis of motion ; and by determining in which
of these positions the equilibrium is permanent, and in
which of them it is momentary.
Hie whole ot Mr Atwood’s valuable paper relates to
the theory ot naval architecture, in so far as it is de¬
pendent on the laws of pure mechanics. If the propor¬
tions and dimensions adopted in the construction of
individual vessels are obtained by exact geometrical
measurement, and observations are made on the per¬
formance ot these vessels at sea ; a sufficient number of
experiments of this nature, judiciously varied, are the
proper grounds on which theory may be effectually ap¬
plied, in reducing to system those hitherto unperceived
causes, which contribute to give the greatest degree of
excellence to vessels of every description. Naval ar¬
chitecture being reckoned among the practical branches
of science, every voyage may be viewed in the light of
an experiment, irom which useful truths are to be de¬
duced. But inferences of this nature cannot w’ell be
obtained, except by acquiring a thorough knowledge of
all the proportions and dimensions of each part of the
ship, and by making a sufficient number of observations
on the qualities of the vessel, in all the varieties of situ¬
ation to which a ship is commonly subject in the prac¬
tice of navigation.
The following is an ingenious investigation of the same
Subject by Mr English, which we give in his own words.
“ However operose and difficult (says he) the calcu¬
lations necessary to determine the stability of nautical
vessels may, in some cases, be, yet they all depend, says
this author, upon the four following simple and obvious
theorems, accompanied with other well known stereome-
trical and statical principles.
“ Theorem 1. Every floating body displaces a quan¬
tity of the fluid in which it floats, equal to its own
weight j and consequently the specific gravity of the
fluid will be to that ol the floating body, as the magni¬
tude of the whole is to that of the’part immersed.
“ Theorem 2. Every floating body is impelled down¬
ward by its own essential power, acting in the direction
of a vertical line passing through the centre of gravity
ot the whole $ and is impelled upward by the re-action
of the fluid which supports it, acting in the direction
of a vertical line passing through the centre of gravity
of the part immersed j therefore, unless these two lines
are coincident, the floating body thus impelled must
revolve round an axis, either in motion or at rest, un¬
til the equilibrium is restored.
“ Theorem 3. If by any power whatever a vessel be
4 II2 deflected
Floating
Bodies.
FLO [ 684 ] FLO
deflected from an upright position, the perpendicular
distance between two vertical lines passing through the
centres of gravity of the whole, and of the part im¬
mersed -respectively, will he as the stability of the vessel,
and which will be positive, nothing, or negative, ac¬
cording as the metacentre is above, coincident with, or
below, the centre of gravity of the vessel.
“ Theorem 4. The common centre of gravity of any
system of bodies being given in position, it any one of
these bodies be moved from one part of the system to
another, the corresponding motion of the common
centre of gravity, estimated in any given direction,
will be to that of the aforesaid body, estimated in the
same direction, as the weight of the body moved is to
that of the whole system.
“ From whence it is evident, that in order to ascer¬
tain the stability of any vessel, the position of the
centres of gravity of the whole, and of that part im¬
mersed, must be determined •, with which, and the di¬
mensions of the vessel, the line of floatation, and angle
of deflection, the stability or power either to right it¬
self or overturn, may be found.
“In ships of war and merchandise, the calculations
necessary for the purpose become unavoidably very
operose and troublesome ; but they may be much facili¬
tated by the experimental method pointed out in the
New Transactions of the Swedish Academy of Scien¬
ces, first quarter of the year 1787, page 48.
“ In river and canal boats, the regularity and sim¬
plicity of the form of the vessel itself, together with the
compact disposition and homogeneal quality ot the
burden, render that method for them unnecessary, and
make the requisite calculations become very easy. Ves¬
sels of this kind are generally of the same transverse
section throughout their whole length, except a small
part in prow and stern, formed by segments of circles
or other simple curves •, therefore a length may easily
be assigned such, that any of the transverse sections be¬
ing multiplied thereby, the product will he equal to
the whole solidity of the vessel. The form of the sec¬
tion ABCD is for the most part either rectangular,
as in fig. I. Plate CCXV1II. trapezoidal as in fig. 2.
or mixtilineal as in fig. 3. in all which MM represents
the line of floatation when upright, and EF that when
inclined at any angle MXE •, also G represents the
centre of gravity of the whole vessel, and li that of
the part immersed.
“ If the vessel be loaded quite up to the line AB,
and the specific gravity of the boat and burden be the
same, then the point G is simply the centre of gravity
of the section ABCD ; but if not, the centres of gra¬
vity of the boat and burden must be found separately,
and reduced to one by the common method, namely, by
dividing the sum of the momenta by the sum of weights,
or areas, which in this case are as the weights. The point
R is always the centre of gravity of the section MMCD,
which, if consisting of different figures, must also be found
by dividing the sum of the momenta by the sum of the
weights as common. These two points being found,
the next thing necessary is to determine the area of the
two equal triangles MXE, MXF, their centres of gra¬
vity 0, 0, and the perpendicular projected distance n n
of "these points on the water line EF. This being
done, through R and parallel to EF draw RT = a
fourth proportional to the whole area MMCD, either
triangle MXE or MXF, and the distance nn; through Floating
T, and at right angles to RT or EF, draw TS meet- Bodies,
ing the vertical axis of the vessel in S the metacentre; ' " ~
also through the points G, B, and parallel to ST, draw
NGW and BV ; moreover through S, and parallel to
EF, draw WSV, meeting the two former in V and
W j then SW is as the stability of the vessel, which
will be positive, nothing, or negative, according as the
point S is above, coincident with, or below, the point
G. If now we suppose W to represent the weight of
the whole vessel and burden (which will be equal to
the section MMCD multiplied by the length of the
vessel), and P to represent the required weight applied
at the gunwale B to sustain the vessel at the given
angle of inclination $ we shall always have this propor¬
tion : as VS : SW :: W : P; which proportion is ge¬
neral, whether SW be positive or negative •, it must
only in the latter case be supposed to act upward to
prevent an overturn.
“ In the rectangular vessel, of given weight and
dimensions, the whole process is so evident, that any
farther explanation would be unnecessary. In the tra¬
pezoidal vessel, after having found the points G and R,
let AD, BC be produced until they meet in K. Then,
since the two sections MMCD, El DC are equal j the
two triangles MMX, EFK are also equal j and there-
fore the rectangle EK X KF — K.M X KM 1= KM1 j
and since the angle of inclination is supposed to be
known, the angles at E and F are given. Consequent¬
ly, if a mean proportional be found between the sines
of the angle at E and F, we shall have the following
proportions :
“ As the mean proportional thus found : sine 2. F ::
KM : KF, and in the said mean proportional : sine
Z. F :: KM : KE j therefore ME, MF become known j
from whence the area of either triangle MXE or MXF,
the distance n n, and all the other requisites, may be
found.
“ In the mixtilineal section, let AB =9 feet =108
inches, the whole depth = 6 feet == 72 inches, and
the altitude of MM the line of floatation 4 feet or 48
inches $ also let the two curvilinear parts be circular-
quadrants of two feet, or 24 inches radius each. Then
the area of the two quadrants == 904*7808 square
inches, and the distance of their centres of gravity
from the bottom = 13*8177 inches very nearly, also
the area of the included rectangle able — 1440 square
inches, and the altitude of its centre of gravity 12
inches j in like manner, the area of the rectangle AB
c d will be found = 5184 square inches, and the alti¬
tude of its centre of gravity 48 inches; therefore we
shall have
Momentum of
lh» two quad.
Moment, of the
rcctan. a b i e
Moment, of the
rectan. AB erf
=904-;So7 X I3*SI77
ni44o* X 12
^ ~5iS4* X 4S
— 12501*98966016
— 1-7280.
— 248832*
7528*7808 278613*98966016
“ Now the sum of the momenta, divided by the sum
of tl,e areas, w.l! g.ve ^^8^8 _ “ 37 6
inches, the altitude of G, the centre of gravity of the-
FLO [ 685 ] * FLO
Floating section ABCD above the bottom. In like manner,
Bodies the altitude of R, the centre of gravity of the section
"looking. MMCD, will be found to be equal
—. 493^7808
= 24,934 inches; and consequently their difference, or
the value ofGR = iz‘0^2 inches, will be found.
Suppose the vessel to heel 150, and we shall have
the followiner proportion ; namely, As radius : tangent
of 150 :: MX=r54 inches : 14-469 inches = ME or
MF; and consequently the area of either triangle MXE
or MXF = 390 663 square inches. Therefore, by
theorem 4th, as 4936-7808 : 390*663 :: 72 = nn=z
T : 5-6975 inches =: RT; and, again, as radius :
sine of 150 :: 12-072 =: GR : 3.1245 inches RN ;
consequently RT—RN = S'^91 S — 3’1245 = 2'573
inches — SW, the stability required.
“ Moreover, as the sine of 150 : radius :: 5.69*75=:
RT : 22-013 r= RS, to which if we add 24-934, the
altitude of the point R, we shall have 46-947 for the
height of the metacentre, which taken from 72, the
whole altitude, there remains 25-053 ; from which, and
the half width — 54 inches, the distance BS is found
=r 59-529 inches very nearly, and the angle SBV —
8o°—06'—42"; from whence SV =: 58*645 inches.
Again : Let us suppose the mean length of the ves¬
sel to be 40 feet, or 480 inches, and we shall have the
weight of the whole vessel equal to the area of the sec¬
tion MMCD == 4936-7808 multiplied by 480 =
2369654-784 cubic inches of water, which weighs ex¬
actly 85708 pounds avoirdupoise, allowing the cubic
foot to weigh 62.5 pounds.
“ And, finally, as SV : SW (i. e.) as 58-645 : 2*573
:: 85708 : 3760 the weight on the gunwale which
will sustain the vessel at the given inclination. There¬
fore a vessel of the above dimensions, and weighing 38
tons 5 cwts. 28lbs. will require a weight of one ton 13
cwt. 64lbs. to make her incline 15°.
“ In this example the deflecting power has been sup¬
posed to act perpendicularly on the gunwale at B ; but
if the vessel is navigated by sails, the centre veiique
must be found; with which and the angle of deflection,
the projected distance thereof on the line SV may be
obtained ; and then the power calculated as above, ne¬
cessary to be applied at the projected point, will be
that part of the wind’s force which causes the vessel to
heel. And conversely, if the weight and dimensions of
the vessel, the area and altitude of the sails, the direc¬
tion and velocity of the wind be given, the angle of de-
ilohit. flection may be found
•-g-i. Floating Bridge,. See Bridge.
Flock Paper. See Paper.
r LOOD, a deluge or inundation of waters. See
Deluge.
Flood is also used in speaking of the tide. When
the water is at lowest, it is called ebb; when rising,
young flood ; when at highest,/fogv^/Zooc/; when begin¬
ning to fall, ebb water.
FLoon-mark, the mark which the sea makes on the
shore at flowing w-ater and the highest tide, it is also
called high-water mark.
FLOOK of an anchor. See Anchor.
FLOCKING, among miners, a term used to ex¬
press a peculiarity in the load of a mine. The load or
quantity of ore is frequently intercepted in its course
by the crossing of a vein of earth or stone, or some dif¬
ferent metallic substance ; in which case the load is
moved to one side, and this transient part of the load is
called a flanking.
FLOOR, in building, the underside of a room, or
that part we w'alk on.
Floors are of several sorts ; some of earth, some of
brick, others of stone, others of boards, &c.
For brick and stone Floors, see Pavement.
lor boarded Floors, it is observable that the car--
penters never floor their rooms with boards till the
carcass is set up, and also enclosed with walls, lest the
weather should injure the flooring. Yet they general¬
ly rough-plane their boards for the flooring before
they begin any thing else about the building, that they
' may set them by to dry and season, which is done in
the most careful manner. The best wood for flooring
is the fine yellow deal w'ell seasoned, which when well
laid, will keep its colour for a long while ; whereas the
white sort becomes black by often washing, and looks
very bad. rIhe joints of the boards are commonly
made plain, so as to touch each other only ; but, when
the stuff is not quite dry, and the boards shrink, the
water runs through them whenever the floor is washed,,
and injures the ceiling underneath. For this reason
they are made with feather edges, so as to cover each
other about half an inch, and sometimes they are made
with grooves and tenons : and sometimes the joints are
made with dove tails ; in which case the lower edge
is nailed down, and the next drove into it, so that the
nails are concealed. The manner of measuring floors
is by squares of 10 feet on each side, so that taking the
length and breadth, and multiplying them together,
and cutting off two decimals, the content of a floor in.
square will be given. Thus 18 by 16 gives 288 or 2
squares and 88 decimal parts.
Earthen Floors, are commonly made of loam, and
sometimes, especially to make malt on, of lime and
brook sand, and gun dust or anvil dust from the forge.
Ox blood and fine clay, tempered together, Sir Hugh
Plat says, make the finest floor in the world.
The manner of making earthen floors for plain coun--
try habitations is as follows : Take two-thirds of lime
and one of coal ashes well sifted, with a small quan¬
tity of loam clay ; mix the whole together, and temper
it well with water, making it up into a heap: let it
lie a week or ten days, and then temper it over again.
After this heap it up for three or four days, and re¬
peat the tempering very high, till it becomes smooth,,
yielding, tough, and gluey. The ground being then
levelled, lay the floor therewith about 2^ or 3 inches
thick, making it smooth with a trowel: the hotter the
season is the better ; and when it is thoroughly dried,
it will make the best floor for houses, especially malt
houses.
If any one would have their floors look better, let
them take lime made of rag stones, well tempered with
whites of eggs, covering the floor about half an inch
thick with it, before the under-flooring is too dry. If
this be well done, and thoroughly dried, it will look
when rubbed with a little oil as transparent as metal or
glass. In elegant houses, floors of this nature are made
ot stucco, or of plaster of Paris beaten and sifted, and
mixed with other, ingredients.
Floor.
Flocking:,
F loor.
FLO [
FW Fioon of a Ship, strictly taken, is only so much of
|! her bottom as she rests on when aground*
Vloiales. Such ships as have long, and withal broad rloors, lie
* ' on the ground with most security, and are not apt to
heel, or tilt on one side*, whereas others, which are
narrow in the floor, or in the sea phrase, cranked bij
the ground, cannot be grounded without danger ol be¬
ing overturned. . . ,
Floor Timbers, in a ship, are those parts otaship s
timbers which are placed immediately across the kee ,
and upon which the bottom of the ship is framed •, to
these the upper parts of the timbers are united, being
only a continuation of floor timbers upwards.
FLORA, the reputed goddess of flowers, was, ac¬
cording to Lactantius, only a lady of pleasure, who
having gained large sums of money by prostituting
herself, made the Roman people her heir, on condition
that certain games called might be annually
celebrated on her birth-day. Some time afterwards,
however, such a foundation appearing unworthy the
majesty of the Roman people, the senate, to ennoble
the ceremony, converted Flora into a goddess, whom
they supposed to preside over flowers j and so made it
a part of religion to render her propitious, that it
might be well with their gardens, vineyards, &c. But
Vossius (de Idol. lib. i. c. 12.) can by no means al¬
low the goddess Flora to have been the courtezan
above mentioned : he will rather have her a Sabine
deity, and thinks her worship might have commenced
under Romulus. His reason is, that Varro, in his
fourth book of the Latin tongue, ranks Flora among
the deities to whom Tatius king of the Sabines ot-
fered up vows before he joined battle with the Ro¬
mans. Add, that from another passage in Varro it
appears, that there were priests of Flora, with sacri¬
fices, &c. as early as the times of Romulus and
Numa. .
The goddess Flora was, according to the poets, the
wife of Zephyrus. Her image in the temple of Castor
and Pollux was dressed in a close habit, and she held
in her hand the flowers of pease and beans : but the
modern poets and painters have been more lavish in
setting off her charms, considering that no parts of na¬
ture offered such innocent and exquisite entertainment
to the sight and smell, as the beautiful variety which
adorns, and the odour which embalms, the floral crea-
tion. „ .
FLORALES ludi, or Floral Games, in anti¬
quity, were games held in honour of Flora, the god¬
dess of flowers.—They were celebrated with shameful
debaucheries. The most licentious discourses were not
enough, but the courtezans were called together by
the sound of a trumpet, made their appearance na¬
ked, and entertained the people with indecent shows
and postures : the comedians appeared after the same
manner on the stage. Val. Maximus relates, that Cato
being once present in the theatre on this occasion,
the people were ashamed to ask for such immodest re¬
presentations in his presence ; till Cato, apprised of
the reservedness and respect with which he inspired
them, withdrew, that the people might not be disap¬
pointed of their accustomed diversion. There were se¬
veral other sorts of shows exhibited on this occasion ;
and, if we may believe Suetonius in Galba, c. 6. and
2
686 ] FLO
Vopiscus in Carinas, these princes presented elephants
dancing on ropes on these occasions.
The ludi florales, according to Pliny, lib. xviii. c. 29.
were instituted by order of an oracle of the Sibyls,
on the 28th of April', not in the year of Rome
13XVI. as we commonly read it in the ancient editions
of that author j nor in laxiv. as F. Hardouin has
corrected it, but, as Vossius reads it, in 513: though
they were not regularly held every year till after 580.
They were chiefly held in the night time, in the J a-
trician street: some will have it there was a circus for
the purpose on the hill called Hortulorum.
FLORALIA, in antiquity, a general name for the
feasts, games, and other ceremonies, held in honour of
the goddess Flora. See Flora and 1 lorales Ludt.
FLORENCE, the capital of the duchy of Tuscany,
and one of the finest cities in Italy. It is surrounded
on all sides but one with high hills, which rise insen¬
sibly, and at last join with the lofty mountains called
the Apennines, lowards Pisa, there is a vast plain of
40 miles in length', which is so filled with villages and
pleasure houses, that they seem to be a continuation of
the suburbs of the city. Independent of the churches
and palaces of Florence, most of which are very
magnificent, the architecture of the houses in general
is in a good taste', and the streets are remarkably clean,
and paved with large broad stones chiseled so as to
prevent the horses from sliding. The city is divided
into two unequal parts by the river Arno, over which
there are no less than four bridges in sight of each
other. That called the Ponte dell Trinita, which is
uncommonly elegant, is built entirely of white marble,
and ornamented with four beautiful statues represent¬
ing the Seasons. The quays, the buildings on each
side, and the bridges, render that part of Florence
through which the river runs by far the finest. Every
corner of this beautiful city is full of wonders in the
arts of painting, statuary, and architecture. 1 he stieets,
squares, and fronts of the palaces, are adorned with
a great number of statues ; some of them by the best
modern masters, Michael Angelo, Bandinelli, Dona¬
tello, Giovanni di Bologna, Benvenuto Cellini, and
others. Some of the Florentine merchants formerly
were men of vast wealth, and lived in a most magni¬
ficent manner. One of them, about the middle of the
fifteenth century, built that noble fabric, which, from
the name of its founder, is still called the Pala%%o Pittu
The man was ruined by the prodigious expence of this
building, which was immediately purchased by the
Medici family, and has continued ever since to be the
residence of the sovereigns. The gardens belonging
to this palace are on the declivity of an eminence. On
the summit there is a kind of fort,.called
From this, and from some of the lug her walks, you
have a complete view of the city of r lorence, and the
beauteous vale of Arno, in the middle of which it
stands. This palace has been enlarged since it was
purchased from the ruined family of Pitti. Ihe fur¬
niture is rich and curious, particularly some tables ot
Florentine work, which are much admired. Ihe
most precious ornaments, however, are the paintings.
The walls of what is called the Imperial Chamber, are
painted in fresco, by various painters ; the subjects are
allecorical, and in honour of Lorenzo ot Medicis, di-
* stinguished
[ 68? ] FLO
stinguished by the name of the Magnificent. The fa¬
mous gallery attracts every stranger. One of the most
interesting parts of it, in the eyes of many, is the se¬
ries of Roman emperors, from Julius Cgesar to Gal-
lienus, with a considerable number of their empresses,
arranged opposite to them. This series is almost com¬
plete ; but wherever the bust of an emperor is wanting,
the place is filled up by that of some other distinguish¬
ed Roman. The celebrated Venus of Medici, which
has been removed to Paris, is thought to be the standard
of taste in female beauty and proportion, and stood for¬
merly in a room called the Tribunal. The inscription
on its base mentions its being made by Cieomenes an
Athenian, the son of Apollodorus. It is of white marble,
and surrounded by other masterpieces of sculpture, some
of which are said to be the works of Praxiteles and
other Greek masters. In the same room are many va¬
luable curiosities, besides a collection of admirable pic¬
tures by the best masters. There are various other
rooms, whose contents are indicated by the names they
bear; as, the Cabinet of Arts, of Astronomy, of Natu¬
ral History, of Medals, of Porcelain, of Antiquities ;
the Saloon of the Hermaphrodite, so called from a sta¬
tue which divides the admiration of the amateurs with
that in the Borghese villa at Rome, though the excel¬
lence of the execution is disgraced by the vileness of
the subject ; and the Gallery of Portraits, which con¬
tains the portraits of the most eminent painters (all ex¬
ecuted by themselves) who have flourished in Europe
during the three last centuries. Our limits will not ad¬
mit of a detail of the hundredth part of the curiosities
and buildings of Florence. We must not, however,
omit mentioning the chapel of St Lorenzo, as being
perhaps the finest and most expensive habitation that
ever was reared for the dead ; it is incrusted with pre¬
cious stones, and adorned by the workmanship of the
best modern sculptors. Mr Addison remarked, that
this chapel advanced so very slowly, that it is not im¬
possible that the family of Medicis may be extinct be¬
fore their burial place is finished. This has actually
taken place: the Medici family is extinct, and the
chapel remains still unfinished.
Florence is a place of some strength, and contains
an archbishop’s see and an university. The number of
inhabitants is calculated at 80,000. They boast of the
improvements they have made in the Italian tongue,
by means of their Academia della Crusca; and seve¬
ral other academies are now established at Florence.
Though the Florentines affect great state, yet their nobi¬
lity and gentry drive a retail trade in wine, which they
sell from their cellar windows, and sometimes thev even
hang out a broken flask, as a sign where it may be bought.
They deal, besides wine and fruits, in gold and silver
stuffs. The Jews are not held in that degree of odium,
or subjected to the same humiliating distinctions here,
as in most other cities of Europe ; and it is said that
some of the richest merchants are of that religion.
As to the manners and amusements of the inhabi¬
tants, Dr Moore informs us, that besides the conver-
sa%ionis, which they have here as in other towns of
Italy, a number of the nobility meet every day at a
house called the Casino. This society is pretty much
on the same footing with the clubs in London. The
members are elected by ballot. They meet at no par¬
ticular hour, but go at any time that is convenient.
Ihey play at billiards, cards, and other games, or con¬
tinue conversing the whole evening, as they think pro¬
per. riiey are served with tea, coffee, lemonade, ices,
or what other refreshments they choose; and each per¬
son pays for what he calls for. There is one material
diflerence between this and the English clubs, that wo¬
men as well as men are members. The company of
both sexes behave with more frankness and familiarity
to strangers as well as to each other, than is customary
in public assemblies in other parts of Italy. The opera
is. a place where the people of quality pay and receive
visits, and converse as freely as at the Casino above
mentioned. 1 his occasions a continual passing and re¬
passing to and from the boxes, except in those where
there is a party of cards formed ; it is then looked on
as a piece of ill manners to disturb the players. From
this it may be guessed, that here, as in some other
towns in Italy, little attention is paid to the music by
the.company in the boxes, except at a new opera, or
during some favourite air. But the dancers command
a general attention \ as soon as they begin, conversa¬
tion ceases ; even the card-players lay down their cards,
and fix their eyes on the ballette. ’Yet the excellence
of Italian dancing seems to consist in feats of strength,
and a kind of jerking agility, more than in graceful
movement. There is a continual contest among the
pel formers, who shall spring highest. You see here
none of the sprightly alluring gaiety of the French
comic dancers, or of the graceful attitudes and smooth
flowing motions of the performers in the serious opera
at Paris. It is surprising that a people of such taste
and sensibility as the Italians, should prefer a parcel of
athletic jumpers (0 elegant dancers. On the evenings
on which there is no opera, it is usual for the genteel
company to drive to a public walk immediately without
the city, where they remain till it begins to grow dusk-
ish. E. Long. 12. 24. N, Lat. 43. 34.
Florence, an ancient piece of English gold coin.
Every pound weight of standard gold was to be coined
into 50 Florences, to be current at six shillings each ;
all which made in tail 15 pounds; or into a propor¬
tionate number of half Florences, or quarter pieces, by
indenture of the mint: 18 Edvv. III.
FLORENTIA, in Ancient Geography, a town of
Etruria, on the Arnus ; of great note in Sylla’s wars.
Now called Florenza or Firen’za by the Italians ; Flo¬
rence in English. E. Long. 11. Lat. 43. 30.
FLORENTINE marble. See Citadinesca.
FLORESCENTIA (from floresco, “ to flourish or
bloom”) ; the act of flowering, which Linnseus and
the sexualists compare to the act of generation in ani¬
mals ; as the ripening of the fruit in their opinion re¬
sembles the birth. See Flower.
FLORID style, is that too much enriched with
figures and flowers of rhetoric.
FLORIDA, a province of North America, bounded
on the south by the gulf of Mexico, on the north by
Georgia and Alabama, on the east by the sea, and on the
west by Alabama. It was first discovered, in 1497, by
Sebastian Cabot, a Venetian, then in the English service;
whence a right to the country was claimed by the kings
of England ; and this province, as well as Georgia, was
included in the charter granted by Charles II. to Ca¬
rolina. .
Flore; ee
11
Honda.
■1'lorida.
L O [ 688 ] FLO
however Florida was more fully miles. The country is in general flat, and without hills. Fiend,
- - " • ■ • The soil is good, but overrun with pines and brush- 11
wood. The climate is considered better than that of.
roliiia. In 1512, ,
’ discovered by Ponce de Leon, an able Spanish naviga
tor, but who undertook his voyage from the most ab
surd motives that can he well imagined.—The Indians
of the Caribbee islands had among them a tradition,
that somewhere on the continent there was a fountain
whose waters had the property of restoring youth to all
old men who tasted them. Ponce de Leon, who set
out with this extravagant view as well as others, re¬
discovered Florida-, but returned to the place from
whence he came, visibly more advanced in years than
when he set out. For some time this country was ne¬
glected by the Spaniards, and some Frenchmen settled
in it: But by orders of Philip II. of Spain, a force
was fitted out j the French intrenchments were forced,
and most of the people killed. The prisoners were
hanged on trees-, with this inscription, “ Jsot as
Frenchmen, but as Heretics.” . .
This cruelty was soon after revenged by Dominic de
Gourgues, a skilful and intrepid seaman of Gascony,
an enemy to the Spaniards, and passionately fond ot
hazardous expeditions and of glory. He sold his estate 5
built some ships -, and with a select band of adventurera
like himself embarked for Florida. He drove the
Spaniards from all their posts with incredible valour
and activity defeated them in every rencounter : and
by way of retaliation, hung the prisoners on trees,
with this inscription, “ Not as Spaniards, but as
Assassins.” This expedition was attended with no
other consequences-, Gourgues blew up the forts he
had taken, and returned home, where no notice was
taken of him. It was conquered in 1539 by the
Spaniards under Ferdinand de Soto, not without a
great deal of bloodshed -, as the natives were very war¬
like and made a vigorous resistance. The settlement,
however, was not fully established till the year 1565 J
when the town of St Augustine, the capital of the
colony while it remained in the hands of the Spaniards,
was founded. In 1586, this place was taken and
pillaged by Sir Francis Drake. It met with the same
fate in 1665, being taken and plundered by Captain
Davis and a body of bucaniers. In 1702, an at¬
tempt was made upon it by Colonel More, governor
of Carolina. He set out with 500 English and 700
Indians: and having reached St Augustine, he be¬
sieged it. for three months -, at the expiration of which,
tlm Spaniards having sent some ships to the relief of
the place, he was obliged to retire. In 1740 another
attempt was made by General Oglethorpe ", but he be¬
ing outwitted by the Spanish governor, was forced to
raise the siege with loss ; and Florida continued in the
hands of the Spaniards till the year 1763, when it
was ceded by treaty to Great Britain.—During the
American war, which terminated in 1783, it was again
reduced by his Catholic majesty, and it remained sub-
icct to Spain till 1818, when General Jackson alleg¬
ing that support had been given by the Spaniards to
some hostile Indian tribes, seized Pensacola and St
Marks, the only fortified posts in the country except St
Augustine. The province was since ceded by treaty
to die United States, and the treaty after much delay
has at length been ratified (1821.)
Florida is about 400 miles in length, from north to
south, and occupies an area of about 30,000 square
3
C
the neighbouring state of Georgia. The whole white
inhabitants and slaves probably do not exceed 12,000
or 15,000. See Florida, Supplement.
FLORILEGIUM, Florilege, a name the La¬
tins have given to what the Greeks call otnioMyioi, an-
thology; viz. a collection of choice pieces, containing
the finest and brightest things in their kind.
Florilege, is also particularly used for a kind of
breviary, in the Eastern church, compiled by Arcadius,
for the conveniency of the Greek priests and monks,
who cannot carry with them, in their travels and pil¬
grimages, all the volumes wherein their office is dis¬
persed. The florilegium contains the general rubrics,
psalter, canticles, the horologium, and the office of the
feriae, &c.
FLORIN, is sometimes used for a coin, and some¬
times for a money of account.
Florin, as a coin, is of diflerent values, according
to the different metals and diflerent countries where it
is struck. The gold florins are most of them of a very
coarse alloy, some of them not exceeding thirteen or
fourteen carats, and none of them seventeen and a
half. See Money Table.
Florin, as a money of account, is used by the Italian,
Dutch, and German merchants and bankers, but admits
of divisions in different places. Ibid.
FLORINIANI, or Floriani, a sect of heretics,
of the second century, denominated from its author
Flonnus, or Florianus, a priest of the Roman church,
deposed along with Blastus for his errors. Floriuus
had been a disciple of St Polycarp, along with Irenmus.
He made God the author of evil ; or rather asserted,
that the things forbidden by God are not evil, but of
his own appointing. In winch he followed the errors
of Valentinus, and joined himself with the Carpocra-
tians. They had also other names given them. Phi-
lastrius says, they were the same with xheCai'pophorians.
He adds that they were also called soldiers, milites, quia
de militanbusJuerunt. St Irenaeus calls them Gnostics 1
St Epiphanius Phibionites; and Theodoret, Barborites,
on account of the impurities of their lives. Others
call them Zaccheans ; others Coddians, &c. though for
what particular reasons, it is not easy to say, nor per¬
haps would be worth while to inquire.
FLORIS, Francis, an eminent historical painter,
was born at Antwerp in 1520. Pie followed the pro¬
fession of a statuary till he was twenty years of age ;
when preferring painting, he entered the school of
Lambert Lombard, whose manner he imitated very
perfectly. He afterwards went to Italy, and completed
his studies from the most eminent masters. The great
progress he made in historical painting, at his return
procured him much employment} and his countrymen
complimented him with the flattering appellation ot
the Flemish Raphael. He got much money, and might
have rendered his acquaintance more worthy of the at¬
tention of the great, had he not debased himself by
frequent drunkenness. Pie died in 1^0, aged 50.
FLORIST, a person curious or skilled in flowers :
their kinds, names, characters, culture, &c. It is al*o
applied to an author who writes what is called the
• flora
Florist.
FLO [ 689 ] FLO
florist
II
lotton.
flora of any particular place, that is, a catalogue of the
plants and trees which are found spontaneously growing
there.
FLORUS, Lucius ANN.a£us, a Latin historian, of
the same family with Seneca and Lucan. He flourish¬
ed in the reigns of Trajan and Adrian j and wrote an
abridgement of the Roman history, of which there have
been many editions. It is composed in a florid and po¬
etical style ; and is rather a panegyric on many of the
great actions of the Romans, than a faithful and cor¬
rect recital of their history. He also wrote poetry,
and entered the lists against the emperor Adrian, who
satirically reproached him with frequenting taverns and
places of dissipation.
FLORY, Flowery, or Fleury, in Hei'aldry, a
cross that has flowers at the end circumflex and turn¬
ing down ; different from the potence, in as much as
the latter stretches out more like that which is called
patee.
FLOS, Flower. See Flower, Botany Index.
Fotmineus Flos, a flower which is furnished with the
pointal or female organs of generation, but wants the
stamina or male organ. Female flowers may be produ¬
ced apart from the male, either on the same root or on
distinct plants. Birch and mulberry are examples of
the first case, willow and poplar of the second.
Masculus Flos, a male flower. By this name Lin¬
naeus and the sexualists distinguish a flower which con¬
tains the stamen, reckoned by the sexualists the male
organ of generation j but not the stigma or female or¬
gan. All the plants of the class dioecia of Linnaeus
have male and female flowers upon different roots j those
of the class monoecia bear flowers of different sexes on
the same root. The plants, therefore, of the former
are only male and female : those of the latter are an¬
drogynous j that is, contain a mixture of both male
and female flowers.
Flos, in Chemistry, the most subtile part of bodies,
separated from the more gross parts by sublimation in a
dry form.
FLOTA, or Flotta, fleet; a name the Spaniards
give particularly to the ships which they send annually
from Cadiz to the port of Vera Cruz, to fetch thence
the merchandises gathered in Mexico for Spain. It
consists of the captains, admiral, and patach, or pin¬
nace, which go on the king’s account j and about 16
ships, from 400 to 1000 tons, belonging to particular
persons. They set out from Cadiz about the month of
August, and are 18 or 20 months before they return.
Those sent to fetch the commodities prepared in Peru
are called galleons.
The name flotilla is given to a number of ships
which get before the rest in their return, and give in¬
formation of the departure and cargo of the flota and
galleons.
FLOTSON, or Flotsom, goods that by ship¬
wreck are lost, and floating upon the sea ; which,
with jetson and lagan, are generally given to the lord
admiral : but this is the case only where the owners
of such goods are not known. And here it is to be
observed that jetson signifies any thing that is cast
out of a ship when in danger, and afterwards is beat
on the shore by the water, notwithstanding which the
ship perishes. Lagan is where heavy goods are thrown
Vol. VIII. Part II. f
overboard, before the wreck of the ship, and sink to Flotson
the bottom of the sea. B
FLOUNDER, Fluke, or But. See Pleuronec- :
tes, Ichthyology Index. —y—-
Flounders may be fished for all day long, either in a
swift stream, or in the still deep water ; but best in the
stream, in the months of April, May, June, and July:
the most proper baits are all sorts of worms, wasps, and
gentles.
FLOUR, the meal of wheat-corn, finely ground and
sifted. See Meal.
The grain itself is not only subject to be eaten by
insects in that state j but, when ground into flour, it
gives birth to another race of destroyers, who eat it
unmercifully, and increase so fast in it, that it is not
long before they wholly destroy the substance. The
finest flour is most liable to breed these, especially when
stale or ill prepared. In this case, if it be examined
in a good light, it will be observed to be in continual
motion, and on a nicer inspection there will be found
in it a great number of little animals of the colour of
the flour, and very nimble. If a little of this flour
is laid on the plate of the double microscope, the in¬
sects are very distinctly seen in great numbers, very
brisk and lively, continually crawling over one an¬
other’s backs, and playing a thousand antic tricks to¬
gether ; whether in diversion or in search of food, is
not easy to be determined. These animals are of an
oblong and slender form $ their heads are furnished
with a kind of trunk or hollow tube, by means of
which they take in their food, and their body is com¬
posed of several rings. They do vast mischief among
magazines of flour laid up for armies and other public
uses. When they have once taken possession of a par¬
cel of this valuable commodity, it is impossible to drive
them out j and they increase so fast, that the only me¬
thod of preventing the total loss of the parcel is to make
it up into bread as soon as can be done. The way to
prevent their breeding in the flour is to preserve it
from damp: nothing gets more injury by being put up
damp than flour 5 and yet nothing is more frequently
put up so. It should be always carefully and thorough¬
ly dried before it is put up, and the barrels also dried
into which it is to be put j then, if they are placed in
a room tolerably warm and dry, they will keep it well.
Too dry a place never does flour any hurt, though one
too moist almost always spoils it.
Flour, when carefully analyzed, is found to be com¬
posed of three very different substances. The first and
most abundant is pure starch, or white fecule, insolu¬
ble in cold, but soluble in hot water, and of the nature
of mucous substances •, which, when dissolved, form
water glues. The second is the gluten, most of whose
properties have been described under the article Bread.
The third is of a mild nature, perfectly soluble in cold
water, of the nature of saccharine extractive mucous
matters. It is susceptible of the spirituous fermenta¬
tion, and is found but in small quantity in the flour of
wheat. See Bread, Gluten, Starch, and Sugar,
Chemistry Index.
FLOWER, Flos, among botanists and gardeners,
the most beautiful part of trees and plants, containing
the organs or parts of fructification. See Botany
Index.
4 S Flowers,
FLO [690
Flower?. Flowers, designed for medical use, should be jilnck-
k — ■ / gj when they are moderately blown, and on a clear
day before noon : for conserves, roses must be taken in
the bud.
Flowers, in antiquity. We find flowers in great
request at the entertainments of the ancients, being pro¬
vided by the master of the feast, and brought in before
the second course; or, as some are of opinion, at the
beginning of the entertainment. They not only a-
dorned their heads, necks, and breasts, with flowers,
but often bestrewed the beds whereon they lay, and all
parts of the room with them. But the head was chiefly
regarded. See Garland.
Flowers were likewise used in the bedecking of
tombs. See Burial.
Eternal Flower. See Xeranthemum,
Everlasting Flower. SccGnaphalium, |
FLOWER-Fencc. See I’oiNClANA, | g0TANY
Sur-Flower. See Helianthus, ^ Index.
Sultan-FLOWER, See Cyanus,
Trumpet-Flower. See Bignonia,
Wind-Flower. See Anemone,
Flower-de-lis, or Flower-de-luce, in Herald}"!/, a
bearing representing the lily called the queen of flowers,
and the true hieroglyphic of royal majesty *, but of late
it is become more common, being borne in some coats
one, in others three, in others five, and in some semee
or spread all over the escutcheon in great numbers.
The arms of France are, three flowers-de-lis or, in
a field azure.
FLOWER-de-Luce. See Iris, Botany Index.
Flowers, in Heraldry. rJ hey are much used in
coats of arms j and in general signify hope, or denote
human frailty and momentary prosperity.
Flowers, in Chemistry. By this name are gene¬
rally understood bodies reduced into very fine parts, ei¬
ther spontaneously, or by some operation of art; but
the term is chiefly applied to volatile solid substances,
reduced into very fine parts, or into a kind of meal by
sublimation.—Some flowers are nothing else than the
bodies themselves, which are sublimed entire, without
suffering any alteration or decomposition and other
flowers are some of the constituent parts of the body
subjected to sublimation.
Colours of Flowers. See the article Colour {of
Plants.)
Colours extracted from Flowers. See Colour-
Making.
Preserving of Flowers. The method of preserving
flowers in their natural beauty through the whole year
has been much sought after by many people. Some
have attempted it by gathering them when dry and not
too much opened, and burying them in dry sand but
this, though it preserves their figure well, takes oft
from the liveliness of their colour. Muntingius pre¬
fers the following method to all others. Gather ro¬
ses, or other flowers, when they are not yet thorough¬
ly open, in the middle of a dry day; fill the vessel up
tc the top with them ; and when full sprinkle them
over with some good French wine, with a little salt in
it; then set them by in a cellar, tying down the mouth
of the pot. After this they may be taken out at plea¬
sure ; and, on setting them in the sun, or within
•reach of the fire, they will open as if growing natural-
] FLO
lv ; and not only the colour, hut the smell also will be ^lowen,
preserved.
The flowers of plants are by mucb the most difficult
parts of them to preserve in any tolerable degree of per¬
fection ; of which we have instances in all the collec¬
tions of dried plants, or horti sicci. In these the leaves,
stalks, roots, and seeds of the plants, appear very well
preserved ; the strong texture of these parts making
them always retain their natural form, and the colour#
in many species naturally remaining. But where these
fade, the plant is little the worse for use as to the know¬
ing the species by it. But it is very much otherwise in
regard to flowers; these are naturally by much the
most beautiful parts of the plants to which they belong ;
but they are so much injured in the common way of
drying, that they not only lose, but change their co¬
lours one into another, by which means they give a
handle to many errors ; and they usually also wither up,
so as to lose their very form and natural shape. The
primrose and cowslip kinds are very eminent instances
of the change of colours in the flowers of dried speci¬
mens ; for those of this class of plants easily dry in their
natural shape ; but they lose their yellow, and, instead
of it, acquire a fine green colour, much superior to that
of the leaves in their most perfect state. ’Ihe flowers
of all the violet kind lose their beautiful blue, and be¬
come of a dead white : so that in dried specimens there
is no difterence between the blue-flowered violet and
the white-flowered kinds.
Sir Robert Southwell has communicated to the world
a method of drying plants, by which this defect is
proposed to he in a great measure remedied, and all
flowers preserved in their natural shape, and many in
their natural colours.—For this purpose two plates of
iron are to he prepared of the size ol a large half sheet
of paper, or larger, for particular occasions ; these
plates must be made so thick as not to he apt to bend ;
and there must be a hole made near every corner
for receiving a screw to fasten them close together.
When these plates are prepared, lay in readiness seve¬
ral sheets of paper, and then gather the plants with
their flowers when they are quite perfect. Bet this be
always done in the middle of a dry day ; and then lay
the plant and its flower on one of the sheets of paper
doubled in half, spreading out all the leaves and petals
as nicely as possible. If the stalk is thick, it must he
pared or cut in half, so that it may lie flat ; and if it
is woody, it may he peeled, and only the bark left.
When the plant is thus expanded, lay round about it
some loose leaves and petals of the flower, which may
serve to complete any part that is deficient. When all
is thus prepared, lay several sheets of paper over the
plant, and as many under it ; then put the whole be¬
tween the iron plates, laying the papers smoothly on
one, and laying the other evenly over them ; screw
them close, and put them into an oven after the bread
is drawn, and let them lie there two hours. After
that, make a mixture of equal parts of aquafortis and
common brandy ; shake these well together, and when
the flowers are taken out of the pressure of the plates,
rub them lightly over with a camel’s hair pencil dipped
in this liquor; then lay them upon fresh brown paper,
and covering them with some other sheets, press them
between this and other papers with a handkerchief till
FLO [ 691 ] FLO
the wet of these liquors is dried wholly away. When
the plant is thus far prepared, take the bulk of a nut-
meg of gum dragon 5 put this into a pint of fair water
cold, and let it stand 24 hours ; it will in this time be
wholly dissolved : then dip a fine hair pencil in this
liquor, and with it daub over the back sides of the
leaves, and lay them carefully down on half a sheet of
white paper fairly expanded, and press them down
with some more papers over these. When the crum-
water is fixed, let the presser and papers be removed,
and the whole work is finished. The leaves retain their
verdure in this case, and the flowers usually keep
their natural colours. Some care, however, must be
taken, that the heat of the oven be not too great.
When the flowers are thick and bulky, some art may
be used to pare oft’ their backs, and dispose the petals
in a due order ; and after this, if any of them are
wanting, their places may be supplied with some of
the supernumerary ones dried on purpose j and if any
of them are only faded, it will be prudent to take
them away, and lay down others in their stead : the
leaves may be also disposed and mended in the same
manner.
Another method of preserving both flowers and fruit
found throughout the whole year is also given by the
same author. Take saltpetre, one pound j Armenian
bole, two pounds ; clean common sand, three pounds :
mix ail well together. Then gather fruit of any kind
that is not fully ripe, with the stalk to each ; put these
in, one by one, into a wide-mouthed glass, laying
them in good order. Tie over the top with an oil¬
cloth, and carry them into a dry cellar, and set the
whole upon a bed of the prepared matter of four inches
thick in a box. Fill up th ■ remainder of the box with
the same preparation ; and let it be four inches thick,
all over the top of the glass, and all round its sides.
Flowers are to be preserved in the same sort of glasses,
and in the same manner : and they may be taken up
after a whole year as plump and fair as when they were
buried.
Artificial Flowers of the Chinese. See Tong-
tsao.
Flowers, in the animal economy, denote women’s
monthly purgations or menses.—Nicod derives the
word in this sense, from fl-aere, q. they may all have the same benefit
|| of the water. Narcissuses and hyacinths do well toge¬
ther j as also tulips and jonquils, and crocuses and snow-
J drops.
FLUDD, Robert, a philosopher and physician of
some celebrity in his time, was the son of Sir Thomas
Fludd, treasurer of war to Queen Elizabeth 5 and was
born at Milgate in Kent, in the year 1574. He re¬
ceived his education at St John’s college, Oxford, and
afterwards spent six years in travelling through Europe.
He acquired a strong attachment to the Rosjcrucian
philosophy, which chiefly consisted of mysticism and
jargon, and such as were admitted among them had
certain secrets analogous to those of free masonry. On
his return home, he took the degree of M. D. settled
in the city of London, and was chosen a fellow of the
college of physicians. His piety wras of an enthusiastic
nature, and the seeming depth of his knowledge pro¬
cured him much admiration, and gave him a temporary
fame. It is said that he employed a kind of unintel¬
ligible cant when speaking to his patients, which some¬
times contributed to their recovery, as it operated on
their faith. He is chiefly known as a sectary in philo¬
sophy, and not as a physician. He blended the incom¬
prehensible reveries of the Cabalists and Paracelsians,
forming a new physical system replete with mystery and
absurdity. He believed in two universal principles, the
northern or condensing, and the southern or rarefying
power. Innumerable geniuses he conceived to preside
over these, and committed the charge of diseases to le¬
gions of spirits collected from the four winds of heaven.
In his estimation,.a harmony subsisted between the ma¬
crocosm and the microcosm, or the world of nature
and of man. All his fancies and whims it is impossible
to enumerate, yet they attracted the notice of the phi¬
losophers of that age, being supported by mysterious
gravity and the shadow of erudition. Even Kepler
himself thought his extravagant jargon worthy of refu¬
tation, and Gassendi for this purpose wrote his Examen
Philo soph ice Fluddtunce. One of Fiudd’s performances,
entitled Nexus utriusque Cosmi, is illustrated by some
prints of a very singular and extraordinary nature.
FLUENT, or Flowing Quantity, in the doc¬
trine of fluxions, is the variable quantity which is con¬
sidered as increasing and decreasing: or the fluent of
a given fluxion, is that quantity whose fluxion being-
taken, according to the rules of that doctrine, shall
be the same with the given fluxion. See Fluxions.
See also FLUENTS, SUPPLEMENT.
FLU! D, an appellation given to all bodies whose par¬
ticles easily yield to the least partial pressure, or force
impres-ed. lor the Laws and Properties of Fluids,
see Hydrodynamics in this work ; and Fluids, Ele¬
vation of, in the Supplement.
There are various kinds of animalcules to be dis¬
cerned in different fluids by the microscope. Of many
remarkable kinds of these, a description is given under
the article Animalcule. All of these littht creatures
are easily destroyed by separating them from their na¬
tural element. Naturalists have even fallen upon shorter
methods. A needle point, dipped in spirit of vitriol,
and then.immersed into a drop of pepper water, rea¬
dily kills all the animalcules : which, though before
frisking about with great liveliness and activity, no
sooner come within the influence of the acid particles,
] FLU
than they spread themselves, and tumble down to all
appearance dead. The like may be done by a solution 1
of salt ; only with this difference, that, by the latter
application, they seem to grow vertiginous, turning
round and round till they fall down. Tincture of
salt of tartar, used in the same manner, kills them still
more readily ; yet not so, but there will he apparent
marks of their first being sick and convulsed. Inks
destroy them as fast as spirit of vitriol, and human
blood produces the same effect. Urine, sack, and sugar
all destroy them, though not so fast ; besides, that there
is some diversity in their figures and appearances, as
they receive their deaths from this poison or that. The
point of a pin dipped in spittle, presently killed all the
kinds of animalcules in puddle water, as Mr Harris
supposes it will other animalcules of this kind.
All who are acquainted with microscopic observa¬
tions, know very well, that in water, in which the
best glasses can discover no particle of animated mat¬
ter, after a few grains of pepper, or a fragment of a
plant of almost any kind, has been some time in it, ani¬
mals full of life and motion are produced; and those
in such numbers, as to equal the fluid itself in quanti¬
ty.—When we see a numerous brood of young fishes
in a pond, we make no doubt of their having owed
their origin to the spawn, that is, to the eggs of the
parents of the same species. W?hat are we then to
think of these ? It we will consider the progress of na¬
ture in the insect tribes in general, and especially in
such of them as are most analogous to these, we shall
find it less difficult to give an account of their origin
than might have been imagined.
A small quantity of water taken from any ditch in
the summer months, is found to be full of little worms,
seeming in nothing so much as in size to differ from
the microscopic animalcules. Nay, water, without
these, exposed in open vessels to the heat of the wea¬
ther, will be always found to abound with multitudes
of them, visible to the naked eye, and full of life and
motion. These we know, by their future changes,
are the fly worms of the different species of gnats, and
multitudes of other fly species; and we can easily de¬
termine, that they have owed their origin only to the
eggs of the parent fly there deposited. Nay, a closer
observation will at any time give ocular proof of this;
as the flies may be seen laying their eggs there, and
the eggs may be followed through all their changes to
the fly again. Why then are we to doubt hut that the
air abounds with other flies and animalcules as minute
as the worms in those fluids ; and that these last are on¬
ly the fly worms of the former, which, after a proper
time spent in that state, will suffer changes like those
of the larger kinds, and become flies like those to
whose eggs they owed their origin ? Vid. Reaumur,
Hist. Insect, vol. iv. p. 431.
The differently medicated liquors made by infusions
of different plants, afford a proper matter for the
worms of different species of these small flies: and there
is no reason to doubt, but that among these some are,
viviparous, others oviparous ; and to this may be, in a
great measure, owing the different time taken up for
the production of these insects in different fluids.
Those which are a proper matter for the worms of the
viviparous fly, may be soonest found full of them ; as,
probably the liquor is no sooner in a state to afford
them 1
Fluid.-
FLU [ 694 ] FLU
fluid. them proper nourishment, than their parents place
—■ i them there : whereas those produced from the eggs of
the little oviparous flies, must, after the liquor is in a
proper state, and they are deposited in it in the form of
eggs, have a proper time to be hatched, before they
can appear alive.
It is easy to prove, that the animals we find in these
vegetable infusions were brought thither from else¬
where. It is not less easy to prove, that they could
not be in the matter infused any more than in the
water in which it is infused.
Notwithstanding the fabulous accounts of salaman¬
ders, it is now well known, that no animal, large or
small, can hear the force of fire for any considerable
time £ and, by parity of reason, we are not to believe,
that any insect, or embryo insect, in any state, can
bear the heat of boiling water for many minutes. To
proceed to inquiries on this foundation : If several
tubes be filled with water, with a small quantity of vege¬
table matter, such as pepper, oak bark, truffles, &c.
in which, after a time, insects will be discovered by
the microscope ; and other like tubes be filled with
simple water boiled, with water and pepper boiled to-^
gether, and with water with the two other ingredients
all separately boiled in it; when all these liquors come
to a proper time for the observation of the microscope,
afi, as well those which have been boiled as those
which have not, will he found equally to abound with
insects, and those of the same kind, in infusions of the
same kind, whether boiled or not boiled. ’I hose in the
infusions which had sustained a heat capable of destroy¬
ing animal life, must therefore not have subsisted either
in the water or in the matters put into it, hut must
have been brought thither after the boiling j and it
seems by no way so probable, as by means of some
little winged inhabitants of the air depositing their eggs
or worms in these fluids.
On this it is natural to ask, how it comes to pass,
that while we see myriads of the progeny of these
winged insects in water, we never see themselves? Hie
answer is equally easy, viz. because we can always
place a drop of this water immediately before the fo¬
cus of the microscope, and keep it there while we are
at leisure to examine its contents 5 hut that is not the
case with regard to the air inhabited by the parent
flies of these worms, which is an immense extent in
proportion to the water proper for nourishing these
worms 5 and consequently, while the latter are clus¬
tered together in heaps, the former may be dispersed
and scattered. Nor do we want instances of this, even
in insects of a larger kind. In many of our gardens,
we frequently find vessels of water filled with worms
of the gnat kind, as plentiful, in proportion to their
size, as those of other fluids are with animalcules.
Rvery cubic inch of water in these vessels contains
many hundreds of animals : yet we see many cubic
inches of air in the garden not affording one of the
parent flies.
But neither are we positively to declare that the pa¬
rent flies uf these animalcules are in all states wholly
invisible to us j if not singly to he seen, there are some
strong reasons to imagine that they may in great clus¬
ters. Every one has seen in a clear day, when look¬
ing stedfastly at the sky, that the air is in many places
disturbed by motions and convolutions in certain spots.
2
These cannot be the effects of imagination, or of Fluid,
faults in our eyes, because they appear the same to allj Fluidity,
and if we consider what would he the case to an eye • —
formed in such a manner as to see nothing smaller than
an ox, on viewing the air on a marsh fully peopled
with gnats, we must be sensible that the clouds of
these insects, though to us distinctly enough visible,
would appear to such an eye merely as the moving
parcels of air in the former instance do to us : and
surely it is thence no rash conclusion to infer, that the
case may be the same, and that myriads of flying in¬
sects, too small to he singly the objects of our view,
vet are to us what the cloud of gnats would be in the
former case.
Nervous Fluid. See Anatomy Index.
Elastic Fluids. See Chemistry Index.
FLUIDITY, is by Sir Isaac Newton defined to
he, that property of bodies by which they yield to any
force impressed, and which have their parts very easily
moved among one another.
To tliis definition some have added, that the parts
of a fluid are in a continual motion. This opinion ia
supported bv the solution of salts, and the formation of
tinctures. If a small hit of saffron is thrown into a
phial full of water, a yellow tincture will soon be com¬
municated to the water to a considerable height ;
though the phial is allowed to remain at rest ; which
indicates a motion in those parts of the fluids which
touch the saffron, by which its colouring matter is car¬
ried up.
With regard to water, this can scarce be denied j
the constant exhalations from its surface show, that
there must he a perpetual motion in its parts from the
ascent of the steam through it. In mercury, where in¬
sensible evaporation does not take place, it might be
doubted ; and accordingly the Newtonian philosophers
in general have been of opinion, that there are some
substances essentially fluid, from the spherical figure of
their constituent particles. The congelation of mer¬
cury, however, by an extreme degree of cold, de¬
monstrates that fluidity is not essentially inherent in
mercury more than in other bodies.
That fluids have vacuities in their substance is evi¬
dent, because they may be made to dissolve certain
bodies without sensibly increasing their bulk. For ex¬
ample, water will dissolve a certain quantity of salt;
after which it will receive a little sugar, and after that
a little alum, without increasing its first dimensions.
Here we can scarce suppose any thing else than that
the saline particles were interposed between those of
the fluid ; and as, by the mixture of salt and water, a
considerable degree of cold is produced, we may thence
easily see why the fluid receives these substances with¬
out any increase of bulk. All substances are expand¬
ed by heat, and reduced into less dimensions by cold ;
therefore, if any substance is added to a fluid, which
tends to make it celd, the expansion by the bulk of
the substance added will not he so much perceived as
if this effect had not happened *, and if the quantity
added be small, the fluid will contract as much, per¬
haps more, from the cold produced by the mixture, than
it will he expanded from the bulk of the salt. This
also may let us know with what these interstices be¬
tween the particles of the fluid were filled upj namely,,
the dement of fire or heat. The saline particles, up¬
on
Viuke.
F L U [ 695 ]
on their solution in the fluid, have occupied these
___ spaces ; and now the liquor being deprived of a quan-
~ ‘ity of this element equal in bulk to the salt added,
feels sensibly colder.
As, therefore, there is scarce any body to he found,
but what may become solid by a sufEcient degree of
cold, and none but what a certain degree of heat will
render fluid 5 the opinion naturally arises, that fire is
the cause of fluidity in all bodies, and that this ele¬
ment is the only essentially fluid substance in nature.
Hence we may conclude, that those substances which
we call fluids are not essentially so, but only assume
that appearance in consequence of an intimate union
with the element of fire ; just as gums assume a fluid
appearance on being dissolved in spirit of wine, or salts
in water. *
Upon these principles Dr Black mentions fluidity
as an effect ol heal. The different degrees of heat
which are required to bring different bodies into a
state of fluidity, he supposes to depend on some parti¬
culars in the mixture and composition of the bodies
themselves : which becomes extremely probable, from
considering that we change the natural state of bodies
in this respect, by certain mixtures ; thus, if two
metals are compounded, the mixture is usually more
fusible than either of them separately. See Chemis¬
try Index.
ft is certain, however, that water becomes warmer
by being converted into ice ; which may seem con¬
tradictory to this opinion. To this, however, the doc¬
tor replies, that fluidity does not consist in the degree
of sensible heat contained in bodies, which will affect
the hand or a thermometer j hut in a certain quantity
which remains in a latent state. This opinion he
supports from the great length of time required to melt
ice $ and to ascertain the degree of beat requisite to
keep water in a fluid state, he put five ounces of water
into a Florence flask, and converted it into ice by
means of a freezing mixture put round the flask. Into
another flask of the same kind he put an equal quantity
of water cooled down nearly to the freezing point, by
mixing it with snow, and then pouring it off. In this
he placed a very delicate thermometer ; and found that
it acquired heat from the air of the room in which
it was placed : seven degrees of heat were gained
the first half hour. The ice being exposed to the
same degree of heat, namely, the air of a large room
without fire, it cannot be doubted that it received heat
from the air as fast as the water which was not frozen j
hut, to prevent all possibility of deception, he put his
hand under the flask containing the ice, and found a
stream of cold air very sensibly descending from it,
even at a considerable distance from the flask ; which
undeniably proved, that the ice was all that time ab¬
sorbing heat from the air. Nevertheless, it was not
till 11 hours that the ice was half melted, though in
that time it had absorbed so much heat as ought to
have raised the thermometer to 140°$ and even after
it was melted, the temperature of the water was found
scarce above the freezing point: so that as the heat
which entered could not be found in the melted ice,
lie concluded that it remained concealed in the water,
as an essential Ingredient of its composition.
FLUKE, or Flounder. See Pleuronectes,
Ichthyology Index.
FLU
See Fasciola, Helminthology
Fluke TForm.
Index.
Fluke of an Anchor, that part of it which fastens
in the ground. See Anchor.
FLUMMERY, a wholesome sort of jelly made of
oatmeal.
The manner of preparing it is as follows. Put three
large handfuls of finely ground oatmeal to steep, for
24 hours, in two quarts of fair water : then pour off
the clear water, and put two quarts of fresh water to
it j strain it through a fine hair sieve, putting in two
spoonfuls of orange flower water, and a spoonful of su¬
gar: boil it till it is as thick as a hasty pudding, stir¬
ring it continually while it is boiling, that it may be
very smooth.
FLUOR, in Physics, a fluid; or, more proper¬
ly, the state of a body that was before hard or solid,
but is now reduced by fusion or fire into a state of
fluidity.
Fluoe Acid. See Fluoric Acid,Chemistry Index.
I Luo R Albus, a flux incident to women, commonly
known by the name of whites. See Medicine Index.
Fluor Spar or Blue John, called also fluxing spars,
vitrescent or glass spars, are minerals composed of cal¬
careous earth united with fluoric acid. See Mineralo¬
gy Index.
1 LUSHING, a handsome, strong, and considerable
town of the United Provinces, in Zealand, and in the
island of Walcheren, with a very good harbour, and a
great foreign trade. It was put into the hands of
Queen Elizabeth, for a pledge of their fidelity, and
as a security for the money she advanced. It was taken
by the British in the memorable and ill conducted ex¬
pedition of 1809, ail^ kept some months. E. Long. 3.
32. N. Lat. 51. 26.
FLUTE, an instrument of music, the simplest of all
those ol the wind kind. It is played on by blowing it
with the mouth ; and the tones or notes are changed by
stopping and opening the holes disposed for that pur¬
pose along its side.
This is a very ancient instrument. It was at first call¬
ed the flute a bee, from bee an old Gaulish word signi-
fying the beak of a bird or fowl, but more especially
oi a cock ; the term flute a bee must therefore signify
the beaked flute ; which appears very proper, on com¬
paring it with the traverse or German flute. The word
flute is derived from fluta, the Latin for a lamprey or
small eel taken in the Sicilian seas, having seven hole#
immediately below the gills on each side, the precise
number of those in the front of the flute.
By Mersennus this instrument is called the fistula
dulcis sen Anglica; the lowest note, according to him,
for the treble flute, is Cfa ut, and the compass of the
instrument 15 notes. There is however, a flute known
by the name of the concert flute, the lowest note of
which is I'. Indeed, ever since the introduction of the
flute into concerts, the lowest note of the instrument,
of what size soever it is, has been called F ; when in
truth its pitch is determinable only by its corre¬
spondence in respect of acuteness or gravity with one
or other of the chords in the scala maxima or great
system.
Besides the true concert flute, others of a less size
were soon introduced into concei ts of violins ; in which
case the method was to write the flute part in a key
correspondent
Flake
worm
II
Fiiue.
rlule.
FLU [ 696 ] FLU
correspondent to its pitch. This practice was introdu¬
ced in 1710 by one \Voodcock, a celebrated performer
on this instrument, and William Babell organist of the
church of All-Hallows, Bread Street, London. They
failed, however, in procuring for the flute a reception
into concerts of various instruments ; for which reason
one Thomas Stanesby, a very curious maker of flutes
and other instruments of the like kind, about the year
1732, adverting to the scale of Mersennus, in which
the lowest note was C, invented what he called the new
system; in which, by making the flute of such a size
as to be a fifth above concert pitch, the lowest note be¬
came C sol fa ut. By this contrivance the necessity of
transposing the flute part was taken away •, for a flute
of this size, adjusted to the system above mentioned,
became an octave to the violin. Io further this in¬
vention of Stanesby’s, one Lewis Merci, an excellent
performer on the flute, published about the year 1735*
six solos for this instrument, three of which are said to
be accommodated to Mr Stanesby’s new system } but
the German flute was now become a favourite instru¬
ment, and Stanesby’s ingenuity failed of its effect.—
One great objection indeed lies against this instrument,
which, however, equally affects all perforated pipes j
namely, that they are never perfectly in tune, or can¬
not be made to play all their notes with equal exact¬
ness. The utmost that the makers of them can do is
to tune them to some one key; as the hautboy to C,
the German flute to D, and the English flute to F j and
to effect this truly, is a matter of no small difficulty.
The English flutes made by the younger Stanesby came
the nearest of any to perfection ; but those of Bressan,
though excellent in their tone, are all too flat in the
upper octave. For these reasons some are induced to
think, that the utmost degree of proficiency on any
of those instruments is not worth the labour of attain-
ing it.
German Flute, is an instrument entirely different
from the common flute. It is not, like that, put into
the mouth to be played ; but the end is stopt with a
tompion or plug, and the lower lip is applied to a
hole about two inches and a half or three inches di¬
stant from the end. This instrument is usually about
a foot and a half long j rather bigger at the upper end
than the lower $ and perforated with holes, besides that
for the mouth, the lowest of which is stopped and open¬
ed by the little finger’s pressing on a brass or some¬
times a silver key, like those in hautboys, bassoons, &c.
Its sound is exceeding sweet and agreeable j and serves
as a treble in a concert.
Flute, or Fluyt, is a kind of long vessel, with
flat ribs or floor timbers, round behind, and swell¬
ed in the middle j serving chiefly for the carrying of
provisions in fleets or squadrons of ships*, though it is
often used in merchandise. The word flute, taken for
a sort of boat or vessel, is derived, according to Borel,
from the ancient flotte, a little boat. In the verbal
process of the miracles of St Catherine of Sweden, in
the 12th century, we read Unus equum suum tinacum
mercibus magniponderis introduxit super instrumentum
de Itgnis fabric alum, vulgariter dictumfluta. U pon which
the Bollandists observe, that in some copies it is read
fiotta, an instrument called by the Latins ratis; and
that the word or flotta arose from flatten or viat¬
ica, “ to float.”
Flutes, or Flutings, in Architecture, are perpen- Fialeti
dicular channels or cavities cut along the shaft of a co- Flux,
lumn or pilaster. They are supposed to have been first ‘ v1—
introduced in imitation of the plaits of women’s robes;
and are therefore called by the Latins striges and ruga.
The French call them cannelures, as being excavations ;
and we, flutes ox flutings, as bearing some resemblance
to the musical instrument so called. They are chiefly
affected in the Ionic order, in which they bad their
first rise ; though they are also used in all the richer
orders, as the Corinthian and Composite; but rarely
in the Doric, and scarce ever in the Tuscan.
FLUX, in Medicine, an extraordinary issue or eva¬
cuation of some humour. Fluxes are various, and vari¬
ously denominated, according to their seats or the hu¬
mours thus voided ; as a flux of the belly, uterine flux,
hepatic flux, salival flux, &c. The flux of the belly is
of four kinds, which have each their respective deno¬
minations, viz. the lientery, oxfluxus lientericus; the
cceliac, orfluxus chylosus; the diarrhoea; and the dy¬
sentery, or bloody flux. See Medicine Index.
Flux, in Hydrography, a regular periodical motion
of the sea, happening twice in 24 hours; wherein the
water is raised and driven violently against the shores.
The flux or flow is one of the motions of the tide;
the other, whereby the water sinks and retires, is called
the reflux or ebb. There is also a kind of rest or ces¬
sation of about half an hour between the flux and re¬
flux ; during which time the water is. at its greatest
height, called high water. The flux is made by the
motion of the water of the sea from the equator to¬
wards the poles ; which, in its progress, striking against
the coasts in its way, and meeting with opposition from
them, swells, and where it can find passage, as in flats,
rivers, &c. rises up and runs into the land. This mo¬
tion follows, in some measure, the course of the moon ;
as it loses or comes later every day by about three quar¬
ters of an hour, or more precisely by 48 minutes ;
and by so much is the motion of the moon slower than
that of the sun. It is always highest and greatest in
full moons, particularly those of the equinoxes. In
some parts, as at Mount St Michael, it rises 80 or 90
feet, though in the open sea it never rises above a foot
or two; and in some places, as about the Morea, there
is no flux at all. Its runs up some rivers above 120
miles. Up the river Thames it only goes 80, viz. near
to Kingston in Surry. Above London bridge the wa¬
ter flows four hours and ebbs eight; and below the
bridge, flows five hours and ebbs seven.
Flux, in Metallurgy, is sometimes used synony¬
mously with fusion. For instance, an ore, or other
matter, is said to be a liquid flux, when it is com¬
pletely fused.
But the word flux is generally used to signify cer¬
tain saline matters, which facilitate the fusion ot ores
and other matters, which are difficultly fusible in es¬
says and reductions ot ores ; such are alkalies, nitre,
borax, tartar, and common salt. But the word
flux is more particularly applied to mixtures of diflerent
proportions of only nitre and tartar; and these fluxes
are called by particular names, according to the pro¬
portions of these ingredients, as in the following ar¬
ticles.
White Flux, is made with equal parts of nitre and
of tartar detonated together, by which they are alka¬
lized.
3
Flat*
FLU
lized. The residuum of this detonation
/ composed of the alkalies of the nitre and of the tartar,
both which are absolutely of the same nature. As the
proportion of nitre in this mixture is more than is suffi¬
cient to consume entirely all the inflammable matter
of the tartar, the alkali remaining after the detonation
is perfectly white, and is therefore called white flux ;
and as this alkali is made very quickly, it is also called
extemporaneous alkali. When a small quantity only of
white flux is made, as a few ounces for instance, some
nitre always remains undecomposed, and a little of the
inflammable principle of the tartar, which gives a red
or even a black colour to some part of the flux; but
this does not happen when a large quantity of white
flux is made $ because then the heat is much greater.
This small quantity of undecomposed nitre and tartar
which remains in white flux is not hurtful in most of
the metallic fusions in which this flux is employed : but
if the flux be required perfectly pure, it might easily
be disengaged from those extraneous matters by a long
and strong calcination, without fusion.
Crude Flux. By crude flux is meant the mixture of
nitre and tartar in any proportions, without detonation.
Thus the mixture of equal parts of the two salts used
in the preparation of the white flux, or the mixture of
[ <597 ] FLU
is an alkali one part of nitre and two parts of tartar for the pre¬
paration of the black flux, are each of them a crude
flux before detonation. It has also been called white
flux, from its colour; but this might occasion it to be
confounded with the white flux above described. The
name, therefore, of crude flux is more convenient.
Crude flux is detonated and alkalized during the
reductions and fusions in which it is employed $ and is
then changed into white or black flux, according to
the proportions of which it is composed. This detona¬
tion produces good effects in these fusions and reduc¬
tions, if the swelling and extravasation of the detona¬
ting matters be guarded against. Accordingly, crude
flux may be employed successfully in many operations j
as, for instance, in the ordinary operation for procuring
the regulus of antimony.
Black Flux. Black flux is produced from the mix¬
ture of two parts of tartar and one part of nitre deto¬
nated together. As the quantity of nitre which enters
into the composition of this flux is not sufficient to con¬
sume all the inflammable matter of the tartar, the al¬
kali which remains after the detonation contains much
black matter, of the nature of coal, and is therefore
called black flux.
Flu*.
FLUXIONS.
INTRODUCTION.
r | ’HE branch of mathematical analysis which is called
in this country the Method of Fluxions, but on the
continent the Differential and Integral Calculi, was in¬
vented near the end of the 17th century ; and Sir Isaac
Uewton, and Mr Leibnitz, two of the greatest philoso¬
phers of that age, have both claimed the discovery.
It will appear very possible that two such men should
both fall upon this method of calculation nearly about
the same time, if it be considered, that from the begin¬
ning of the 17th century its principles were gradually
coming into view, in consequence of the united labours
and discoveries of a number of mathematicians, such as
Napier, Cavallerius, Roberval, Fermat, Barrow, Wallis,
and others. And considering the number of men of
the first abilities engaged at that time in the study of
mathematics, we may reasonably suppose, that the flux-
ional or difterential calculus, would very soon have
been found according to the ordinary progress of human
knowledge, even although a Newton, or Leibnitz, had
not by the force of superior genius anticipated perhaps
by a few years that event. The first intimation that
was given of the discovery of the calculus was in the
year 1669, when through the intervention of Dr Bar-
row, a correspondence was begun between Sir Isaac
Newton (then Mr Newton), and Mr Collins, one of the
secretaries to the Royal Society. Dr Barrow commu-
nicated to the latter a paper by Newton, which had for
its title, De analysiper cequationes numero terminorum
infinitas. In this paper, besides shewing how to resolve
equations by approximation, Neivton teaches how to
square curves, not only when the expression for the or¬
dinate in terms of the abscissa is a rational quantity,
Vol. VIII. Part II. f
but also when it involves radical quantities, by first re¬
solving these into an infinite series of rational terms by
means of the binomial theorem, a thing which had
never before been done. Newton in this paper gives
some rather obscure indications of the nature of his cal¬
culus, which however serve to shew, beyond all doubt,
that he was then in possession of it j and indeed there
is good reason to believe that he knew it as early as
the year 1665, or even sooner.
These analytical discoveries of Newton were imme¬
diately circulated among mathematicians both in this
country and abroad, by Dr Barrow, and by Collins
and Oldenburg, the two secretaries to the Royal So¬
ciety.
About the end of the year 1672, Newton commu¬
nicated to Collins, by letter, a method of drawing tan¬
gents to curve lines, illustrated by an example, from
which it again plainly appears, that he now possessed
his method of fluxions.
In the course of the following year, Leibnitz came
to London, and communicated to several members of the
Royal Society, some researches relating to the theory of
differences. It was however shewn to him, that this
subject had been previously treated by Mouton an astro¬
nomer of Lyons *, upon this Leibnitz directed his atten¬
tion to the doctrine of series, which was now consider¬
ably advanced, in consequence of the discoveries of the
English mathematicians.
The first direct communication that passed between
Newton and Leibnitz, was by a letter, which the former
addressed to Oldenburg, about the middle of the year
1676. In the beginning of this letter, which was in¬
tended to be shewn to Leibnitz, Newton speaks of him
with much respect. The letter itself chiefly refers to
4T the
Introduc¬
tion.!
FLUXIONS.
698
,ntiodue- the theory of infinite series. In a second letter, written
tiou also with a view to its being communicated to LabmlZ,
' v Newton, after bestowing deserved commendation on him,
proceeds to explain the steps by which he was led to the
discovery of the binomial theorem. He afterwards,
among other things, delivers several theorems which
have the methofl of fluxions for their basis j but he does
not give their demonstrations, and only observes, that
they depend on the solution of a general problem, the
enunciation of which lie conceals under an anagram
of transposed letters, but the meaning of it is this:
An equation being given containing any number of flow¬
ing quantities, to find their fluxions ; ami the contrary.
This letter affords another proof that Newton was now
in full possession of his calculus.
In the end of June 1677, Leibnitz sent to Oldenburg,
for the purpose of being communicated to Newton, a
letter containing the first essays of his Differential
Calculus. The death of Oldenburg, which happened
soon after, put an end to the correspondence, and in
the year 1684, Leibnitz published his method in the
Leipsic jdcts for the month of October 1684. The title
of the memoir which contained it was, Nova mcthodus
pro maximis et minimis, itcmque tangentibus, quce nec
fractas, nec irrationales quantitates moratur,et smgulare
pro Hit's calculi genus. Thus, in whatever way Leibnitz
came by his calculus, whether he discovered it solely
by the force of his own genius, or founded it on the me¬
thod of fluxions, previously invented by Newton, both
of which hypotheses are possible, his method was cer¬
tainly published before Newton's, which, except what
transpired in consequence of the circulation of his
letters and manuscripts, became only known to the
world in general for the first time, by the publication of
the Frincipia in the end of the year 1686.
It seems at first to have been allowed, that Leibnitz
had invented his calculus, without having any previous
knowledge of what had been done by Newton; tor in
the first edition of the Pnncipia, Newton says, “ In the
course of a correspondence which ten years ago I carried
on with the very learned geometrician Mr Leibnitz,
having intimated to him that I possessed a method of de¬
termining maxima and minima, of drawing tangents,
and resolving such problems, not only when the equa¬
tions were rational, but also when they were irra¬
tional •, and having concealed this method, by transpo¬
sing the letters of the following sentence—An equation
being given, containing any number of flowing quan¬
tities, to fluid their fluxions ; and the contrary; this ce¬
lebrated man answered that he had found a similar
method, which he communicated to me, and which dif¬
fers from mine, only in the enunciation, and in the no¬
tation.” To this, in the edition of I7r4 added,
“ and in the manner of conceiving the quantities to be
# . generated*.”
lib H lem* There is reason to suppose that Leibnitz might have
3 »«hol. * continued to enjoy undisturbed the honour of being
considered as one of the inventors of the fiuxional or
differential calculus, if he had not manifested a disposi¬
tion to attribute the invention too exclusively to him¬
self. This called forth some remarks respecting the
priority of Newton's claim to the discovery. In parti¬
cular M. Facio asserted, in a treatise on the Line of
swiftest descent, published in 1699, “ that he was obli¬
ged to own Newton as the first inventor of the diffe¬
rential calculus, and the first by many years ; and that introdiu.
he left the world to judge whether Leibnitz, the second Uon.
inventor, had taken any thing from him.” “
On the other hand, when Newton's treatise on the
quadrature of curves, and on the enumeration of lines
of the third order was published, which was in 1704,
the Leipsic journalists insinuated, in a very liberal ac¬
count which they gave of the work, that Leibnitz was
the first inventor, and that Newton had taken his method
from Leibnitz's, substituting fluxions for differences.
In consequence of tins attack on Newton, Dr John
Keill asserted, in the Philosophical Transactions for
1708, that Newton was beyond a doubt the first invent¬
or of the arithmetic of fluxions, and that the same arith¬
metic having its name and notation changed, was
afterwards published by Mr Leibnitz in the Peipsic
Acts. In answer to this, Leibnitz replied, in a letter
to Hans Sloane, secretary to the Royal Society, that no
one knew better than Newton himself, that the charge
against him implied in Keill's assertion was false •, and
he required Keill to retract what he had said. To this
request, however, Keill would by no means accede ; but
on the contrary, he wrote a long letter to the secretary
of the Royal Society, in which he endeavoured to prove,
not only that Newton had preceded Leibnitz in the in¬
vention, but that he had given to the latter such in¬
dications of the nature of his calculus, as made it easy
for him to fall upon the same. This letter was sent to
Leibnitz, who replied, that Keill, although learned, was
too young a man to be fit to judge of what had passed
between him and Newton, and he requested the Royal
Society to put a stop to Keill's clamours.
The Royal Society being thus appealed to as a judge,
appointed a committee to examine all the old letters,
papers, and documents, which had passed among the
several mathematicians, relating to the question. The
judgment of the committee was to the following et~
feet: “ That Mr Leibnitz was in London in 1673,
and went thence to Paris, where he kept a correspond¬
ence with Mr Collins by means of Mr Oldenburg, till
about September 1676, and then returned by London
and Amsterdam to Hanover 5 and that Mr Collins was
very free in communicating to able mathematicians
what he bad received from Newton, lhat it did not
appear, that Mr Leibnitz knew any thing of the differ¬
ential calculus before his letter ot the 21st ol June 1677?
which was a year after a copy of Newton's letter of the
10th of December 1672 had been sent to Paris, to be
communicated to him, and above four years after Mr
Collins began to communicate that letter to his corre¬
spondents •, in which letter the method of fluxions was
sufficiently described to any intelligent person. That
Newton was in possession of his calculus before the year
1669, and that those who had reputed Leibnitz the first
inventor, knew little or nothing of his correspondence
with Mr Collins, and Mr Oldenburg, long before, nor of
Newton's having that method above 15 years before
Mr Leibnitz began to publish it in the Leipsic Acts.
That for these reasons, they reckoned Newton the first
inventor, and were of opinion, that Mr Keill in assert¬
ing the same had been in no ways injurious to Mr
Leibnitz."
It is deserving of remark, that the committee deliver¬
ed no opinion upon the advantage which Leibnitz was
accused of having taken of the hints furnished to him
in
FLUXIONS.
utroduc- the course of his correspondence with Neivton; they
tion. left the decision of this point to the world in general ;
I*~v——' and to enable every one to judge for himself, the Koyal
Society ordered the opinion of the committee to be
printed, together with all the documents upon which
it was founded. These appeared in 1712 under the
title of, Commerctum Epistolicum de Analysis promota.
This work was carefully circulated over Europe, to vin¬
dicate the title of the English nation to the discovery.
The Commercium Epistolicum having appeared, Leib¬
nitz expressed great dissatisfaction, and threatened to
reply in such a manner as to confound his adversaries.
There seems no reason however to suppose, that any
thing he could have said, would have affected Newton's
claim to the honour of being the first inventor; for on this
point there cannot be any doubt. With respect, however,
to the other question, whether Leibnitz took his calcu¬
lus from Newton, or found it himself, it is impossible
to decide with such certainty. Mr Montucla, in his
JoUi. p. History of Mathematics *, says, “ There are only three
, places of the Commercium Epistolicum, which treat of
the principles of fluxions in so clear a way, as to prove
that Newton had found it before Leibnitz, but too ob¬
scurely, it seems, to take from the latter the merit of
the discovery. One of these is in a letter from Newton
to Oldenburg, who had signified to him, that Slusius
Gregory had each found a very simple method of draw¬
ing tangents. .Nmjfo/z replied, that he conjectured what
the nature of that method was; and he gave an example
of it, which shews it to be in effect the same thing as
those geometricians had found. He adds, that it is
only a particular case, or rather a corollary to a method
much more general, which, without a laborious calcu¬
lation, applies to the finding of tangents to all sorts of
cQrves, geometrical or mechanical, and that without
being obliged to free the equation from radicals. He
repeats the same thing without explaining himself far¬
ther, in another letter, and he conceals the principle of
the method under transposed letter’s. The only place
where Newton has allowed any thing of his method to
transpire, is in his Analysis per cequationes numero ter¬
minus infinitas. He here discloses, in a very concise
and obscure manner, his principle of fluxions, hut there
is no certainty of Leibnitz's having seen this essay. His
opponents have never asserted that it was communica¬
ted to him by letter, and they have gone no farther than
to suspect, that he had obtained a knowledge of it in
the interview which he had with Collins, upon his se¬
cond journey to London. Indeed, this suspicion is
not entirely destitute of probability, for Leibnitz ad¬
mitted, that in this interview, he saw a part of the
Epistolary Correspondence of Collins. However I think
it would be rash to pronounce upon this circumstance.
If Leibnitz had confined himself to a few essays of his
new calculus, there might have been some foundation
for that suspicion ; but the numerous pieces he inserted
in the Leipsic Acts, prove the calculus to have receiv¬
ed such improvements from him, that probably he owed
the invention of it to his genius, and to the efforts he
made to discover a method, which put Newton in pos¬
session of so many beautiful truths. This is so much
the more likely, as, from the method of tangents dis¬
covered by Dr Barrow, the transition to the differen¬
tial calculus was easy, nor was the step too great for
such a genius as that with which Leibnitz appears to
have been endowed.” Such is the opinion of Montucla,
who being a foreigner, cannot be supposed to have
been too partial towards Newton, an Englishman. The
British mathematicians have hitherto, with few excep¬
tions, entertained an opinion still more decidedly in fa¬
vour of the claims of their celebrated countryman.
It has been said that Newton took no share in the
controversy ; this however seems not to have been ex¬
actly the case, for besides suppressing in the third edi¬
tion of his Brincipia (printed in ih]’l6') the passage we
have already quoted, which seems to admit that L.cib-
nitz invented his calculus for himself, he is known to
have written the notes which accompany the edition of
the Commercium Epistolicum, printed in 1722. Leib¬
nitz had also begun to prepare a Commercium Epistoli¬
cum, but he died before it was completed.
Besides the disputes that have happened respecting
the inventor of the method of fluxions, the accuracy of
the method itself has been the subject of controversy,
both in Britain and on the continent. The differential
calculus was attacked abroad by Nieuwentat, a writer
of little or no reputation as a mathematician, and by
Nolle, who was an expert algebraist, and an indefatiga¬
ble calculator, but rash, and too confident of the just¬
ness of his own opinions, and jealous of the inventions
of others. To the first of these writers Leibnitz him¬
self replied, and afterwards Bernoulli &nA Herman ;
the attack from Rolle was successfully repelled by Va-
rignon, who was as zealous and intelligent, as his ad¬
versary was warm and impetuous.
The very concise manner in which the great inventor
of the method of fluxions thought proper to explain its
principles, gave occasion to the celebrated Dr Berkley
bishop of Cloyne to call in question, not only the logical
accuracy of the reasoning employed to establish the
theory of fluxions, but also the faith of mathematicians
in general, in regard to the truths of religion. The
bishop commenced the controversy first in'a small work
entitled The Minute Philosopher ; but his principal at¬
tack made its appearance in 1734, under the title of
“ The Analyst, or A Discourse addressed to an Infi¬
del Mathematician'' (understood to be Dr Halley,)
“ wherein it is examined whether the object, principles,
and inferences of the modern Analysis are more dis¬
tinctly conceived than religious mysteries and points of
faith." One of the best answers which was made to
this work came from the pen of Benjamin Robins, and
is entitled, “ A discourse concerning the nature and
certainty of Sir Isaac Newton's methods of fluxions,
and of prime and ultimate ratios." Other mathemati¬
cians likewise attempted to defend Newton, and the
method of fluxions, against the very cogent and well-
directed arguments of the bishop: but the most satis¬
factory way of removing all objections to the method,
was to abandon those obscure and inaccurate modes of
expression, of which Berkley had, not without some rea¬
son, complained, and to substitute in their place, others
more intelligible, and more consonant to the common me¬
thods of mathematical reasoning. This was accordingly
done by the celebrated Maclaurin, who, in the year
1742, published his Treatise of Fluxions, a work which,
although in some respects rather diffuse, placed the prin¬
ciples of the method beyond controversy, by establishing
them on the firm basis of geometrical demonstration.
It would extend this introduction to too great a
4X2 length
699
Introdve.
tion
700
Direct
Method.
FLUX
length werG we to enter into a detailed account of the
various improvements which the calculus has received
’ from its first invention to the present time. We shall
just briefly observe, that among those who contributed
the first and the most effectually to its improvement, we
may reckon Newton and Leibnitz themselves, the two
illustrious rivals for the honour of its discovery ; these
were followed by the two brothers Jat?ies and John Ber¬
noulli, by the Marquis de Vliopital, and many other
foreign mathematicians j and in this country we may
reckon Craig, Cheyne, Cotes, Taijlor, and De Moivre,
as among the earliest of its improvers. It is to Cotes
in particular that we are indebted for the discovery of
the method of finding the fluents of certain rational frac¬
tions, a discovery which was extended by De Moivre,
so as to form one of the most beautiful and complete
branches of the theory of fluxions.
Besides innumerable memoirs on particular branches
of the fluxional calculus, which are to be found in aca¬
demical collections, many distinct treatises have been
written on the subject. Some of the most valuable of
these are as follow. The Method of Fluxions and In¬
finite Series, by Sir Isaac Newton. This work was
written in Latin, but was not published till the year
when it was translated into English, and given to
the world, along with a comment, by Mr Colson. Har-
monia Mensurarum, by Cotes, a most valuable and ori¬
ginal work, published in iqid. A Treatise on Fluxions,
in two books, by Maclaurin, published in 1742. Many
parts of the writings of the celebrated Euler have a re¬
ference to the theory of fluxions, or the differential and
integral calculi. He has, however, three works in par¬
ticular that relate to that subject 5 the first is his In-
troductio in AnaJysin Infinitorum, the second his In-
stitutiones Calculi Dijfcrentialis, and the third his In-
stitutiones Calculi Integralis.
There is a work on this subject which deserves to be
particularly mentioned, both on account of its excellence,
and the singular circumstance of its being composed by a
lady. Its title is, Analytical Institutions, in four books,
originally written in Italian, by Donna Maria Gaetana
Agnesi. This lady was Professor of Mathematics and
IONS. Parti.
Philosophy in the University of Bologna ; her work was Direct
originally published in 1748, and has been styled by her Method,,
countryman Frisi, Opus nitidissimum, ingeniosissimum,
et eerie maximum quod adhuc ex fcemince alicujus cala-
mo prodierit. A part of this work has been published
in the French language by Bossut. An English transla¬
tion was prepared for the press many years ago by the
late Professor Colson ; it remained, however, unpublish¬
ed, and might still have continued so, but for the libera¬
lity of Baron Maseres, who, after satisfying some pe¬
cuniary claims upon the manuscript, caused it, in 1801,
to be published (we believe at his own expence), in
two volumes quarto. The Doctrine and Application of
Fluxions, by Thomas Simpson, is a work deservedly in
high estimation. The Doctrine of Fluxions, by Emer¬
son, is also very generally read by the British mathe¬
maticians. We are sorry, however, to observe, that
there is no work in the English language that exhibits
a complete view of the theory of fluxions, with all the
improvements that have been made upon it to the pre¬
sent time. We cannot at present acquire any tolerable
acquaintance with the subject, without consulting the
writings of the foreign mathematicians. There are se¬
veral excellent works in the French language ; we may-
mention in particular a Traite de Calcul Dijferentiel et
de Calcul Integral, by Cousin, in 2 vols. 410. *, another
by Bossut, in 2 vols. 8vo. y and another by La Croix,
in 3 vols. 4to. This last deserves particular notice, as
the author intended it to comprehend the substance of
the various valuable treatises by Euler, as well as of
the most important academical memoirs that relate to
this subject. The author has also published an abridg¬
ment of his work, in one volume octavo. Principiorum
Calculi Differentialis et Integralis, by VHuilier, pub¬
lished in 1795, contains a very clear exposition of the
principles of the calculus. The writings of our coun¬
trymen Landen and Waring, and of these foreign ma¬
thematicians La Grange, Le Gendre, La, Place, and
many others, abound with improvements in the calcu¬
lus. Having given this sketch of the history of this
very important branch of mathematical science, we pro¬
ceed to explain its principles.
PART I. THE DIRECT METHOD OE FLUXIONS.
Sect. I. Principles and Definitions.
1. IN the application of algebra to the theory of
curve lines, we find that some of the quantities which
are the subject of consideration, may be conceived as
having always the same magnitude, as the parameter
of a parabola, and the axes of an ellipse or hyperbola y
while others again are indefinite in respect of magni¬
tude, and may have any number of particular values,
such are the co-ordinates at any point in a curve line.
This difference in the nature of the quantities which are
compared together, has equally place in various other
theories, both in pure and mixed mathematics, and it
naturally suggests the divisions of all quantities what¬
ever into two kinds, namely, such as are constant, and
such as are variable.
2. A constant quantity is that which retains always
the same magnitude, however other quantities with
which it is connected may be supposed to change y and
a variable quantity is that which is indefinite in respect
of magnitude, or which may be supposed to change its
value. Thus, in the arithmetic of sines, the radius is a
constant quantity, while the cosine, sine, tangent, &c,
of an arch, also the arch itself, are variable quantities y
and in the conic sections the axes and the parameters
of the axes are constant quantities, and any abscissa
and its corresponding ordinate are variable quantities.
Constant quantities are usually denoted by the first
letters of the alphabet a, l, c, &c. and variable quan¬
tities by the last letters z, y, x, &c.
3. Any expression of calculation, containing a va¬
riable quantity, along with other constant quantities, is
called a Function of that variable quantity. Thus, sup¬
posing a; to be variable, and the other quantities constant,
any
3art I.
Direct c it . ,, a-\-bxm
Method. any one °* these expressions a xnf —^—-—
cos. x. sin. x, &c. is a function of x; and in any such
equation as yrra + the quantity y is
called a function of x. Even although the variable
quantities x and y, should not be separated as in the last
example, but should be related to each other as in the
following,
a x*y -j- 6 w y* -|-y3:=o,
as, setting aside the consideration of the constant quan¬
tities, the value ofy depends on that of «, and on the
contrary the value of x depends upon that of y, the
quantity y is said to he a function of xt and on the other
hand x is said to be a function of y.
4. If a variable quantity be supposed to change its
value, then a corresponding change will take place in
the value of any function of that quantity. Let us
examine the nature of this change in the magnitude of
a function.
First, let us suppose that, x denoting any variable
quantity, the function to be considered is any integer
power of that quantity, as x*, or x3, or a?4, &c.; then,
x being supposed to be increased by an indefinite quan¬
tity h, and thus to become a?-}-/z, the function will
change its value j if it be it will become or
701
power of the indefinite quantity h multiplied by some Direct
function of the variable quantity # as a co-efficient. Method.
The second term of the series consists of the second v -r
power of /i, multiplied also by a function of a; as a co¬
efficient $ and, in like manner, the third and following
terms are composed of the third and higher powers of /i
(the exponents forming the arithmetical series 1, 2, 3,
4, &c.) each multiplied by a function of a? as a coeffi¬
cient; and it appears, that the particular form of the
function which constitutes the coefficient of any assigned
term of the series depends entirely upon the particular
form of the original function. Thus, when the original
function is x*, the function which is the coefficient of
the first term is 2 x ; when the original function is a?3,
the co-efficient of the first term is 3 a?* ; when the ori¬
ginal function is a:4, the coefficient of the first term is
4A;*, and so on. It also appears that the functions of x,
which are the co-efficients of the powers of h, are com¬
posed only of the variable quantity x and given quan¬
tities, so that they are entirely independent of the inde« -
finite increment h.
6. These observations may be extended to a function^
that is any power whatever of a variable quantity, by
the application of the binomial theorem. Let x be
supposed to become A?-fA, then A;nwill become
but by the binomial theorem, (see Algebra, Sect,
xvii.) (Af + A)" when expanded into a series, is
FLUXIONS.
, fl*, log. A?,
#*-{- 2 x h-\-h* ;
and if it be x3 it will become (a?-{-/*)3 or
a;3-{-3 x* /z-}-3 x ;
and if it be x* it wfill become (A?-f-^)4, or
#4-f-4 a:3 ^4-6a:* A*-f-4A?A3-}-/i4}
and so on, for other integer powers.
If we compare the new value of the function in each
of these cases with its former value, it will immediately
appear, that the new value may be resolved into two
parts, one of which is the original value of the function,
and, therefore, the other is the increment which the
function has received, in consequence of the change in
the value of the variable quantity a;. Thus, the function
being a?*, we have found its new value to be a?*-{-2a?A-f-A2,
of which expression, the first term x* is the original value
of the function; therefore the other part of the expression
viz. 2xli-\.J? is its increment. In like manner the ex¬
pression a;3=3a:Vj-{-3 a?A*-{-AV which is the new value
of the function jc3, may be resolved into a;3, the original
value of the function, and 3 x^h-J-3 at A*-f- A3 its incre¬
ment j and a;4 4-4 *3 A+ 6 a;* 7^4-4 at A3A4, the new
value of the function x4, may be resolved into a;4, the
function itself, and 4 a?3 A-}-6 a:2 A*4-4 a? A3+A4 its in¬
crement.
5. Having seen that, by conceiving the variable
quantity x to receive the indefinite increment A, the
functions a;s, .v3, x4 receive the increments
2 x A-j-A*,
3 a:2 A4-3 a? A24-A3,
4 A?3 A 4-6 #2 A24-4 x A3 4-A4,
respectively, we next observe that each increment is ex¬
pressed by a series, the first term of which is the first
„ . , , . 72(72—O
xn+ - A?”-1 A 4- — xn
I 1*2
A*1
n(n—i) (72—2)_ n_,
A? J
I • 2 * 3
A3 4- &C.
where it appears, that the first term of the series is the
original value of the function, and the following terms
are the first, second, and following powers of the incre¬
ment A, each multiplied by a new function of a;, that
is independent of the increment. Let us denote the
functions nxn~xt ^ a;"”1, and — - .
1-2 1 ‘ 2 • 3
Ar"-3, &c. hyp, q, r, &c. respectively ; and it is to be
observed that, in the present case, as well as in the case
of any other function of x we may hereafter consider,
by the letters p, q, r, &c. or the same letters with ac¬
cents over them, or lastly the capital letters P, Q, R,
&c. we do not mean to denote functions of x of any
particular form, but functions of a? in general, that con¬
sist only of x and given quantities. This being kept in
view, it appears that the variable quantity x being sup¬
posed to change its value, and to become A?-j-A, the
function xn changes its value, so as to become
Af-f-pA-f-yA*4-7’A34-sA44-> &c.
a series, the terms of which have the properties already
explained in the two preceding sections.
7. Every rational and integer function of a variable
quantity x is necessarily of this form,
A#1*-}- B «/34-CA;y4- dccojj
where A, B, C, &c. and «, /3, y, are supposed to denote
constant quantities.
Let us examine what is the form which the function
assumes when the variable quantity a; changes its value
to -
FLUXIONS.
Direct to n-\-h; and to avoid complicated expressions, let us
MeltloJ- suppose the function to consist of these two terms A** +
We have already found in last section that * being
up posed to become a?-[-A, will become
j^-p h-\-q A2+r 7i3+ &c.
where p, q, r, &c. denote functions of * independent
of hf as explained in the last section, and consequently
Ax" will become
' Ax*-\-Ap h+Aq/i*-{-Arh3+&c.
■ In like manner Ba?!3 will become
B*'3 +B p' h+B q' Aa+B r* h3 + &c.
q', tJ, &c. denoting also functions of x independent
ofh; therefore taking the sum of the two series, it ap¬
pears that, supposing x to change its value, and to be¬
come x + h, the function Ax* + B ^ becomes
A** 4- Ba;£ +(A- p+B/?0 ^ + (A
+ (Ar+Br0^3 + &c.
now p and p' being functions of x, Ap + Bp' will also
be a function of#, and may be denoted more simply by
P, and for the same reason Aq-\-Bq\ Ar-j-Br', &c.
which are functions of #, may be denoted by Q, B, &c.
thus the expression for the new value of Ax* +B is
Ax* + B +P + &c.
a series, the form and properties of which are in all
respects analogous to those of the series that expresses
the new value of the function and although we have
supposed the function to consist of but two terms, still
the form of the series and its properties will be the same ;
that is, it will consist of two parts, one of which is in¬
dependent of h, and;is the original value of the function,
and the other is a series, the terms of which are the suc¬
cessive powers of the increment h, each multiplied by a
function of the variable quantity # as a coefficient.
This conclusion may be expressed in symbols concisely
thus. Let u denote any rational and integer function
of a variable quantity #, let * be conceived to change
its magnitude, and to become and let.« denote
the new value which the function acquires in conse¬
quence of the change in the value of #, then
u!zzu-\-p 7i~\-q h*-\-rh} &c.
where p, q, r, &c. denote functions of x as already
stated.
8. Suppose next the function of x to be of this
form
(A** + E^ + C*> H-&c.)%
that is, suppose it to be the «th power of a polynomial,
consisting of any number of terms whatever. Let the
expression between the parenthesis be denoted by v,
then we are to consider the function v". Now when x
becomes x+ht we have already found that v becomes
v-^-ph-^qh^-^r h3 -f- &c.
therefore t;“ will become
(v+ph + qh'Jfr/i3-^ See.)",
or, putting p h-^-qh?-\-r h3 -f- &c. r:M,
t>" will become (u-f-M)",
and this expression when expanded into a series by the
binomial theorem is
aT-fat;"-1 M9+t;c"-J M4 &c.
where a, r, See. express numbers.
Now from the form of the series denoted by M, it is
manifest that its square, cube, or any power of it what¬
ever, will be a series proceeding by the powers of h, and
having for the coefficients of its terms certain combina¬
tions of the quantities jo, q, rt &c. which being func¬
tions of #, any combinations of them will also be func¬
tions of x. Therefore, each of the terms of the above
series, expressing the developement of ex¬
cepting the first term v", will itself be a series proceed-
ing by the powers of h, and having its terms multiplied
by functions of #, and consequently their sum will be a
series of the same nature. Let us as before denote the
functions of x, which are the coefficients of the succes¬
sive powers of 7i by P, Q, R, Sec. and we shall hav*
upon the whole
(v-\-p h-\-q ^-{-rh3 -{- &c.)"
expressed by a series of this form
t;"-fP^-f-Q/P + RA3 -f &c.
therefore, putting the single letter u for the function*
or for
(A** -f B -}- C ** -f &c.)"
and u' for the new value which u acquires by x chang¬
ing its value to x-\-h,
u'=t/ + PA + QA9-fR//3+ &c.
a series of the same nature as before.
p. Let us now consider a fractional function of
and let us suppose it to be
A' x*f + B' xP -f- C' xy' -f &c.
A ** + B #£ + C + &c.
Where A', A, B', B, &c. also *8, &c- de¬
note constant quantities. Let v denote the numerator
of the fraction, and its denominator, then the func¬
tion is
v t
—, or vw~lt
w
now when x becomes #4“v becomes
v~{-p h-\-q h?-\-r h3 4* &-c*
and to~l becomes
U)~l-\-pt h-\-q' h3 -f*
and consequently v iv~* becomes
04-f A-H A*4-8cc.) («r* x/ A-f / A*4-&c.)
and
Fart I. Pi
Dircet j
Method.
Part I.
Direct and the product of these two factors, by actual multi-
Method. plication is
-f- v p*
p
7 4-^' 1
5“ +PP' fh*
3 +W-'q J
-J* &c»
Now, here as before, it appears that the coefficients of
the powers of h are functions of x, therefore, denoting
these functions by P, Q, R, &c. and observing that
vw~x is —, we have the new value of— expressed
w w
by the series
-f PA-fQ /i*+R/i3 4. &c.
or, substituting the single letter « for —, that is, for
A' -f-TV -f-C'
A xK -j-B x^ -f-C x7 -f-&c.
and putting u! for the value that u acquires when x be¬
comes tf-j-A,
t/=M-|-P A+Q 7i3-f-Scc.
a series in all respects analogous to those already found
for the other functions of x.
io. In the functions which we have hitherto consi¬
dered, the exponents of the powers of x were constant
quantities. Let us now' consider a function in which
the exponent is the variable quantity x itself.
Suppose then the function to be «*, where a de¬
notes a given number ; then, by supposing x to become
#-{-/*, the function will become
x4-h x h
a 1 —a a .
Now it has been shown in the article Algebra,
§ 295, that if A be put to denote the quotient arising
from the division of a logarithm of a by the logarithm
of 2’7i828i8‘” the exponential quantity oh is expres¬
sed by the series
/i*_j A3-j-&c.
FLUXIONS.
change that takes place in the magnitude of the vari¬
able quantity from which the function is formed, we
may conclude the truth of the following general pro- 1
position to be sufficiently established.
Let x denote a variable quantity^ and u any function
whatever of that quantity, let x be supposed to receive
any increment h, and thus to become x-j-h, and let u'
be the new value which the function acquires by the
change in the value of x; then, the new value of u may
in every case be expressed thus :
u'=zu-{-p h-\-q h*-\-r h3
where p, q, r, &c. denote quantities that are quite in¬
dependent of h, and consequently can only involve the
variable quantity x, and given quantities.
7°3
, A 7 , Aa
14 hA
I 1 1’2
l-2'3
therefore, ox^"A, the new value of the function, is
'A A* As
ox (1 4- — A4- --- £*4- -1— A34-&c.
I 1*2
this series, by multiplying all its terms by and put-
ting p, q, r, &c. for that part of each term which is
independent of h, becomes
0*4-/' ^*4-r A34-&c.
so that denoting the function o* by «, and its new
value by u',
u'—uA^-p h-\-qh*-\-r h3+&.C.
a series of the same form as the others.
11. From a due consideration of what has been
shewn relating to the changes that take place in the
magnitude of a variable function, corresponding to the
Direct
Method.
12. Having examined what is the general form that
any function of a variable quantity acquires by a change
in the value of that quantity, and found it to be a series,
the first term of which is always the function itself, it is
evident that the remaining terms will express the incre¬
ment that the function receives, in consequence of the
change in the magnitude of the variable quantity from
which the function is formed. Let us now compare
the simultaneous increments of a variable quantity and
its function with each other, and that vve may at first
avoid general reasoning, and fix the mind more com¬
pletely, let us suppose the functions to have a determi¬
nate form, as a*, x3, a4, &c.
Putting u and u' as before to denote the two succeed¬
ing values of the function, first let it be supposed that
7/=a*, then x being supposed to receive the indefinite
increment h, and thus to become xA^-h, and «to change
its value to a'=:(x4-Aa)> we have
u'—x'14- 2 .r A 4-^*,
or, u'—u 4- 2 a; A 4-7^,
and consequently
u'—'ll— ix h4-A*.
Thus it appears, that the simultaneous increments of
x and x* (or u') are A and 2 a? 774-77*, respectively. Let
us now compare these increments, not in respect of their
absolute magnitudes, but in respect of their ratio to each
other, thus vve shall have the increment of u to the in¬
crement of a;, as 2x hAf-h2 to h, that is, (dividing the
two last terms of the proportion by A) as 2 a?4-^ to 1.
Or, instead of employing an analogy, let us, for the
sake of brevity, and in conformity to the algebraic no¬
tation, rather express each of these ratios by the quo¬
tient arising from the division of the antecedents of the
ratio by its consequents, and put the results equal to
each other. Then, observing that the symbol u'—ut
which expresses the difference between the succeeding
values of the function, may be employed to denote its
increment, vve have
u'—u 2 x A-I-A* .
—a—=“+/S-
Hence it appears that the expression for the ratio of
the increment of the function u to the increment of the
variable quantity x is made up of two parts, one of
these, viz. 2 a x, is quite independent of A, the incre¬
ment of x, and the other is in the present case that incre¬
ment itself. In consequence of this peculiarity in the
form
704
Direct
Method.
FLUXIONS.
form of the expression for the ratio, it is evident that
if the increment h be conceived to be continually dimi¬
nished, the part of the expression which consists of h
will continually diminish, so that the whole expression,
viz. 2.v-f-A, may become more nearly equal to its first
term 2X than by any assignable difierence ; therefore,
. v'—u
2 x may be considered as the limit of the ratio ^ •
that is, a quantity to which the ratio may approach
nearer than by any assignable ditference, but to which
it cannot be considered as becoming absolutely equal.
Let us next suppose that M=:tf3, then * being suppo¬
sed to become x+h, we have u'=:(x+hy, =a;3+3
+ 3 * or
-J-3 a?* A+3 # A*+
Part I.
denote any function of a variable quantity, as for ex- Direct
ample, axn} or a xm-\‘bxn-\- &c. or tMethod.
axm-\-bxn-\- &c. ,1 J
a' A?m, + ^n/+&c.
and u' being put for the new value which the function
acquires when x becomes a?+A,
A-}-^ A2-f-r A3-f &c.
where p, y, r, See. denote functions of x that are inde¬
pendent of A, therefore,
v!—«=/> A+g'A*-j-r A3 Sec.
and
-—A-f-r A3-f-&c. ^
and consequently
u'—«=3 A+3 x A*+AJ,
and -^—3 #*+3 x A+A2.
n
or,
—-—=rp-}-A (g-J-r A-{- &c.)
n
Thus it appears, that whatever be the form of the
Here it is evident, as in the former case, that the ex¬
pression for the ratio is composed of two parts,
one of these, viz. the first term 3 is a function of*
that is independent of the increment A, but the other,
viz. 3 * A+A», or A (3 *+A), is the product of two
factors, one of which is the increment itself. From the
particular form of this latter part of the expression tor
the ratio, it is plain, that A being supposed to be con¬
tinually diminished, that part will also dimmish, and
may become less than any assignable quantity, .there¬
fore in this case, as well as in the former, the ratio
u u has a limit, and that limit is the first term of the
general expression for the ratio, namely the quantity
3 A?*.
Suppose, next, that «=*♦, and consequently
«,=(*+A)4= that u being supposed to
function, the ratio is always expressed by a quan-
/i
tity which may be resolved into two parts j one of these,
viz. p, is independent of the increment A, and the other,
viz. A(9-fr A -|-&c.), is the product of A by a series,
the first term of which is a function of.#, and the re.
maining terms also functions of x multiplied by the first,
second, third, and higher powers of A. Now from the
particular form of this last part of the general expression
for the ratio, it is manifest, that A being conceived to
be continually diminished, the quantity A (y-j-r A+
&c. will also be continually diminished, and may be¬
come less than any assignable quantity j therefore, the
limit of the ratio is simply p, that is, the function
/i
of x, which is the coefficient of the first or simple power
of A in the general expression for the increment.
14, From what has been just shewn, we may infer
the truth of the following general proposition relative
to the simultaneous changes that take place in a varia¬
ble quantity and its function.
Zet x denote a variable quantity and u any function
of that quantity, let x be conceived to change its value
and become x-f h, where h denotes an arbitrary mere-
ment, and let u' denote the new value that the function
acquires, in consequence of the change in the magnitude
ofx. Then, observing that h and u'—u are the simul¬
taneous increments of the variable quantity and its
function, if h be conceived to be continually diminished,
the ratio —continually approach to a certain
Limit, which will be different for different functions, but
always the same for the same function, and in every case
quite independent of the magnitude of the increments.
The ratio which is the limit of the ratio ot the in¬
crements, when these increments are conceived to be
continually diminished, may be called the limiting ratio
of the increments.
1 c. The analytical fact contained in the preceding
proposition, affords the foundatien for a mathemati¬
cal theory of great extent, and which may be divided
3
'’art I.
Direct
Method.
FLUX
into two distinct branches j one having for its object the
resolution ot the following problem. Having given the
relation oj any number of variable quantities to each
other, to determine the limiting ratios of their incre¬
ments; and the other the converse of that problem,
namely, Having given the limiting ratios, to determine
the relations oj the quantities themselves,
I he theory to which we have alluded constitutes the
Method of Fluxions, and in explaining the founda¬
tion of the method, we have endeavoured to show, that
it rests upon a principle purely analytical, namely, the
theory of limiting ratios 5 and this being the case, the
subject may be treated as a branch of pure mathema¬
tics, without having occasion to introduce any ideas fo¬
reign to geometry.
16. Sir Isaac Newton, however, in first delivering the
principles of the method, thought proper to employ con¬
siderations drawn from the theory of motion. IJut he
appears to have done this chiefly for the purpose of il¬
lustration, for he immediately has recourse to the theory
of limiting ratios $ and it has been the opinion of several
mathematicians of great eminence (a) that the consi¬
deration of motion was introduced into the method of
fluxions at first without necessity, and that succeeding
writers on the subject ought to have established the
theory upon principles purely mathematical, independent
of the ideas of time and velocity, both of which seem
foreign to investigations relating to abstract quantity.
17. That we may conform to the usual method of
treating this subject, we proceed to show how the
theory of motion is commonly applied to the illustration
of the nature of variable quantities, and of the relations
that result from their being conceived to change their
value.
As quantities of every kind, if we abstract from their
position, figure, and such aflections, and consider their
magnitude only, may be represented by lines, we may
consider a variable quantity x, and u any function of
that quantity, to be represented by two lines AP, BQ,
which have A, B, one extremity of each given, and
which vary by the points P, Q, their other extremities,
moving in the directions AC, BD, while the equation
expressing the relation between x and u, or their repre¬
sentative lines AP, BQ, remains always the same.
IONS.
and the measures of the velocities with which the varia¬
ble quantities increase have been called \.\\ewfluxions,
19. To simplify the hypothesis, we may suppose that
the point which generates the line AP, or x, moves uni¬
formly 5 thus the measure of its velocity, or the fluxion
of will be a given quantity, with which the measure
ol the velocity ol the point Q, or the fluxion of w, may
be continually compared. To determine then the flux¬
ions, or rather the ratio of the fluxions of x a variable
quantity, and u, any function of that quantity, is in
eflect to resolve the following problem.
Having given an equation expressing the relation at
every instant between the spares passed over by two
points, one of which moves with an unjorm velocity:
It is required to Jind an expression Jor the ratio that
the measures of the velocities have at every instant to
each other.
20. Now it is a fundamental principle in the theory
of motion, resulting indeed from the very nature of a
■variable velocity, that when two velocities are compar¬
ed together, whether they be both variable, or one of
them uniform, and the other variable, the measures of
their velocities are any quantities having to each other
the ratio that is the limit of the ratio of the spaces de¬
scribed in the same time, when those spaces are con¬
ceived to be continually diminished. And hence it fol¬
lows that the ratio of the fluxions of two variable quan¬
tities is no other than the limiting ratio of their simul¬
taneous increments.
I hat the theory of motion may be applied to the
generation of variable algebraic quantities, we have sup¬
posed them to be represented by lines \ this, however,
is not necessary, if the variable quantities are them¬
selves geometrical magnitudes j for like as a line is
conceived to be generated by the motion of a point, so
a surface may be considered as generated by the mo¬
tion of a line, a solid by the motion of a surface, and
an angle by the rotation of one of the lines which con¬
tain it j and the fluxions of those quantities at any in¬
stant, or position, will be the measures of the veloci¬
ties, or degrees of swiftness, according to which they
increase at that instant or position.
But in every case the ratio of the rates of increase,
or fluxions of two homogeneous magnitudes, will be the
limiting ratio of their simultaneous increments.
7^5
I)ir< et
Mtthod.
P
B j D
Q
18. The lines AP, BQ being thus conceived to vary,
the relation that is supposed always to subsist between
them, in respect of their magnitudes, will necessarily
give rise to another relation, namely, that which will
constantly subsist between the velocities of the moving
points P and Q, by which the lines are generated.
With a reference to this particular mode of conceiv¬
ing variable quantities to exist, the quantities them-
■elves have been called flowing quantities or fluents,
21. Having thus found that by conceiving variable
quantities as generated by motion, and taking their ve¬
locities, or rates of increase, as an object for the mind
to contemplate and reason on, we are in the end led to
the consideration of the limiting ratios of their incre¬
ments, a subject which is purely mathematical, and in¬
dependent of the ideas of time or velocity, we shall ex¬
change the definition of a fluxion given in § 18, which
involves those ideas, for another that rests entirely up¬
on the existence of limiting ratios.
By the fluxions then of two variable quantities hav-
ing any assigned relation to each other, we are in the
following treatise always to be understood to mean any
indefinite
^ a3 Lagrange, Cousin, La Croix. &c. abroad, and Landen in this country*
Vox.. VIII. Part II. f 4 U
7oS FLU X
Direct indefinite quantities which have to each other the limit-
Method. ing ratio of their simultaneous increments. . _
,,u—* In conformity to this definition of fluxions, it is evi-
dent that we are to consider them, not as absolute, but
as relative quantities, which derive their origin from the
comparison of variable quantities with each other in re¬
spect of their simultaneous variations ot magnitude.
22. Sir Isaac Newton employed different symbols at
different times to denote the fluxions of variable quanti¬
ties. It is now however common in Britain to denote
them by the same letters employed to express the quan¬
tities themselves, and each having a dot over it. Thus
x denotes the fluxion of the variable quantity expres¬
sed by tf, and in the like manner «, v, z, denote
the fluxions of the variable quantities », v, as, respec¬
tively.
23. Suppose now that u is any function of a variable
quantity x, and that the limiting ratio of the simultane¬
ous increments of u and x is the ratio of^? to I, where
j9 denotes some other function of x ; then, from the de¬
finition just given of fluxions, we have
and u—px.
x
Hence it follows as a consequence of the preceding
definition, that the fluxion of u, any function of a vari¬
able quantity x, is the product arising from the multi¬
plication of that function of x which is the expression
for the limiting ratio of the increments by the fluxion of
the variable quantity x itself.
SECT. II. Investigation of the Rules of the Direct
Method of Fluxions.
IONS. Parti.
of this change in the magnitude of x, u also changes its D'rect
value, and becomes u\ then, observing that u'—u and Method.
h are the simultaneous increments of wand*, the limit-' '—r'—'
ing ratio of - ~~J~ is 2 Ijet expression for the
limiting ratio be put equal to the ratio of the fluxions of it
and x, that is to , thus we have —r—zx, and u=2xx.
x x
Hence it appears, that whatever be the magnitude of
the quantity that expresses the fluxion of x, the fluxion
of u or x* will be expressed by the fluxion of x multi¬
plied by 2 X.
Again, when u~x^t and u\ u'—u and h denote the
same as before, it has been shown, § 12, that the limiting
ratio of U is 3 #*, therefore, § 21, — = 3 **, and
h x
11=3 x*x; that is, the fluxion of x3 is expressed by the
fluxion of x multiplied by 3X*.
And when «=«4, it has been shown, § 12, that the
limiting ratio of ^ is 4#*. Therefore, § 2X,
~—^x* and uzz/[x3x.
x
To resolve the problem generally, or when u—x"*
let us suppose x to become a;-J-4, and u to become u\
then u,z=.(x-\-h')n. But this last quantity, when ex¬
panded into a series by the binomial theorem (ALGE¬
BRA, Sect. XVII.) is
#”-}-/) A-{-g'^* + 7,/$3+ &c-
or u^p h\-h* + &c.)
where p, q, r, See. denote functions of at, independent
of h. Therefore,
u'—u=ph-\-h} (q-\-rh-\- &c.),
24. The method of fluxions naturally resolves itself
into two parts, as.we have already observed, § 15. We
proceed to explain the first of these, which is called the
Direct Method, and which treats of finding the ratio of
the fluxions of variable quantities, having given the re¬
lations of those quantities to each other.
25. We shall begin with investigating the ratio of the
fluxions of two variable quantities in that particular
case, when one of them is any power of the other.
Let us suppose then, that u is such a function of a
variable quantity x, that u=xn, where n denotes any
number whatever, it is required to determine the ratio
of the fluxions of u and x.
If we recur to the definition which has been given of
the fluxions of variable quantities in § 21, it will appear
that we have in effect resolved the problem just proposed
in three particular cases, when treating of the limiting-
ratios of the increments of variable quantities. For it
has been shown in § 12. that when u—x1, and x is sup¬
posed to change its value, so as to become x-\-h, where
4 denotes an indefinite increment, and in consequence
and —j—~p-\-h h-\- &e.).
n
Therefore, supposing h to be continually diminished,
the limit of is p ; but, whatever be the nature of
the exponent n,p is always nxn~*, (Algebra, § 26J.),
therefore, the limit of —f —IS nxn~t, and consequently
~-=znxH~l, and u—nxn~'tx.
x
26. As ive shall have frequently occasion to employ
the result of this investigation, it will be proper to ex¬
press it in the form of a practical rule. thus.
To find the fluxion of any power of a variable quan¬
tity. Multiply the fluxion of the variable quantity it¬
self by the exponent of the power, and by a power of
the quantity whose exponent is less by unity than the
given exponent, and the product will be the fluxion re¬
quired.
27.
'"'b) We are here to be understood to mean the ratio of the values of the increments, which may al¬
ways be compared with each other, whether the variable quantities be of the same kind, as both lines, or both
surfaces, &c.-or of different kinds, as the one a line, and the other a surface.
Part I:
Direct 27. In determining the limit of the ratio of the simul-
Method. taneous increments of x and xn, we have referred to the
* ' binomial theorem j but the only application we have
had occasion to make of that theorem was to determine
the numeral coefficient of the second term of the deve-
lopement of when n is supposed to be any
number whatever, which is an inquiry of a more simple
nature than the general investigation of the theorem.
We shall now show how that coefficient may be deduced
from the first principles of algebra. Thus the investi¬
gation of the fluxion of xn will be rendered independent
of the general demonstration of the binomial theorem ;
and we shall hereafter show that the theorem itself is
easily investigated by the direct method of fluxions.
707
Let both sides of this equation be raised to the power Direct
n, having first substituted M v for A v-f-Bi;**}- &c. Method.
Now as we have just found that m and n being integers,
(1 -f-u)m= 1 ^-mv + &c.
and (1 1 -f See.
Here we stop at the second term, that being the only
one whose coefficient is required. Substitute now for
M v its value A t; -j- &c. then, stopping again at the
second term, we get
(r + Mv)"=:i-|-»At;-|- &c.
FLUXIONS.
28. Since x-\-h is equal to x (i + -J it follows that
(sr+/i)" zzxn thus the developement of {x-\-h)
h\ h
is reduced to that of or putting to
(i-j-f)". Now if we give particular values to and
suppose it to be I, 2, 3, &c. or —I, —2, &.c. or last¬
ly it T> &.c. we can find the series that express the
powers of l-f-u, whose exponents are those numbers, by
the operations of multiplication, division, and evolution,
and in every particular case we shall find, that the
powers of 1 -j-p are expressed by a series of this form,
i_|_At;-f-B -j-Du4-f- &c.
Where A, B, C, D, &c. denote numeral coefficients
which depend for their value only on n the exponent
of the power, and not on the quantity v ; and as the
form of the series will be found to be the same what¬
ever particular values we may give to the exponent, we
may conclude that it is the same, whether the exponent
be positive or negative, whole or fractional.
29. First let us suppose that the exponent is a whole
positive number, then, because
(l-|-i;)"=:I + At;-{-Bt>*-f- &c.
if we multiply both sides of this equation by l-f-tf, and
collect together the like powers of v, it will appear
that
+4v+^r+
&c.
Hence it appears that the coefficient of the second term
of any power of 1-f-t; exceeds that of the next less
power by unity. Now in the case of the first power of
l+v, the coefficient of the second term is obviously 1,
therefore, in the second power it is 2, in the third power
3, and universally, in the nth power it is n ; so that n
being a whole positive number,
-f-v) —— 1 See.
30. Let us next suppose the exponent to be a frac¬
tion denoted by^> so that
J n
(1 4- At;*+ Bt;4^. See.
therefore,
-f-^At; -{- &c.
and making the coefficients of v in each series equal to
each other,
nAzzm, and A~ —.
n
31. In the last place, let us suppose that the expo¬
nent is a negative quantity either whole or fractional,
so that
m
(i-f-t;)--"—i-f A'^-f BV-f See.
or —^-=i+A,t;+BV+&c.
(1+1/)^
then, multiplying both sides by (i-f-u)^ we get
i=(i+^)n (1-|-A'v-j-BV 4. See.)
or, substituting 1 -}-At;-f B^-f- &c. for (l+v)7 and
actually multiplying the two series,
i=i-f-A 7 4-B *)
-f-A' ^ -f-A A' >i>* -f- Sec.
+B' ^
Now that this equation may subsist, whatever be the
value of v, it is necessary that
A+A'=ro,
B-j-AA'-f-B'zzo,
See.
and by these equations we may determine A', B', C',Sec.
It is, however, only required at present to find the first
of these, viz. A', now we have A'rr -—A •, but A being
the coefficient of the second term of the series expressing
(1 -f i0~”, the exponent of which is positive, we have al¬
ready found it to be therefore, A'=: —. —
n ' n
32. As we have found that the coefficients of the se¬
cond term of the developements of (i-ftO",
and (i-f- t>) » are and— respectively, it ap¬
pears that whatever be the number denoted by «, the two
first terms of the series expressing'(i -J-r)'1 are, 1 -f-nt;,
and therefore, substituting for v its value and multi¬
plying by xn, the two first terms of the series expressing
(v+^)* are xn-irnx1'-'h, agreeing with what we
have
FLUXIONS.
Part L
have assumed in § 25* as g‘ven by the binomial theo¬
rem (c).
33. The mode of reasoning employed to determine
the ratio of the fluxions of u and x, when the former is
a function of the latter of the form xn, will apply equal¬
ly when the function has any other assigned form. But
instead of investigating in this manner the fluxion of
every particular function, it is better to consider a com*
plex function as the sum, or diflerence, or product or
quotient, &c. of other simple functions, and to inves¬
tigate rules for each of these cases, supposing that the
fluxions of the simple functions are previously known.
34. Let us first suppose that u, a function of a varia¬
ble quantity x, is equal to the sum of v and w, two
other functions of x* It is required to find the fluxion
of u, having given the fluxions of v and w.
Let x be conceived to change its value, and to be¬
come #4-A, then, as ti-and w will also change their va¬
lues, § 11, the one to
v'zzv-^-ph-^qJi1&c.
and the other to
w’—w -\-p' h-\-qr h*-\-r' A*-}- &c.
if u' as usual denote the corresponding new value of
we have
u=.v-\-w
, C u-j-jo A<7/i*-j- &c.
u £A-f-*/A*-f-&c.
u'-—&c*
U~~T~~ ~P “f"P* + (?+?'>* + &C*
where a. A, e, denote any given numbers, positive or
negative, then, by reasoning as above, it is evident that
s~au-\-bv-\-cw-\-
Direct
Method,
Therefore, to find the fluxion of (he sum of any number
of functions, each multiplied by a constant quantity.
Multiply the fluxion of each function by its constant
coefficient, and the sum of the products is tfte fluxion
required.
36. If c denote a constant quantity, and u, v, be
functions of a?, such, that u—c-\-v; then x being sup¬
posed to become x-\-h, and consequently v to become
v\ or tt-j-pA-f-^A3-}-&c. and c-\-v to become c-f-v
ph^qh1 &c. we have
u~c-\-v,
and u'~c-\-v-\~p h-\-q h"1 &c.
and hence u'—u~p h-\-qhx-\- &c.
or u'—u— vf—vy
u'—u v' V
therefore, —= —,
and these ratios being always equal, their limits must al¬
so be equal; therefore, substituting for the limiting ra¬
tios those of thefluxions,we have t=:—r and uzsv; that
X X
is to say, the fluxion of c-\-v is v, from which it appears.
That the fluxion of any variable function is the very
same as the fluxion of the same function, increased or
diminished by any constant quantity. This is a remark
of great importance in the theory of fluxions, as will ap¬
pear hereafter.
If we now conceive A to be continually diminished, we
shall have the limit of —j— expressed by^-f-/?'. But
h
p is the limit of § M* antl manner p* is
'lift 'IV
the limit of —^—, therefore,
». • r «'—u >. r ^—v ii**. e w>—w
limit of —;—= limit of — f- limit of ?—.
hah.
Substitute now the ratio of the fluxions instead of the
limited ratios, and we have
u v '
xxx
therefore, u~v-\-w.
35. If we suppose s to be a function of x, and u, v,
w, &c. other functions of a’, such that
szzau-\-bv-{-c'W-\-&tc.
37. Let us now suppose that #, v, w, are functions of
x, such, that tfzzvw, it is required to find the fluxion
of v, supposing the fluxions of v and w to be given.
By supposing that x, u, v, and w, change their va¬
lues as usual, we have
U—VW
u'=:(v-\-p h^-qh* &c.) (w-\-pf Jih*&c.)
and this last expression by multiplication becomes
u'zzvw+vp'h+v q' A21 c
-\-tvph-\-w q h1}
therefore
u'—u—(vp'-\-wp)h-f{vq’-\-pp'-\-w qfld-^ &c.
. . u'—u
dividing now by A, and taking the limit of —^—, we
have that limit expressed by vp'-\-wp ; but p' is the li¬
mit of H>- j -1 § 14, and in like manner/? is the limit of
V— : V ; therefore
the
(c) In this investigation we have supposed n to be a rational number. If, however, it were irrational, still the
result would be the same; for, corresponding to every such number, two rational numbers, one greater, and the
other less than it, may be found, which shall differ from each other by less than any assignable quantity. There*
fope, the general properties of these numbers must belong also to the irrational number which is their limit.
’art I.
Direct
! Method
FLUXIONS.
709
the limit of
r ft’X limit of
w—~~w
W—u
h
-f-W’X limit of
Hence, by substituting for the limiting ratios the ratios
of the fluxions, we have
~ — —- -} r and uzz v tv tvv.
XXX
Therefore^ tojlncl the fluxion of the product of any
two functions,, multiply the fluxion of each function by
the other function, and the sum of these products is the
fluxion required.
38. We have just now seen that when
then u—wv-\-vw.
Let each side of the latter equation be divided by the
corresponding side of the former, thus we get
« v vw
— ■+- — J
u v w
suppose now that the function w is the product of two
other functions s, t, so that
u—v s t,
then, because w~s t, from what has been shewn it fol-
w s t
lows that—:= — j therefore, substituting this va-
• • • •
lue of — in the equation —{- —, it becomes
w ^ u v ' w
u u *
In general, if we suppose that
u—v s t r &c.
by reasoning as above it will be found that
« r j?
u v ' s
+ y + 7- + &c-
whatever be the number of factors.
Suppose the number of factors to be three, so that
u—v s t
u v ' s ^ t ’
then substituting v s t for u in this last equation, and
taking away the denominators, we find
uz=stv^-vt s-\~vst.
And as a similar result will be found, whatever be
the number of factors, we may conclude that The
fluxion of the product of any number of functions is
equal to the sum of the products of the fluxion of each
function by all the othei'functions.
39* •Let us in the last place suppose that n——, and
that it is required to find the fluxion of u, having given Direct
the fluxions of v and w. Method.
From the given equation we have vzzw u, and there-
fore 37.) v—wu-\-uw, let —be substituted for u
in this equation, and it becomes vxzwu 4- from
w
which we easily obtain
w v—v w
u—
Hence we have the following rule for finding the
fluxion of a fraction.
Multiply the fluxion of the numerator by the denomi¬
nator, andfrom the product subtract the fluxion of the
denominator multiplied by the numerator, and divide the
remainder by the square of the denominator ; the result
is the fluxion required.
40. It will now be proper to shew the application of
these general rules for determining the fluxions of va¬
riable functions to some particular examples.
Example 1. Suppose wna-J-iy/'A?—Required
x
the fluxion of u.
Here a being a constant quantity, the fluxion of
a-t-b^/x— — is the same as the fluxion of by/x— —,
Z
§ 36, or bx^—cx~x. Now, by J 26, the fluxion of
. 1 *
x* is \ «T-T x, which expression is equivalent to
2 7*
of x~~x is
x, or—A?“a x, or
therefore,
multiplying the fluxion of x* by b, and the fluxion of
x * by c, and taking the sum of the products, agree¬
ably to the rule in j 35. we have
• bx cx
,2)fx
Ex. 2. Suppose k=o+ 3-^ —
By writing the function thus
_,4
c ,
ar3y'ar ' a:1*
u—a-\-bx T—cvw ^Jrdx—*
the application of the same rules employed in the last
example gives us
u~ —flx ^ xc x 7 x—2dx x
or, exchanging the fractional indices for the radical
sign, and otherwise reducing,
• —2 b x . Acx
2x3\/x* ' 3xt3fx
2 dx
xa
Ex. 3.
7iO
Direct
Method.
FLUXIONS. Part [
Ex. a. Suppose and this equation, by reducing all the terms on the lat- Direct
In order to find the fluxion of this function by the ter side of it to a common denominator, is more simply Method,
rules already laid down, it will be necessary to consider expressed thus, . l—Y-—H
it first as a function of a variable quantity that is itself . (o^-f-oV1—4 x*) x
a function of x. Let us then put a+bxm equal to v, « — ^ (aa—#a)
and thus the proposed equation becomes uz=:vn 5 then, u
being considered as a function of v, we have by § 26.
u~nvn~1v. Again, v being considered as a function
Ex. 7. Suppose «=—
0 + *
a*-)-**
of#, from ^ the equation v—a + bxm, we find by § 36. j£ere we employ the rule given in § 39. for finding the
tinU § 26. v—mbx"1-1 x. Let this value of t; be substi "
tuted in the expression for «, and it becomes
fluxion of a fractional function $ thus we find
(a*.|_v^w)7=^.
t/—-
2\/V
uzzsj (a—,y4-!z)J = (a—y+2;)7.
Now as the fluxion of a—y+as, is—y-}*z (§ 35),
which, by restoring \f (a*—»*) for v, and leaving out we find from § 26. that by considering a—y+!3 as a
the number common to the numerator and denominator, single variable quantity,
becomes
y/ (a*-#*) •
Ex. 5. Suppose ms= y/ (a+i x-\-c #*).
By proceeding in the same manner as in last ex¬
ample, we find
(b>{-2cx')x
1 (—y-fa)
=Kq—y+«)7 C—y+^)
=ix/ a—(—y+s)
b T
but, since we have, § 26,
,=—lbx~
b x
#=-
u~
y/(a+bx-\rcx1')
2 X y/X*
Ex. Suppose «=# V (a2—#1).
Here the proposed function is the product of these
three functions, viz. #, a*4-#*, and y/ (a*—#*). There¬
fore, its fluxion will be found by proceeding according
to the rule in § 38.
Now the fluxion of x is #, and the fluxion of as4-*3
is 2xx, and the fluxion of y/ (a2—#2) has been found
in last example to be —■■■• —. Therefore, multi-
\/(o—#*)
plying the fluxion of each function by the product of
the other two functions, and taking the sum of all these
products, we find
and since (c*—#*),=r by consider¬
ing c*—#* as a single variable quantity, and observing
that its fluxion is —2x#, we find by § 26, that
vf—1 * —4##
ZZZj (c*—#e)
X — 2 # x=
3 VO’-*’)
Instead of y, 25, y, z, substitute now their values in
the expression for the fluxion of u, thus it becomes
«=i ✓ (a- -^+ V C^—»■)•)
/ bx
XX \
0*—#*)/’
f (o24-a;2) y/a*—x1 x
t* =: ] + 2A?* \/«*—x* *
1 x1 (0*4-^*) V
\2 X y/ X 3 3 -0 0
Ex. 9» Suppose urzavm yn, where i» and y denote
any fumtims of a variable quantity.
Then, $ 37,
y/ (o’—#1)
«« $ayn'X' of 7
+ avm x fluxion of y" J
But
3
irt I.
Direct But fluxion of vmzznivm~*vi § 26,
^e^10tK and fluxion of yn~nyn~Ty ;
therefore,
uzzamy*vm-tv+a n vmy* 1 y,
— av'n~tyn~l(my v n v y).
F L u x I o N s.
where q denotes a new function of x, derived from j»,
the former function, by the same kind of operation as
that by which p was deduced from u.
Suppose now q to denote the particular function
«(«—1 then,
71
Direct
Method.
Lx. 10. Suppose where vt 2$ and y denote
any function of a variable quantity. Then, because
fluxion (v-\-%)zzv-\-‘z, § 34, and fluxion 3^=3 3^ y,
§ 26, we have, § 39,
;r_y3(v +x)—3(v+x)y*y,
y6
_yp+gQ—sP+gQy
y4
41. As when u denotes that particular function of x
which is xn, we have (§ 25.)
u
~—nxn-' }
x
80, m general, whatever be the form of the function de¬
noted by «/, we have always
4-=«p—1) (n—2) a"-3,
<7
or ~—r.
where r denotes a function of x derived from y, as q
was derived from p, or p from the original function 11,
And it is evident that we may proceed in this manner
as far as we please, unless it happen that in finding the
series of functions y, y, r, &c. we at last arrive at a re¬
sult that is a constant quantity, and then the series of
operations will terminate. Thus if the function was
ax*, vve should have
wzzax*
11 *
—■^z^ax^zzp,
—=4-3°*s=y,
X
4-=4.3.2ax=rt
■=p,
—=4.3 • 2.1 .a—2.4a.
x
where p denotes a new function of x, resulting from the
analytical process employed to find the fluxion of the
function u, and depending for its form upon the par¬
ticular form of that function : just as in involution, or
any of the other operations of algebra, a result is ob¬
tained depending upon the particular nature of the ope¬
ration, and the quantities operated upon.
Let us put p to denote the particular function tixn~1,
Here the expression for — is a constant quantity, which
x
has no fluxion.
Hence it appears, that relatively to any function of a
variable quantity, there exists a series of limiting ratios,
deducible from that function, and from each other, by a
repetition of the operation of finding the fluxion of a
variable function.
or the expression for -v the ratio of the fluxion of u to
x
the fluxion of x when z/=*B, then, supposing that «—1
is not equal to o, (for in that case nx’^* would be sim¬
ply n, a given number,) we may reason concerning the
ratio of the fluxions of the variable quantities p and x,
in all respects as concerning the ratio of the fluxions of
wand x; and accordingly, from the equation
p—?ixn~%,
we get, by taking the fluxions,
~zzn{n—i)*"-2,
x
or, considering p as denoting generally the function of
x that results from the operation of finding the fluxion
of the original function u} whatever be the form of that
function, we have
42. In treating of the fluxion of a function, we have
hitherto regarded the fluxion of the variable quantity x,
from w'hich the function is formed, merely as one of
the terms of a ratio, without considering whether it was
a constant or a variable quantity.
Now as we may assume any hypothesis respecting the
nature of the fluxion of .r, that is not inconsistent with
what has been already delivered, we shall suppose it to
be constant. This assumption, if we consider the fluxions
of variable quantities as the measures of their respective
velocities, or rates of increase, is in eftect the same thing
as to suppose that the variable quantity x increases uni¬
formly. Then, as in the expressions
n p q *
~r~p, —=y, —=r, &c.
XXX
or these others, which follow from them,
uzzpx, p—qx, q—rx, &c.
the symbol x is to be understood as denoting a constant
quantity, it follows that if p be variable, then px, or
u will be variable $ and if y be variable, then qx, or
Pt will
7T2
Direct p wlll be variable j and if r be variable, then r *, or
Mctfa<>l!l“ , q wiH be variable, and so on.
43. Let us now recur to the relation in which the
succeeding functions /*, §cc» stand to the original
function u.
By performing that particular analytical opera¬
tion upon the function z/, which consists in fuiding its
fluxion, we obtain p a? as the expression for its fluxion,
that is, we get uz=px; and by repeating the operation
on the function p, we get pz=qx ; and therefore p x=
qx*j but, x being regarded as a constant quantity,
p x is deduced from px, considered as a function of x,
just in the same manner as pa* is derived from the ori¬
ginal function u; therefore the expression q x2 is dedu¬
ced from the function u by performing the operation of
taking the fluxion twice ; that is, first upon the function
u itself, and then upon u or px, the expression for its
fluxion and in this second operation x (or the fluxion
of the quantity from which the function is formed) is
considered as a constant quantity.
The expression qx*, obtained in this manner from
the function n, is called the second fluxion of the func¬
tion * and to express its relation to the function u, it is
denoted by ii, that is, by the letter denoting the func¬
tion itself with two dots over it. Thus, like as u=px,
vre have
.. «
u—qx* and -r-—
**
Again, since q~rx, it follows that qx3~rx* j but,
as x is constant, q x' is derived from qx%, by the ope¬
ration of finding its fluxion, considering ^ as a func¬
tion of x, just in the same manner as qx*, or « is de¬
rived from p x, or u, and in the same manner as u is de¬
rived from the original function u ; therefore, like as
pxovu is the first fluxion of the function, and q x*
or « is its second fluxion, so rx* is called its third
fluxion, and is denoted by «, that is, by the letter ex¬
pressing tile function itself, having three dots placed
over it, so that
u ~r x3 and ~ —r.
ars
The fourth fluxion of a variable function u is denot-
Part I
second and higher orders of fluxions of a function, let Direct
us suppose u to denote the particular function a xn *, Method,
then, proceeding agreeably to what has been laid down ’
in last section, we obtain, by the rule for finding the
fluxion of any power of a variable quantity ($ 26.)
«=« a x”"'1 x
u—n(ii—i)a xn’‘2xt,
u—n{n—1) {n—2)0 xn~*x3,
u —n{n—1)(«—2)(«—3) axn~4x*, &c.
Here we have exhibited the first, second, third, and
fourth fluxions of the function ax»-, the law of con¬
tinuation is obvious, and it appears that when « is any
positive integer, the function a xn will have as many
orders of fluxions, as there are units in n, and no more ;
for if n were supposed ~3, then, as the fourth fluxion,
and all the subsequent ones, are multiplied by n—3, or
in that case by 3—3=0, they consequently would va¬
nish, and a similar observation may be made when n is
any other whole positive number.
45. That we might be able to apply the rules of
§ 26, § 34, &c. to the determination of the fluxion of
a complex function of a variable quantity, we have
found it convenient in some cases to consider such a
function as composed of other more simple functions of
the same quantity, and we have expressed its fluxion by
means of the fluxions of those other functions. In find¬
ing the fluxion of any higher order than the first of such
a complex function by those rules, we must keep in
mind, that it is only the fluxion of .v, the variable quan¬
tity from which the functions are all formed, that is to
be considered as constant, and that the fluxions of the
functions themselves are in general variable quantities 5
so that each of them may have a second, third, &c.
fluxion, as well as the function which is composed of
them.
Let us suppose, for example, that
u=x/(a2+x*) j
then, considering as a function of x, and putting
t; to denote it, we have uzn^vzzv*, andtt=^ii~7v
: but since vz^a'-X-x2, it follows that vzz2x x\
therefore, substituting for v and v their respective va¬
lues, we have
• xx
V C°*+**)
FLUXIONS.
ed by «, that is, by the letter u with four dots over it,
and is derived from the third fluxion, in the same man¬
ner as the third is derived from the second, or the se¬
cond from the first, or the first fluxion from the variable
function itself $ observing, that in repeating the opera¬
tion of taking the fluxions, the symbol x (or the fluxion
of the variable quantity from which the function is
formed) is considered as a constant quantity. And the
same mode of notation and deduction is to be under¬
stood as applying to a fluxion of any order whatever of
a variable function.
44. To illustrate what has been said respecting the
Now, to find the second fluxion of u, we may either
. . • xx
take the fluxion of this last expression, viz. " ■ - tt,
V(° +* )
and consider the symbol x, which is found in it, as de¬
noting a constant quantity $ or we may recur to the
• * *v
equation and take the fluxion of this other ex-
* */v
pression for u ; and in this case, we must consider that
both v and v denote variable functions of x, and there¬
fore that the fluxion of
\rv
may be found by the rule
for
Jart I.
FLUXIONS.
7U
Direct for gnd;ng the fluxion of a fraction 5 observing that v is
_ ^ to be substituted as the fluxion of v. Accordingly, pro¬
ceeding by this last method, and considering that the
kv
fluxion of -y/^jthe denominator of the fraction, is —7-, we
yv
find
. I
u t v
I 2VV—V2
Now from the equation vz=a*-j-x2 we ;have v~ 2 x
and (observing that x is constant) t7=2 x2. Let these
values of t>, v, and v, be now substituted in the ex¬
pression for «, and it becomes
u
’ 4(0*-f. a?*)#2—4#***
v=:a-\-b xm, t=C’{-dxH,
then v=mb.xm~J x t~n dxn~l x,
and considering x as constant,
v=zm(m~—l')b xm~2xs,
&c.
t—n(n—i)dxn
these values of v, t, v, t, &c. being substituted in the
expressions of uf «, &c. will give the successive fluxions
of u in terms of x and x only.
46. If the fluxion of a variable quantity be consider¬
ed as the measure of its rate of increase, if that rate be
uniform, then its measure will be a constant quantity ;
but if it be variable, then its measure will be a variable
quantity, which will also have a certain rate of increase
or decrease ; and the measure of this rate will be its
fluxion, or will be the fluxion of the fluxion of the ori¬
ginal variable quantity j that is, it will be the second
fluxion of the original variable quantity. And if this
second fluxion is not a constant quantity, then the mea¬
sure of its rate of variation will be its fluxion, or will
be the third fluxion of the original variable quantity,
and so on. Thus a quantity will have a successive order
of fluxions till some one fluxion become constant, and
then it will have no more.
The very same expression for u would have been found
if we had employed the other method.
By proceeding as in this last example, the rules al¬
ready delivered for finding the first fluxion of any func¬
tion of a variable quantity will apply to the finding of
the fluxion of any higher order.
Thus if we had uzzv t, where v and t denote each a
function of another variable quantity x, and it were re¬
quired to find the different orders of fluxions of «, con¬
sidered also as a function of x ; then, by the rule of
§ 37* we have
47. We have hitherto supposed the equation expres¬
sing the relation between a variable quantity, and a
function of that quantity, to be of such a form, that
the function was found alone, and of the first degree on
one side of the equation, and some power, or combina¬
tion of powers, of the variable quantity on the other j
as in these examples,
u~axn,
a-\-bxm
c-\-dxu’
u—tv -}-t> ty
and w=:fluxion of ft; + fluxion oi vt ;
but v and t being variable functions of a?, we may con¬
sider -uandtfas denoting also variable functions of a:, the
fluxions of which are to be denoted by v and t re¬
spectively j now by the rule in b 37, we have
fluxion of tv^v \-\-t Vy
and fluxion oivt=.t v-\-v t,
therefore, u-=z2v t-\-v t.
By considering v, v, v, also t, t, as denoting each a
distinct function of x, we may find the third fluxion of
u from the second, in the same manner as the second has
been found from the first, and so on for the other orders
of fluxions of u. If. it be now required to express the
successive orders of fluxions of ti in terms of x and its
fluxion, we must find the values of y, f, &c. also of f,
t &c. in terms of x and its fluxion, and these values,
also the particular functions of x denoted by v and /,
being substituted in the expressions found fora, a, &c.
will give to these expressions the form required.
If for example we suppose that
Vol. VIII. Part II. +
In such cases as these, u is said to be an explicit function
of x. We are now to consider how the ratio of the
fluxions is to be found when the relation between the
variable quantity and its function is expressed by an
equation, the terms of which involve different powers,
both of the function, and the variable quantity j as in
the following example,
y*—-a a: y-{-£ a?*—cr:0,
where we are to consider y as a function of a:; but from
the particular manner in which its relation to x is ex¬
pressed, it is said to be an implicit function of that quan¬
tity.
Now in this example, by the resolution of a quadra¬
tic equation, we find
axz±z>sj * (a*—4^) ^* + 40 £
and as y is here an explicit function of .v, its fluxion or
the ratio of its fluxion to that of at, might be determi¬
ned by the rules already laid down. Bi)t it is to be
observed that it is only in the particular case of the
proposed equation being of the second degree that we
can effect the solution generally in this manner. If it
4‘!X were
71 4-
Dircct
Method.
FLUXIONS.
were of a higher order, this particular mode of solution
would be often impracticable, for want of a general
method of resolving equations.
Part I.
. V i i* • r ^ V Direet
putting —■ equal to the limit of v, we have there- Method.
fore
48. We may however in all cases resolve the pro¬
blem, without a previous resolution of the equation, by
reasoning as follows.
Whatever be the degree of the equation, by giving
particular values to x, we can, by the theory of equa¬
tions, obtain corresponding particular values ofy ; there¬
fore, we may be assured that in every casey is expres¬
sible by means of x in some way or other, if not in
finite terms, at least in the form of a series, the terms
of which shall involve powers of x. Hence we may
infer, as in the case of explicit functions, that when x
changes its value, and becomes x-\-h, y will also change
its value, and become
y-f-p/i-f-y A*-|- &c.
where p, q, &c. denote functions of A?, that are indepen¬
dent of the arbitrary quantity h. Let us denote p h
&c. the increment of y, by the single letter
k; then y-J-& is the new value of y, corresponding to
x-\-h, the new value of x. Let these new values be sub¬
stituted instead of x and y in the proposed equation
y’—a x y -\-b x'—c—O)
and as the result must still be =0, we have
(y_j_&)*—a{x-\-K) erro $
which equation, by actually involving its terms, substi¬
tuting for k its value p h-^-q ^a-j- &c. and arranging the
result in the form of a series proceeding by the powers
of /*, becomes
y*—a xy-\-b —c’
y—a(p x+y^ + lb x)h >• =0.
+Q/i2+R^3+ &c.
Here Q, R, &c. denote quantities independent of
and involving x, y, p, q, &c. that is to say, x, and func¬
tions of at, and therefore Q, R, &c. are also functions
of x. Now as this equation must subsist whatever h
may be, which is a quantity quite arbitrary and inde¬
pendent of the coefficients by which its powers are mul¬
tiplied, it follows (as has been observed when treating
of the method of indeterminate coefficients, Algebra,
§ 261.) that the coefficients of the different powers of
h must be each equal to o.
Therefore,
y*—a x y-\-b x*—c=o
2 p y—“{px+y) + 2 ^ *=°»
&c.
From the first of these equations we can infer nothing,
as it is no other than the proposed equation itself j but
from the second we find
a y—2 b x
p .
2y—a x
Now A, and kzzp A-f-y Ae-{- &c. being the simulta-
k
neous increments of x and y, we have ^zrp-|-yA-}-&c.
therefore, supposing A to be continually diminished, and
1=
y __av-
•ibx
x 2y—ax
thus we have obtained an expression for the ratio of the
fluxions of y and x} from which we find
2yy—a{xy-\-yx') -\-2bxxzzOy
and this is precisely the expression we should have ob¬
tained, had we taken the fluxion of each term ef
y2—a x y-\-b x*—c=o,
the proposed equation, and put the result equal to o.
49. But to see that this will always be the case,
whatever be the degree of the equation, we have only
to observe, that, by the very same process employed to
deduce from the original equation
y*—a xy-\-b x*—c~o,
these two others
2yp—a (xy-}-y)-j-2 A arrzo,
^y y—flOvy-f-y*)!"2 bxxzz.0 $
if we suppose the equation to be generally expressed
thus,
y^-}-fly,”*,,• • •+a;r-f-c=0,
where the exponents /, m, n, and r denote constant
quantities, we shall obtain
? I—1 • /• *
ly p+a(my
xnp+n y’
v-> 7
r0’
and hence, by substituting for y its value -4-, and bring-
x
ing x from the denominator,
l yl *y-\-a(mym 1 AWy-{- n ym xn 1 x)l
1 r—* * f ~0‘
From which it appears that, when the relation be¬
tween x, a variable quantity, and y, a function of that
quantity, is expressed by an equation, the terms of
which are brought all to one side, so as to produce an
expression mo 5 the relation of the fluxions will be
found, by taking the fluxion of each term of the equa*
tion {considering y as a function of x), and putting the
sum of these fluxions equal to o.
50. Having from the equation
y*-_a # y-f-A a;2—-emo,
found that
• (a y-—2 b x') x
y=— »
J 2 y—a x
if it be required to find the second fluxion of y, we
have only to take the fluxion of the latter side of this
equation,
«
Part I.
FLUXIONS.
si T'r:-™^;::e*a!“"eunt’miy»(■+<>)-i=A;+2B^+3c«>t;+4Dt,.i+&c.
— ' or leaving out the quantity v, common to each term,
J (2y—axXay—2bx)x 1 "C1+^)“~I=A + 2 B ^+3 C t;*+4 +&c.
y ■— ~ C^.y ^ ^ X J TjPt Lnttl Gf/Ioc Apfliia l 1. • 1*
lbx~) (ay—a*)
( 2ij—ax)*
1l$
TL'r.ct
M etisod.
Let both sides of this equation be multiplied by 14-f,
and divided by nt thus we shall have
an equation which abbreviates to
(46—a'Xxy—yx)x
V (2y—axy ’
and from which we may exterminate y by means of the
equation
Thus, by performing on the quantities the analytical
process of taking their fluxions, we have obtained a
new expression for (1 Let tf,e quantities that are
independent of v in each expression be put equal to
each other, and also the coefficients of like powers of
v ; thus we obtain
(ay—zbx') x
ly—ax
By the same mode of proceeding we may determine
the third or any higher fluxion of the function y.
51. As far as we have yet gone in explaining the
principles of fluxions, we have had continually occasion
to employ the rule for finding the fluxion of the parti¬
cular function where x denotes a variable quantity,
and n any constant number ; and we may therefore in
respect of other functions, consider xn as a simple func¬
tion. Besides the function at", writers on Analysis have
considered each of the following as also constituting a
simple analytic function of a variable quantity j viz.
a*, where a is constant, and x is variable.
Log. x, that is the logarithm of a?, a variable number.
Sin. #, that is the sine of at, a variable arch of a circle,
radius being unity.
Cos. #, that is the cosine of at, a variable arch of a
circle, radius being as before unity.
1 ~ —, and hence A—«
n
a = Ahi? B_ A
n 2
B-£g.+3c c-^2b
n 3
C_3C±4D
n 4
&c. &c.
Or, substituting successively the expression for each
coefficient in that which follows it,
A ~ n,
p —”P—1)
2 ’
c __ wP—OP—2)
2*3 »
52. We have already found the fluxion of a;", and we
proceed to find the fluxions of the other simple functions
of x ; and, as in the investigation of these we shall have
occasion to employ the binomial theorem, it will be
proper to show how that theorem may be deduced from
the principles already explained. We are then to find
the series that expresses p-j-A;)n, when ^ is any number
whatever. Or, since (a-J-Ar)’* is equal to o° (i+u)",
where v denotes the fraction -, we may leave the quan¬
tity o" out of consideration, as has been formerly ob¬
served, J 28, and seek the series that expresses (i-j-i;)".
As we have already pointed out (§ 28.) the process of
induction by which we may find the general form of the
series, we shall not here repeat it, but assume
(1 1 -j-At;-f-Bt>*-f-C
where A, B, C, D, &c. denote numbers that are inde¬
pendent of v.
Now, as the fluxion of a variable function must be
the same, whether that function be expressed by one
term, or developed into a series of terms j by performing
the operation of taking the fluxion on each side of the
above equation, the results must be equal, that is, § 26.
D= ”P—OP—2)p—3)
&c. 234
Hence it appears that
+,.,+ ^=L>.. 1 2)
2 2*3
1 ”P—OP—OP—3)
_r 2'3’ 4
t>4+ &c.
and therefore, substituting — - for v, and multiplying
by o",
(a-\-x'yi—anar-f- o"~g y*
2
. *P—OP—O „
H — « 5 &c.
2*3
where the law of continuation is evident.
53. We now proceed to investigate the fluxion of
the function u=a*, a being supposed constant, and m
the variable quantity, to which the function is referred.
4X2 Let
7,6 FLUX
Direct Let x be supposed, as formerly, to change its value,
Method, and to become x + h, and put u' for the new value that
v— the function acquires by this change in the magnitude
of x} then we have
ft
:a —a X« ,
IONS. Parti.
wise than by a general symbol. Therefore, we have Direct
now got
/= i + A /*+B /**+C-f- &c.
and consequently,
and, taking the difference between the two succeeding
values,
(AA+BA*+CA*+ &e.)
and
x h
u’—li=. a Xa ■
x x , h
-U =a {a ■
.1).
it’ 71
=A.ax-\-^axh-\-Cax 4*-{-&c.
. h • ,
We must now develope the expression a I into a
series, the terms of which are arranged according to the
successive powers of the increment h. T-O effect *-^s»
let us put b=a—I, so that a= I -f-6, and a —i
but by the binomial theorem, this last expression may
be expanded into the following series :
i jt.kb+hsi=iib'+h{’‘-'^ c*-2W*e.
Therefore,
+ &c.
As the terms of this series are not arranged accord¬
ing to the powers of 4, but according to the powers of £,
it is necessary that we transform it into another having
the required form ; now this may be effected by actually
multiplying together all the factors that constitute each
term, and arranging the series anew in such a manner,
that each of its terms may be a power of 4, multiplied
liy a coefficient composed only of the powers of 4, and
given numbers $
Accordingly we have
4 4 . • • — 4 4,
4(4—i) ^
2 2
^-0 (*-2) i3= H
o • o 2 ^2 0
&c.
2- 3
Therefore, by taking the sum of all the quantities on
each side of these equations, we get the series,
i +W+ b‘ + Kh~^ 4>+ &o-
otherwise expressed thus,
i4-A4-j-B4* + C4J-f- &c.
where A is equal to the infinite series 4-
Hence, when 4 is conceived to be continually di¬
minished, we have the limit of —^— expressed by An ,
and therefore, § 21,
~= Aa*, and u=Aaxx.
x
54. In the preceding investigation, we have had oc¬
casion to develope the exponential expression a into a
series of this form,
i+A4+B4*-f-C4*4- &c.
that is, a series the terms of which are the successive
powers of the exponent, each multiplied by a coefficient,
which is independent of the exponent.
We have however only determined the coefficients
of the first two terms of the series, these being the only
ones we had occasion to employ.
The result of the investigation however may be ap¬
plied to determine all the coefficients by the very same
kind of process as that which we have employed m
§ 53, to determine the coefficients of the terms of the
series which constitutes the other expansion of o .
Instead of denoting the exponent by 4, let us consi¬
der it as a variable quantity, and express it by *, then,
from what has been shewn, it appears that
0*=i _j_Aa;-{-B a:*-J-Ca;j4-Da;4+ &c.
where A, B, C, &c. express constant quantities. Let
the operation of taking the fluxions be now performed
on both sides of this equation, (observing that the flux¬
ion of
«'—K=sin. x >
but by the arithmetic of sines (see Algebra, \ 353), ^
sin. (A?4^)=:sln' x cos« ^4cos. x sin. h, therefore,
wrrsin. a? cos. /i4cos* x sin. h—sin. x
=cos. x sin. A—sin. a? (1—-cos. h).
In this case, as" when treating formerly of other
functions, we might consider the above expression for
u'—u, as resolvable into a series ph^-qh*-\- &c. pro¬
ceeding by the powers of the increment, and thence we
might
7iS
Method. m;ght the 1ImIt aS bef°re* But We may
discover the limit otherwise, by proceeding as follows :
Because
Sin.*/j=ri —cos.*/i=(i-f-cos. h) (i—cos. h)
sin.* h
therefore, I—cos. fizz ; 7- >
7 1 -^-cos. h
Let this value of 1—cos. h be substituted in the ex¬
pression for u'—iiy and it becomes
. sin. x sin.* h
wzzcos. x sin. h— ; j— j
i-j-cos. h
And hence, dividing by and arranging the terms
..... • s*n* ^
so as to exhibit the ratio —7—, we get
FLUXIONS.
Now, it has been just shewn that
u'—u
sin. h r
— —-— 4 cos. X-
h i
sin. h sin. x
l-}-cos
n. x~t
7XS
Conceive now h to be continually diminished, and we
. -tJ—u , .• •. »sin. A
shall have the limit of —^— equal to the limit ot —
multiplied by the limit of the following expression
cos. x-
sin. h sin. X
14-cos. h
Now, the sine of an arch being less than the arch
itself, we have *7^^1. Again, the arch being less
sin. h sin. h . . , , sin. h
than its tangent, —;—77^—7 J but tan. n—
and therefore
flux, of sin. (c—«)=5C0S. (c—x~) X^ux. of (c-—
but cos. (c—»)=:sin. x, and the fluxion of c—a; is —x,
therefore
—x sin. x.
Thus it appears, that the fluxion of the sine of a va¬
riable arch is equal to the fluxion of the arch multiplied
bij its cosine ; and that the fluxion of the cosine is equal
to the fluxion of the arch {taken with a negative sign)
multiplied by the sine.
60. We can now very readily find the fluxion of any
other function of an arch of a circle. Thus, suppose
, , sin. x ,
u=tan. x ; then, because tan. xzz , we have uzz
cos. x
—This expression being considered as a fractional
cos. x
function of x, we have, by § 39, and what has been just
now shewn
_x cos.* a; 4-# sin.* x
COS.* X *
Part I„
Direct
Method,
u~-
a?(cos.*a; 4-sin.* x)
cos.* X
or, since cos.a A?-|-sin.* xm, and
• x
h " tan. h7 " cos. h*
sin. h , sin. h
j-ncos. h ; consequently —7— ^
tan. h h
cos. h. Hence it appears, that the expression for the
ratJ0 .S1‘n — is less than I, or radius, but greater than
h
cos. h. But h being conceived to be continually dimi¬
nished, cos. A continually approaches to I, and may
come nearer to it than by any assignable difference $
therefore, the limit of ^— is I. As to the other ex-
h
sin. h sin. x , , . , ,
pression, cos. x 1 jpCos h 7 'vlien A18 suPPoset*to be
, j. . sin.^sin..v
continually diminished, its second term, to wit, ~_j_CQS j
may become less than any assignable quantity} there¬
fore the limit of the expression is simply cos. x : thus,
upon the whole we have found that the limit of —^—
is cos. x, and therefore
u • •
—r=rcos. x, and uzzx cos. x.
x
The fluxion of the other function, «=rcos. x^ is ea¬
sily deduced from that which we have just found, by
proceeding thus:
Put c to denote a quadrant, then pos. tfrrsin. (c—*),
and therefore
strain, (c—x).
♦ u ~ —
cos.' x
Hence also we have xzz-
cos. x
■— x sec.* x.
rr secant x,
u
sec." x 14-w*
In like manner, if we suppose u zz sec. x, then, be¬
cause sec. xzz-
, we have uzz—-—, and
cos. x
cos. x
• u sm. x
u— —,
COS. X
. sin. x , I
or, since == tan. x, and =r sec. x,
cos. x cos. x
u zzx tan. x sec. x.
Proceeding in this manner, we find that when *cr
cotan. x, then
tan .* a? cos.* * sin. * * 7
And when uzz cosec. at, then
-zz—x cotan. x cosec. x.
61. Let us now consider the fluxions of geometrical pjBle
magnitudes: And first let it be required to find the gcXlXi
expression for the fluxion of BDPC the area bounded fig, i.
by CP, a curve line, and by CB, PD, the ordinates at
its extremities, and BD, the portion of AE, the line of
the abscissas, which lies between those ordinates. Let
the numerical measures of AD and PD, the co-ordi¬
nates at the point, be denoted by a: and y, and
the numerical measure of the are BDPC by s;
'art I.
Direct
lethod.
then y and / may both be considered as functions of the
abscissa x.
Let x, or AD, be supposed to change its value, and
to become AD', and let D'P', and BD'P'C be the cor¬
responding new values of y and s; then DD', and
DD'P'P will be the geometrical expressions for the si¬
multaneous increments of the abscissa and area. But,
as one of these quantities is a line, and the other a
space, they cannot be compared in respect of their ratio.
Therefore, let us consider a as denoting a line whose
numerical value is unity, and then the numerical va¬
lues of the increments of the abscissa and area may be
considered as analogous to the geometrical quantities
and the area DD'P'P respectively, which
quantities being homogeneous may now be compared
with each other. We are now to investigate the limit
. area DD'P'P
0* —JTx DD'—’ 116 &eneral expression for the ratio
of the increments of s and x. Draw PM and P'N pa¬
rallel to AE, meeting the ordinates in M and N. The
curvilineal area DD'P'P is greater than the rectangle
DD'MP, that is, greater than PD X DD'; but less than
the rectangle DD'P'N, that is, less than P'D'xDD',
therefore
FLUXIONS.
719
63. We proceed now to find the fluxion of an arch Direct
of a curve. Let APP' be a curve line of any kind, Method,
and AB, BP any two co-ordinates at a point P in the
curve. Put x for AB, the abscissa, y for BP, the or-^*®'
dinate, and a for the curve line AP, then z and y may
be considered as each a function of x. Draw P'B'ano¬
ther ordinate, and draw PM parallel to AB, meeting
P'P' in M, and draw the chord PP'; then PM, MP',
and the arch PP', are the simultaneous increments of x,
y, and z respectively. Now we have *
arch PP'_ arch PP' _ chord PP'
PM ~ chord PP' X PM
But chord PF= V(PM>+MP'*)=PMv/(i+^-^ ;
therefore,
arch PP' arch PP' _ MP7*
PM
chord
PP' „ . MF*\
PpX vXi+ pjvpj-
Suppose now the increments to be continually diminish-
*2 arch PP' 7v2
ed, then, as — = limit of —, and ~ = limit of
X x
and
area DD'P'P
a X DD'
area DD'P'P
aX DD'
PDxDD' PD
aXDD' ^
P'D x DD' ^ P'D'
axDD' ^ a
But the increments being supposed to be continually
. . PD . . P'D' PD 7
diminished, is the limit of , therefore is
a a a
1 f r area DD'P'P
also the limit of —» an“ hence fS 21D
aXDD' J
x a x an s—yx.
That is, the fluxion of a curvilineal area is equal to
the product of the ordinate, and the fiuxion of the ab¬
scissa.
62. Before we proceed to investigate the expression
for the fluxion of an arch of a curve, it is necessary that
we should inquire what is the limited ratio of an arch
of a curve to its chord.
Let APB be any curve line, all the parts of which
are concave towards its chord AP. Let AQ, QP be
tangents at the extremities of the arch, and let apq be
a triangle similar to APQ, but having its base of a
given magnitude, then
AQ-f-QP : AP :: aq-\-qp : ap.
Suppose now the point P to approach to A, then the
angles at A and P, and consequently the angles at a
and p, which are equal to them, will decrease, and
may become less than any assignable angles ; therefore,
the limit of the ratio of aq-\-qp to ap is evidently a
ratio of equality ; hence also the limit of the ratio of
AQ-f-QP to AP is the ratio of equality; and since the
arch AP is less than AQ-j-QP, but greater than its
chord AP, the limit of the arch AP to its chord AP
must also be the ratio of equality.
MP'* arch PP'
21.), and 1—limit of (last §) we have
x x1'
Hence it appears that the square of the fluxion of a
curve line of any kind is equal to the sum of the squares
of the fluxions of the co-ordinates.
64. The expression for the fluxion of a solid may be
found by the same mode of reasoning as that which we
have employed, § 61, to find the fluxion of a curvilt-
neal area. Let APQ/; be a portion of a solid generat-^
ed by the revolution of APB, a curve line, abont AC, a
line taken in the plane of the curve, as an axis. Let
PD/;, P'D7/;' be the lines in which BA b, a plane pas¬
sing along the axis AC, meets PQ/), P'Q'/)', the planes
of two circles formed by sections of the solid perpendi¬
cular to its axis. Draw PM and P'N parallel to AD.
Put AD=«, DP=y, let s denote the solid APQ/),
having y for the radius of its circular base, and x for
its altitude ; put w for the number 3•I4I59••• viz. the
circumference of a circle having its diameter zr I, and
let a denote an area, having its numerical measure ex¬
pressed by unity; then DD', or «xDD' being con¬
sidered as the increment of x, the portion of the solid
comprehended between the parallel planes PQ/), P'Q'/)'
will be the corresponding increment of s, which we are
•
to consider as a function of x; hence (§ 21.) is
x
equal to the limiting ratio of the portion of the solid,
comprehended between the planes PQ p, and P'Q' pi
to the solid axDD'. But the former of these solids
being evidently greater than a cylinder Pot, having
the circle PA/) for its base, and DD' for its altitude,
that is greater than wPD*xDD', and less than a cy¬
linder N/)', having the circle P'Q'/)' for its base, and
DD' for its altitude, that is less than w P'D'* X DD';
it:
720
Direct
Method.
F‘g< 5-
flux:
it follows, that as long as DD' has an assignable mag-
nitude,
4-^xPD* x DO' X -xId5T»
,rPD'*
and -4-^.?rP'D,a X DD' X aX j)i)/'»
ttP'D'2
^ a 5
but the increment DD' being continually diminished,
"P-, the greater limit of—, approaches continual-
a x
ttPD* w V*
ly to its lesser limit = —= (because g= i) ?ry*,
so as to come nearer to it than by any assignable differ¬
ence, therefore4-=7r/, and 's—Tcijx K. Now, if we ob-
x
serve that wy* is the area of the circle PQp, it will
appear, that the fluxion of a solid generated by the revo¬
lution of a curve about its axis is equal to the fluxion of
the axis multiplied by the general expression for the area
of a circle formed by supposing the solid to be cut by a
plane perpendicular to its axis.
65. To find the fluxion of the surface of the solid,
let us denote that surface by s, and let x and y denote
as before $ then the surface contained between the cir¬
cles PQjj and P'Qy will be the increment of s, cor¬
responding to DD' the increment of x. Draw the
chord PP' j then the curve line PP' being supposed to
revolve about the axis AC, and thus to generate the
increment of the surface of the solid, the chord PP'
will generate at the same time the convex surface of a
frustum of a cone $ now the limiting ratio of the curve
line PP' to its chord PP' being the ratio of equality,
the limiting ratio of the surfaces generated by the re¬
volution of those lines will also be the ratio of equality $
"B • •
therefore -r» which is equal to the limit of
x
surf, gener. by arch PP'
ixDD'
will also be equal to the limit of
surf, gener. by chord PP'
but the convex surface of a frustum of a cone is equal
to the product of its slant side into half the sum of the
circumference of its two bases (see Geometry), and in
the present case these circumstances are equal to 2 PD
Xw, and 2P'D'Xw, therefore i*- is equal to the limit
x
O N S. Part 11
limit of this expression (if we consider that P'M and
DD' are the simultaneous increments of 3/ and«) is evi¬
dently equalto C1 + |r)» therefore
4 = 27ry/(**+?*).
If we now observe that 2Try is the circumference of
the circle PQp, and f (**-{-y*) is the fluxion of the
curve line AP, § 63, it will appear, that the fluxion of
the surface of a solid generated by the revolution of a
curve about its axis is equal to the fluxion of the curve
line multiplied by the general expression for the circum¬
ference of a circle formed by supposing the curve to be
cut by a plane perpendicular to its axis.
SECT. III. The Application of the Direct Method »f
Fluxions.
Having explained the principles of the direct me¬
thod of fluxions at as great a length as we think suit¬
able to the work of which this treatise forms a part, we
proceed to shew how the calculus may be applied to
the resolution of some general problems in Analysis and
Geometry.
Investigation of a general formula for expanding a
Function into a Series.
66. In treating of the principles of the method of
fluxions, we have, from an examination of particular
functions, inferred by induction, that u being any func¬
tion of a variable quantity x, which was either actually
expressed, or capable of being expressed by a combina¬
tion of the powers of a?, then, x being supposed to change
its value, and to become x+h, the new value which
the function u will acquire when x-\-h is substituted in
it instead of x will always be capable of being expand¬
ed into a series of this form,
u -j-j9 h-\-qh%^-rh?Jf &c.
where/), q, &c. denote functions of x that are quite in¬
dependent of h.
We have shewn that, from the particular form of
this developement, it happens that jhe ratio of/)/*-f-
q A2-f-r + &c. the increment of the function, to h the
increment of the variable quantity x itself, admits of a
limit, which is always expressed by/), the coefficient of
its second term $ and as we have defined this limit to
be the expression for the ratio of the fluxions of u and tf,
so that/)=:—, the new value of the function may also
of
*r(PD-f-P'D') PP'
DD»
=w(PD+P'D')
PP'
DD' *
but the point D' being supposed to approach to D, the
x
be expressed thus,
»
3
And
!lart I.
Direct
Method.
FLUX
Anti this expression may he considered as indicating not
only the general form of the series, but also the particu¬
lar relation subsisting between «, the original function,
and p, the coefficient of the second term of the series,
the latter being in every case that function of x which
results from the operation of taking the fluxion of the
former, and dividing by x
We are now to investigate the relation that subsists
between each of the remaining coefficients and the ori¬
ginal function.
67. First let us suppose the function u to have the
particular form xnt n being a constant number. Then
x changing its value to x-\-h, u changes to
therefore, by the binomial theorem (§ 52.)
u'—xK-\-nxn~'1 h-ir-7^——— at”-* h*
2
n(n——2) , .
-f A h* -f See.
But since u—xn, by taking the successive fluxions of
and considering x as constant, we have
u
X
t- = —1) («—2) at*-5,
JV*
u
— = n(«—1) (»—2) (n—3)
X*
&c.
Let «, -r- &c. be now substituted for nx*-1,
X X*
n{n—1) a"-*, &c. respectively, in the series for
and we have
u . , u 7i% u h3 u A4
—A-f-v- f.-r-
X X* 2 x3 2.3 x* 2,3.4
+ &c.
68. This manner of expressing the developement of
uf, or (Af-f-A)B, indicates directly the relation that each
of the coefficients of the successive powers of A has to
the original function.
The first term of the series is the original function
w, or xn, itself, or it is what the function (a;-f-A.)" be¬
comes upon the supposition that A=o. The second
h ti
term is A, or —, multiplied by the coefficient -r-, which
1 x
coefficient is a function of x derived from the original
function by the operation of taking its fluxion, and di-
• , . .A*
viding the result by x. The third term is mul¬
tiplied by the coefficient—, that is, by a function of x
x*
derived from the preceding coefficient ~ by the same
Vol. VIII. Part II. f
IONS.
operation as that coefficient was derived from the
original function, namely by taking the fluxion of
x
considering x as constant, and dividing by x. The
fourth term is —multiplied by that is, br a
1.2.3 xl J
function of .r deduced from the third coefficient by the
very same operation as that by which the third was de¬
rived from the second, or the second from the first.
And so on with respect to all the other terms of the
senes, the «th term being the product of
1.2.3...O—1)’
and the (?z—l)th fluxion of the function u divided
by a,'"'-1.
i
69. Let us now suppose that u denotes any other
function of x, then, whatever be its nature, it may al¬
ways be conceived as capable of being expressed by a
series, the terms of which are powers of x, in tlm
manner 5
A^+B^+C/+D^-f &c.
where A, B, C, &c. a, A, c, &c. denote constant num¬
bers. Thus we have
KzrAa’^-J-Ba’^-l-Ca^ -}- &c.
Then, x being supposed to become ^-f-A, and (in con¬
sequence of the change in the value of x) u to become
we have
«'=A(*+Ay7-fB (x+7i)b+C(x+hy+ &c.
Let us now denote Axa by P, Ba^ by Q, C** by
R, &c. then by last §
A(x+A)‘=P+?A+i.- +? - + &c.
X X* 2 X° 2.3
B ^+A)>=Q+$.4+3—+ ^- + &c.
X X* 2 x3 2.3
C (*+A)'=B+^+^.—- + &c.
* X* 2 X3 2.3
&C,
Therefore, substituting these developements in the
series expressing
P+Q+R+Scc
p Q »
+ (-r + -+“ +&c.)A
XXX
+4+|+ft+&c0£ ,
Xa X* X* 2
-f -r-- + &C.) —
X3 X3 x3 2.3
-f- &c.
4 T But
722
Direct
Method.
FLUXIONS;
But
aA= I + AA + — /** + —7*3 + &c.
*=P+Q+R+ &c.
4-Z.+2-+5-+ &c.
X X x X
2-3
Parti
Direct
Method,
X2 X
&c.
X' X2
or, exchanging h for x,
. A3.r* A3 ^3
o =i + A.r + H — + &c.
the same result as we formerly obtained in § 54.
Ex. 2. Suppose «=log. x. Then, x becoming
x-\-hy u becomes w'zrlog. Now from the
Therefore, substituting t/, 4, See. for the series to c j au x a “
’ & X equation «=log. .r, we find (by J 57.) -r = —.
x x
which they are respectively equal, Here M denotes the modulus of the system. Again, sup-
hx
h>
u H un-tun~ % . c j t. r /r w ^ 7
+— — + — ^ posing x constant, we find by $ 26, r-=
7* 1 xx 1,2 •■^•3 ^ a
M n
before the values of u’ », -4-, &c. in the general fo^-
x
u u
mula «'= « 7i + — + &c. it becomes
Hence it appears that u being any function of x 2M " 2-3^ &c. Therefore, substituting as
whatever, if x+h be substituted in that function in- .x3 » w— tX4
stead of x, the series expressing the developement of
this new value of the function will have the general
properties which have been shewn, in last §, to belong
to it in the case of the function having the particular
value xn.
The very general theorem which we have just now
investigated is one of the most elegant and important in
analysis. It was first published by Dr Brooke Taylor
in a work entitled Methodus Incrementorum, which
made its appearance about the year 17x6. The theo¬
rem itself is generally known by the name of Taylor's
theorem. It is more general than the celebrated Bino¬
mial theorem, inasmuch as this last, and innumerable
others, are comprehended in it as particular cases.
M
log. (*+A)=:log. x-\- — h—
If we suppose xssi, and change h into y, we have,
because log. .r=log. 1=0,
log. (1+y) =M (y— f*+y — &c.)
Tor the particular method of applying these two
series to the calculation of logarithms, see ALGEBRA,
§ 285 to § 291. See also Logarithms.
Ex. 3. Suppose now tt=sin. x. TLhen r^rrsin.
(a -f-A). From «=sin. x, by the application of the
u u . w
rule in $ CQ, we deduce ■ . — cos. a, -7- = — sin. a, -r-
5 X X* X3
M
70. We shall now give some examples to shew the
manner of applying Taylor’s theorem, as well as its great
utility as an instrument of analysis.
Example 1. Suppose —uaxt a being constant and a
variable. Then a becoming a-f-^., u becomes u'zzax+h‘
Now from the equation u-=zax we derive (§ 56.)
^r—A.ax1 (here A denotes Again, consider¬
ing a as constant, and repeating the operation of tak-
U Ax 1 u
■ = A ax, we get
x x
— cos. x, — sin. x. &c. Therefore, substituting
— . ..
. « * , » a . - j, for a' . &c. their values in the general formula
ingthe fluxion — = A ox, we get -r- =: A* o , and hence lor «,«/, , occ. 5
0 .7! a*
again -^-=A 3 a*, &c. Therefore, substituting for
a
u'f u, ^r, 4—* &.c« their values in the general theorem
a a
1- &c. it becomes
a x* 2
as before, we have
7. . A . A*
sin. (a-f-A)=sin. a-j-cos. a — sin. a
h3 . h* > oT
—cos. a f- sin. a -- + oie.
1.2.3 i.2*3*4
or sin. (a+A) is equal to
sin. a (1-
Hx+h zza* (1 Kh + ■“ h* + ^ + &c.)
Suppose now, that aso, then, as in this case a*=:i,
we have
-j-cos. a (4—
2 i.2.3«4
h3 h*
■ &c.)
1.2.3 I-2-3-4-5
&c.)
If we suppose azro, then, as in that case sin. a—0,
the preceding formula becomes
sin.
art I.
Direct
Method.
FLUX
sin. h—h-
As
A*
1.2.3 1.2.3.4.5
or, substituting x instead of A,
■&C.
sin. xz=x-
*s
'&c.
1.2.3 I.2.3.4.J
Ex. 4. Suppose «=cos. x, then u'zzcos. (.r+A),
1 • 1. X w .11
and since tf=cos. x, by $ 59, — = — sm. ftr, —= —.
cos. —= sin. at, &c. Therefore, substituting as be¬
fore these values in the general formula —A-J-
x
&c. it becomes
r \ .A A*
cos. rAf-j-/i)“cos, Af—sm. x cos. a;
I 1.2
-f-sin. x
or cos. (a?-}-A) is equal to
A*
A3
cos. X (-
1.2.3
A*
&c.
sin. a? (A-
1 . 2 1 1 . 2.3 . 4
A3 A?
1 . 2.3 1 . 2 . 3.4.5'
&c.)
■ &c.)
which expression, when x~o, and therefore cos. Afzri,
sin. Ar=o, becomes simply
cos. A~l-
A*
A4
1 . 2 ’ 1 . 2.3.4
or substituting at for A,
at2 a;4
1 &c
cos. «=:l-
I . 2 1 I . 2 . 3.4
&c.
71. It may be remarked that in each of these ex¬
amples, from the developement of uf the new value of
the function of w, ve have been able to deduce a deve¬
lopement of u the function itself. But it is easy to see,
that by proceeding in the same manner with the gene¬
ral formula as we have done in these particular exam¬
ples, we shall obtain a general expression for the deve¬
lopement of any function whatever.
The general formula is
A3
u h u A*
« = «4- + -
^ I . 2^at3 I • 2. 3
f &c.
IONS.
Now, v! being the value that u assumes when A?-fA
is substituted in it instead of a?, if we suppose x—O, then
uf becomes the very same function of A, that u is of x.
.Let us denote the values which each of the functions
723
Direct
Method.
u u o
u, -7-, -V-, &c. acquire, when A?=o, by U, IP, U", &c.
x a;3
respectively.
I hen (f) u' (considered as the same function of A
that u is of at) is equal to
^—{-IP"—— f- &c.
1 1.2 1.2.3
Let x be now supposed to be substituted both in u\
and the series which is its developement instead of A,
then u' becomes «, and we have
«=U+U'- +U"— O-U'"—— U See.
I 1.2* 1.2.3“
and in this formula it is to be considered, as already
stated, that U, IP, IP', &c. denote the particular values
which the functions u, t-, —, —, &c. acquire respec-
X X* X3
tively, by supposing that in each of them x is taken
=0.
72. As an example of the application of this series
let us resume the equation u— —=A2 or, ~=A3 a*, &c.
x x3
Suppose now that A;=ro, then u, or a* becomes a°—xf
^-=Aax becomes A, A* a* becomes A*, &c. so
x x*
that Um, U'zrA, \J"=A2, &c. substituting therefore
these values in the general formula, it becomes
o*=i + A--fA2 —+ A3—+&c.
I I. 2* I.2.3r
Let us next suppose that u is an arch of a circle of
which the sine is a?4(radius being unity), then A:=:sin. u.
Now the ratio of the fluxion of u to the fluxion of x
will be the very same whether we consider « as a
function of x, or A? as a function of «; therefore (§ 59.)
xzzu cos. v, and 4-= » but since sin. cos.
x cos. u
—a?*), therefore,
u 1
T” ■v/(I—-A?*)'
Taking
(f) For the sake of illustration let us take a particular example. Suppose «=(o+a?)’‘, then-T-=»(i7-f .r)”-1,
a*
~—n(n O (a+a-)’1-1, &c. Suppose now that ^=0, then u becomes on, becomes wo"”1,becomes
^2 v v x or
n(n—1) a"”2, &c. so that in this particular case we have U=2an, U,=»o’,“I, U^rrn(«— 1)a"-2, &c.
4 Y 2
Fi*. 6.
Takins now the fluxion of
ion of the result, &.c. we have
\/(i—»*) ’
FLUX
and the flux*
Ci—
3**
*3 (i—(i—w1)!
3-3x
3-Sxi
x*
&c.
(I—^)t
Suppose now that a;=:0, then u becomes O, -^—becomes
X
— becomes o, —
x* x3
becomes x, — becomes O, &c. so
that U=o, U'—i, U"=o, U'"=:x, U""=0, &c.
Therefore, substituting in the general formula, we find
* — -f- &c.
u~x-\-
1.2.'
By prosecuting the computations farther, we may find
i 3*xs 3*5*^7
u—x-
2.3 ^ 2-3-4-i 2-3-4-5-6-7
+ &c.
Part I,]
Direct
Method.
Application of the Method of Fluxions to the Drawing
of Tangents.
73. The theory of tangents to curve lines furnishes a
good illustration of the truth of the principle which we
have considered as the foundation of the method of
fluxions, namely, that whatever be the form of a func¬
tion, the ratio of its increment to the increment of the
variable quantity from which the function is formed, is
in every case susceptilde of a limit.
Let AB, the abscissa of a curve, be the geometrical
expression of a variable quantity and let BP the cor¬
responding ordinate, be the expression for y, any func¬
tion of .r; then the curve line itself is the locus of the
equation expressing the relation between x and y. Let
PT, a tangent to the curve at P, meet AB the abscis¬
sa in T *, through P draw any straight line meeting the
abscissa in D, and the curve in p; draw the ordinate
p b, and from P draw P n parallel to the abscissa, meet¬
ing the ordinate bp in n.
The triangles DBP, Y np are similar j therefore
pn\nY :: PB : BD.
Now p n, and nY, or B b, are the increments of PB
and BA, or of y and x respectively, therefore the ratio
of the simultaneous increments of PB and BA, or y
and x, whatever be their magnitudes, is equal to the
ratio of PB to BD. Conceive now the point p to ap¬
proach continually to P, then the angle contained by
the straight line PD, and the tangent PT, will de¬
crease, and the point D will approach to T; at the
same time n p, and n P, the increments of y and x, will
be continually diminished} still, however, they will have
IONS.
to each other the ratio of PB to BD, but this ratio ap¬
proaches continually to the ratio of PB to BT, and be¬
comes at last more nearly equal to it than any assign-^
able ratio } therefore the ratio of PB to BT is the limit
of the ratio of PB to BD, and consequently is also the
limit of the ratio of pn, the increment of y, to nY, the
increment of x. And as this conclusion does not de¬
pend upon the particular nature of the curve, or upon
any particular relation supposed to subsist between x
and y, we may conclude, that whatever be the form of
the function, the ratio of the simultaneous increments of
the function, and the variable quantity from which it is
formed, has a limit to which it approaches when the in¬
crements are conceived to be continually diminished.
It is now easy to see how the method of fluxions may
be applied to the determination of tangents to curves,
for since the ratio of the ordinate PB to the subtangent
BT is always the limiting ratio of the increments of the
ordinate and abscissa, it is equal to the ratio of their
fluxions, that is
y : x :: y : subtan. BT.
Hence in any curve whatever, referred to an axis, the
subtangent, (that is, the segment of the abscissa between
the ordinate and tangent) is equal to -r- y where x
V
denotes the abscissa, and y the ordinate at the point of
contact; and the subtangent being found, the position
of the tangent is thereby determined.
Let us apply the above general formula to some ex¬
amples.
Example 1. Let the proposed curve be a circle. It Fig. 7.
is required to determine the position of PT, a tangent
at any point P in its circumference.
Put 2a for AE the diameter, also x for AB the ab¬
scissa, and y for BP the ordinate at the point of con¬
tact.
From the nature of the curve, we have
ABxBErrBP*, that is
w(2a—.r)=ry2.
Hence taking the relations of the fluxions of x and y,
we have
2a x-—2xx-=z2yy,
therefore — =: —-—»
y a x
and BT
y "—a
from which it appears that BT the subtangent is a
third, proportional to a—x and y, that is, to CB the
distance of the ordinate from the centre, and BP the
ordinate, agreeing with what is known from the ele¬
ments of geometry.
Ex. 2. Let the curve be a parabola, required the j'jg. 1.
same as before.
Put x for AB, the abscissa, and y for BP the ordi¬
nate at P the point of contact} also a for the parame¬
ter } then, from the nature of the curve
PB2=axAB, that is
axz=f
therefore,
■’art I.
Direct
Method.
ig- 9-
S-to.
FLUXIONS.
therefore, taking the fluxions, we get axzzZyy, and
4=^, and
y a
g-p __ jv 2 y* lax ^
CT=a+x—~a*+x'
725
a-\-x a-j-»
therefore CB : CA :: CA : CT.
Direct
Method.
from which it appears that the sub-tangent BT is double
the abscissa BA.
Ex. 3. Let the curve be an ellipse.
Put AB=#, BP=y, AC the semi-transverse axis — causes
x
also to vanish, without at the same time making
x*
•!— to disappear, then we have
x3
y h*
y h3
ty-y—
y h3
y=y+-
X3 2.3 X* 2.3.4
y *
X3 2.3 X4 2.3.4
■ Sec.
-f- &c.
y
and as by giving a proper value to h, 4 may be
x3 2.3
rendered greater than the sum of all the following terms
in each series, it follows, that -L being supposed to be
x3
any quantity either positive or negative, because of its
sign being different in the two values, the one of them
will be greater, and the other less than y, the maximum
or minimum value, which result is inconsistent with the
V
nature of a maximum or minimum. If however 4- be
x3
assumed =0, then
y—
(0) If this should not appear sufficiently obvious, let
A £+B Aa-f C h3+T> h*+ &c.
be such a series, where A, B, C, D, &c. denote quantities either positive or negative, but which are independ¬
ent of A. Then, writing the series thus,
A(A-j-B A+C A*+D A»-f &c.)
it is obvious that if h be conceived to be continually diminished, and at last to become =0, the part
BA+CA*-J-DA3+&c.
will also become =0, therefore before it vanishes it will be less than A, or any other assignable quantity, there
fore B A*+C A4-f &c. may become less than A h.
?»rt I.
FLUXIONS.
Direct
Method.
A*
, y h*
V=y+ v~ ■—
2-3-4
+ &c.
Sj- We shall conclude this theory by applying it to
an example. Let y be such a function of x that
then by § 49,
y'—Zmxy+x'—a'zzo,
(y—m x)y—(my—Ar)«f=o,
and hence —■ — —-
y—mx
=0,
therefore my—xzzo, and y~
To find the value of let this value of y be substituted
in the original equation, it thus becomes
- _**—oW0,
hence we find
m a
and ^ 1/(1—m*)'
We must now examine what is the nature of the ex¬
pression for 4-^. Taking the fluxion of the equation
y _ m—yx ,
r fix' an^ cons*^er^ng & is constant, we have
x y—mx
y_ — C1—m*) (.xy—yx)
** (y—mxy ’
therefore, dividing by *,
v _ I y 7
hut as in the present case 4- =0, and yrr—, this ex-
x x
y_ __ —m
~ (l_w*)’
>/(!—m*)’
here again, the coefficient of the second term having in
both values the same sign, the conditions of the maxi-
y __
O-v/C1—w*2)’
mam or minimum are fulfilled, and the sign of shews
A4
when the one or the other is to have place.
It must now be sufficiently evident, without proceed¬
ing any further, that a function can only admit of a
maximum or a minimum when the frst of its fluxions
that does not vanish, is of an even order (or is its second
or fourth fluxion, <^c.), and that the sign of that fluxion
is negative in the case of a maximum, but positive in the
case of a minimum.
Of the values of fractions, the numerators and deno¬
minators of which vanish at the same time.
86. There are some fractional functions of such a
nature, that by giving a particular value to the variable
quantity, both the numerator and denominator of the
fraction vanish, and thus the fraction is reduced to this
^°rm o’ an exPress*on fr°m which nothing can be con¬
cluded. We have an example of this in the fraction
which, by supposing x—a becomes— 2 •
x —“ or—o* o
we must not however conclude that the fraction has no
determinate value in this particular case, for if we con¬
sider that its numerator and denominator have a com¬
mon divisor, viz. x—a, it is evident that by taking this
x—a
(x—a){x+af
becomes —-—, an expression, which in the case of x~a
divisor out of both, the fraction-^—
x-\-a
is equal to 1
2a
87. In general, if we make x=za in an expression of
this form-— it becomes however its true va-
—a) o
lue is either nothing, or finite, or infinite, according as
mr^n, or m—n, or m^Ln ; for by taking out the fac¬
tors common to the numerator and denominator, the
fraction becomes
Q
• Jl
■ in the first case, — in the
second, and
3 in the third j here we sup-
Q(at—a)"
pose that P and Q are such functions as neither become
nothing, nor infinite, by the supposition of x—a.
88. Therefore, when by giving a particular value to
x a function of that quantity assumes the form-,todis
o
cover the true value of the function in this particular
case, we must disengage the factors which are common
to the numerator and denominator. This may be done
in most cases by finding their common measure (Alge¬
bra, § 49.) but the direct method of fluxions furnishes
us with another method.
In the expression P (#—a), where P denotes any
function of x that is independent of x—a, if we suppose
x—a, then the expression vanishes ; the fluxion however
of the expression, viz. (x—a), P -f- P a?, is a quantity
which does not vanish when x=za, but is then reduced
to its last term, that is to P x.
4Z
729
Direct
Method.
which equation, by putting instead of x its value
m a
-, becomes
and as this result is negative, we conclude that the va¬
lue which we have found for y is a maximum.
pression becomes simply
Vox.. VIII. Part II,
Again,
73°
Direct
Method.
FLUXIONS.
Part I.
Again, the function P (tf—fl)3 vanishes by supposing
x—a, but if we take its fluxion, viz. (,r—a)2 V + 2 («
p and again the fluxion of this quantity, we
get
(*—«)2P + 4 (x—a) P x + i . 2 P ^e,
an expression which does not vanish upon the hypothesis
0f .r—but is reduced to its last term, viz. 1.2P#2.
By proceeding in tills manner, it is easy to see that by
taking the fluxion of a function of the form P (x «)
m times successively (m being a whole number) we shall
finally obtain an expression, all the terms of which, ex¬
cept the last, vanish by supposing that x=a ; and that
the last term will be I . 2.3 • • • P#'1', an expression
free from the factor (x—a)m, and involving only the
function P.
denominator is 2x x, neither of which quantities vanish Direct
when x=zl, therefore, in this particular case, the value ,
. . 3 #2 3
of the fraction is
2 X
Ex. 2. Suppose the fractipn to be
which vanishes when x~c.
ax2—2 ac x-\-ac*
bx2—2 be A’-J-^c2’
By taking the fluxions of the numerator and denomi¬
nator we obtain
2 a xx—2acx a ,v—ac
, a fraction,
2 b xx—2bcx k x ^ c
the numerator and denominator ol which still vanish
upon the hypothesis oi x—c, we therefore take the flux¬
ions a second time, and get r — r for fhe value of
2 bx2 b
the proposed fraction in the particular case of x—c.
89. It is not necessary that we should know the num¬
ber n, nor that we should exhibit the factor (a?—a)\
in order to determine when the expression P (x—-a)'\ is
freed from that factor. We have only to ascertain alter
each operation of taking the fluxion, whether the result
vanishes or not, when we substitute a instead of x ; for
in the last case the operation is finished, and the result
is the quantity 1 . 2.3 . . P a?”1. Suppose for example
the function to be x3—ax2-\-a2x-ya3, which vanishes
when x=a, its first fluxion also vanishes when x—a, but
not its second fluxion, which is (6 x—2 fl)A;2, hence we
, may conclude that the function has the form P (at—a)*,
which is besides obvious, because
a?3—ax2—n* a?-{-«3—(Ar-f-a) (x—a)2.
90. In applying these observations to the fraction
p (AT—a)_ it rs> t]iat by repeating the operation
Q (a#—a)n
of taking the fluxions of its numerator and denominator,
they will he freed at once from the factor x—a, if m—n.
If a result, which does not vanish, be obtained first from
the numerator, then we may be assured, tliat the factor,
a) is found in the numerator raised to a less power
than in the denominator, and in this case the fraction is
infinite when x—a. If on the contrary the first result
that does not vanish is found from the denominator,
then the numerator contains a higher power of (a?—a)
than the denominator, and in this case, when X—a, the
fraction vanishes.
The rule for finding the value of a function which
becomes - by giving a particular value to #, may there¬
fore be expressed thus. Take the successive fluxions of
both the numerator and denominator until a result
which does not vanish be obtained from either the one
sr the other, or from both at the same time ; in the first
case the function is infinite, in the second it is equal to
p, and in the last case its value is finite.
91. We proceed to illustrate this rule by a few ex¬
amples.
#3_i
Ex. 1. The value of the function — is required
at—1
when a:=:i.
The fluxion of the numerator is 3** x, and that of the
Ex. 3. Suppose the fraction to be
a?3—ax2—a2x -j- a*
which vanishes when a=«. In this example, by taking
the fluxions of the numerator and denominator once,
we get
3A?2i— 2a x x—a2x _ 3 A'2—2 ax—a2
an expression, of which only the numerator vanishes
upon the supposition of x—a; hence we may conclude
the true value of the fraction in this case to be O.
The contrary happens in the fraction
a x*—x
a4—2 a3 x —[- 2 a x}—a4 ’
we may therefore conclude that when xzza, this last
fraction becomes infinite.
92. The rule § 90. can only be applied when the
factors common to the numerator and denominator are
integer powers ol x—a, for as by taking the fluxions,
the index of (a-—a)m is diminished by an unit at each
operation ; when m is a fraction we shall at last arrive
at a result containing negative powers of x—a,, which
therefore, when x—a, will become infinite. rIlie fol¬
lowing mode of proceeding will however apply to all
cases Avhatever.
V
Let ~ be a fraction of which the numerator and
X'
enominator both vanish when xr=a; by substituting
1 it c-J-A instead of x, the functionsX and X; may be
A h* + B// + &C. A' A* + B7/ + &c.
which are ascending, that is, having the exponents of
the powers positive and increasing} because the series
must become o, upon the hypothesis that h—O. W e
have therefore
AA*+B/^+&c.
A'r+B7/ + &C..
instead of the proposed fraction.
Now
art I.
Direct
itethod.
F L U X I
Now If «s a', by dividing the numerator and de¬
nominator of this expression by the factor, /t“, which
is common to all the terms of each, it becomes
A h*-*' + B &c.
A'+ B-f&c.
O N S.
Let a thread be fastened to it at H, and made to paos
along the curve, so as to coincide with it in its whole
extent from H to F. Let the thread be now unlapped
or evolved from the curve, then its extremity F will de¬
scribe another curve line FAPP'. The curve HCF is
called the EvOLUTE of the curve FAP; and the curve
FAP is called the Involute of the curve HCF.
/
31
Diiect
Method.
£• IS.
a quantity which, by supposing //—Q, is reduced to
-2-, that is to 0. If again the expression for the
A
fraction, after dividing the numerator and denominator
by /<*, is
A + BAg-*'-f.Scc.
A'+BA/s'~a'-f&c. ’
A
which, by supposing h to be =0, becomes simply —, a
finite quantity. If, however, at «sil then the expres¬
sion for the fraction is
A+ B//-”+&c.
Ar'-a+B//~a -f&c. ’
which, when h~0, becomes —, an expression which
may be considered as infinite. Thus it appears that in
each case the true value of the fraction depends only on
A and A', the first terms of the series.
The following rule is applicable to every function
that can appear under the indeterminate form
Find the first term of each of the ascending series
which express the developements of the numerator and
denominator when a + h is substituted in them instead
of x. Reduce the new function formed of these first
terms to its most simple form, and make b—o $ the re¬
sults shall be the different values of the proposed func¬
tion when x is made equal to a.
Example. Suppose the function to be
f x— f af (x—a)
f (**—a1) ’
which, when x—a, becomes By substituting a-j-A
instead of x, and developing the results into series, the
> &
numerator becomes /*'-{-—7—+ &c. and the deno-
1 2fa 1
h*
minator s/2al^ ^ ^-y=. + &c. Taking now the first
2V2o
term of each series, we have-
A*
V
2 a
expression in which A is not found j therefore the value
of the function is 1 ■, when A'rro.
V
Of the Radii of Curvature.
93. Let HC'CF represent a material curve, or mould*
94. From this mode of conceiving the curve to be
generated, we may draw the following conclusions.
1st. Suppose PC to be a portion of the thread de¬
tached from the evolute, then PC will be a tangent to
the evolute at C.
2dly. The line PC will be perpendicular to a tan¬
gent to the curve FAP at the point P, or will be a
normal to the curve at that point. For the point P
may be considered as describing at the same time an
element ot the curve Fi\P, and an element of a circle
q P cf, whose momentary centre is C, and which has PC
for its radius.
3dly. That part of the curve between F and P,
which is described with radii all of which are shorter
than CP, is more incurvated than a circle described on
C as a centre, with a radius equal to CP. And in like
manner PP', the part of the curve on the other side of
P, which is described with radii greater than PC, is
less incurvated than that circle.
4thly. The circle q R q' has the same curvature as
the curve APP' itself has at P: hence it is called an
Equicurve circle, and its radius PC is called the
Bai>ius of Curvature at the point C.
95. We are now to investigate how the radius of
curvature at any point in FAP any proposed curve may
be found.
Let AB and BP be the co-ordinates at P any point
in the curve, and PC its radius of curvature 5 and let
PC meet AB in E. Put the abscissa ABrrr.r, the or¬
dinate BP—y, the arch APzra, the angle AEP (that
is, the arch which measures that angle, radius being
unity) zzr, the radius of curvature PCrsr. Take P'
another point in the curve, and let P' C' be the radius
of curvature at that point. Let P'C' meet AB in E',
and PC in X), and on I) as a centre, with a radiusm,
describe an arch of a circle, meeting the radii PC, P'C'
in m and n. Then the arch PP' will be the increment
of k; and since the angle PDP' is the difference of the
angles PEA, P'E'A, the arch m n will be the corre¬
sponding increment of v.
Suppose now the point P' to approach continually to
P, then the points (7 and I) will approach to C, and
the ratio of the arch PP', the increment of 25, to the
arch m n the increment of v, will approach to the ratio
of CP to C m, that is to the ratio of r to I ; therefore
the ratio of r to 1 is the limit of the ratio of AP' to m n,
PP'
or r~ limit of , and passing to the ratio of the
m n °
fluxions, r—--, thus we have obtained a formula ex-
v
pressing the radius of curvature, by means of the fluxion
of the arch of the curve, and the fluxion of the angle
which a normal to the curve makes with the line of the
abscissas. We proceed to deduce from this formula
4Z2 other
732 FLUXIONS.
Direct other expi’esslons which may involve the fluxions of * ^
Method, and y only.
Part I,
fore, yz
96. Because PE is a normal to the curve at E, the
tangent of the angle PEA or v is equal to — (§ 75*)»
y
put then because tan. vzzty we have, by taking
y. t .
the fluxions (§ 60.), v stc.* v=:ty but sec.* w=l+tan.a t;
(§ 63.), therefore = t
y* y y% y
t if
and vzz-
Substituting now this value of v in the formula r=-r
v
it becomes
25*
‘y%
If we now recollect that and that
y
it will appear that this other expression which we have
found for r involves in effect the fluxions of x and y
only.
97. In computing the values of , and —t-t—
y * if
we
may consider any two of the three quantities x, y, 25,
as a function of the remaining quantity ; and upon that
hypothesis compute their fluxions.
Thus if we suppose that y and 2! are functions of x,
then, as in taking the fluxions of y, t, and a, we must
considers as a given or constant quantity, from the
. x , • x y t . , ,
equation — we have t— (j 390> and sub-
y . y*
stituting this value of t in . ■ the value last found for
ty'
r, it becomes
r=-4^r:
—* y —at y
If again, instead of considering y and 2; as functions of
r, we consider x and % as functions of y, then from the
equation t (as y must now be reckoned constant),
y
• x %
we get t s=—r, thus the formula rr=—r- becomes
y 1 y%
* y
* y
3 * -i—.01*
a xx _——, and, making x constant,
2X*
Direct ^
Method. I1'1,
— *^2 v» $ a X
-—, therefore, ^
^xi
x /(Ax-\-a\ . . - ,
^—~—]» an“> putting r for the radius of
curvature,
r_ _Ca+4^)4
— xy 2
If in this general expression, we put .vrro, we find
. a for the radius of curvature at the vertex of
2 y o
the curve.
Ex. 2. Suppose the curve to be an ellipse, required
as in the last example.
Putting a and c to denote the two axes, the equation
of the ellipse is a'y'z^^^ax—x*). Hence taking the
first and second fluxions, we have 2a*yy—ci x (a—2x')t
and 2 c? y1 2 or y y — — 2 c* x* \ whence y
ca x(a—2x) . •• <2*v*-I-c*x% , . ,
and —y— ■~—7 , which expressions,
2 o*y " a'y
by substituting the values of y and y, become
• c x (a—2 w)
2a{a x—a?*)
f o*c*A;,(a—2 a:)*
•• j 4o*c(aA:—a:2) ^{ax—x1)
y 1 * 1
t
a\/{ax—.v*)
_ca:* (a—2a:)*-{-4(oa:—A:*)
«—• “ S\ *
a 4(nA?—Af*)v'(cAf—a:1)
c a a:*
4 (or a:—a:*)t
therefore,
2s=v/ («*+ y1)
fc*x*(a—2A:)a \
^'4a1 (a a?—) +* )
_ * //c2a2-j-(a2—ca) (4ax—4x*)
\ ax—a* '
and
—v y
(oV + 4(ol—fa) (ax—x2y
2 a* c
Wc shall now apply these formulae to some examples.
which expression, when a=o, becomes simply the
98. Example 1.—It is required to find the general
expression for the radius of curvature of a parabola.
The equation of the parabola is ynza* xl, there*
radius of curvature at the vertices of the transverse
axisL, but when zzArfa, it becomes , the radius of
2 c
curvature at the vertices of the conjugate axis.
PART
FLUXIONS.
art II.
[averse
olethod.
PART II. THE INVERSE METHOD OF FLUXIONS.
733
Inverse
Method.
99. AS the Direct Method of fluxions treats of
finding the relation between the fluxions of variable
quantities, having given the relation subsisting between
the quantities themselves 5 so the Inverse Method
treats of finding the relation subsisting between the va¬
riable quantities, having given the relation of their
fluxions.
Whatever be the relation between variable quantities,
we can in every case assign the relation of their fluxions ;
therefore the direct method of fluxions may in this re¬
spect be considered as perfect. But it is not the same
with the inverse method, for there are no direct and
general rules, by which we can in every case determine,
from the relation of the fluxions, that of their flowing
quantities or fluents. All we can do is to compare any
proposed fluxion with such fluxions as are derived from
known fluents by the rules of the direct method, and if
we find it to have the same form as one of these, we
may conclude that the fluents of both, or at least the
variable parts of these fluents, are identical.
100. In the direct method we have shewn, that by
proper transformations, the finding of the fluxion of any
proposed function is reducible to the finding of the
fluxions of a few simple functions, and of the sums, or
products, or quotients of such functions. In like man¬
ner, in the inverse method we must endeavour to trans¬
form complex fluxionary expressions into others more
simple, so as to reduce them, if possible, to some fluxion,
the fluent of which we already know.
Sect. I. OJ the Fluents of Fluxions involving one
variable quantity.
101. As when y is such a function of a variable
quantity .r, that yrr Ax'"-f-C, where A, m and C de¬
note constant quantities, we find by the direct method
(J 36. and $ 26.) that y—m A xm~x x% or (putting a
instead of m A, and n instead of m—1), y—ax" x; so,
on the contrary, as often as we have the fluxional equa¬
tion
y—ax" .v,
we may conclude that the relation of the fluents is ex¬
pressed by the equation
axnJrI
W-J-I
fCi
inquiry in which the fluxional equation y=axHx oc¬
curs. If it be known that y=0, when x acquires some
known magnitude, which may be denoted by b, then
abn+t
the general equation y~ —jC, becomes in that
»-j-1
particular case
abn+t
71F1
C;
Hence, by subtracting each side of this last equation
from the corresponding side of the former, we get
y=
o(x’,+1 A,+'‘)
»+1
an equation that is independent of the constant and ar¬
bitrary quantity C#
103. By giving particular values to n in the fluxion¬
al equation y—axnx, and in that of the fluents w—
a (x"*1—£"+I) . ^
— , we may obtain particular fluxional
equations, and corresponding equations of the fluents.
There is however one case which requires to be no¬
ticed ; it is when n is =—1 j then the equation of the
fluxions \sy=2ax-'xz=~, and that of the fluents, ac¬
cording to the general formula y~
o(*—I+I—i-1*1)
—i-f-i
_ a(*°—b°) _c(i—1) o ,
— ^ ~t——^ ““o’ r°m “usexPresSl0n
it is manifest, that nothing can be concluded. The value
g(x"+i b'"*1')
ol the function * *n Par^cu^ar case
of //-f-izro may be found by the rule given in $ 90
lor determining the value of a function when it assumes
the form -; but it may be otherwise found by pro¬
ceeding thus. Put and let » =r and
log. e
loir, b , , , ,
?= 1^17 J tben’ by 1 >e formula of § 54.
m , . p*»i* „
xm=i -\-pmF f- &c.
for by substituting m A instead of a, and m—1 instead
of » in this equation, it becomes y=rA x^-j-C, the same
equation as that from which the fluxional equation was
derived.
102. The value of the constant quantity C, which is
generally called by writers on fluxions, the correction
®f the fluent, is to be determined from the particular
+ &c-
and therefore
O-?)
xm—bm ■
and
Thus
734-
FLUXIONS.
Inverse
Method.
Thus we have
*»+* &n+i
or expressed ge-
n-\-\
Part II,
So, on the contrary, if we have any fluxional equation invert®
of this last form, we may conclude that Method,
nerally by a series, all the terms ot which, except the
first, being multiplied by m or w-f-i, will vanish when
n -j- i~o, or when n— —I j hence it appears that the
. o(.rn+I—. . ,
general equation y— ’ “ecomes in t‘ie
particular case of n ~—I, (/?—y)’ which, substi¬
tuting for p and q their values, and observing that
y=a t-\-b v-\-c u ^-&c. -}-C.
And since that when wdiere u, v and t de¬
note any function of a variable quantity, and C a con¬
stant quantity, we have, § 37, u^vt-\-t v, so, on the
contrary, if
log. x log. b
log. e log. e log. e
X log. becomes
a , x
—y-. X log. 7
J]og. 45 b
where log. e and are ^a^en according to the
we may conclude that
uz=.v
and in like manner if we have
tv—vt v vt
t ?
same system, which may be any system of logarithms
whatever. So that if we take the Napierean system in
which log. e—i, then
we may infer from § 39. that
«= -+C.
t 1
y—a 1. *- =a 1. .r—a 1. b~a 1. ar-f-C,
where C denotes a constant quantity, and where the let¬
ter 1, in this formula, and in others in which it may oc¬
cur, is put as an abbreviation of the words Napierean
logarithm, so that by o l.a is meant a multiplied by the
Napierean logarithm of x, &c.
This expression which we have found for the value
of y in the particular case ef y — ax~x x, or —, co¬
incides with what we might have found by considering
that when yz=\. x, it has been shewn (§ 57.), that
• x 9 (tx
y— - so that conversely, when —, we may con-
x %
elude that yzrol. ff-f-C, where C denotes a constant
quantity, to be determined from the particular question
in which the fluxional equation may occur.
106. It is often convenient to denote the fluent of a
fluxional expression without actually exhibiting that
fluent. For this purpose we shall employ the sign^j
putting it before the fluxion whose fluent we mean to
denote. Thus, by the expression J'a xn x, is to be un¬
derstood the fluent of a xn x; and as this fluent has been
found to be - + ~i+C\ we may express this conclusion
in symbols shortly thus,
fa
ax x =r
ax
.n-J-X
71 —I
-C.
104. It must now be evident that if
y—axmxJrbxnx-\- ca:pa:-{- &c.
where ot, n, p, &c. are constant numbers, then
107. Suppose we have y—(ax-$-b)m x, we may ex¬
pand (a x-{-b)m into a series, and multiply the series by
x, and find the fluent of each term of the result. But
we may also find the fluent of this expression without
employing the developement of (a x-\-b)m, by proceed-
bxn+t C*p+,
OT-f-I ‘ W+I P+I
-j- &C. -fC }
here C denotes a constant arbitrary quantity that may
he considered as the sum of the constant quantities which
*+* bxn+'
ought to be added to the terms , —-—, &c.
6 OT-fl
each being regarded as a distinct fluent.
%—0 J
ing thus. Put ax-^b—%, then X— ——, and
Substitute now these values of and #, in the ex-
pression for y, and it becomes y~ — j hence we have
2,m+I
(§ ioi.)y=
a(OT-{-i)
stituting (o.v + i) for
105. In general, since that when
yzza t-\-b v-^c u-{- &c. -|- C,
where t, v, u, &c. denote any functions of a variable
quantity, and C a constant quantity, we have (§ 35.
‘ 36-)
y=
-j-C, and consequently, by suh-
{.C.
o(OT-j-l)
io8. Suppose that yzz(axn-\-b)”'x*r~*x. By putting as
before axn-\-b=zz, we have n a xr'~x x=z, and x”-1*
and
ysoi-f-i &c.
— . hence y rzr ——^ snd y— r . x
-j-C, and
substituting foi: « its value cxn~f-&,
t.
y=
art II.
Inverse
fJIetbod.
log. Let us now consider fractional functions, and
to begin with a simple case let us suppose that
Axm.V -p x—b . 25*
V—7 —r—Jrut a074-^7725, then »= , .v=
’ a a
and consequently,
FLUXIONS. < 735
place, we remark that the greatest exponent of the Inverse
powers of x in the numerator may be supposed to be Method,
less than that of its powers in the denominator. For if v’"
it were not so, by dividing U by V, and calling Q
the quotient, and II the remainder, we should have
U * • R a?
'^r=(e^+-y-> and
y~~ am+125’‘
We have now only to find the developement of (2;—&)”,
to multiply each of its terms by 2; and divide it by 2;",
and take the fluent of the result.
Let us take for example the case of m = 3, and
n—2t then
/ _ A(«—by%
V~ 0*7? ’
2; 2;—3 bK-y^tf-TT'T,— b*%-x7i\^
Hence, taking the fluents of the several terms, as in
§ 105, we have
+C.
Let us now restore the value of z, and then it appears
that when
/t-=/
y=7:
A x3x
(ax+by*
A I i
A'xm'4Wxn/ +TW+&cc. ’
which, by putting U to denote the expression between
the parentheses in the numerator, and V the denomina-
tor, may be represented by Now in the first
Now, Q being a rational and integer function J"Q x may
be found, as in § 104, and it only remains to find
f*Rx . . . ,
/ y j an expression in which the highest exponent of
the powers of x in R is less by unity than in N ; so that
R x
the fraction ——— may be generally expressed thus,
(A at’-^B A7"-24CA:n-J +T)«
-j-AV-1 + ii'x*-2+UXn-i . . .4/f''
Ihe general method of finding the fluent of a frac¬
tional expression of this form consists in decomposing it
into a series of other fractions, the denominators of
which are more simple. These fractions may be found
by proceeding as follows: By putting the denominator
of the proposed fraction equal to o, we get this equa¬
tion,
a;B-{-AV~t 4-BV-a . .. -fT'rro.
Suppose now that the roots of this equation are found,
and that they are denoted by
—a, —a', —a", —a'", &c.
which quantities we shall suppose, in the first place, are
all unequal. Then the expression which has been as¬
sumed as equal to o, may (Algebra, Sect. X.) be
considered as the product of n factors
#40, x4a\ x-J-a”, x-j-a'", Sec.
Let the proposed fraction -y be now assumed as equaSi
to the sum of the simple fractions
N
N'
N"
-, &c.
x-J-a’ x-ya'’ x-j-a
having for their denominators the simple factors of the
denominator of the proposed fraction, and for their
numerators quantities which are constant, but as yet are
indetermined.
T-hat we may avoid complicated calculations, and
present a determinate object to the mind, let us suppose
that the fluxion of which we are to find the fluent is
(A a?24B a?4C)a;
x3-j-A'xi~j-B'x-j-C/*
and that we have by the resolution of the cubic equa¬
tion x34A'x* 44 C'— o found
x3 4 AV 4 B'a? 4 Cf:= (* 4 °) (* 'h0') •
The
A*
The fractions
N * N'.v N"*
FLUXIONS.
~1. ^ («+^>y' (ai+o'O^ ^ -^-const.
Part II. FJ|
Inverse
Method.
i"1
x-\-a x+a x-^-a”
when reduced to a common denominator are
N(*-fa')0+a"> N'C^+aKHha^
(.f + o) (ar+a') (A?4-a'>)’ (*-|-o) (ar-f-o') (tf-f'0")
N'T^ + a) (x-j-a')x
(x+a) (x-f-a') (x-j-a") *
The common denominator of these fractions is the same
as that of the proposed fraction, and each ot the nume¬
rators, as well as their sum, is a function of x of a de¬
gree lower than tire denominator, that is, in the present
case, it is a function of the second degree. By taking
the actual products of the factors in the numerators,
and adding the results, we find the sum of the fractions
equal to
. r(N-fN'+N">a 1
y I + £ N(«’+ __ (KAf-fL>
x*-^-2xx-^~clx-^-/3,’ °r (x-f-xy-f/S1 ’
Put x-j-x~zf then it becomes
(K.
and this again, by putting L—K«=M, is resolved in¬
to these two fluxions
K z « ( Mz
z’-j-p2
We can immediately find the fluent of the first of these,
by putting z^+^zrv, for then zz= —, and
2
==T'Ss=K*Lv’ ^ I03-)'
=K IV 0*+£2).
With respect to the other fluxion, if we put z~/iy, we
have
M z M y
5s*+0a~ /3 i+ya’
A
z'+f
but we have seen ($ 60.) that —- is the fluxion of
1+S
an arch of which the tangent is y, therefore
/M y M , ‘
J — T-T-71 = a»-c (tan. =y) + const.
M
/3 l+y1
=r—arc Ctan.zz1—f- const.
t K /3 ‘
It is proper to remark, that if - be the tangent of a»
p
arch, then the sine of that arch is
cosine is
4
7-r, and it*
—, thus we may express the fluent
V(*2+/3X)’
3 A under
738 FLUX
Averse under ilifferent forms, by Introducing the sine or cosine
Method, of the arch instead of its tangent,
V If instead of » we substitute in these two fluents *-}-«
IONS.
Part II, ^
and another function, which is the fluent . 1 w!
+ ’ Method.
of
we find that the fluent of
*2+ 2c6X-\-x'i-\-fi1
K 1. ,v/02 + 2*.r + *t + /32)
C«2+/32)5
Mi
, that is, let us assume
f—X 1 1
Gz . r Ha
C^+/32) (*2+A
■y-1 "h/l
(ts24-/32j*-1 ’
L—K
(flf1 -j- 2x-V +«2 + /3*)
:* 1
and from this equation, by rejecting what is common to
each term, we find
(K"'....v-fL"/...>
X% -\-2XX-\-x*
where K, L, K', L', &c. denote indeterminate but
constant coefficients, which may be determined by re¬
ducing these fractions to a common denominator, and
proceeding as in the two preceding § §. Then the whole
difficulty is reduced to the finding of the fluxion of the
expression
(Kat-(-L)a;
(*,4-2<*y + *,+/S,)?
(Ky-f-L).r
where q denotes some integer number. To simplify this
expression put x-\-et—zy and L—KarzM, then it be-
(K% X wh;ch we shall now shew may be
M=G(s2+/S2)—?(?—i)G*9+H(*9+0*)>
and hence
M=G/32+H£2-J-(G—i)G-f-H)*a j
Therefore by comparing together like terms, we find
M=G/32+H/S2, G—2(5-—i)G-j-H=o 5
and from these equations we get
M
G=
(22—2)/3
~(2?—2)/3* '
comes
(*2+^),
reduced to
Hs;
we decompose its fluent into two parts
may
To effect this reduction
Let these values of G and H be now substituted in our
assumed equation, and it becomes
M;
M
(*2+£’)J (2?—(»2+^*)?~1
J r^-k.a2)* “t/ 1
Ms
(32+£2)* T/ (22 + /32)*
The fluent of the first part may be immediately found
• V
by putting »1 -|-/32 = t; ; for then %% —— and
M(2q—3) r 2
Thus we have reduced the determination of the fluent of
Mz - - *
to that of-
C Kzz r Ku __ Kt;-7**
(i*y 2v'1 2(1—(/V
’ —, and by proceeding in
(5&3+/82)* ” 7 ” (is1+/3*)J *
the same manner with this last fluxion, its fluent may
2(1—?)
be made to depend on that of-.' ■ J hut this
Let us now suppose that the fluent of the second part
-, is equal to the sum of the algebraic function
(*2+/32r
(s’+^r
will be more readily effected by simply substituting q—i
instead of?, and supposing M=i in the preceding equa¬
tion.
Thus
art II.
Inverse Thus we shall obtain
tVlethod.
f(?,'
(a’ + ZS3)^1 (2^—4)/32 O2-^*) 3
C2<7—iLyi 25
+
(2^—4)/32) «-2 *
which expression will consist of two terms, one an alge¬
braic function of 2:, and the other f ■■■■■ . 25 ;—multi-
’ J {* +& )
plied by a constant and given coefficient. This value of
when substituted in the last equation
O’+ZS2)
wiil produce an expression for^/—consisting of
algebraic quantities and /By continu-
e A ^ (z*-f/33)3-3 J
ing this process it is evident that we shall at last have
M;s
f-_ 3 expressed by a series of algebraic quan*
titles, andy^-—-—■, and here we must stop, for if WB
repeat the process with a view to make the fluent de¬
pend on / —~—— that is on f si, or z, we shall find
' (!s2-f/a*) J
that the coefficient of this quantity becomes infinite. As
to the fluent of —^—- we have exhibited the expres-
S5s + /3* r
sion for it in last §.
In comparing together the results which have been
obtained in the preceding articles, it must appear that
when a fluxion is expressed by a rational fraction, if we
grant the resolution of equations, the fluent may always
115. Let us recur to the fraction and suppose
that A'-J-ff is one of the unequal factors of the denomi¬
nator V, so that we have V— (x-\-a) Q. Let us now
u A P A , .
put -37 = — (--tt, A being supposed a constant
V x-\-a (4 o ir
quantity, and P an indeterminate function of ,r, but
such as not to he divisible by .r-\-a. Then we have
U=AQ-j-P (.v-f-ff), and hence P=r As P
is an integer function with respect to x, it follows from
this equation that U—AQ, which is also a rational and
integer function of#, is divisible by .v-\-a, and conse¬
quently has #-f-a for a factor j therefore, the function
U—AQ will vanish when we substitute —a in it instead
of #, seeing that —a is the value of x that makes the
factor #-J-a~o. Let us denote by u and q, what U
and Q become by this substitution, which however will
not affect the indeterminate quantity A, because it is
independent of x. We have therefore u—Aqzzo, and
consequently A=r-.
This value of A requires that we should know the
function Q given by the equation Vrr(#+a) Q, and
we may always find it by dividing V by «+#. The
direct method of fluxions affords also a very simple
method of determining it. For by taking the fluxion
of the above equation we have
^ 6
-r-=Q+(^+r0 ~i
x x
if in this result we make .v-j-azro, or x=—a, and de-
V
note by v what — becomes by that substitution, we
shall have vzzq, and consequently A=r —.
The expression A=- has always a finite value, for
the numerator and denominator can never become r:o,
because we suppose the fraction reduced to its lowest
terms, and consequently, that the numerator U has not
for a factor ar-f-a, which is a factor of the denominator,
but which being contained in it only once does not enter
into Q.
116. Let us now consider how the numerators of the
fractions, into which the proposed fraction i» to be
decomposed, are to be found in the case of the denomi-
< A 2 nator
1
74°
FLUXIONS.
Inverse nator V having equal factors of the first degree. In this
Method, case we have V=;Q (#+«)’', and we assume
U A B c
V” (*+a)n_,”(-v+0)"~I + (*+a)"“2
N P
*
By reducing to a common denominator, we find U
equal to
Q ja+b (*+.^+++p(*+«)"
and P equal to
U—Q(A + B(^+rO + C(A;4-a)2...4.N (^4-«)"~:)
C^+°)n
Part 11,
The direct method of fluxions facilitates greatly the
preceding operations. For the numerator of P being
divisible by (Ar-f-0)”is necessarily of this form X(a+#)'>» *
X being an integer function of ar, but which does not
contain the factor x-\-a. Now agreeably to what has
been shewn in § 88, the successive fluxions of this nume¬
rator, as far as the n—l order inclusive, vanish when
x—a is supposed =:0. By giving to the numerator the
following form,
Q -A-B O+o) -C (*+«)’... )
and observing that the function Q does not contain the
factor x-f-fl, it is manifest that it is only the part of this
expression between the parentheses which ought to be
divisible by (tf+a)". Let us put — —Z,then the suc-
cessive fluxions of that part are
luverie
Method,
and as P ought to be an integer function of at, the nume¬
rator of its value is necessarily divisible ii times suc¬
cessively by x—a ; therefore, that numerator ought to
be equal to o, when —a is substituted in it instead of
x. Now this substitution being made, each of the
terms of the numerator which is multiplied by x-^-a
vanishes, so that there remains only U—AQ, but that
this quantity may be divisible by a?-fa, it is necessary
that u—q A=o, where u and q denote the same as in
last 6, hence A
q
This value of A changes U—Q A into U — - Q,
q m
which must be divisible by A:-f-a. Let us, with a view
Zl
to abridge, put U Q=U' (Ar-fa), then substitut¬
ing this quantity in the value of P, and dividing both
numerator and denominator by Ar-fa, we have P
equal to
U' Q (B-fC (Ar-f-a) . . . -f-N (j-fq')"-*)
O-fa)"-1
Now to obtain B we make x-f-arro, then, putting
to denote what U' becomes by substituting —a in it
in place of a;, we have v!—q B=o, and B=r —
Instead of B let its value be substituted in {l'—QB,
ii!
/ and this quantity becomes 1? Q, which vanishing
when a;-|-az=o, will have at-fa for a divisor y therefore,
'll!
we may put U' — Q^U" (a;-fa), then, substituting
this last quantity instead of the former in the value of
P, and dividing the numerator and denominator by
x-fa, we have P equal to
U"—Q(C-f !)(*-fa) fN (*-fV->)
(a;-fa)'-2
Bv continuing the same mode of reasoning, and the
u"
same notation, we find u"—q C~o, and C=—. And
q
so on with the remaining quantities.
Z—Bat—2 C (x-fa)#—3 D (x-fa)ax ...
Z—2 Cx*—2.3 D (x-f x)x*. ..
Z—2 . 3 D xJ...
&c.
and these results ought all to vanish when we put
x-f a=o. Thus we have
Z—A=o, and A= -,
q
• • Z
Z—B x=o, B=: —,
x
.. . Z
Z— 2 C x2—0, C~ ——,
2 a.’2
Z—2.3 D x3=o, D= —,
2.3X*
&c. &c.
Z Z
observing that in each of these functions —r, , we
x 2 x3
must substitute —a instead of x.
The most simple way to find the value of Q in this
case is to divide V by (x-f a)”, but we may also find
it by the direct method of fluxions, as in the preceding
§ ■, for, since V=Q (x-fo)”, if we take the fluxion
of each side of this equation « times, and then make
.r-f a=o, we shall find, § 88, the wth fluxion of V
equal to 1 . 2 . 3 . •. « Q x", and consequently
^ nth flux, of V
t£= r-
I . 2 . . . » «n
117. Let us now consider how we are to find the
U
numerator of the fraction which forms a part of y
when it has this form
A x-f B
x2-f 2«x-f*a-f
Assume
F L U X I O N S.
art II.
Inrerse
(Hetbod.
Assume
_U
V:
A*4-B
then, reducing the latter part of this equation to a
common denominator, we find
U=Q (A*-f B)-t-P (Ar*2«1r4-«,+^)-
Hence we deduce
p_ tJ—Q (Aa;-}-B)
—4~2 * *+542+z3 2 *
As P Is supposed to be an integer function with
respect to x, it follows that U—Q (Ax-J-B) is divisi¬
ble by x2-±- 2 x x-j-x*-t-(32 •, therefore, the former of
these two quantities must contain among its factors those
of the latter, and the quantities, which, being- substitu¬
ted for x, cause the latter to vanish, must also make the
former vanish. But the factors of x2,-\-2xx
are Ar4-«-f-/3 —I, and —/J —1, and these,
being put each =0, gives us x—— (a+Z3 sj—0> an^
— (x—/3 a^/—1), therefore, each of these values of
.v being substituted in U—Q (A#-f-B) ought to make
that quantity vanish. Let us denote by xr=±zu!—1,
and by qz+zq'^/—1 what U and Q respectively be¬
come when — —1) is substituted in each in¬
stead of at, then, after this transformation, we have
f/zfc:?/ .J —1 ~l
—(?—?' >/—O |—A (x+p sj—i) +B|=0.
This equation is twofold, because of the sign with
which several of its terms are affected, and it is equiva¬
lent to those which would be formed by putting the
real part equal to o, and the imaginary part ~0.; from
this consideration we have
x A—q' fi A—q Br=0,
Z3 A-\-q’ x A—q'li—O,
two equations which give us the values of A and B.
The function Q may be found as in § 115. For, if
we take the fluxions of each side of the equation
Q (#*+2*tf + «l-p/8*)=V,
and afterwards make
-f 2 * .v -f- x2 -f-/3*=ro,
we find Q (2.va;-F 2xx)—Vt and hence
Q= r-J
2X X-\-2 X X
Let the two values of .v, to wit —(xztr/i be
substituted instead of it in this equation, then, putting
I to denote wiiat the expression — becomes
x
by that substitution, and writing q:
of Q, we have
, /—- v+v'is/—:
q—q v
•q’ v"/—x instead
sj—1
which, by multiplying the terms of the fraction on the
latter side of the equation by ^—i, becomes
1 / irqrrty
z±z 2/3
741
Jnverje
Method.
Hence by putting the real parts of each side of this
equation equal to each other, and also, the imaginary
parts equal to each other, we find
v' f v
2/3' * ^ 2/3*
118. If the factor x2-\-2xx-\-xx is found several
times in the denominator of V, so that
^ =Q then, § 113, we assume
in this case equal to
Ar+B A'*-|-B'
fx2 4- 2xx 4-** 4-0*)»+(*? 4- 2«« 4-«v4^3*)”-*
A"a?4-B" P
^"(zifI4-2^4-«,4./3,)”-a f-Q
reducing this expression to a common denominator, and
so ordering the equation as to bring P to stand alone on
one side, we find P equal to
U—Q IAar-j-B 4- (A'*4-B') (** 4- 2*# 4-«2 4-Z3*) 7
^1+ (A"a;4-B") (a?i4-2«a;4-^4-i3»)2. . . \
(AJi4-2fl{A;4-*24-(32)n
By reasoning in this as in the preceding case, it may
be concluded that the numerator of this expression
ought to vanish when —(xz±zfl —1 is substituted in
it instead of x; therefore putting »dhz«'and
q—q' s/—1 t0 denote the same things as before, we
deduce from that substitution
UZ+Ztl' AaJ—1
—(?—?'n/—-O (—A v/:=i4-B)
the very same equation for the determination of A and
B, as we have already found in last
Having found the values of these quantities, they may
be substituted in the numerator of P, and the terms
U—Q (Aa'4-B) becoming divisible by at*-f-2 <* ^ + ***
4~/32, the whole expression becomes divisible by the
same quantity. Calling therefore IP the quotient
arising from the division of U—Q (Aa?4-B) by
at* 4“ 2 x xx2(i2, we have P equal to
U'_Q[A/a?4-B,4-(A"a;4-B")C^,4-2«a?4-**4-/3*) ...]
a;24-2«a?4-*,4-/3,)'^x
If in this numerator we substitute instead of x its va¬
lues deduced from the equation Ar24-2* Ar4-«24-/32z=C,
and put the result —o, we may determine A' and B' in
the very same way that we have already determined
A and B, and by proceeding in this manner we shall
find the remaining coefficients A", B", &c.
This case is quite analogous to that which has been
already treated in § 116, and the direct method of
fluxions applies to it in the same manner as to the other.
For since Q does not contain the factor .r a4-2f«‘4-«a-f-i6*t
if the numerator of P be divided by the function Q, the
result, which may be denoted by r, ought to be of this
form:
74.2 FLUXIONS.
Inverse form >~X (.i;,+2 »* + «’+/3‘)*, ami consequently
Method, ought to vanish, as well as all its fluxions, from
' v—~tl,e first order to the n—I order, inclusively, when
** + 2*#-4-fl*+/32=0 ; this being the case, we have
these equations,
Part II,
* \ _ Inverse
—X*—Xs, therefore-v=8^7 +7^—4>;S—3A:*. If in Method.
x •
i |i«<
sl«t
this expression we substitnte -f-1 instead of x (viz. the
value of a? deduced from the equation x—i=ro) we find
the result to be 8, therefore v~S. So that
r=0, rz=o, r=o,...
and so on to the n—I fluxion of which ought also to
berzo each of these equations becomes twofold when
we substitute, instead of x, the values of which it is sus¬
ceptible in consequence of the equation a;1-f-2
-f/3*=o. By putting the real and the imaginary parts
separately =0, we shall obtain as many equations as
are sufficient to determine A, B, A', B', &c.
It may also be remarked, that from the equation
V=Q (x22ctxct*
wc find Q equal to the quotient arising from the di¬
vision of the wth fluxion of V by the wth fluxion of
#*+2«.r-f-«2-f-/32, observing to assume
x*-{-2ctx^-uz-]-/2t=o.
119. We shall now give some applications of what
has been said relative to the fluents of rational fractions.
Suppose the fraction to be
A _ « _ 1 ^ A I 1
x—1 8 a?—1’
Let us next investigate the values of B and C in the
B C
fractions and —, by means of the rule of
(a?-}-1)1’ .r-j-i’
§ 116, and that we may make the symbols expressing
the quantity under consideration agree with those em¬
ployed in that formula, let us exchange the letters B
and C for A and B, so that we are to consider
o+iy
B
J . In the first place we have
Q=
A;* -f- x1—xA—3
:xs—-J- .r4—x3 5
O + i)2
Put then j?=:i *, substituting now this value
of x in the value of Q, the result is 4=?, therefore
Let this value of A be substituted for A
4
x* -f-AT7—-a:4—Ai3
The factors of its denominator are easily found, for it
may be put under this form
A:3(Ars-J-A:4—x—l)=ra’3 (a?-|-i) (a;4—1),
the factor x*—I may be decomposed into a:1—1 and
a:*-J-i, or a;—I, A’-f-i, and A:*-f-i, thus we have the
denominator equal to
x3 (*—0 («+1)1 C^+O
in the expression forll' in the § above cited, and we have
U'=
U—A Q _ 4—Ag-f-x3—r4+x3
therefore (§ in, § H2, and § 113.) the proposed
fraction is to be decomposed as follows
Af + I 40+0
—xs + 2x*—3A:3 -f-4A?2—4X 4- 4
_ _
Hence putting —1 instead of x in the expression for U',
have u' = - and B=—= % Thus the two fiac-
and
we
1 1
tions under consideration are found to be -. ^7—-j.
Kx
+
B x
(.r-f t):
+
Ca;
Ho? E AT
v3 * X1
+
F x
+
X —f- I
(Ga?4-H)a:
1 -f-Ar*
E 1 , We might have deduced the value of B from
8* #4-1
25 . U
the formula B=§ n6, where Z is put for
x
Q
for we have
By reducing these fractions to a common denominator,
and comparing the numerator of their sum with that of
the proposed fraction, we might determine the unknown
quantities A, B, C, &c. we shall, however, rather em¬
ploy the methods that have just been explained.
7 u
Q (a:6—xs4-a:4—a:3)’
, Z 6xs—5at4 -j- 4A;3—3a:4
and -j_ — •
x
By comparing this particular example ^—-
if in this expression we substitute —I instead of a?, it
becomes —7= ?, the same value for B as before.
16 o
Ua-
with the general expression it appears thatUzzi,
and V=a^4-a:7—xA—x3. First let us investigate the
A
numerator of the fraction , and for this purpose
x— I
, • , E , F
Let us now consider the fractions + 0ir
exchanging the symbols 1), E, i for A, B, C,
a
we employ the formula A= — (§ 1 ijO- As we have
U=i, it is evident that w=i ; and since V=A«4-Ar7
The numerators A, B, C may all be found from these
formulas of § 116.
2
art II.
FLUXIONS.
I in verse
iMetliod.
A=l=Z'B=fC:
L
743
1.2-v
observing that in this case —s—i j and that
we must substitute o instead of x in each formula) after
taking the fluxions. Now we have
7-E- *
Q - .rs+x<—.v—l’
Z —i
X ~ —.r—i)1’
Z 20rr3-f-T2.F*
i2 (a-’s + ^4—x—i)*
4-2(5 a,4-f 4 a?3—
(.t’S-j-.r4—x—1 )3 '7
Hence putting x=zo, we find
A=r—I, B — 41 > C——1,
so that
-+- +E=_2.+L_I.
X3 X* X X3 ■ X* X
There yet remains the fraction or to
at* 41 at*4i
be considered. It may be found by subtracting the sum
of all the others from the proposed fraction j we proceed,
however, to find it directly by the formulas of § 117.
In the first place we have Q=ra64a'5—at4—a’3 $ next,
the factor a?24I being put —o gives -i
x—o, ft—o. Hence we find
n n
and by the second
2 cos. 3 2!=(t;-f-^) —(v+~)
y'=-~h
_ (2 m-f Qtt | f—
( 2 m 4-1) *■
=-J+T;-
Proceeding in the same way with the third and follow-
ing equations, we find
2 COS. 4 25 = 1^ + -^,
122. By means of the indeterminate number m, each
of these expressions for y furnishes all the values of
which this quantity is susceptible, for we may take suc¬
cessively
m—Q, m=l, m=2, w=3, &c.
The first formula gives
s . i
2 COS. 5 2!=L’s-{-—J
so that we may conclude in general that
i
2 COS. n 2S=t>" + — >
hence we have this quadratic equation
v“"—2 cos. n«Xvn+I:=0»
from which, by completing the square, we find
l'B=COS. /I53=±=^/ (cos.* 71 25 1) j
therefore, by substituting for v the quantity it was put
to represent, and observing that \/ (cos.* nz —l)
=rv/ (—sin.*«2s)=x/—i sin. n%, we have
(cos. i sin. 25)” = cos. n %z±=K/—i sin. n z,
as was to be proved.
121. The function #"=+=: a” is transformed to
a" (y”z±zi) by putting a'sray, and to discover its
factors, we must resolve the equation
?fztzi=o.
The expression y=rcos. 25-fV-I sin. z satisfies this
equation, by a very simple determination of the arch z ;
for we have y"=(cos. —1 s’10* «)" = cos. nz
W”1 sin. nz, and as by putting ct to denote halt
the circumference, and m any whole number, we have
(Algebra, § 352.)
sin. m irzzo, cos. m ir^zzizl.
where the sign is to be taken, if m be an even num¬
ber, but — if it be odd, we have only to suppose « z
—m v, in order to obtain ynzz'ztzi.
That we may distinguish the case in which m is even,
3
y—cos. 0.ir=l
2ar , / . 2*-
w=:cos. f-v—1 sin*—»
y=cos.^+N/=Isb.^,
^ 71 ^
&c.
It is evident that we shall always have different results
as far as mzzn—i. If, however, we suppose m=n ;
then we have y—cos. 2 7r=J, which is the same as the
first of the values already obtained, and if we suppose
m~n~l~i, then (Algebra, § 25O
Cos.
(2n-}-2)
n
2*-
»
71
Sin.
(2» +
n
which is the same as the second value, and so on with
respect to the others.
By this mode of proceeding we shall not only obtain
the n roots of the equation yB=i, or y"—1=0, but,
with a little attention, we shall discover that these roots
may be arranged in pairs, by bringing together those
that only differ in the sign of the radical V-1 ’ for
since.
Cos. (27r—-p)=cos. p, and sin. (2*-—/>)=—sin. p,
it follows that
(2n—2mjtt / . (2n—
V—cos. f-V—1 sin- ‘ “
rrcos.
2m *
—yy/—“-i sin.
2 m #
n
Hence it appears that we may comprehend all the roots
of the equation yB—1=0 in the single expressions
?art II,
Inverse
Method.
y—cos.
2m w
n
~sj—i sin.
2 m v
by giving to m only these values
o, j, 2, ..
if « is even, and these values
FLUXIONS. 74s
'Ihe first and the last of the factors of the second de- Inverse
gree are the squares of y—i, and y + l, factors of the Method,
first degree, each of which only enters once into the' v" '
proposed function j it will therefore be necessary, when
we employ the factors of the second degree, to reject
the first and last, and take instead of them
(y—1) (y+i)=y2—i.
The factors of the first degree of the function y*—i
o, i, 2,
are
if n is odd j and it may be observed that in the form¬
er case the last value of y is
yrreos. w— —i,
because that then the equation y*—irro has two real
roots.
The two values comprehended in the formula,
2 m 7T . , . 2 m w
y-CoS.__ + >/_I
give for factors of the first degree of the quantity y”—I,
the two imaginary expressions
y-(cos- —— + v—I s.n. ),
, 2m-x .— . 2m v\
and the product of these is the expression
2 nix
y*—2y cos.
+ i,
which comprehends all the real factors of the second
degree.
As an example of the formula
2m x , ; . 2m x
yrzeos. —v—1 s,n* ,
the simple factors, or those of the first degree, contain¬
ed in the function y5—i will be
y—
, 2 x . 2x\
(COS. I sin.—J,
r 4 ^ /
y—(cos.
y+I*
The formula
2m x
y2—2ycos.
f i
gives as factors of the second degree
y*—2y-f i,
y—2y cos.—g—1,
ya—2ycos.-^-+i,
y3+2y+i.
Vol. VIII. Part II.
y-
. 2 w / . 2x \
y-—(cos.- =±=\/ —i sin. )
5 5 '
)•
y—(cos.
Those of the third degree are
y*_2y+i,
2 2
y —-2y cos.— }. i,
4 x
~T
ys—2y cos.
5
4 ^
+ 15
but it is to be observed that the first factor of the se¬
cond degree is the square of y—i, which enters only
once into the proposed function.
* 23* When the function to be decomposed into fac¬
tors is y”-f i, the formula
y=cs. ti>+v/^Isi„. j£S±l>
n n
which corresponds to that case (§ 121.) is also suscep¬
tible of the double sign =±r, provided we stop at the
value of m, which gives
2m-\. i~n, or 2m-{-i~n-—i,
according as n is odd or even j hence it follows that
n—1 n—2
2 * l~ ' 2 ’
the factors of the first degree are
(2m-{.i)x_A_ /—- . (2m + i)x
y—(cos.
/I
and those of the second
y2—2y cos.
—1 sin.
(2 w-f-i)*-
)■
+ 1.
When among these last there is found some which are
squares, we must take only one of their simple factors,
in the same way as in the two preceding examples.
When the function is y^-j-i,
the factors of the first degree are
y—(cos. ^—\/—1 sin.
y—(cos,
y+i 5
3 *\
Zy/ 1 sin.
5B
and
746
Inverse and those of the second,
Method.
if—2y COS. “ 4* r>
, Qir ,
y —zy cos. — 4-T>
r+2y+i*
The function has for factors of the first de
FLUXIONS.
and we immediately find
yn——cos. sin. 5 $
we now assume, as in § 121,
^=cos. sj—i sin. x ;
then we find (§ 120),
^"=cos. n ,zc±zsJ —i sin. n 2,
Fart II.
Inverse
Method.
gree
y—(cos. —i sin.
y—(cos. 2?=S=V—' sin.
r 57r-
y (cos. ~6-:
which expression for yny being compared with its other
value, gives
cos. wtsrzcos. sin. MBzzsin. 5.
These relations will be satisfied if we suppose n %
—2m + ^> —m being any whole number whatever, for
cos. (2 m 7r-j-^)=cos. sin. (2 m w-j-^)=sin. IJ
we have therefore
and those of the second,
y*—2y cos. ^ +r>
3*
2 m 7r-f-^
^2—2y cos. or
2^ cos.f +1.
2 m 7r4-^
?/—cos. ;
J n
--si-
i sin.
2WW+S
The factors of the first degree of the function
cos. 5-f-i
124. Such functions as are of this form x^-\-2pxn will consequently be comprehended in this formula,
Jf-q may be treated in the same manner as those which .
consist of only two terms. By putting the function —o, cos,
and resolving the equation which is thus produced, in n
the same manner as if it were of the second degree, we
find the factors to be
~sj—
1 sin.
2m7r-\-'h
\
*” + 0—v'
if exceed <7, the second term of these factors is real,
and by making
z±ian=pz±zj O1—7),
we have the functions of the form
xz±zan
to decompose into factors.
When p'^-q, then we put q^zb**, x=byy
and the function becomes
If p the coefficient of the second term of the pro¬
posed function be negative, the only change necessary
is to make p——a”, and to take the arch 3 greater than
a quadrant.
Of the Fluents of Irrational Functions.
125. When a fluxionary expression involves irrational
functions, we must endeavour either to transform it into
another that is rational, or to reduce it to a series of
m #
irrational terms of this form Axn x, and then, in either
case its fluent may be found by the rules already deli¬
vered.
Letustake forexample the fluxion
b2’ly‘zn+2an bn7f+bn
, 2 a’* \
but the condition p**zLq, or a2
(1.4- -v/x—3 Vx*) x
I-J-SV#
It is evident that by putting .r—5s4, all the extractions
indicated by the radical signs may be effected, and the
6s$s(i-J-z3—z4)z ,. .
fluxion may be transformed to ’
b , makes an^Lb , ^ dividing the numerator by 1 +z2, may be otherwise
And therefore L may be represented by the expressed thus,
cosine of a given arch and the proposed function will
be reduced to
b2tt ffn + 2if cos. 5-f-i),
we have then only to resolve the equation
cos. §4-1=0,
—6(z7z—z6z——zxz 4-2-
14~»
The fluent of which is
i
"S ~6 ' 5 3
4-z—arc (tan. rrz)
}
4- const.
12C.
Part ir. FLUX
Inverse 126. VVe sfuiU first consider such fluxions as contain
Method, the irrational function v'(A + T>*-f-G*?2), and which
have necessarily one or other of these forms,
X ^/(A-j-B.r-J-CA?2), ——-— —
X being put for any rational function of and it may
be remarked, that the latter form comprehends the for¬
mer, which may be written thus :
X ary/fA-fB.r-fCa:2) X y'(A--f-Ba'-fCX*)
(A-j-Ba'-J-Ca1)
_ X(A-t.BA?-f-Ca?*)a’
“ VCA + Bx+C^y ’
and here the numerator of the fluxion is a rational func¬
tion of x.
Before we transform the expression ^/(A -f-Btf-f (X*)
into another that is rational with respect to the va¬
riable quantity it contains, we shall put the quantity
A-fB^-fCa2 under this form,
^/'A B \
C(‘C + C'*+*
and, in order to abridge, we shall put
~ , A B _
G —c , ^ — 17,
then we have
VC A+B*+Gv2)^
Let us now assume y' + =a-f-z, then,
squaring both sides of the equation, we find a-\-bx—
(7—.<2*
2XX+Z1, hence we get -, and consequent-
2iz—o
h
•J (A-f-B«-f-C.r2)=c (x-\-'z)—c(-—
\ 2Z~~o J
■ 2 (a—
(22;—by
X
By means of these values the fluxion —-— ——
^(A-fBa-f-Ga*
is transformed into another fluxion Z 8, where Z de¬
notes a rational function of 3, which is real when C or
e* is positive ; but as when C is negative c becomes
imaginary, the fluxion Z 3 which involves c becomes
also imaginary.
In this case we have to consider ^(A-f-B-r—L*2),
and making
r s A B
it becomes c v'(«+6#—z*2)- The quantity a?*—bx—«
may always be decomposed into real factors of the first
degree j let us represent these factors by x—«, and
then it is evident that
a~\-b x—a‘2= — (a:2—b.v-+-a')
~ (.r—«) («'—.r).
ions.
Let us now assume
s/(•!'—«) («'—x) — (x—ot)*t
then squaring both sides of the equation it becomes
divisible by x—and we have «'—x~{x—at) 3*, from
which we find
«3*-}-«' %
X=—5— , (.F x) 3= ^;
»2+I V 3*-f-I ’
values which
X x
^ 2(«—«')3 3
~\^+iy ’
render the
rational.
proposed fluxion
\/ (A-f-B ar-f-Ga?*)
127. Let us now take for example the fluxion
X
ViA+Ux+Cx*5 by aPP1yinS t0 5t the first of the
preceding transformations it becomes
— 2 a
c(23—b)
, the
fluent of which is—-1. (2a—b) + co)ist. Substitut¬
ing now for 2 its value -x-\- ^ (a + bx+x2), and for
x, b, and , the quantities they severally represent, the
fluent becomes
{*(
B
“V'v-2a:'/c
+ 2X/(A+Bx+Cx
-{-const.
a result to which we may also give this form,
—li i S —77; — x VC
^/G ’ I 2VC
(. + \/(A-fBar-f Car5
-f- 1. — -f- const.
By uniting the constant quantities into one, and ob¬
serving that the radical quantity y/C may have the sign
prefixed to it, we have at last
fz
— equal to
y^ A-j-Bar-|-Cr*
7C L { + ITC +X^C+ v/(A+Bar+Car*) |
const.
128. Let us take for the second example
By employing the latter trans-
2 3
of which the
y^A-f-Ba;—Car*)*
formation of $ 126, we have —
^ 6(a2+l)
fluent is
2
arc (tan. =3) -f const.
Substituting now instead of 3 its value
^(V—*)
. — 7 deduced from the equation x'—xz^Cx—x)**,
V (^—«)
5 B 2 and
747
Inverse
Method.
748
FLUXIONS.
Part II
Inverse , P
Method, and putting ior we ge'- / “
equal to
v/C
/ v/(es'—x \
[ tan.=—7 }
V ^{x—xj
without aftecting the generality of the expression, inverse
„ Method.
T • I — t _
For if we had .t To;(rt-{-^T) 9, we may assume xrrfc6,
then x=:6%5z, and the fluxion becomes 6 z1 z
x and «' being the roots of the equation
B
(a + fo.3) 11. We may also suppose/z to be positive, for
if it were negative, so that the fluxion were .'vm_r .v
C C
I.et us suppose that A=C=:i and B=ro, then the pro-
(a^-bx~n)^ we have only to assume ,r=r J_, and the
posed fluxion becomes in this particular case
the
a/(i—O’
and the preceding formula gives for its fluent — 2 arc.
(tan.rr ^ 4- const, for x and «' being
roots of the equation a2—l=ro, we must take x~ —x,
and a.'— i.
We may, however, give this fluent another form
by proceeding thus : Let v be the arch whose tan.
fluxion is transformed to —z~m~'sz(a-\-zn) \
Let us inquire in what case the fluxion a”1-1 x
p_
(a4-^") may become rational. Assume a + b
V vn
%q—fir
then (a-f.&r")s —%r, arn—xm—
v C1 'v)^ tjien ^ \ anj x —
^/(x+a’)
2
l+x
i — tan. 2 v and .v”1-1 x—^-.z1 1
no
l tan. 2 v
I =
1 = 2 cos.2 v —— i, but
/ —a\ n T ’
^^—J ssj hence
posed fluxion is transformed to
q-a m
(V)’ 1
the
pro-
„ . g—a. m
q r+3-» . /a; \—~
* " %
nb
i -|~tan. tr sec.‘t;
2 cos.2tr—i=cos. 2 ir (Algebra, § 358-), therefore,
a?=cos. 211. Put s for the arch whose sine is and n
for half the circumference, then a‘=:cos (4 ^—s)» . ... . m .
therefore 2vz=.\tf—and since it has been shewn that expression is evidently rational as often as ~ is
f x = —2ti+ const, therefore / whole number
-/ —*r2) VC1 ■z'2) Tliere are >
yet other cases in which the fluxion may
—s—\7r-\-c0nst. or, by including the arch ^ tt in the become rational, and which may be determined by as-
r x . suming a^-bxn—xnu'1^ thus we have
constant quantity,y—^ = s + const. This con¬
clusion agrees with what has been shewn in § 59.
Instead of finding the fluent of
an
’u'—V X ~ {u‘—b) "'
^/(A+Bai—Ca;2) cy/(a-\-bx—xx)
by first transforming it to a rational expression, we may
reduce it directly to an arch of a circle by proceeding
m ,_x .
xm~J x——~a U- and because that (a—bx™) *
m 1 1? ^ y
n(uq—b') n
as follows. Put a; =2, then xzz%, and the fluxion
2
the fluxion a?”1 ia? (a + frvn)9 is trans-
is transformed to
$ again, put a+4^a
cv/(o+|62—ss2)
=g2, and X—g u, then x=:g «, and this last fluxion is
K—)
formed to
,rt lj:
transformed to
tl I
—— —, the fluent of which is -
c^Ci—z/2)’ c
—qa
” 5 P+5-1
2.+ £-t X
arc (sin.—11) const.
z(zz9—
, on p 9
an expression which is rational “ 4* “ 93 a 'v^°^e
Of the Fluents of Binomial Fluxions.
129. Let us now consider such fluxions as have this number,
form,
F
A?*-1 ^(a+AO 5,
and which are sometimes called binomial fluxions. We
may here suppose m and n to be whole numbers, may try to reduce it to its most simple case, as we have
130. As it is not possible, in every case, to express
in finite terms the formulaJxm~Tx(aJrbx') ’, we
done
Ijif
jlftli
art II.
n verse
FLUXIONS.
nverse , . . r x . * ( m—n \ n .
letbod. done wilh respect toUj ^ II4> which we fx™’-'x {a-\.bxnyz=
749
Inverse
Method.
have succeeded in reducing to/—^—. To eftect this
i % (r>4-
reduction, we remark, that, since when u and v denote
any Junctions of a variable quantity, the fluxion of uv jience afc jast we
\stiv-\-vu, (J 37.) therefore /*av—iiv—•J'vu. Now
J f xm-n(a->rbxny*'
l)nb[——n)'x (a+bxn)v\
(A)
J« (a+^n)p=
or one
other 1 x '!(c+^a?")p+x ^ 1
ade to b(pn+m) f xm-n~t x f
it we can decompose the expression xm~z x (a-\-b xH) ®
into two factors, such that we can find the fluent of one
of them, then denoting that factor by v, and the
by u, the fluent of the proposed fluxion will be made
depend on that of v uf which in some cases will be
more simple than the proposed fluxion. That we may It js easy to see that> as we have> by thls formula?
abridge a little the results, we shall writep instead of-, reduced the determination of the fluent of xm~l x
*l„f -n . r ^1 {a-\-b xny to that of xm~n~z x (a-\-bxny. vie may re-
so that p will represent any fraction ; the proposed , . . , , . v ‘ re
fluxion thus simplified in its form is ^uce t ns a.s^ °f 2" *x («+£#")'’ by writ¬
ing m—n in place of m in equation (A), then by
a?’"-1 x (aJrbx',y. changing m into m—2 n we may reduce the fluent of
^ 1 * Cc + At”)p to thatof «(o-}-^")p,and
so on.
Among the different ways of resolving this fluxion T 1 1 ^ 1 , - , .
into two factors, we shall choose that which diminishes , }} £e”era ’ 1 r ( en0 e 116 nuRl er of reductions, we
the exponent of x without the parentheses, we therefore s ia a as corRe 0
write the fluxion thus x (a-^b xny, and the last formula will be
xm~n ^ xn—t # A:n),’,
now the fluent of the factor ar"-1 x (a-{-ia;n)p may
always be determined, whatever be the value ofp, by
§ 108; let us denote this factor by v, then
(a■4-&a;”')I’+, ,
:—-—-—7-, and u—xm~n,
(p-\-l)nb
thus the formula f u vzxlu v—J'v u give us /*»-.*
(a-f-Z'a;n)p equal to
xm~n (fl-^-ia:n)p+,
j'xm-[r-r)n-X x
_jr^j:a+bxny+x
b(pn -\-m—(r— i)n)
a(jn—rn}J'xn~rn~zx{a-\-bxny
b^pn^m—fi'—1)»)
It appears by the last formula, that if m is a multiple
of «, theny'xm~xx{a-\-b x^y will be an algebraic quan¬
tity, for in that case the coefficient m—rn will be
=0, and therefore the term containing J' xm~rn~'t x
(a-j-£a?n)p will vanish. This result coincides with what
we have already found, § 129.
131. We may also obtain a reduction, by which the
exponent p will be diminished by unity. For this pur¬
pose it is sufficient to observe that j*xm~l x (a-f-&v"),s
is equal to
J'x'n~x x (o + ^'n)p~I (a-f-ix") =:
a J*xm~s x (a-\-bxny~z
+bj'xn*n-'x (a+bx*y-z,
Substituting now this last value in the preceding , , . ,
equation, and collecting into one the terms involving an^ ^1.a^ ^oriyiu a (A) clanging tn in 0 w+w,
the fluent f x”1-* x (a-j-Z»xn)p, we find ani* ^ ‘ntoi, 1 gives
{p+i)nb
■ ■ — m—^—j- fxm~n~1 x (a-i-ix")p+I.
(p+l) nbj v 1 y
But J*xm~n~z X (a-\-bxnyirt~
J'xm~n~1 x (a-f-bxny (a-j-ixn)=
a f xm~n~z x (a-\-bxny
b J'xm~z x (a-{-bx”y.
750 FLUXIONS,
xm(a-\-bxny’zzamJ‘vm_I A? (a-f i'.vn)r *
b(pn-\-m)
Inverse
Metliod.
a vi
Part II.
Inverse
Method
b(jn°\-n-\-7ii))Jxm^n~lx (a4-hxny
Substitute now this value in the preceding equation,
we have
(B)
J”Km~l x{a-\-bxny—
xm (a 4. bxny -\-p n ax{a -f-Zw”) v~r
pn-\-m
By means of this general formula we may take away
successively from p as many units as it contains, and by
the application of this formula, and formula (A), we
may cause the fluent j*x™ 1 x (ji-\-b x'^1 to depend on
rxm~rn~r x (a4 bxny~\ rn being the greatest mul¬
tiple of r contained in ??i—I, and s the greatest whole
number contained in p.
• 5
The fluent J'x"1 x ^a-\-bx3)^, for example, may, by
the application of formula (A), be reduced successively
to
/ x*x (a4 At3) 5, and f XX (a-\-bx3)r,
and by formula (B) f xx (a+bx*)^ is reduced succes¬
sively to
f x x ^a-^-bx2')1 y and /1 xx (a-^-bx3^,
1^2. It is evident, that if m and 11 were negative,
the formulas (A) and (B) would not answer the pur¬
pose for which they have been investigated, because, in
that case, they would increase the exponents instead of
diminishing them. If, however, we reverse them, we
shall find that they then apply to the case under consi¬
deration.
From formula (A) we get
J'xm-n~1 x (a+bxny=z
xm~n (a-\-bxny+l
a (rn—n)
b(m-\-np')J' xm~'tx(a-\-bxny
a(m—n)
Substitute now m-\-n in place of 7w, and it becomes
(C)
f xn~xx (a 4 bxny~
This formula diminishes the exponents without the
parentheses, because m-\-n—I becomes
when —~m is substituted instead of m.
To reverse formula (B) we first take
j"xn~'1 x(a -\-bxr,y~x~
— xm(a-\-bxny
pna
(m-\-np) Cx™*1 x (a-\-bxy
H »
p n a
Then, writing ^41 instead of p, we find
(D)
J'xm~'tx (a°ybxny~
xm(a-t-bxny+t ■
(m+n^-np') j'xm-lx(a-\.bxny+x
^ (/;4l) « a
This formula answers the purpose we have in view,
because p+i becomes -441 when p is negative.
These formulas (A), (B), (C), (D), are inapplica¬
ble when their denominators vanish. This is the case
with formula (A) ; for example, when m=—np; but,
in every such case, the proposed fluxion may have its
fluent determined either algebraically or by logarithms.
X y
-, m being a whole
y(l—x)
positive number. Formula (A) immediately applies to
this case, so that by putting a— 1, b= — 1, t?=2, p= —
we have
f x'^1 \f (1—*■*)
fx'*-' x ]
V(i—*2)_ | m—2 rxm~3"
1 ~Tt
^ ' m—1 \/(i—jcl)
or, substituting m in place of m—1,
xm~1^/ (1—xz)
fxm x
m—1
J'xm~2x
Z1" m J v'C1—^2)
Let
jrt II.
llnTerse
[lethod.
FLUX
Let us suppose, for example, that w=J, then
sf (i_*=)+c««(.
Let us now suppose that ^=3, then
• f—f*V(i—^z)
p X% X 1
Jinj^T)- l+rf-
y** A? A?
or, substituting for J ^ ^^ its value,
/vfr-b3=“(^+|)
If we suppose mz=2i then
—%**/ (1—»*)
But we have already found, § 128, that
farc (sil1-=^’
therefore, putting A for arc (sin. =a?),
f-7 (I ^ C1—fl?2)+H+ const.
In the very same way we find that
/x*x
■v/c—o ■seillall°
—*•+ I*) v/(l—+ 3 A+ const.
134. In the case of m, a negative number, we must
have recourse to formula (C), from which we find
r x~m (\—a;*)
p x~n~'t X 3 m
J J , ^zzlrx-^x
L m J ^
which formula, by writing —m instead of m—I, be¬
comes
r \/ (1—
r x \ (;»—1) a?”'-*
a?m y' (1—x%) I _j_ w—2 r x
L m—xm~* ^(1—a?1)
We cannot here suppose m~it for that value would
render the denominator =0 j therefore, before we can
apply this formula, it is necessary to investigate the
IONS.
or otherwise thus, put I—»a?*s:b*, then
751
Inverse
Method.
x=za/(i^z2), A?=-
V (!--»*)*
Therefore
x —^ fss
a? y' (1—-a;*) ~i—2, — i -f-2 1—%’
The fluent of the right hand side of this equation is
evidently (§ 103.)
—* ]- U+z) + i 1. (I-S!)=—i 1. (I±|) •
. (r-l-s;)2
or, since -—- = — the same fluent may be ex-
* /O 1 5S
pressed thus,
—x 1. Cl+25^-—1. I+a -
1—a'
V(I—O’
therefore, by substituting y' (1—.a?*) for a, and a? for
V (i—-O, we have
f _ , /i+y/O—**)\ ,
JXx/ (I—A?1)" * ( ^ J + C°nSL
If we suppose ?w~2, the formula becomes
v/(i—»2)
-j- const.
fluent of
^ We may easily find it from § 126,
X
x*^ (1—a;*)
If we suppose then
r \/ (1—
r X _ J 2Af2
J*W{I—*') ~~] + xr x
Xy/(i—A;*)5
which expression, by substituting for
j 77 7— its value, becomes
^ X(x—xj
r >y/(l—O
a? 3 2 a*2
7a;V(i-0~ ^ + j 1,
-f- const.
Of Finding Fluents by Series.
I35* We can always easily find an expression for the
fluentJx. x, where X denotes any function of a;; when
that function is expanded into a series, each term of
which is some power of x multiplied by a constant quan¬
tity ; thus suppose
X=rAa?”*-|-BArm +”-f-Cxm +lB-{- &c.
then X x is equal to
&x’r-
FLUXIONS.
Part II.
A xm tf + B A?m+nA? + C *w+SnA? Sec.
and taking the fluent of each term by § IOI,
A *w+, B
x*. x6
a*-\-x2
See.
Inverse
Method.
fxi=
w-f*1
0 «c*»+ *«+ *
‘ m-\-n -j-1
Sec. + const.
Hence, multiply both sides by a?, and taking the fluent
of each term, we get
If in the develepement of x there be any term of this
s* ax , *\ .
/— r= arc (tan. =r - ) + co«.s^.=:
Ja' + x* y a/^
form :_5 the fluent corresponding to that term will be
Al. * (§ 103).
136. The most simple function of x that can be ex¬
panded into a series is —, which becomes
I xx2 *3 . Ct
r + -T r + &c*
*1 n* * rt 5 fA •
X X X ^ t Q 1 /
-j 4- &c. + const.
a 3 a3 5 as 7 a7 1
If we wish to deduce from this equation, the value of
, X . .
the least arch whose tangent is —, it is necessary to
suppress the arbitrary constant quantity, for when that
arch =0, then j thus we have the arch whose
X
tangent is — expressed by the infinite series
a
Hence we find
x
X x JL x * 4 &c
a 3 o3 5 as 7 a7 *
X XX X*X X*X
a-\-x~ a a2 ** a3 a4
and taking the fluents
/•i_= 4 +&«•
J a-\-x a 2 a* 1 3 a3 4 a4
Let 7T denote the circumference of a circle whose dia¬
meter is unity, or half the circumference of a circle
whose radius is unity, then, as the sine of 30 degrees,
or is and its cosine y^C1—3> we *iave
7T — sm.^?r — Let be substituted instead
tan. - ~
O COS. ^5T
4 const.
Now we know thaty^-^ = 1. (a4^) (§ 57-) therefore
of x in the above series, and a be supposed =1, thus
we get
x*
1. — — —j 4 —3 — ^ “H
v ^ ^ a 2 a* 1 3 a3 4 a4 ‘
’=>/ixil-±-3+-±-1lp + bc.)
4(
4 const.
and therefore
To find the value of the constant quantity we have only
to make ^=0, for then the equation becomes 1. a—
const, therefore
r=x/l2X (l —V
3-3 5*3
7*3:
;4&c>) ?
x x" x*
a 2 a* 3 a3
hence, if we subtract 1. a from each side, and observe
1. (04#)—!• — —- 4 &c.
by taking the sum of about fifteen terms of this series,
we shall find *-=3.1415927. The determination of
this number is of great importance in every branch of
mathematics.
that 1. (*4o)—1. a=l. = (I+^) we get
x\ x x* x3 ft?4 -
^1 a J aj 2 a2 3 a3 4 a4
X X
138. By proceeding with the fluxion in
the
same manner as we have done with
1'rom this conclusion we may deduce rules for comput¬
ing the logarithms of numbers.
0*4 A?*’
we get
137. Let the fluxion be which may be put un~
rxmx _ *m+I
J an4«~ (m41)ore (?»4«4l)a:
Vm+1*+I
4. _Z__&C.
^ (/»42w4I)a3"
der this form and which consequently belongs to Adiis series proceeds by the positive powers of 01
is an ascending series, but we may also expand
the arch of which the tangent = — (j 60.). By re¬
ducing — into a series, we find
& a* 4**
into a series proceeding by the negative powers of ar,
and which will therefore be called a descending series.
Thus because
1
»let
“art II.
Inverse
IMethad.
I I a*" atH a**
— -r *5' *«'■ + &c-
FLUXIONS.
We may easily find an expression for the fluent of
xn-\-an
U x
the rational fraction (§ ur.) by expanding the
. U .
quantity — into a series, but the result thus obtained
is in general very complicated, and seldom convergent 5
besides, this manner of finding the fluent is hardly of
any use, since it may be expressed by means of arches
of a circle and logarithms, both of which are readily ob¬
tained from the common trigonometrical tables.
139. The fluent of xm-t x is easily ob-
tamed by first expanding the quantity into
a series by the binomial theorem, then multiplying each
term of that series by xn~x x, and taking the fluents
of the results by § 101. Thus we have
V
(a-f- b x*J
f y xm+H K p—q) b* xm+**
xm+3a t ?
w'e have, after multiplying both sides by xmx, and tak¬
ing the fluents
/'xm x
J xn-\-an
(n—m—1 )«’*-"
(2«—m—(3«—m—
+ &c. const.
This last series will be convergent, when x is greater
than o, and at the same time m^.n, and But
besides, that it may contain algebraic terms only, it is
necessary that none of the divisors n-—m—1,2n—tn—I,
3?i—m—I, &c. become =ro ; this circumstance will
take place as often as is a multiple of », and in
which case
the series
4" &cc* which is the develope-
X'~"' X1
ment of the fluxion, will contain a term of this form,
-, the fluent of which is arn 1. x.
x
get
If in this result we put m=2, nzzo, and o=ri, we
1.2.3 ? 0
w+3»
■f- f -f- const.
r x 1 . 1 1 . o
/ ; =— ; — ;4- &C. 4- Const.
^i+a;4 x ' •xx3 ex1 '
+A?4 x " 3 a<3 J i
But although the expression
I -J-A?4
is the fluxion of
the arch having x for its tangent, we must not conclude
that this series is the developement of that arch, for x
being supposed =0, each of the terms of the series be¬
comes infinite.
The consideration of the constant quantity added to
the fluent will remove this apparent difficulty, if we re¬
mark, that to know the true value of a series, it is al¬
ways necessary to begin with the case in which it is
convergent. Now the series
This is an ascending series, but to get a descending
\i? nP
series we must divide (a+bxnJ ? by a? ? and multiply
Arm“, x, the remaining part of the fluxional expression,
by the same quantity, thus the fluxion is transformed
to
A?’”’' q 1 x (b^-ax—n) ?
the fluent of which, by proceeding as in the former
case, is
^ r , , o—g)*
Lq>qxn+T pa gx^ q
l_mq~\-np qb m q^-{p—-q) n
X 3 Af®
5 xs
-f- &c.
, O—*g)"
X p—y) «2 q ^ q
1.2 b* tnq-\-(p—2 q)n
&c. f + consU
converges so much the faster as x is greater, and it va¬
nishes when x is infinite j but in this extreme case the
equation
arc (tan. =*)=—
5 x*
const,
4* &c.
becomes simply arc - = const, where denotes half the
2
Circumference of the circle $ therefore, substituting this
value of the constant quantity, we have
. *— &c*
2 a: ■ 3 at* 5 a:* 1
Vol. VIII. Part II. +
either of these series may be employed if a and b are
both positive, or q an odd number, but if q be an even
number, the first formula becomes imaginary on ac-
1
count of the factor a ? if ap be negative, and the same
thing happens to the second formula if a* be nega¬
tive.
140. Let it be required to express by a series the
fluent of •
x . T I I
arc (tan. =*)=- — —+ —
dical quantity
V C1—**)
That we may develope the ra-
we put it under this form,
iC
754
biTeise ^ ao expression which when expanded by
,Method- the binomial theorem, is
FLUXIONS. Part II. %
eary that the terms of the series should be each simply inrene I*'
a power of a? multiplied by x and constant quantities.
If for example we have this fluxion
Methed,
1 2 '2.4 2.4.6
therefore, multiplying each term of this series by x,
and taking the fluent, we get
X\/(l—e1*8)
in which e is supposed to denote a small constant quan¬
/:
l.X3
—x-
1 . 3 . AT*
tity, we may expand y^C1—e*»*) or (1—into a
series, which will thus become
2.3 2.4*5
1.3*5**7
const.
1 . * A A
I—— ta- — —t4.*4-
2
i*i*3
2.4
2.4.6
e6x6— See.
' 2.4.6.7
If we suppose x to denote the sine of an arch, then
*/ (1—«*') is its cosine, and —rr —r\ is the fluxion
and the fluxion -— rH- will be transformed to
of the arch itself (§ 59.)} therefore the series which
we have just found, expresses the length of the arch of
a circle, radius being unity, and the sine of the arch x.
If we suppose the series to express the smallest arch
that corresponds to the sine Af, then, as when the sine
of that arch =0, the arch itself =0, the series ex¬
pressing the arch must vanish when ajsso, therefore
we must suppress the constant quantity added to com¬
plete the fluent j or suppose it =0. The same series
has already been found by the direct method of fluxions
in § 72.
Let n denote the same as in § 137, then, as the sine
) i 2
1.1
-y/(l—-AT
2.4
e4*4
2.4.6
&c. J.
the series will converge very fast when e is small, for
that y'( 1—x*) may be a real quantity, x* must be les»
than 1. We must now multiply each term of the series
by the common factor
and take the fluents,
v/fi—#*)
which being all contained in the general expression
f-4
-, will be found by $ 133. Thus, putting
of 30 degrees, or *f is 4, we have, by substituting
4 instead of x in the preceding series, and multiplying
**)
A to denote an arch of which x is the sine, we have
both sides by 6,
»=3(i+TT^ +
*-3
i*3*5
&c
•)
/
•X\/(l—etx*')
1/(1—^)
2.3.2* * 2.4.5.24 ‘ 2.4.6.7.26
by means of this series, which involves only rational
numbers, we may compute (but with more labour), the
value of sr as before.
Suppose the fluxion to be x ^/(ax—x*)t which may be
. x 1
otherwise expressed thus, xa*x* CImmm
nomial theorem (1—
x 1.1 ac* 1.1.3 AC3
+i* *9)—
+r?{(ra+^)^-^-r4A}
G*'+ 'ik>+
2.4.6
i*3*5,
&c.
+ &c.
2.4.6
4- const.
2a 2.4 a* 2.4.6 o3
« r .
Let each term of this series be multiplied by aTArT at, and
the fluent taken by § 101, thus we gitf)c4/{ax—x*)z=:
Of the Fluents of such Fluxions as involve Logarithmic
and Exponential functions.
3 r 7
* / 2A?t I 2*t I. I 2 x’
° \ 3 2 5a ““ 2.4 70*
I.I . 3 2X* 0 N ,
§ ; —- &c.) 4- const.
2.4.6 9a3
142. Let it be required to find the fluent of x (1.#),
where 1. a: denotes the Napierean logarithm of x. In
this case, as well as in some following examples, we
shall have recourse to the principle already employed in
§ 130, namely, that if v and 25 denote any functions of
a variable quantity x, then
141. By resolving a fluxion into an infinite series,
the object in view is to transform it into a series of
other fluxions, each of which may have its fluent deter¬
mined by known methods 5 but it is not always neces-
J'% V — V% —•J'V 25*
Let us therefore assume xmxxzvt and 1. x~2-, then,
cr”,+*
(§ 10,*)>'^p7=v» and (§ 57)i -^ = ^erefore»
substituting
art II.
Iimnc gubstituting these values of v, %, v, sa in the formula
Metkod. becomes
Inverse
Method.
. /'XmX xm+t
or, since /—— = ;;—-—r + const.
(1- Urr- c^T-}+M’K‘-
FLUXIONS. 75S
If we suppose n~I, then observing that J*o* *=r~
« J<-).
fa*xx—j
If n=2, then
*"+*
, § 101, and zrr (l.fl)o’AT, (§ 56.) therefore,
substituting these values of v, 2, v, k in the formula
/* vs:t;a! as, we get
Cm « * o" *"'*'* /^(l. o) a**H+,Ar
/ £f" AC* Af — “■“ / . r
»-{-I ^ »-{-I
therefore, substituting n—I everywhere instead of «,
/- . • a*** /'(l. a) c*#"*
a* jk’1-* x— /- ;
« n
hence, bringing J'cfx'x to stand alone one side of the
equation
y-» • a* xn r» /»«.*
a’x% xzz — fa* x 1 x.
i. a. 1. aj
y'aTx,xz
1. a (1. a)»
fa* xx.
const.
1. a
1. a*
Iset us next suppose that the proposed fluxion is
xm x (1. x)\ Put xmx~v, and (1. A?)“=a!, then (§ 101.)
Af'“+t n x (\. a;'!"'-*-
—V, and (S cy.) rrss, therefore,
gubstituting as before these values in the formula
.A v—v 25 —f » z, we get
r • , . *-+1(l. «)" r»A;mA: (I.ac)*-*
/ xmx (1. xyzs * — / : .
J ' J m-\-l J 771-J-1
It is evident, that by this formula the determination
of the fluent of xm x (1. #)* is reduced to that of
x™ x (1. A?) which we have already found, and in
like manner that the determination of the fluent of
x (1. #)3 is reduced to that of Acm a? (1. x)*, and so on,
from which it appears that the fluent of Acm x (1. #)*
is expressible in finite terms when » is a whole positive
number. The formula, however, will not apply when
tn—— 1, because of the denominator *7* -f-1 rr—I 1 =0.
But in this case we have
f— (1. Af)"= —I— (1. at)"** 4- const.
.f x 7*-j-l
If n be negative or fractional, the fluent of xmx (1. Ac)*
can only be expressed by an infinite series.
143. As an example of an exponential function, let
it be required to find the fluent of o’ x* x. Here we
may put v for *" x, and z for a*, then we have
In this expression we substitute the value of f a* x x
just found, thus it becomes
ix
(1.0)*
2 I .
rr y 4* const.
. o)»J
(1
Proceeding in this way, we may find the fluent when
77—3, or when 77=4, or in general, when t» is any in¬
teger number whatever, the number of terms in the
fluent being in this case always finite; it is not so how¬
ever when n is either negative or fractional.
Of the Fluents of such Fluxions as contain functions
related to a circle.
144. Let suppose that =r , § 101, and since
77-J-l’ 5
arsrsin. *, we have *=25 cos. (1—-**) (§ 59.),
and therefore i thus we have
V (i-**)
/n • 2S A? ’r* I p * t»x
a xn ac= — —- j
774-1 774-1 —* )
hence the determination of the proposed fluent is re-
f #*+**
duced toy ■' — n which we have already consider-
v'(I—**)
ed in § 133. By the same mode of reasoning we may
determine the fluent when * denotes the cosine of the
arch 2;.
145. It appears from § 59. that n 2$ being put to
denote any arch of a circle to radius unity, the fluxion
of the sine of that arch is t» a cos. 77 a; therefore, on
the contrary,
/25 cos. 77 *= - sin. 77 z 4* const.
77
?•
iC2 yi.i
In like manner, from the formulas of J 59. and § 60.
we find
sin.
75*5
Invesse
Method.
FLUX
y'js sin. n %— cos. ti « + const.
y-—^—=r - tan. w 4- const.
cos. n% n
C-— n —- cot. w z 4- const.
^ sin.2 n% n
/z sin. n% \
= - sec. » a 4- const.
cos.* n% n
/% cos. n z —I , .
nr cosec. n z + const.
sin.2 n% n
146. By the second of these expressions we find the
fluent of
IONS.
Part II,
% (A-fB sin. 58+C sin. 2Z-{- &c.)
to be
A 53——B cos. z—Hi C cos. 2Z—&c» const.
and from the first expression we find the fluent of
2; (A-J-B COS. 53 + C COS. 2 58 + &C.)
to be
A z + B sin. z+f C sin. 2 z-f- &c. + const.
147. It has been shewn in the Arithmetic of Sines,
(see Algebra, § 356.) that
sin* z=-f- (—cos. 2z-f-0»
therefore, by what has been shewn in § 145.
J*z sin.* cos. 2Z-}-z)
= £ (— f sin. 2 z-f-z) + const.
It has also been shewn that
sin.3 z~-J (—sin. 3 *4*3 s^n* 2:)>
therefore, multiplying each term of this expression by
z, and taking the fluents,
fz sin.3 9t=l (f cos. 3 z—3 cos. z)-j- const.
In the same manner may the fluent of z sin."z be
found, n being any positive integer number whatever.
Again, it has been shewn (Algebra, § 356.)»
that
cos.* z=^ (cos. 2 z-j-1) i
therefore
f 58 COS.* co9*
zzl (i sin. 2Z+*) + const.
and because
cos.* z=! (cos. 3 z+3 cos. z)
therefore, multiplying by z, and taking the fluents, Jnveise
Methed.
J'z cos.3 zzrf (f sin. 3Z-J-3 sin. z) + const.
and proceeding in this way we may find the fluents of
z cos." », n being any positive integer number.
• •
148. The fluents of z sin." z, and z cos." z may be ex¬
pressed under another form, by proceeding as in § 142.
Thus, beginning with z sin.” z, and resolving it into z
sin. zxsin."-* z, if we put z sin. z=:t;, and sin.7*”1
zz=tt we have by § 145, fr:—cos. z, and (by § 26
and § 59) <=(72—1) z cos. z sin."“*z, therefore, sub¬
stituting in the formulaJ't'Oz=.i)t—-j'vtt we have
/z sin." z=:—cos. z sin."“Iz
4-(«—cos.*z sin."-*z,*
but cos.* z=i—sin.* z, therefore f z sin."z is equal to
—cos. z sin."-* z-{-(«—1)/z sin."~*z
—(«—1)J*« sin."z;
which expression, by bringing together the terms con¬
tainingf z sia.nz becomes
/z sin."z=r— - cos. z sin."”* z
n
4- -—- f« sin."-* z.
n J
By giving particular values to n we have
/,. . 1 . 1 /»*
z sin.*=:—- cos. z sin. z-f- -y z
=— - cos. z sin. z4- - z 4- const.
2 1 2 1
/z sin.3z=:— - cos. z sin.* z4- - fz sin. z
3 r
—— - cos. z sin.® z— - cos. z 4- cows#.
3' 3
We may proceed in this way as far as we please, dedu¬
cing the fluent of z sin.4z from that of z sin.* z, and
the fluent of z sin.* z from that of z sin.3 z, and so on.
If in the general formula we substitute every where
2—ft instead of ft, it becomes
/*
z sin.
z=
«—2
cos. z sin. z
ft—1 . .
1 / z sin.^z,
r ft—2*'
an
inverse
| Method
Tart II. FLUX
s an expression which, by bringing ^% sin. ~n% or
/» %
—-— to stand on one side of the equation, becomes
sin. 1
rJ-
J sin." *
cos. 2;
(«—1) sin.
n—2 n as
IONS. Ih*}
n z've substitute 4 *•—3 instead of (where \ * MeThod.
denotes a quadrant), and observe that
sin. (4sr—»)=cos. 25, cos. ({tt—2;)=sin. z,
and that the fluxion of (£*-—2:) is —ss, we shall imme¬
diately obtain
n—2 /» 2S
w-—I J sin.
'53
This formula is not applicable to the case of «m, be¬
cause then each of the terms of the fluent is divided by
«—1=0, and therefore becomes infinite. In order to
obtain the expression for the fluent in this particular
case, we proceed thus. It is evident that -r^ =
sin.2 23
1 , , 1 1 1
—, but — L
I—cos « I—C0S*2! 2(1—C0S.23 ^ 2(1-j-COS. k)
as will be found by reducing the fractions to a com¬
35 COS."53= — sm. 2; cos."
n
n—i r-
H / 2
n J
/V 25 I
^C0S.”25 n 1
ss COS."-* %
sm. %
cos.
25
, 2 r g
n—1^/ cos.”-!
mon denominator, therefore
sin. 25 2(1—cos. z)
+
and in like manner from the formula expressing the flu¬
ent
sm. 2;.
25
2(i+c.s. an<1 c»”sc4“e"lly.
25 I 25 sin.2: ( I y*2! sin. z
J sin. 25 ~ 2*^1 —COS.25 ' 2'' I+C0172S’
butif it be considered that the fluxion of cos. 2 is —25
sin. 2 (§ 59.) it will appear by 5 103 that /I.^ ?in‘_z
=1. (1—cos. 2), and that—l(i-fcos.2),
therefore
r 25 1. . . 1
J ~ — '• (x—-co3’ *) r. (i-fcos. 2) + const.
2
_£l /I—cos^N
2 \I -}-cos. z/ '
sin. z
we deduce
=u.(i±^) +consl.
™ cos. 2 \I—Sin. z/ ~
__1. \/(i + sin.z)
\/(i—sin. 2)
const.
sin.«
const.
_ j y/(i—cos. 2)
* \/( t -f-cos. 25)
const.
V'Ci+cos. 2)
If in the general formula for n— we suppose 2
we have
cos.2
sin. 2
150. It has been shewn in Algebra, § 357, that
16 cos. * 2 sin. — sin. 52-f.sin. 3Z + 2 sin. 2, there¬
fore
j* z cos. * z sin. ,z= cos* J 25— - cos. 3 2;
—■2 cos. 2) -J- const.
The same mode of finding the fluent will apply to any
fluxion of this form z sin.^z cos.” 2 ; or by resolving
the fluxion into two parts, the determination of its
fluent may be reduced to that of a fluxion in which the
exponents m and n are less than in the proposed flux¬
ion, by the method of proceeding already employed in
§ MS-
ISI‘ T•
ABzta?, BP=y, the diameter AD=a, the area ABPnj.
From the nature of the circle y*=:a x—x*, therefore
y—\f{ax—a;*), and
s—fy Z =y* v' (a*—AT1).
In this case the fluxion is not of such a form as to
admit of an algebraic fluent in finite terms, we must
therefore have recourse to the method of series, but we
have already found the fluent in this way in § 140,
therefore, from the series there brought out we have
/ (2X 1 2**__1,1 2x1
s_ V ° ^ g 250 2.4 7 a*
1.1.3 2x*
m-j-n
fore the two values of the expression "
be considered as indicating the two areas APB, AP'B,
zc 3 ““ &c0
2.4.6 9 a3
this expression does not require a constant quantity to
be added to it, because wiien a?=zo we must also have
5=0.
If we suppose the arch AP to be f of the quadrant
AE, then it is known that PB=£ the rad. AC=f a,
therefore, if we suppose the radius = 1, we have in this
case BC=V3. and ^=1^^3 = 0.1339^6
nearly. If this number be substituted instead ot x, and
a few terms of the series computed, we shall find the
area ABP =*0452931} to this add the triangle
CBP=}X v/x=::0*2165063, and we have the sector
A.CP=*26l7994, which number when multiplied by
3 gives *7853982 for the area of the quadrant. This
number also expresses the area of a circle of which the
diameter is I.
Fx. 3. Suppose the curve to be an ellipse.
Pat Fig. a i*
the
Part II.
)n verse the transverse axis AD^o, the conjugate axis
j**c*bod-. 2CEs=3, also AB=a;, BP=:y,- then by the nature of
(lie curve y=-L j (o.v—*•), and ii=yi=L-
FLUXIONS.
~ * v/ (*>_*)_»= al i. £ liViy—°*) |
but if a straight line be drawn from C to P so as to form
(oa’—x*') ; but if a circle be described on AD as a
diameter, and BP the ordinate of the ellipse be produced
to meet the circle, it appears from last example that
* V (a is the fluxion of AQB the segment of
the circle corresponding to the elliptic area APB or s;
therefore, putting v for the segment AQB, we have
the triangle CBP, it is manifest that — x J (x* aM
is equal to 4 CBxBP, that is, to the triangle CBP,
therefore the excess of the triangle CBP above the
area s, that is the hyperbolic sector CAP, is equal to
the logarithmic function
759
Inverse
Melhod.
• b v bv . t
s— and here the constant quantity c must
be suppressed;, because e and v must vanish together.
Hence it appears that the area of any segment ef an
ellipse is to the area of the corresponding segment of
its circumscribing circle as the lesser axis of the ellipse
is to the greater; therefore the whole ellipse must be
to the whole circle in the same ratio.
Ex. 5. Let the curve be a hyperbola, of which C is
the centre. Put the semi-transverse axis CAxro, the
semiconjugate axis =6, CB=.v, BP=ry, the area
APBrrj?. From the nature of the curve ——
J a
V therefore
tf-f- V (a*—a*) *»
~~ i’
Ex. 5. Suppose the curve to be an equilateral by-Fig. 23.
perbola, that is a hyperbola whose axes are equal, and
that it is required to find the curvilineal area DCBP
comprehended between DC, a perpendicular from D
(a given point in the curve) to one of the asymptotes,
and PB, a perpendicular from any other point in the
curve to the same asymptote.
Let A be the centre, put AC=o, CD=6, AB=r*,
ftrea DCBP—From the property of
the asymptotes we have xy=,ab. and therefore y= ~
J x''
hence (§ 103.)
s=fy*~tfx (**—c»).
„ • />abx
S J y *=J— —Cib 1. X-f-Ca.
But it appears from formula B (§ 131.) that
y; v/ (**_a*) =i* S - wf 7, \
v \.x J
and again by $ 127,
// (**_«») =1# {Ar+^ + c,
therefore
3=1 2aXx/ a>) — ~^ V'(**“«*) ^ + c*
To discover the value of the constant quantity c, we
must consider that when xzxat then s=o, and in
this extreme case the general equation just found be¬
comes
hence czs ~ 1. a, and consequently, observing that
— T1"^ *+'/} +
oij r \/ (aj-—o*) T|
~““T * l a $
we get
>= A * ^ (*W) _ ^ 1. j i±^£-=£l> j.
It immediately follows from this formula that
To find the value of c, let us suppose x=:a, then sso
and the general formula becomes in this case
o=ra b 1. o-|-c, and hence err —a b 1. at
therefore
t—a b 1. x—a i 1. a~ab 1. —.
a
If we suppose a=A= i, then s=z\. x, from which it
appears that in this case the hyperbolic area DCBP
represents the Napierean logarithm of the number xf,
it was from the consideration of this property that the
logarithms originally invented by Napier were called
hyperbolical logarithms.
But the logarithms of any other system may also be
represented by areas of the same hyperbola; for this
purpose it is only necessary to determine the magni¬
tudes of o, and £, so that — “M, where M denotes
the modulus of the system, thus we shall have
and j=ra# M1. , or, putting cm, s=M 1. an ex¬
pression for the logarithm of x according to any sys¬
tem whatever of which the modulus is M (Algebra,
$ 287.)
. the curve be the cycloid of which AEFig.
is the axis and A the vertex, let a semicircle be de¬
scribed on AE as a diameter, draw AG perpendicular
to the axis, and from any point in the curve draw PB
perpendicular to AG and PD perpendicular to AE,
meeting the circle in Q, and draw QC to C the centre
of the circle. Put AC=c, ABm, BP=y, the area
ABPnrs, and put v for the angle ACQ, that is for
the arch of a circle which measures ACQ, the radius
ofi
^6
fluxions.
Inverse of that circle being unity, then AD=o (i—cos. v),
Method. DQ=fl sin. vf and arch AQ=av, and since from the
nature of the curve, PD=: arch BQ-j-DQ, therefore
PD=av4-o sin. v=a O-f-sin. v) j hence
Fig. i$-
xzza (u-f-sin. v), x^zav (i-fcos. t>), (§ 59.)
y—a (1—cos. v)
s—J'x y—J*aZ v C1—cos**t’)
=uzJ'v sin.* v—^a* v—f«a sin. v cos. t; (§ 148.)
ACx arch AQ—f CD X DQ}
and here no constant quantity is wanted to complete
the fluent j because upon the supposition that AQ=o
both sides of the equation vanish as they ought to do}
now it is obvious that -J ACxAQ—area of sector
ACQ, and f CDxDQ= area of triangle DCQ, there¬
fore
$=£ area of circ. seg. ADQ.
Let AG be the greatest value of x; complete the
parallelogram AGFE, then from the general expression
for the cycloidal area, it follows that the whole cycloi¬
dal space APFG is equal to the semicircle AQE ; but
from the nature of the curve, EF is equal to AQE,
half the circumference, therefore the rectangle EG is
equal to four times the semicircle AQE j from these
equals take away the external cycloidal space AGF,
and the semicircle AQE, which have been shewn to be
equal, and the remainders, viz. the internal cycloidal
space APFE, and three times the semicircle AQE,
are equal to each other.
Part II.
ion of a curvilinear area, in § 61, is not immediately inverse
applicable when the nature of a curve is expressed in Method,
this way, we shall therefore investigate another formula '■“■“■v——'
suited to this particular manner of considering curves.
Let us suppose that APR is a curve the position of Fig. ts,
any point P of which is determined by PF, its distance
from a given point F, and the angle which PF makes
with AF a line given by position. Let a circle be de¬
scribed on F as a centre with a rad. =1, then FP, as
also the area FAP, may be considered as functions of
BD the arch of that circle which measures the angle
PFA. From F draw FP' to any other point P' of
the curve meeting the circle BD in D\ Put FPnrr,
the area FAPrr.?, the angle AFP, or the arch BDiri',
then the area PFP', and the arch DD', will be the cor¬
responding increments of s and t>, therefore, § 21,
• “ limit of
area FPP'
DD'xFD*
Here DD' the increment of v is multiplied by FDm,
to render the terms of the ratio homogeneous. On F
as a centre, with FP as a radius, describe an arch of a
circle meeting FP' in Q, then, as the sectors FDD',
FPQ are similar, we have
FE
hence DD'xFD
FP*:: FD x HD': FPxFQ=2 sect. FPQ,
2FD
FP!
sect. FPQ sect. FPQ
T
s r* area FP'P
and t- = — x 1,m* Sect. FPQ *
v
153. In some cases it is more convenient to refer a
curve line to a fixed point than to an axis. Thus in¬
stead of expressing the nature of the circle by the equa¬
tion y^—a x—a?*, where y denotes a perpendicular from
any point of the curve upon a the diameter, and x the
distance of that perpendicular from one end of the
diameter, we may otherwise express it by the equa¬
tion fczrrt', where % denotes a variable arch of the
circle reckoned from one end of its diameter, r its
radius, and v the angle contained by a line drawn from
the centre of the circle through the extremity of a, which
angle is measured by an arch of a circle having its radius
unity.
The nature of the different conic sections may be de¬
fined in the same manner. Let P be any point in a
conic section, of which F is one focus, and FA a part
of the axis •, let DC the directrix of the section meet
FAin C, join PF, and draw PB perpendicular to the
axis, and PD to the directrix j then from the nature of
the curve (Conic Sections) PF has a given ratio to
PD, that is to FC—FB ; putFC=a, FP=r, the angle
PFC=u, and suppose PF : PD :: n : 1, then PF=
«.PD=:«.FC—n.FB, hence observing that FB=FP
, a n
X cos. v, we get r~a n—nr cos. t;, and r=:
but the point P' being supposed to approach continually
• HFGcl
to P, it is manifest that the limit of is unity,
’ sect. rPQ
or 1, therefore
L — if, and s =-r*i\
V 2 ^
154. By means of this formula we may find the
areas of that class of curves called spirals. Let us take
for example the spiral of Archimedes, which may be
defined thus. Conceive a straight line FR to revolve Fig. *7*
about F the centre of a given circle, departing from
a given position FB $ conceive also a point P to move
in the revolving line, so that PF its distance from the
centre may be to BD the arch of the circle passed over
by the revolving line, as m to «, then the point P will
generate the spiral.
Put BF=a, the angle BFR=t;, the line FP=r,
and the area generated by the line FP=^, then the
arch BDr=a v, and since from the nature of the curve
71V * 71 V
r \ av V. m \ n. therefore v— , and vzz—, hence
am am
• 1 «* , • n T
the general formula sz= —r*t; becomes 5— ~2 a nT
i + tfcos. 1; therefore,
which equation expresses generally the nature of a conic
section.
/»n r%r n 1
J\zam 6 c
Mr3
m
154. The formula which we have found for the flux-
3
this fluent does not require a constant quantity to be
added*
Part II.
FLUXIONS.
Inrerse added, as both s and r evidently vanish at the same
Method, time.
F.'r i*.
Fir *9-
156. As the general expression for a curvilineal area
BCPD is /V x, where a?“AB the abscissa reckoned
from a given point A in the axis, and y=BC the ordi¬
nate, it follows that X being put to denote any function
of a variable quantity x, the fluent of may always
be exhibited by means of a curvilineal area. Thus let
CPp be a curve of such a nature that AD and DP the
co-ordinates being denoted by x and y, the equation of
the curve is y=X, then, assuming any ordinate BC as
given by position, we have evidently
A arzrarea CBDP.
As the ordinate BC (which is assumed as given by po¬
sition) may be taken any where, the fluxion of the area
being the same wherever it is taken, it appears, as has
been already observed (§ 101) that the function f Xa?
may be considered as indeterminate, for it admits of in¬
numerable values corresponding to any particular value
of x, and in this respect it differs from an algebraic
function, which for a given value of x has always a de¬
terminate number of values. If however x be supposed
to increase from any determinate magnitude c, to any
Other determinate magnitude a', then, taking the ab¬
scissa AD=o, and A d^La\ and drawing the ordinates
DP, dp, we have
when x~a, J'yixzz area CBDP,
and when *=a', xtz area CB dp,
therefore, while x increases from a to a', or receives the
increment a'—a, the function increases from area
CBDP to area CBJp, and thus receives the increment
area YD dp, which is of a determinate magnitude, as
the ordinates PD, p d have both a determinate position.
Hence it appears that in assigning the fluent of X#, we
only determine the change that takes place in the va¬
lue of the function while x passes from one par¬
ticular value to another particular value.
157. As there are general and known methods by
which an approximate value of any curvilineal area may
be found, when a fluent is expressed by such an area,
those methods may be applied to find an approximate
value of the fluent. Let PD ef/? be a curvilineal area,
supposed to represent the fluent f^x between the li¬
mits of #=: AD and x—A. d. Conceive D to be di¬
vided into a number of equal parts DD', D'D", Dd, and
the ordinates P'D', P^D" drawn, and the two sets of
parallelograms DE, D'E', D^E" and DV, D'V, d e" to
be completed, the former constituting a rectilineal figure
circumscribed about the curvilineal space DYY'Y" pd,
and the latter a rectilineal figure inscribed in that space;
then as the circumscribed figure must necessarily be
greater than the curvilineal space, that is, greater than
VOL. VIII. Part II. f
/x x taken between the limits of a;=:AD and «=A a,
and the inscribed figure must be less, it follows that if
we compute the areas of the circumscribed and inscribed
figures we shall obtain two limits, the one greater, and
the other less than J'^Ax. And as by increasing the
number of equal parts into which D J is divided we may
bring the circumscribed and inscribed rectilineal figures
as near to a ratio of equality as we please, it is always
possible to find two limits which shall differ from each
other, and consequently from J'^.x (which lies be¬
tween them), by less than any assignable quantity.
158. If we join P, P', P", py the tops of the ordinates,
the rectilineal space formed by the trapeziums DPP'D'
D'T" p d will be more nearly equal to the
curvilineal area, than the circumscribed rectilineal fi¬
gure formed by the parallelograms DE, D'E', D"E" ;
therefore, the sum of those trapeziums being found, it
will be equal to the fluent f^'x nearly.
Suppose, for example, that it is required to find the
-lue °f/-rrrr between the limits of #=0, and xzz 1.
I-f-iV
In this case X= —so that the equation of the
curve Yp is let us suppose Dd the distance
between the extreme ordinates to be divided into ten
equal parts, then putting o, *1, -2, &c. to I instead of 4;
76l
in the formula
i-f-w3’
we obtain eleven successive
values oiy, or eleven equidistant ordinates, the nume¬
ral values of which will be as follows,
The first =r i*ooooo
the 2d =r
the 3d rr
the 4th rr
the 5th —
the 6th r:
•99010
•96154
*9I743
•86207
•80000
the 7 th zr *73529
the 8th = '67114
the 9th ~ *60975
the 10th *55249
the nth oz *50000
By the elements of geometry the area of the recti¬
lineal figure formed by the trapeziums is found by
adding together all the ordinates except the first
and last, and half the sum of the first and last, and
multiplying that sum by the breadth of one of the tra¬
peziums j now the sum of the ordinates, with the ex¬
ception of the first and last, together with half the sum
of the first and last, is 7*84981, and the common breadth
of the trapeziums is *1, therefore■ ; = 7.84981 x
•1=,758y+'/0*+4ya)|j
Ex. 2. Suppose the curve to be a circle, and that Cfjg, *o.
is its centre, and AE a quadrant of the circle. Put
CBrntf, BPrry, the arch EP=«, the radius of the
circle =a, then a;*-}-y*zr«*, and y=f (a*—**), and
• —xx
(*!+
--f-
\/(l—e*xx')
^/(l—,r3)
-r*= limit of
v
chord PP'
chord DD'
?
now the limit of the angle PQD being evidently a right
angle, we have
This fluent can only be expressed by means of an infi¬
nite series, and it has been already given in this form
in j 141.
If we take l,then all the quantities in that series
which are multiplied by v/(i—*2) will vanish, but in
this particular case 55 is the elliptic quadrant EA, and
A is a quadrant of the circumscribing circle, or f sr,
therefore the elliptic quadrant is equal to
r PP' ,• -v/CPQ’+QP'1)
lm' do7~ lm‘ ' un- J
= lim. \/ ^
PQ* , P'Q
i3D,J ' 13D'
■}
but
PQ3 __ FP*
2 an<^
p'Qa __
DD'3~
r*
v*
therefore
ig. it.
w Cl-— e*
2.2
1.1.3
r‘T»3-3-5
2.2.4*4" 2.2.4.4.6.6
e6—&c.)
This series converges very fast if e be a small fraction.
Ex. 4. Suppose the curve to be a cycloid. Let a
circle be described on its axis meeting the ordinate PB
in Q, and draw CQ to the centre of the circle. Put
ABzsat, BP~y, the cycloidal arch AP~ss, the radius
AC—a, the angle ACQ—t;, then AB=a (1—-cos.«;),
BQ—a sin. v, the circ. arch AQ—a so that x~o
(1—cos. ti), and from the nature of the curve y~~~n
(%»+ sin. ti), therefore (§ 59.)
xzza v sin. v, y~a n (i -f cos. v),
** +y*=o*^,|sin.s v-f(i -f cos.
rra*i;a(2-f-2 cos. v);
but 2+2C0S. i;=4cos.*f t! (Algebra, § 356.) there¬
fore
3—j*C*s+^*)—2 0^ucos. f v
= 40 sin. fu+C, (§ 145.)
but as when v=.oy then 2S—o, therefore C=o, and %
—4 ° s*ni‘ \ v > but if the chord AQ be drawn, 2 a sin.
f tizr chord AQ, therefore chord AQ.
, and 2; rry' v*7'2 j.
Let us apply this formula to the spiral of Archimedes, Fi* 17
the equation of which (§ 155.) is amv-nr, and
therefore
nr •
, and v1 —
am
«2 r9
2 2 i hence
a* ni1
*=fv <>+r'v')=\j-rS(a'+
This fluent may be found by formula B, § 131, and it
is worthy of remark that the fluxion has the same form
as that which we have found in § 161 for an arch of a
parabola j thus the length of any portion of the spiral
of Archimedes may be exhibited by means of an arch
of a parabola.
To find the Contents of Solids.
163. If AX3 the abscissa of a curve be denoted by xt Fig- 4.
and PD the ordinate by y, and the solid generated by
the revolution of the curve APD about AD as an axis
by s, it has been shewn, in § 64, that s=vy2x, there¬
fore the general formula for finding the content of a
solid is
I. Jot
162. The formula %— yv(*. -f-#*) not being ap¬
plicable in its present form to curves of the spiral kind,
we shall here investigate another suited to that parti-’
eular class of curves.
Let APR be a curve of such a nature that the posi¬
tion of any point P jn the curve is determined by PF,
its distance from a given point F, and by the angle
which PF makes with AF a line given in position.
We shall employ the same construction and notation
here as in § I54> with the addition of drawing the
chords DD', PQ, PF, and putting the arch AP—<3;
then it is manifest that the simultaneous increments
and r will be the arches DD' PP', and the
Straight line P'Q respectively. Hence
*
£$
V
5= limit of
arch PP'
arch DD'
Ex. I. Suppose the solid to be a paraboloid, or that
which is generated by the revolution of a parabola about
its axis j in this case^a=a«, and taking the fluent so
that when xzzO, then srro,
•*=»/ya a f x Jr a x2,
or but nxf is the content of a cylinder
having y for the radius of its base and x for its altitude, •
therefore the content of a paraboloid is half that of a
cylinder having the same base and altitude.
Ex. 2. Suppose the solid to be a parabolic spindle,Fig
which is generated by the revolution of APB an arch *
of a parabola about AC an ordinate to its axis. In this
case let AD—DP=y, AB—£, the parameter of the
5 1^ 2 axis
*'»K- 3*-
and 3.3.
Kig. 4.
axis rrcr, then from the nature of the parabola AD X DB
(b—.v^x
and
rraxPD, thatis»(i—*)=:ay, hence y=
taking the fluent, so that s and x may vanish together j
rr irj'y2 x — ~x%(b—a^x
— - {b* x*x—2bx3x -\-x* x}
it f h~.v3 b x* | X* \
r+y/
or, since a‘
(h—,r)2*
'
b'x
1
•xy1 ^ b'x bx* | x9
= (^y>iT“^' + 7r
which expression (by supposing x—ACzz^b, and put
for
ting d for CE, the greatest value of y) gives
the content of half the solid generated by the curve
, . . 8 * a* £ ,
AEB, therefore the entire spindle is ——, or (by
observing that xd*b is the content of a cylinder having d
for the radius of its base and b for its length) it is x\-
of the circumscribing cylinder.
Kt. 3. Suppose the solid to be a spheroid produced
by the revolution of an ellipse about either of its axes j
put a for ^AB the axis round which the curve revolves,
h for fEF the other axis, x for AD the height of any
segment made by a plane perpendicular to the axis of
♦he solid, v for PD the radius of its base, and s for its
‘ b'
content. Then from the nature of the curve ya =~
(2aa‘—a'*), therefore taking the fluent upon the sup-
position that s and x vanish together,
s—x J'x — X'laxx—x* x)
xb% x*\
= —T* (ax )•
^ v 3 /
FLUXIONS. Part II.
perpendicular to that axis, meeting it in D, and put- Inverse
ting AD=j?, and the variable solid APQp (considered Method,
as a function of x') =:$, by proceeding as in § 64, we » 1 —J
,,, . , , . . increment of« ,
would have found the limit ot ^ 7—, and con-
mcrement or x
sequently A-, equal to the area of the section of the solid
x
made by the plane PQ/?; therefore putting V for that
function of x which expresses the area of the section, we
have szzVx, and szz J'V x.
Let us suppose for example that AEFG is a solid Tig. 34,
bounded by any plane figure EFG as a base, and by
the surface which will be generated if we suppose a
straight line drawn from A any given point above that
plane to revolve in the circumference of the base.
Let AC be a perpendicular drawn from the vertex
of the figure to its base, and let PQ/> be a section of
the solid by a plane parallel to the base, meeting the
perpendicular in D. Put arr the area of the base of
the solid, V= the area of the section PQ/J, £=AC the
altitude of the whole solid, #=:AD the altitude of the
part cut off by the plane PQ/>, and sz=. the content of
that part $ then, as from the nature of the solid it is
pretty evident that the part of it cut off by the plane
PQp is similar to the whole, and as the bases of similar
solids are as the squares of their altitudes, we have
d* b
7T
V &a : #a, hence V:
a x
~~9r
and
✓\r * a r 1' axi
s=fVx = ¥fx*=j^,
To find the content of the whole spheroid, we have
only to take x 2 a, thus the formula becomes
and as 2r>rb%a expresses the content of a cy-
3
linder having 2b for the diameter of its base, and 2 a
for its height, it follows that the contents of a spheroid
is x that of its circumscribing cylinder.
It is obvious that what has been found for the
spheroid will apply also to the sphere, by supposing
the axes equal, or a—b>
164. If instead of supposing the solid APQ/> to be
formed by the revolution of a curve round its axis (in
which case it is called, a solid of revolution) we had sup¬
posed it to have any figure whatever, then by referring
the solid to some straight line AC, given by position,
as an axis, and in which A is a given point, and sup¬
posing PQ^ to be a section of the solid made by a plane
this expression for s does not require the addition of any
constant quantity, for by putting *=0, we have 5—0 as
it ought to be. Suppose now x-=zb, then 5= -^=\aby
from which it appears that the content of the whole
solid is4»of the product of the base by the perpendi¬
cular. It is evident that pyramids and cones are solid#
of the kind we have been considering.
To find the Surfaces of Solids.
165. The altitude AD of a solid, generated by re-yig. ^
-volution of a curve about AD as an axis, being as be¬
fore denoted by x, and PD the radius of its base by y,
let us now put s to denote the curved surface of the
solid, then, as it has been shewn, § 65, that*' = 2 try
*J (** we have
s=2«-J'yv'C^+y2)
as a general formula for the surface of a solid.
Ex. I. Suppose the solid to be a sphere, generated by Fig. 35*
the revolution of a circle about its diameter AB, put
the radius of the sphere =a, then AD being denoted
by x, and PD by y, we have from the nature of the
curve y*=2ax—x*, therefore
, • (a—x)x
yrry'C20 *—"* )>
*f(2ax—x') ’
and
x
art II*
1 averse
.lethod.
FLUXIONS.
I'S- ?•
«8+^* = ^(I +
(a—*)1
lax-
2\
i'J
a11 x*
ax
2 a x—x'1 y*
therefore, y v/C^'1+ya) = o^v, and taking the fluent, so
that when x—o7 then 5=0,
:2w J1] ix*-\-y*') — 2 va x ;
now If It be considered that 2-x a\$ the circumference
of a great circle of the sphere, It will Immediately ap¬
pear that the surface of a segment of a sphere is equal
to the circumference of a great circle of the sphere
multiplied Into the height of the segment. Hence it
follows that the whole surface of the sphere Is equal to-
four times the area of one of its great circles.
ting
we "
Ex. 2. Suppose the curve to be a parabola, then put-
g AD =x, DP=y, the parameter of the axis =za,
have (§ 161. example 1.)
nple I.)
VO* y V C«9+4 y*)» therefore
*=2*yv \/(**+ya)
=-£/*;
(aa+4yaF+c» $ Io8-
To discover the value of the constant quantity C, we
must observe that when x~o, then y=o, and ^=o,
therefore putting o Instead of s and y, the above equa¬
tion becomes 0= -7- +C, hence C—
and
'=*{
(Qa + 4ya)T—Q3
6 a
l
(having weight), of which the centre of gravity is re¬
quired, and that PB, PD are co-ordinates drawn from
any point in the curve perpendicular to AB, AD two
axes at right angles to each other j let the arch AP re¬
ceive any increment P p, let C be the centre of gravity
of AP, G the centre of gravity of P p, and C' the centre
of gravity of APp. From C and G draw CE, CF,
GH, GK perpendicular to the axes AB, AD. Put
PD=.r, PB=y, CF=X, CE=Y, AP=3, also let the
arch APp—x', and let the distances of C' its centre of
gravity from the axes AD, AB be denoted by X',
and Y' respectively; then, observing that the arch
Pp by the proposition in last $,
sX-Ks'—sOxGKzzs'X';
hence -GK.
55'—a
If we now suppose the arch P p to be continually di¬
minished, and observe that ss'X'—aX, and ss'—a, are
the simultaneous increments of aX and a, it will ap¬
pear (§ 23.) that
7G5
Inverse
Method.
flux, of (aX)
limit of GK.
By the very same way of reasoning we find
flux, of (aY)
limit of GH,
but the point p approaching to P, it is manifest that
the point G will also approach to P, so that the limit
of GK is PD or .v, and the limit of GH is PB or y,
hence
flux. (aX) flux. (aY)
. —x. .
=y.
To find the Centre of Gravity of any Line, Surface, or
Solid.
166. It belongs to the theory of Mechanics to ex¬
plain what is meant by the centre of gravity, and to de¬
monstrate its general properties, and here it is only ne¬
cessary to shew how the method of fluxions may be ap¬
plied to deduce from some one of those properties rules
for finding that centre in any proposed case.
The properties of centres of gravity which we shall
employ as the foundation of the application of the
method of fluxions to its determination may be enun¬
ciated shortly thus.
ig. jd. Let C be the centre of gravity of a mass of matter
denoted by M, and c the centre of gravity of another
mass m, and D the centre ,of gravity of the two masses
M and m, from these points let perpendiculars CA, c a,
DE be drawn to any straight line PQ, then
M X CA-f-m xc o=r(M-f»0 X DE.
g.37' 16’]. Let ui now suppose that AP is any curve line
flux. (*X)=:afx, flux. (jzY^zzyz.
Taking now the fluents of each side of these equations,
and dividing by »,
X=
it is evident that by these two equations the position of
C the centre of gravity is determined.
168. Let us next suppose that it is required to find C fig- 37>
the centre of gravity of the plane area APB. As the
arch AP was in last § supposed to receive the incre¬
ment Pp, so let the area APB now receive the incre¬
ment BP pi, and let C, C' and G (which in the former
case were supposed to be the centres of gravity of the
arches AP, APp, and Pp respectively) now be sup¬
posed to be the centres of gravity of the areas APB,
Apb, and BPpb; put the area APB=r.y, the area
A p b—sf, and let X, Y, X', Y', denote as before.
Then, reasoning exactly as in last case, we have
(by § 116.),
* X-f-(^—s) x GKrryX'
xY-K*'—*)xGH:=:yY'
hence
766
Inverse hence
Method-
Fig. 37'
yX' S X ^ s’Y'-
5= UK, -7-
SJ—S
=GH j
and the point being supposed to approach to P, so
that ^X'—sX, «'Y'—sY, and .s'—the simulta¬
neous increments of X, Y and s, may be continually
diminished,
£i±4£2Q=lira. GK, ----I^Ulim. GH j
, 5 S
but as the ordinate pb approaches to PB, it is manifest
that the ultimate position of G, the centre of gravity of
the area BP/?£, will be the middle of PB, therefore
the limit of GK is x, and the limit of GH is fy, thus
we have
flux. (sX) flux. (sY)
—; =*, : =Ty>
s s
and consequently,
V-fXS
FLUXIONS. Fart ll
jyo. If instead of the centre of gravity of the sur- Invem
face generated by AP, the centre of gravity of the Method
solid generated by the revolution of the plane figure '"—"V'*
APB about AB as an axis be required, the reasoning
will be the very same as in last §, substituting the
solid generated by the plane figure instead of the sur¬
face generated by the curve line $ so that putting i
for the content of the solid, and X for AE the di«
stance of its centre of gravity from the vertex, we have
also
or since srry .v, and $ zz fy x 6i.),
fy** Y_/y‘*
~fy* ’ 2/y*
169. Let it next be required to find the centre of
gravity of the surface of a solid generated by the revolu¬
tion of the curve AP about AB as an axis. Let the
surface of the solid be conceived to receive an incre¬
ment generated by Py) an arch of the curve. In this
case it is evident that the centres of gravity of the sur¬
face generated by the curve AP, the surface generated
by the curve APy>, and the surface generated by the
arch Pyj, will each be in AB, the axis of the solid j
suppose them to be at E, E' and H respectively. Put
AE=X, AE'—-X', also put s for the surface generated
by AP, and s' for the surface generated by APpt then
as before (from § 166.) we have
* X+ {s'—s) AHrrs'X',
hence ^=1?™ AH,
s'—s
and = lim. AH j
s
but the point p approaching to P, the limit of AH is
manifestly AB or #, therefore
flux, (s X) , v
\ ■ ~ and Xrr
fX
or since szzl-xyz (§ 65.),
Z*JxyK
X=
ivj'yz
X=
-fx
but in this case s—vifx (§ 64.) therefore
V* *xx Jy*xx
X=
171. We shall now apply these formulas to some ex¬
amples.
Example 1. Let it be required to find the centre of Fig. 37,
gravity of AP an arch of a circle. Suppose AB to
be a part of the diameter, and in addition to the nota¬
tion of § 167. put a for the radius of the circle, then
from the nature of the curve, y*zz2ax—x*, hence
• CIX
(proceeding as in § 165. Ex. I.) we have 2= 'j'
therefore zyziax, and %xz: but from the equa-
tion y*—2ax—x9, by taking the fluxions we get
• XX Cl X • •
y 'yzzax—xx, and hence = ■— — ySs;—y,
. y y
therefore i x—a{%—y) 5 substituting now the values
of kx and ssy in the formula of § 167, we have
x=
= ■Ly'(s-y)
%
(2—y+c)
/yZ. = iL/*
Y~
To discover the values of the constant quantities
c, c', we have from the equations in which they occur,
a crrXs;—a 2+ay, a c'—Y%—a x ;
but when 0, then x, y, X and Y are each
-therefore cs=o, and c'=o, thus we have simply
Xsr Y—
Ex? 2*
art II. FLUXIONS.
ifnverse Ex. 2. Tjet it be required to find the centre of
'VIethod. gravity of APB an area bounded by AP an arch of a
circle and PB, Bx\ its sine and versed sine. Let a de-
•57’ note the radius, and let the remaining notation be as
in § 168. Then, because At, we have xs'xzyxx,
but from the equation a x—xz (which expresses
the nature of the curve), we find
fy' XX fx (^2 ax*—x3~)
fy'x fx (2 ax—at*)
xxzzax—yyt therefore
a; s—ayx—y* y~a s—if y.
We have also y s=.yxx=z (2 a at—at*) at, therefore,
f xs f(a s—y*y)
X =: ~~ ■
767
Inverse
Method.
4«at3—j-at4 -|-ct
axz—4*3 +c'’
and reasoning as in the last example, we find c=:o, and
c'—o, and therefore
8 OAt—3 at2
12 a—4 x *
If the segment be a hemisphere, in which case x~at
then Xz=|o.
Sect. Ill, Of Flux tonal Equations.
Ty3+C)
Y=
fy* A 2ax—at*) at
2s
2 S
=— (aat*—f ati+c').
172. It has been shewn 49,)» how, from an equa¬
tion being given, expressing the relation between at a
variable quantity, and y a function of that quantity, we
may deduce the equation that expresses the relation of
their fluxions. We are now to show how from the lat¬
ter, or jluxional equation, we may return to the equa¬
tion of the fluents, which, relatively to the other, may
be called its primitive equation.
37<
By proceeding as in the last example we find c and ef
each =0, thus we have
a x*—x*
6 s
Ex. 3. Suppose now the figure to be the surface
generated by the revolution of AP an arch of a circle
about the diameter AB, and that the centre of gravity
of the generated surface is required. Then because
from the nature of the circle we have y 53= a x
y
and x y o& “ ax x, therefore, substituting these values in
the formula of § 169. it becomes
\x?-\-c
x-^-c' *
To find the values of the constant quantities ct c', we
have
c=:X (x-j~c')-»-f.r*t
i.r*-fc
but as when x=o, then X=o, it is manifest that c and
c' are each rro, thus we have
X=:4^.
Ex. 4. Let us now suppose that it is required to
find the centre of gravity of the solid generated by the
revolution of AP an arch of a circle about the diame¬
ter. In this case, because y'zzZax—x*, we have
from § 170,
173. As any primitive equation and the fluxional
equation derived from it both hold true at the same
time, and as the constant quantities which enter into
the former retain the same values in the latter, it fol¬
lows that by means of the two equations we may exter¬
minate any one of the constant quantities, and thus from
any proposed primitive equation deduce a fluxional equa¬
tion, in which one of the constant quantities contained
in that primitive shall not at all be found.
For example, let the primitive equation hey+ax
-\-bzzo, by taking the fluxions we have y-^-ax=o,
a fluxional equation in which b is not found; if, how¬
ever, it be required to find an equation in which a shall
be wanting, we have only to eliminate a by applying
the common rules of Algebra (Algebra, Sect, vii.)
to the two equations
y-f-aa’-f-fcrro, y-{-axzzO’,
and hence we have yx—xy-\-bx—o, thus it appears
that from the primitive equation y-J-aA;-j-^=ro we may
deduce a fluxional equation which may be expressed
under either of these forms,
y-j-OAzzo, yx—x y-\.bx=zo\
these hold true at the same time as the primitive equa¬
tion, they are alike related to, and any two of the
three being given the other necessarily follows from
them.
As a second example, suppose the primitive equation
to be .r2—2 ay—o*—irzo, bypassing to the fluxions
we immediately find xx—ay—0, an equation in which
b is not found. If, however, it be required that the
fluxional equation shall want a, we have only to apply
the common rules of elimination to the two equations j
thus from the second we get a— . , and this being
y
substituted in the first it becomes
.r
763
FLUXIONS.
Fart II,
lareise
M fcthoti.
lxy x x*x*
■ b =0,
t'roni which we have
—b') y*—-‘ixyy x-—xixtz:o
and taking the square root, having previously reduced
the equation to a proper form,
yV —yy—xx—o.
174. It is evident that by proceeding in this manner
we shall, in some cases, arrive at a fluxional equation in-
2/
volving the second and higher powers of —rj and when
this happens we can only find the value of by the re¬
solution of an equation ; but this may be avoided by
preparing the primitive equation in such a manner, that
the constant quantity to be eliminated may be entirely
separated from the variable quantities, so as to form one
of the terms of the equation, then, upon taking the
fluxions, this term being constant will vanish, and thus
we shall obtain an equation entirely free from the con¬
stant quantity contained in that term. Thus the primi¬
tive equation y-^-ax-^b—O has already such a form that
by taking the fluxions we gety-J-flwrro an equation in
which b is not found. If it be required, that upon ta¬
king the fluxions, a shall vanish j we must put the equa¬
tion under this form -f- a=o, and then taking the
fluxion, we find immediately
———————— u,
x*
contrary any fluxional equation being given, its primi¬
tive equation may contain one constant quantity more
than the fluxional equation, but it can contain only'
one, for no more than one constant quantity can be
made to disappear by returning from the primitive to
its fluxional equation.
an expression in which a is not found, and which by
rejecting the divisor x* becomes yx—xy-^bx—o, and
these two forms of the fluxional equation are the very
same as have been found in the last §. In the second
example, viz. x*—2 a y—a*—b—ot the equation has
already the form suited to the elimination of £, for the
fluxional equation is xx—oy=o, but in order that a
may vanish, we must resolve the equation with respect
to a, so as to give it this form,
y— ^(>*+y9—£)+«=0 j
passing now to the fluxional equation, a disappears, and
we have
y\/ O* -fy»—6)—yy—y.r
'/(y+y*—
It is evident that we have only to reject the denomina¬
tor to give the equation this form,
y v/0*+y*—£)—y y—**r=o,
the same as was found in the conclusion of last §.
Ib verse
Method.
If
Hei
176. The fluxional equation expressing the value of
which is derived from any primitive equation in¬
volving #, and y a function of x, may be called a flux¬
ional equation of the first order; and as from this equa¬
tion considered as a primitive, we may in like manner
derive an equation that shall involve -^-(§ 50.),thislast
x%
may be called a fluxional equation of the second order,
and the fluxional equation from which it is derived may
be called its primitive equation of the first order, to
distinguish it from the absolute primitive equation, from
which all the others are conceived to be derived. A
similar mode of definition is to be applied to the higher
orders.
177. As any primitive equation and the fluxional
equations of the first and second orders derived from it
must all hold true at the same time, it is evident, that
by means of the three equations, we may exterminate
any two of the constant quantities contained in them
that we please, and thus produce a fluxional equation of
the second order that contains two constant quantities
less than the primitive equation. There are however
two other ways by which we may arrive at the very
same fluxional equation of the second order. For as
from the given primitive equation we may deduce two
diflerent fluxional equations of the first order, one
of which shall contain one only of the two quantities to
be eliminated, and the other shall contain the other
quantity only j we may consider each of these equations
in its turn as a primitive, and, by proceeding in the
manner explained in § 173 an^ § I74» flerive from it a
fluxional equation, in which that particular constant
quantity which remained in its primitive, but which was
to be finally eliminated, shall not be found j thus, from
each of these primitives we shall deduce the very same
fluxional equation of the second order, that shall be
freed from two of the constant quantities contained in
the absolute primitive equation.
Let us take for example the equation
x*—2ay + b'zzO j
by proceeding as explained in § 173* or § 174» we
these two fluxional equations of the first order,
xx—ay—o, (^a+i*)y—ixyx—O,
in the one of these the constant quantity a is wanting,
and in the other b is wanting. Taking the first equa¬
tion xx—ayzzO, and proceeding as in § 50 (observing
that x is constant) we find**—a y z: o, if from this equa¬
XX
175. From what has been now shewn we may infer,
that as from any proposed primitive equation we can
deduce a fluxional equation that shall contain one con¬
stant quantity less than the primitive contains, so on the
3
tion we now eliminate a by putting instead of it -r
(deduced from the equation ar#—ay—.0) we£nd&fter
proper reduction
FLUXIONS.
Part II.
Inverse
Method.
y*—*y=o,
a fluxlonal equation of the second order, in ■which both
a and b are wanting, and having ar*—2ay-\.b* for its
absolute primitive equation.
Let us now take the other fluxional equation of the
first order which involves b, viz. (*2-f£2) y 2 x yx
=zo; by proceeding with this as with the former we
find y—2 y a,2zro } from the first of these
equations we find ;r2-f-£,—
2xy x
• »
y
and from the se¬
cond x^^b'zz
hence we have
2yx*
y
therefore,
2xyx ly x*
y ~ y ’
and
l6g
Now the fluent of is ml. # -f-c' ($ 103.), and in Method.
Inverse
like manner the fluent of —is n 1. therefore
m\. x +«l.y+c,-}-c"=:o,
or, transposing c-f-c", and putting a single constant
quantity for their sum, which, to be homogeneous with
the logarithmic quantities, may be — log. c, or — 1. c,
ml.tf-f/il.yrrl. c, or 1. (»n) +1. (y^ssl. nz /»225 2
i—nz-\-z2~ J (i—z)2~ i—2;’
let the terms be now collected into one expression, then
observing that •§ 1* (1—-2^-f-z2) r=i* (1—2>)2 =
1. (1—z), we have
l.a?+l. (1—z) 4-
= Cj
and, substituting instead of »,
(^) + ^=c>
1 \ x / x—y
or, substituting 1. c instead of C, and collecting the lo¬
garithmic functions into one,
x—7/ —x
!• ' —— 1
c x—y
therefore, passing from logarithms to numbers, by ob¬
serving that, as when a=l.^, we have by the nature of
logarithms eazzp, where e denotes the number of which
the Napierean logarithm is 1, so in the present case we
and taking the fluents of the terms, observing that each
being a logarithm function, their sum may be put equal
to a constant logarithm,
1. 1. ^z+ x/(i + z2) ^ = 1. C,
y
which expression, by substituting for z its value —,
becomes
,.I=1.C+, ^y+^f+r}].
If we now consider that
(y W**+?) (y—x/^+y*) = —*8»
and therefore that
.7+ \/Q* +.r) —x
« y—VO'+y1)’
it will appear, that the above equation may be other¬
wise expressed thus :
l.tf_l.c-fl.
from which, by passing from logarithms to their num¬
bers, we find y—v'(a*+y*) =—•, and hence, by so
ordering the equation that the radical may disappear,
we get «2=c24-2 cy, which is the primitive equation
required.
181. An equation which is not homogeneous, may in
some cases, by proper transformations, be rendered ho¬
mogeneous > this is the case in particular with the equa¬
tion
( o 4-+Hy>+(£+p?=<7y) y=o,
which is general of its kind $ for this purpose we assume
*=*4-*, and y=w4-/3, then x=zi, (and y=u ; by sub¬
stituting these values of a, y, x, y, in the proposed e-
quation it becomes
have
•r—y _
x—y
and hence the primitive equa¬
tion is found to be
(a+mct+nP+mt+nuyt
++Pc6u
Let us now suppose x and fi such that
1”
X
x—y
x—y—c e —O.
As a second example let the fluxional equation be
xy—yx—x—x ^{x2 4-y*)
which is also homogeneous. Assume as before y^.xz}
then y—xz-\-%x, and substituting these values of y
and y in the proposed equation, it becomes
^-^(i 4-a5*)—xz~Ot
a+mx-\-n/3=:o,
b+px+qP=oy
by these two equations the values of x and fi are deter¬
mined, and the transformed equation is reduced to
(mt-\-7iii)i+(pt-\-qu)u=oy
an equation which is homogeneous, and therefore may
be treated in the manner explained in last §.
This transformation will not apply, however, when mq
—fj p=o, because then the values of x and p would be
r infinite.
hrt II.
FLUXIONS.
Method, infinite. In this case we have y— —and therefore
hence the original equation
may be expressed thus,
Let us take a.particular case, and suppose the equa^
tion to be y-f-yar—then we have Psi, Q=:»",
an<^ ,/^> x:=x’ hence in this case the general formula
becomes
77C
Inverse
Method.
ci x y
-f(WA?-f/?y) (x+!Ly)
Assume nowmar-J-wy, theny+^ -l-x; the va¬
ra n
lues of mx-\-ny and y being now substituted in the
equation, and the whole reduced to a proper form, it
becomes
* , {hm-\-p %)%
st -f- — — ~0
a in n—b mt -}- (m n—p m) ss
The fluent of the second term of this expression will in¬
volve logarithms, except that m n—p m—o, in which
case the primitive equation is
2(a in n—b rai2)
182. When a fluxional equation has this form
y-f-Pyaf=Qi
where P and Q denote any functions of ar, the variable
quantities may be separated in the following manner.
Assume yriXsi, then, taking the fluxions, we have
y—zX-f-Xz, and by substitution, the proposed equa¬
tion becomes
z X-J-Xzs-J-PXzx~Q,x;
now as in this equation X and z may be supposed to
denote indeterminate functions of x, we may divide it
into two others, such, that the variable quantities in
each may be separable ; to effect this we assume
Xz-f-PXs x=o, zX=zQx;
hence, dividing the first equation by X, we have
«-j-P z *1=0, and —-J-Parrro, and taking the fluents,
l.*+/Pi=0 , and hence, by passing from logarithms
to their numbers,
—fPx
J8=e $
here no constant quantity is introduced, it being suffi¬
cient to add it at the end of the operation ; let this
value of z be substituted in the second equation, then
by deducing from it the value of X, we have
X=/P^Q^
and X=rf J2x Qx-f-c;
and since y=X z, therefore
y=g fPx Qx+c^*
I
y—e A?" ff-f-c J1.
The fluent J* (f xnx may be found by § 143 j let
us suppose for example that ra=2, then we have
e* a2 x=zex (a;2—2 x—2),
so that the fluxional equation being
y+yA;=A;2 *,
the primitive equation is
y=x*—2 a?-|-2-1-c e~*.
The general equationy-f-Py'A;=QA;, which involves
the simple power only of the variable quantity y, and its
fluxion, has been called a linear equation of the first
order } it has also, with more propriety, been called a
fluxional equation of the jirst degree, and of the first
order.
183. The equation
y+Py x— Qy™
where P and Q as before denote any functions of a?, is
easily reduced to the form we have been considering;
for assume yT-n=r(i—n) then y_ny=», and yrz^y”,
and y= (1—-ra) zyn ; if we now substitute the values
of y and y in the equation, it becomes
y” —n) P « y” x—Qynx ;
let the terms of this equation be divided by y", then,
including the factor (1—.») in the indeterminate func¬
tion P, the result is
z-f-P^Ar=:QA:,
an equation of the very same form as that which has
been considered in last
184. The most general form that can be given to a
fluxional equation of the first order, and consisting of
three terms only, is
yu 2;* b -J- £ ug z1 u — a u %■? u ;
to give this equation a more simple form, let all Its
terms be divided by yu , it then becomes
k—f * . ^ g—i h—f * & e—i •
5 Jz -J—ra6 z J u—-u «.
Suppose now
zk~~f z — -—^
, j/-7' «-=
then = y, ra8 <^'I=rAf,
5 E 2
and
772
Inverse
Method,
h-f
y (5—*+Oy
__ Ot—g_i+7 ^;
(#—'-f-Oy
FLUXIONS. Part II.
visor, which was common to them all, having disap- inverse
I^et us in order to abridge put
(£—t/H*1)/* /
’ U-*'+Oy“
e—g
{g—i+^y
^-/+i
g-—i+r
then the eqviation becomes
peared. In such cases, however, if we can by any
means discover that factor, by restoring it we shall im- ’
mediately have a complete fluxion, the fluent of which,
with the addition of a constant quantity, when put =o,
will be the primitive equation.
For example, if the equation be ary—ya;=o, here
xy—yx cannot be immediately produced by taking the
fluxion of a function of x and y ; but, if we divide the
x y y x
equation by so as to give it this form — =c,
we obtain the expression ■■-a— which is a complete
Method.
yJrhynx~axmx.
If we suppose n—iy the resulting equation y ^y a*
—a xm x may have its variable quantities separated by
the method explained in § 183 j but if we go only one
step farther, and suppose «=2, so that the equation
is
yJrbykx—axm *,
the difficulty of separating the variable quantities gene¬
rally is so great as to have hitherto baffled the utmost
efforts of the most expert analysts. This equation is
commonly called Riccati’s equation, on account of its
having been first treated of by an Italian mathematician
of that name, who succeeded in separating the variable
quantities in some particular cases, namely, when m is
equal to 4-—, where p denotes any whole positive
number.
185. If the separating of the variable quantities ge¬
nerally be a problem of insurmountable difficulty when
the equation consists of only three terms, its solution
can much less be expected, when the equation consists
of four, or any greater number. There are, however,
particular cases in which some of the most skilful ana¬
lysts have, by employing happy and peculiar artifices,
succeeded in resolving the problem, but the methods of
proceeding are, generally speaking, not reducible to
any determinate rules.
186. When the expression which constitutes a flux-
ional equation is such as would be produced by taking
the fluxion of some function of x and y, in which case
it may be said to be a complete fluxion, then, without
attempting to separate the variable quantities, we have
only to add a constant quantity to that function, and
the result put =0, will evidently be the primitive equa¬
tion required.
If, for example, the equation be wy+y tfrro, it is
obvious that the expression xy-\-yx is immediately pro¬
duced by taking the fluxion of the function xy (y being
also considered as a function of x), therefore the primi¬
tive equation is xy-{-c=zO.
From the view which has been given in § 174. of the
origin of fluxional equations it appears, that in passing
from a primitive equation to its fluxional equation, the
terms of the latter in many cases will not constitute a
complete fluxion, by reason of some multiplier^ or di«
3
V p V
fluxion, viz. that of the fraction-, therefore —+ c=c,
’ xx
ory-f-ca?=0, is the primitive equation.
In like manner, the equation mxy^ny x~o, which
does not in its present form express a complete fluxion,
yet becomes so when multiplied by «’*“* ym-1, for then
it is
m x* ym~% y -\-n xn'r't ym x—o,
from which it appears that the primitive equation in
this case must be xn ym -^czzo.
187. That we may be able to discover whether the
terms of any proposed fluxional equation constitute a
complete fluxion, and also from what expression such a
fluxion has been derived, we must attend to the process,
by which we find the fluxion of an expression composed
of two variable quantities, one of which is a function of
the other.
To avoid very general reasoning, we shall take for
granted what is evidently possible, that any function of
x and y may be generally expressed by a formula oi
this nature,
A xm yn=B y*+C xr y'-f- &c.
where A, B, C, &c. denote constant quantities, and
the exponents m, n, &c. given numbers, the number of
terms being supposed either finite or infinite. Now
the fluxion of the whole expression is the sum of the
fluxion of its terms, but in taking the fluxion of each
term, beginning with the first A xm yn, the fluxion of
which is
m Axm~t ynx-\-n A.xm y"-^,
it is evident that the result is composed of two parts,
one of which is the expression we would find for its
fluxion, if x only were considered as variable, and y as
constant, and the other is the expression for its fluxion,
if y only were considered as variable and x as constant j
hence it follows, that the sum of the fluxions of all the
terms will have the very same property j so that, if u
be put for the whole expression, we shall in every case
have
«=:MA;-j«Ny,
where M x denotes the result that will be found if the
fluxion of u be taken upon the hypothesis that x alone
*9
Pa:
In
III
Part II.
FLUX
Inrerse Is variable, and Ny is the fluxion of e/, supposing y alone
Method. t0 be variable> ^
188. Resuming the consideration of the general ex¬
pression
Axm y" +B Xp 7f -f CVy'-f &c.
let the fluxion of any one of its terms, for example,
Axm yn, be taken, supposing x alone variable, and the
result is mAxm~tynx. Again, let the fluxion of this
result be taken, supposing y alone variable, and we find
it to be tn n AKm~x yn~lxy. Now, if we first take
the fluxion of Axm y’\ supposing y variable, we get
n Axmyn~xy, and then, the fluxion of this result, con¬
sidering x alone as variable, we get mnAxm~tyn~tx yt
which is the very same expression as was found by pro¬
ceeding in a contrary order j and as the same must hold
true of all the terms, we may conclude, that if the fluxion
of u any function of at and y be taken, considering^ only
as variable, and then the fluxion of that result, consider¬
ing y only as variable, the very same final result will be
obtained as if we were first to take the fluxion of u sup¬
posing y variable, and then the fluxion of that result,
supposing x variable j but the fluxion of u being ex¬
pressed thus, Mar-f-Ny, it has been shewn that Ma? is
the fluxion of «, if x only be supposed variable and Ny
is its fluxion, if y only be variable, therefore, if we take
the fluxion of Ma; upon the supposition that y only is
variable, also the fluxion of Ny upon the supposition
that x only is variable, the results must be identical.
This property affords the following rule, by which we
may always determine whether any proposed expression
constitutes an exact fluxion or not. Let the expression
he put under this form, M x-j-Ny , let M'y be the flux¬
ion of M, supposing y alone variable, and N'x the flux¬
ion ofN, supposing x alone variable, then, if M/ and
N' are identical, Mx + Ny is a complete fluxion; and
if they are not, Mx-f-Ny is not a complete fluxion.
189. It is easy to see, how, from a complete fluxion
«=rMa;-{-Ny we may determine u its fluent j for as
M x has been deduced from u by considering x as va¬
riable, and y as constant, on which account all the terms
of u that involved y only must have vanished, it follows
on the contrary, that if we put Y to denote those terms,
we shall have
«=y^Mar-fY,
the fluent of Mar being taken, regarding x only as va¬
riable. The function Y may be determined, by com¬
paring the fluxion of the expression thus obtained with
the given fluxion Mar-j-Ny.
owJ- 2xx-\-yx-i-xy
Ex. 1. Let the fluxion be
expression when reduced to the form t/isMar-f-Ny is
* — , (fl-f.r)y
u~ 2V(.ay+*‘+xy)+ 2v'(oy+*'+*x)
IONS.
hence M ~
2*4-y
r, N=.
o+ar
773
Inverse
2 v/(ay+•*“* +*y)’ 2v/C«y-j-«*-f-Xy)’ t Method. ^
the fluxion of M, supposing y only variable, gives us
MrrMVzr (cy2a ^0.7
4(«y+*2+*30i’
and in like manner the fluxion of N, supposing x only
variable, gives
N=N^= Qy-Ky—
4(«y-j-ara+.ry) * ’
hence it appears that M'zzrN', and therefore that the
proposed expression is an exact fluxion. To determine
its fluent, the formula uzz J'Mx-^-Y gives us
u= \f Oy+*a+*30 + Y;
the fluxion of this expression taken, upon the supposi¬
tion that both x and y are variable, is
u—
ay 2x x -\-y x -\-x y
(oy+*2+*y)
+Yj
this result, compared with the original fluxion, shews
that Y==o, and Yire, a constant quantity.
Lx. 2. Suppose the fluxion to be
AT ^/0*-fy*)-fy
(ai-t-xy-}-2yi)
Here M=v/ (a*+y*), N= , and by
proceeding as in last example, we shall find M'rrN'sr
■ ■ rv» hence it follows that the expression is a
complete fluxion, and the formula u =/M x-f-Y shews
that J'x y/ (a*-j-y*)-J-Y
f (o*+y*) + Y.
To determine Y, we take the fluxion of this ex¬
pression, supposing x and y both variable, and find it
to be
u—xf (a*+y*) -f-
*yy
+ Y,
V C“*+y2)
and this compared with the original fluxion
.l.ews lhat Y= > he"ee
Ja'+lff . ((J,+
therefore the fluent required is
774
Invei’s-e
Method.
FLUX
«=.r v/(a9+?8)+C
—(x'Vy) V
where C denotes a constant quantity.
190. It may be demonstrated, that as often as a
iluxional equation does not constitute a complete fluxion,
there is always an infinite-number of factors, such, that
if the equation were multiplied by any one of them,
the result would be a complete fluxion. A general
method of determining some one of these factors, how¬
ever, seems to be a problem of such difficulty, that its
solution, except in some particular cases, is not to be
expected.
IONS,
-4-z=p, we find
x
V'C1 -j-p3)—c />=!*»
Part II, fa
Inverse I"
Method.
y=:bpj(i Jrp*)—\ap«—b fp y/(i -fpl)
the fluent oip may be found by the formulas
given in § 130, and \ 131.
191. When a fluxional equation involves the second
or higher powers of x and y, as in this example,
y2.—o* x2—0.
which may be put under this form,
193. When we cannot by any means find an ex¬
pression for the relation between x andy in finite terms,
then we must, as a last resource, have recourse to ap¬
proximation, that is, we must express the value of y in
terms of x by means of a series.
When the form of the series is known, we may de¬
termine the coefficients of its terms, by substituting the
series and its fluxion instead of y and y in the proposed
equation.
Suppose, for example, that the equation is
X-a-o,
we may, by the theory of algebraic equations, deduce
from it the values of -L, considering this quantity as a
x
root of the equation *, thus, in the. present example, by
y-\-yx—m xnxzzo,
we may assume
y=A x* + B x*+l + C + &c.
. t* 1
resolving the quadratic equation K- —a2—o, we have
a?2
then y—kx* B x*x
"f" C*^I* + &c.
~*~n) so that y—and y-|-a#—0, hence
Substituting now the values of y and y in the equa¬
tion, and dividing the whole by a?, it becomes
y—fl«-|-c=o, y-ftftf + c1
./—,
are two primitive equations, from either of which the
fluxional equation y*--o2*2=o may be derived, and
it may also be deduced from their product
(y—a* + <0 (y+otf+cOzro.
* A**-1 + 0+0 B 1 ** + 0*+2) C1
-mxn+ AJ + B3 X
+ C-+3)^+&c.=o.
192. As often as the equation contains only one of
the two variable quantities, for example xt by the reso-
This equation becomes identical, if we assume
a—Ior «=«+!, and
V
lutiou of the equation we may obtain --=X (where X
x
denotes some function of #)» and hence y^=-f X x> but
if it be more easy to resolve the equation with respect to
x than to which we shall denote by p, then, instead
-m
m
, m p_ —^ c
^ ot* «0+1)’ ®C*+1)0+^-)*
D=
-m
*0+0 0 + 2) Ca+3)
Hence we have
, &c.
of seeking the values of p from the equation, we may
find that of x, thus we shall have A?=rP, some function
of p, and hence a; = P, and since y=y .1?, therefore,
y=p P, and y=J^pV^Vp—J'~Pp. The relation
between x and y is now to be found by eliminating p
by means of the two equations
O+i)
,n+3
0+2)
O+0(»+2)(«+3)
*=rP, y—¥p—fv p.
As a particular example, let us suppose the equation
to be xx + o y~b ^/0*+y*), from which, by putting
In order that a primitive equation may be general, it
ought to contain an indeterminate constant quantity
more than is found in the fluxional equation, therefore,
this series which contains no such quantity, must be
considered as incomplete, or as exhibiting the value of
y, upon the supposition, that, when wrro, then yrro.
However, we may obtain a value of y that shall be ge¬
neral,
FLUXIONS.
Fart II.
nera,» by proceeding as follows. Let us suppose we
kno7 tbat when x==a, then ; assume x=a+t, and
y—b + u, then it is manifest, that if the value of w be
.boy denotes a s-“d i”dete™i"-——■
of x and y the equation
y=:xJfCX’Y c'
775
Inverse
Method.
y-\-yx—mx^x—o
becomes
u+(J’+u)t—m(a+t)tn—o.
Assume now
K=Ar+ Btae'|-r4.C^'l-2+ &c.
then, proceeding as before, we find
«Af* I + (*+i)B^+(«+2)Ct*+r+&c. 1
+5
+
Af“+
B,
w a—m
n n—i. n(n—\') n 2 » o I
—a t—m -a i8——&c. j
^=0.
I . 2
J
It is necessary, in this equation, to assume *—-i=o,
or tt~i, and hence we find
A=m an —b, B=
C-
&c.
m n (n—i)gn—2—mnan—x+man—b
2^3 ’
Asy P A;rrP x—JVx—xf X.%—f Xxx,
we have also
y—xj'x x—J*Xxx-\-c x-\-c'.
Suppose, for example, that the equation is y axx'zzc,
so that ~-—ax} here Xrza#, and therefore
V-vf 8^^—J'2 A?* X-\-C x-^c'
~\a x3—fa a;3-J-c ,r-j-c/
x3-\~c x-\-cJ.
In the very same manner we may deduce from the
equation of the third order
y—Xx3-o, or, -L=X
a-3
its primitive equation $ thus we have
i=x*>I=/x*=p+‘.
If we now substitute a?—-a, and y-—b for t and u re*
spectively, the result will have all the generality that
belongs to a primitive equation, expressing the relation
between r and y.
Of Fluxional Equations of the second or higher orders.
194. Whatever difficulties occur in finding the pri¬
mitive equation of a fluxional equation of the first or¬
der, it will easily be conceived, that these difficulties
must be greater and more numerous when we have to
consider fluxional equations of the second and higher
orders.
One of the most simple cases of an equation of the
second order is this,
y —X ^*=0, or =X,
x*
where X denotes a function of x, the variable quantity
whose fluxion is supposed to be constant $ in this case,
because -L =X x, we have =J'X x. Let P denote
x x
that function of which X# is the fluxion, and c, as usual,
an indeterminate constant quantity, then 4- = P4-Cj
x
and ynPtf+cff, and taking the fluents a second time
where P denotes such a function of x, that its fluxion
is X x, and c represents a constant quantity. Again
~ = P X-\-CXy
X
ri? x-\-cx-\-d;
here Q is put forand c' for a second constant
quantity. In like manner we have
• • • •
y—Q x -\-c x x-\-d xt
and yzx. J'Qx-^-^c x'^c'x-^-c1';
and as P and Q are functions of xt the fluents of P#
and Q# may be found by the methods formerly ex¬
plained.
195. Let us next consider such equations as involve
V V
only -v-j and constant quantities. In order to abridge
x x
V •
let us put =rp, then such an equation may be gene-
x
rally expressed thus -L=rP, where P denotes some
known
77g FLUXIONS. Part II.
jSl known function of f ; now no 4-=p, by taking the 2/V y+c, hence J)= 4-= -J (c^Jy y) and
fluxions, and observing that a? is constant, we have p y
JL=JLt hence ^-=P, and and let
xx x P r
*=r—
+2/Yy)
7-+C',
where c and d denote twe constant quantities,
the value of x be substituted instead of it in the equa- To take a particular example, let us suppose the equa-
tlon y =p and It becomca j = and hence tion to he y-y*'=„>, or 4=u+y, here y=e+y.
y=f£f i thus It appears that If we can find the ^ S/Y^=2ny+y*, hmce (and b, § 127O,
1 j
fluentsy^-and^^-, we shall have the primitive \/(c+2°3'+y ).
equation when we eliminate jp by means of these two
equations
3'=c'+ M
= 1. la+y+VO+^+y*)}+t'*
• * y y
107. When the equation contains -4-, -r- and x, it
y 1 xx*
, . , . may be transformed to a fluxional equation of the first
where c and d denote the two indeterminate constant i , . . . • , • • . . i *
quantities that ought to enter into the primitive equa- order, by substituting in it px and p *, instead ot y
and y; if we can find the primitive of that fluxional
equation, and thence the value of p in terms of x, we
shall have the value of y from the formula y=f y *, or
if we have the value of a? in terms of then, because
f P *=P x—fxp> we shall have
y=p x—fx p.
tion.
Suppose for example that the equation is
• ’ ’ ^ J
—* y
which, by putting p for ~, and for ~ becomes
X XX*
transformed to
hence we have
—ap
xzz
(i+P2)-®
• —app
y-p*- (I+/,.H
Suppose the equation to be
(»■+/)or_H±£.5ii=x,
—xy ’ P
where X denotes any function of xf then,
—P
JL= -P a„d /'*=-
X (i+p1H J x '
X—C
ap
v/(i+rt ’ +!>■)’
when by means of these equations we eliminate wre
obtain (a?—c)*+(y—c0*=o*.
The fluxional equation is evidently formed by put¬
ting the general expression for the radius of curvature
(given in § 97.) equal to a constant quantity, and the eqUat;on evidently expresses the nature of a
primitive equation is accordingly an equation to a circle , v, , ?
having that constant quantity for its radius, as it ought curve gud^ that -——, its radius of curvature
Th-/h’ x-v(i+/y
rx . . v
Let us representy-^ by V, then^_ ^ ^ y»y
*=/
to be.
—(r y
. , .. _ fS q7.'), is equal to X a function of x one of its co-or-
196. Suppose now that the equation has this form dinates.
4=y,
198. If the proposed fluxional equation of the second
order contains 4-, -r- and y, to transform it we must
where Y denotes a function of y, then putting as belore x x*
—■—p, we have 4- —4- == ^4-—, hence the equation exterminate x by means of its value — deduced itom
a? x* x y . . .
4-=:Y becomes 4£. — Y, and jjjp = Yy, and p*
*' y
the equation y=p x, thus we shall have
IPart II.
FLUX
Invaf*e
Uelhad.
V = P = P P
x* x y
and the result will be an equation of the first order con¬
taining only/?,/7 and y ; when its primitive equation
can be found, and thence the value of in terms of y,
we may find x by the formula x-=. and by the
formula x— — when y is expressed by
of p.
a
a* A*v,+4
11 rViAi -L*- **
(n-f 2) + («+2) (2« + 3) (2«+4)
, &c>
"(«+i)(/»+2)(2«+3)(2«+4)(.3«+5)(3;?+6)
Yol. VIII. Part II. t
IONS.
and the other
. aA^v"+,
Ax-
777
+
a*Axta+!
means
199. As an example of the manner in which fluxion-
al equations of the second order are to be resolved by
approximation, we shall take the particular equation
y a xnyxt=zo.
If the value ofy which satisfies the equation be sup¬
posed to have this form
A ** + -f &c.
and that the series of exponents goes on increasing, or
that } is positive, we may, by supposing a? to be a very
small quantity, conceive that the expression for y is re¬
duced to its first term, because in that case each of the
following terms will be inconsiderable in respect of that
term. According to this hypothesis we shall have
y=A a:*, y =CC 0—1) A xa~'i x\
and thus the proposed equation becomes
*0—I) A a?*-"2 + a A x^^—o.
It will not be possible to give to cc such a value that
the two exponents a,—2 and ec-\-n shall become equal,
except in the particular case of —2 j but if we sup¬
pose x very small, the equation may be satisfied in two
ways, namely, by taking «=:0, and ec— I, because upon
either supposition the term x («t—1) A a:* 3, which is
the greatest, vanishes, and therefore A is left indeter¬
minate thus we have two series, one beginning with
A, and the other with A X.
Assuming therefore successively
y—A -{- Bat^-}-Ca?2* &,c.
y=A a;-J-B a?i+*+ C*I+aS + &c.
and substituting these values as well as their correspond¬
ing values of y in the proposed equation, we shall find
by arranging the terms, that 5 ought to be =2; after¬
wards by determining the coefficients A, B, C, &e. in
the usual manner (Algebra, § 261.) we obtain two
series, one of these is
Inverse
Method.
(n+2)(»+3)'r («+2)(«+3X2n+4)(2n-f5)
q3 A a;3"**
(«+2X«+3X2«+4)(2«+5X3 «+6)(3 M+7)
As a primitive equation in its general form ought to
contain two constant quantities which do not appear in
the fluxional equation of the second order derived from
it (§ 177.X the value of y to be complete ought to con¬
tain two arbitrary constant quantities, but as each of
these series contains only one such quantity, namely A,
it must be considered as expressing only a particular-
value ofy. The fluxional equation y+fl A?" y A;*=ro is
however of such a nature that from two particular va¬
lues of y we may deduce its general value ; for let us
denote these values by 2 and Z, then, as each of them
must satisfy the fluxional equation, we have
2J -j-a a?" % A?*=ro, Z-J-aar Za;*=o>
let c and C denote two arbitrary constant quantities,
then we have also
c z-\-c a xn z x*~o} CZ-f CflA?”Z*,=ro,
and as each of these equations is identical, their sum
must also be identical, that is
ciZ-j-CZ-fa A?"(c55-|-CZ)«*=ro j
but the very same result will be obtained if we substi¬
tute c 25-|-CZ instead of y in the proposed fluxional
equation, therefore cz-f-CZ is also a value of y, and as
it involves two arbitrary constant quantities, c and €,
it possesses all the generality of which the value of y
is susceptible. Hence it follows that if c be put in¬
stead of A in one of the two series which we have found
for the value of y, and C instead of A in the other se¬
ries, the sum of the two results will be a general ex¬
pression for the value of y.
200. Having now explained the theory of fluxional
equations at as great length as we conceive to be com¬
patible with the nature of this work, we shall conclude
this treatise by resolving a few problems which produce
fluxional equations.
Pi'ob. I. Having given any hyperbolic, or, as it may
more properly be called, Napiereun logarithm, it is re¬
quired to find a general expression for its corresponding
natural number.
Let the number be denoted by I-{-a;, and its lo¬
garithm by y, then y= ^- (§ 57.), or
V -f*y—*=c,
and the problem requires that from this equation we
deduce an expression for x.
As when y=o, then x=o, we may assume
A?=Ay + By*-f-Cy3+ &c.
then A;=:Ay=2Byy-f3Cy*y-f-8cc.
and our equation becomes
y+Ayy
Tig. 38.
y-f-+ + 7 =0.
—Ay—2Byy—3Cy2y—4Dy3y—&c. $
Hence, by comparing the coefficients of the like terms,
it appears that A=i, 2B=A, 3C=B, 4D=CK&c. so
that A=<, B=i, C= ^ D=—. &c.
FLUXIONS. Part II.
equation p .rm=:yn, where a? denotes the abscissa AB, Inverse
2.3-4
therefore .r=y-J- “ 'i-^CC* ant^
i+«=i4-y+
2*3
3
4-1- 4-
+ 2.3 +
2.3.4
.V4
&c.
and y the ordinate PB, and p an indeterminate quan
tity which is the same for the whole of any one of the
parabolas, but different for different parabolas j it is re¬
quired to find the nature of a curve that shall intersect
them all in a given angle.
Let the curve whose nature is required meet any one
of the parabolas in P, let PT, P t tangents to the two
curves meet the axis in T and t; then, from the nature
of the problem, the lines PT, P t must contain a given
angle, let a denote its numerical tangent.
Because PT touches the parabola, the tangent of the
Direct
2 ' 2.3 ' 2.3.4 '
Frob. 2. Let AB, AC, be two straight lines given
by position meeting each other at right angles in A,
let C be a given point in AC one ol the lines, and let
a straight line PQ meet them in P and Q, and cut off
from them equal segments AP, CQ adjacent to the
given points A, C, it is required to find the nature of 0f ^ two eqUations, we find 1- = ^1-’ therefore tan.
the curve to which PQ is a tangent. x nx
Let D be the point in which the tangent PQ meets
the curve, draw HE perpendicular to AC, and HI' to
AP, put CA=o, CE=.r, EH=y, then AE or HE
angle PTB will be equal to (§ 75.) the value of
x
this expression being supposed deduced from the equa¬
tion pxm—ijn} but taking the fluxion of this equation,
and eliminating the indeterminate quantity p by means
1% 50.
T= my.
Again, by considering x and y as the abscissa and
ordinate of the curve whose equation is sought, and to
which P £ is a tangent, we have the tangent of the
y
—a—*, and since EQ“^r ($ 73’) an(^ : *'
y
DF i FP, therefore FP= ; bence PA = angle ( eV.al to|-(§ 75-)- Now the angle TP< b.-
ing the difference of the two angles PTB, P B, it
follows from the formula for finding the tangent of the
difference of two angles, (Algebra, § 368.) that
(PF+FA=) ^ .x)'y- +y, and CQ= (CE—EQ—)
x
#___ Zf., and as by hypothesis AP=CQ, therefore
y
y
my_
n x
y*_%
y
This expression belongs to a class of fluxional equations
which have the singular property of being more easily
resolved by first taking their fluxion, considering the
fluxion of one of the variable quantities as constant j
thus, in the present case, making x constant, we find
(a—#)y
xy'—yxy
x
(«—_yxy.
or ; —
* y
hence dividing by y, the equation is easily reduced to
y x
Jy
and taking the fluents
v'y=c—-/(c—.v)
but when ^=0, then y=o, therefore and
y'yrr »/a— \/{a—at), or x-zzlsjay—y,
which equation belongs to the common parabola.
Frob. 3. Let APQ be one of any number of curves
of the parabolic kind, having the same vertex A, and
axis AE, and the nature of which is defined by the
my y *
71 X X
hence we have
9 • • ®
a(nxx+myy')-\‘niyx-—nxyz=.o,
a fluxional equation expressing the nature of the curve,
which being homogeneous may be treated according to
the method explained in § 180.
If the curves be supposed to cut each other at right
angles, then, a being infinite, the part of the equation
which is not multiplied by a vanishes in respect of th«
other, which is multiplied by it j hence we have
7ixx-\-myy—o.
and taking the fluents
n x~ -J- m yx~c,
where c is put for a constant quantity. This equation
shews that the curve is an ellipse, the centre of which i#
at A the common vertex of all the parabolas.
The problem which we have here resolved is only a
particular case of one more general, and which has for
its object Tq determine the nature of the curve which
intersects all other cui ves of a given kind in a givett
angle. The problem thus generalised is known by th*
name of the Problem of Trajectories ; it was originally
proposed by Leibnit% as a challenge to the English
mathematicians, and resolved by Newtont on the day he
received it. See Fluents, Supplement.
FLY,
/
4
f
I
fir-
i-n
PltL
foil'
F L Y [ 779 ] FLY
Fly FLY, in Zoology, a large order of insects, the dt-
(J stinguishing characteristic of which is that their wings
B'iy-uee. are transparent. By this they are distinguished from
“ ^ ' beetles, butterflies, grashoppers, &c. Flies are subdi¬
vided into those which have four, and those which have
two wings. Of those with four wings there are several
genera or kinds ; as the ant, the bee, the ichneumon,
&c. Of those with two wings, there are likewise se¬
veral kinds, as the gad-fly, gnat, house-fly, &c. For
their classification and natural history, see Entomo¬
logy.
House Fly. See Muse a.
Pestilential Fly. See Abyssinia.
Fly, in mechanics, a cross with leaden weights at
its ends j 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,
Flies for Fishing. See Fishing Fly.
Vegetable Fly, a curious natural production chiefly
found in the West Indies. “ Excepting that it has no
wings, it resembles the drone both in size and colour
more than any other British insect. In the month of
May it buries itself in the earth, and begins to vegetate.
By the latter end of July, the tree is arrived at its full
growth, and resembles a coral branch ; and is about
three inches high, and bears several little pods, which
dropping off become worms, and from thence flies, like
the British caterpillar.”
?hilTram. Such wTas the account originally given of this extra-
®r ordinary production. But several boxes of these flies
having been sent to Dr Hill for examination, his re¬
port was this : “ There is in Martinique a fungus of
the clavaria kind, different in species from diose hitherto
known. It produces soboles from its sides; I call it
therefore clavaria sobol/fera. It grows on putrid ani¬
mal bodies, as our fungus ex pede equino from the dead
horse’s hoof. The cicada is common in Martinique *,
and in its nympha state, in which the old authors call
it tettigometra, it buries itself under dead leaves to wait
its change •, and when the season is unfavourable, many
perish. The seeds of the clavaria find a proper bed on
this dead insect, and grow. The tettigometra is among
the cicadas in the British museum •, the clavaria is just
now known. This is the fact, and all the fact; though
the untaught inhabitants suppose a fly to vegetate, and
though there is a Spanish drawing of the plant’s growing
into a trifoliate tree, and it has been figured with the
creature flying with this tree upon its back.” See
Edwards’s Gleanings of Natural History.
Fly-Foot, or Flight, a large flat-bottomed Dutch
vessel, whose burden is generally from 600 to 1200
tons. It is distinguished by a very high stern, re¬
sembling a Gothic turret, and by very broad buttocks
below.
FLY-Catcher, in Zoology. See Muscicapa.
Venus^s FLY-Trap, a kind of sensitive plant. See
Dionma Muscipula, Botany Index.
Fl r-Tree, in Natural History, a name given by the
common people of America to a tree, whose leaves,
they say, at a certain time of the year produce^/cj.
On examining these leaves about the middle of sum¬
mer, the time at which the flies use to be produced,
there are found on them a sort of bags of a tough
matter, of about the size of a filbert, and of a dusky Fly-tree
greenish colour. On opening one of these bags with Flyers.
a knife, there is usually found a single full-grown fly, ' *—-
of the gnat kind, and a number of small worms, which
in a day or two more have wings, and fly away in the
form of their parent. The tree is of the mulberry
kind, and its leaves are usually very largely stocked
with these insect bags $ and the generality of them are
found to contain the insects in their worm state 5 when
they become winged, they soon make their way out.
The bags begin to appear when the leaves are young,
and afterwards grow with them ; but they never rumple
the leaf or injure its shape. They are of the kind of
leaf-galls, and partake in all respects, except size, of a
species we have frequent on the large maple, or, as it
is called, the sycamore.
FLYERS, in architecture, such stairs as go straight,
and do not wind round, or have the steps made tapering;
but the fore and back part of each stair and the ends
respectively parallel to one another : So that if one
flight do not carry you to your designed height, there
is a broad half space ; and then you fly again, with
steps everywhere of the same breath and length as
before.
Flyers, the performers in a celebrated exhibition
among the Mexicans, which was made on certain great
festivals, and is thus described by Clavigero in his Hi¬
story of that people. “ They sought in the woods for
an extremely lofty tree, which, after stripping it of its
branches and bark, they brought to the city, and fixed
in the centre of some large square. They cased the
point of the tree in a wooden cylinder, which, on ac¬
count of some resemblance in its shape, the Spaniards
called a mortar. From this cylinder hung four strong
ropes, which served to support a square frame. In the
space between the cylinder and the frame, they fixed
four other thick ropes, which they twisted as many
times round the tree as there were revolutions to be
made by the flyers. These ropes were drawn through
four holes, made in the middle of the four planks of
which the frame consisted. The four principal flyers,
disguised like eagles, herons, and other birds, mounted
the tree with great agility, by means of a rope which
was laced about it from the ground up to the frame ;
from the frame they mounted one at a time successively
upon the cylinder, and after having danced there a
little, they tied themselves round with the ends of the
ropes, which were drawn through the holes of the
frame, and launching with a spring from it, began their
flight with their wings expanded. The action of their
bodies put the frame and the cylinder in motion ; the
frame by its revolutions gradually untwisted the cords
by which the flyers swung; so that as the ropes length¬
ened, they made so much the greater circles in their
flight. Whilst these four were flying, a fifth danced
upon the cylinder, beating a little drum, or waving a
flag, without the smallest apprehension of the danger
he was in of being precipitated from such a height.
The others who were upon the frame (10 or 12 persons
generally mounted), as soon as they saw the flyers in
their last revolution, precipitated themselves by the
same ropes, in order to reach the ground at the same
time amidst the acclamations of the populace. Those
who precipitated themselves in this manner by the ropes,
that they might make a still greater display of their
5 F a agility,
FLY
[ ?8o ]
FLY
Fhen agilityt frequently passed from one rope to another, at
Flying, that part where, on account of the little distance be¬
tween them, it was possible for them to do so. The most
essential point of this performance consisted in propor¬
tioning so justly the height ot the tree with the length
of the ropes, that the flyers should reach the ground
with 13 revolutions, to represent by such number their
century of 32 years, composed in the manner we have
already mentioned. This celebrated diversion is still
in use in that kingdom*, hut no particular attention is
paid to the number of the revolutions of the flyers ; as
the frame is commonly hexagonal or octagonal, and the
flyers six or eight in number. In some places they put
a rail round the frame, to prevent accidents, which
were frequent after the conquest j as the Indians be¬
came much given to drinking, and used to mount the
tree when intoxicated with wine or brandy, and were
unable to keep their station on so great a height, which
was usually 60 feet.
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 tail ; by the first, the bird sustains and wafts
himself along ; and by the second, he is assisted, in
ascending and descending, to keep his body poised and
upright, and to obviate the vacillations thereof.
It is by the size 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 '70th part of the muscles of the
body, in birds exceed and outvyeigh all the other
muscles taken together j upon which IVIr YVhlloughby
makes this reflection, that if it be possible for a man to
fly, his wings must be so contrived and adapted, that he
may make use of his legs, and not his arms, in mana¬
ging them.
The tail, Messrs Willoughby, Ray, and many others,
imagine to be principally employed in steering and
turning the body in the air, as a rudder j but Borelli
has put it beyond all doubt, that this is the least use of
it, which is chiefly to assist the bird in its ascent and
descent in the air, and to obviate the vacillations of the
body and wings j for as to turning to this or that side,
it is performed by the wings and inclination of the
body, and hut very little by the help of the tail. The
flying of a bird, in effect, is quite a different thing from
the rowing of a vessel. Birds do not vibrate then
wings towards the tail, as oars are struck towards the
stem, 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 horizontally, and preserves
the same situation what way soever the bird turns. In
effect, as a vessel is turned about on a centre of gravity
to the right, by a brisk application of the oars to the
left; so a bird, in beating the air with its right wing
alone towards the tail, will turn its forepart to the
left. Thus pigeons changing their course to the left,
would labour Tt with their right wing, keeping the
other almost at rest. Birds of a long neck alter their
course by the inclination of their head and neck j which
altering the course of gravity, the bird will proceed in
a new direction. ,
The manner of Flying is thus. The bird first bends
his legs, and springs with a violent leap from the
ground j then opens and expands the, joints of his
wings, so as to make a right line perpendicular to the Flying.
sides of his body : thus the wings, with all the feathers * y —
therein, constitute one continued lamina. Being now
raised a little above the horizon, and vibrating the
wings with great force and velocity perpendicularly
against the subject air, that fluid resists those succus-
sions, both from its natural inactivity and elasticity, by
means of which the whole body of the bird is protrud¬
ed. The resistance the air makes to the withdrawing
of the wings, and consequently the progress of the bird,
will he so much the greater, as the waft or stroke of
the fan of the wing is longer : but as the force of the
wing is continually diminished by this resistance, when
the two forces continue to he in equilibria, the bird will
remain suspended in the same place $ for the bird only
ascends so long as the arch of air the wing describes
makes a resistance equal to the excess of the specific
gravity of the bird above the air. If the air, therefore,
be so rare as to give way with the same velocity as it
is struck withal, there will be no resistance, and con¬
sequently the bird can never mount. Birds never fly
upwards in a perpendicular line, hut always in a para¬
bola. In a direct ascent, the natural and artificial ten¬
dency would oppose and destroy each other, so that the
progress would be very slow. In a direct ascent they
would aid one another, so that the fall would be too
precipitate.
Artificial Flying, that attempted by men, by the
assistance of mechanics.
The art of flying lias been attempted by several per¬
sons in all ages. The Leucadians, out of superstition,
are reported to have had a custom of precipitating a
man from a high cliff into the sea, first fixing feathers,
variously expanded, round his body, in order to break
the fall.
Friar Bacon, who lived near 500 years ago, not only
affirms the art of flying possible, but assures us that
he himself knew how to make an engine wherein a man
sitting might be able to convey himself through the
air like a bird j and further adds, that there was then
one who had tried it with success. I he seciet con¬
sisted in a couple of large thin hollow copper globes,
exhausted of air j which being much lighter than air,
would sustain a chair whereon a person might sit. Fa¬
ther Francisco Lana, in his Prodromo, proposes the
same thing as his own thoughts. He computes, that a
round vessel of plate brass, 14 feet in diameter, weigh¬
ing three ounces the square foot, will only weigh 1848
ounces 5 whereas a quantity of air of the same bulk will
wei^h 2155T ounces j so that the globe will not only
be sustained in the air, but will carry with it a weight
of 373|d ounces $ and by increasing the bulk of the
globe, without increasing the thickness of the metal^
he adds, a vessel might be made to carry a much great¬
er weight.—But the fallacy is obvious: a globe of the
dimensions he describes, Dr Hook shews, would not
sustain the pressure of the air, but be crushed inwards.
Besides, in whatever ratio the bulk of the globe were
increased, in the same must the thickness of the metal,
and consequently the weight be increased : so that there
would be no advantage in such augmentation. See
Aerostation.
The same author describes an engine for flying, inr
vented by the Sieur Besnier, a smith of Sable, in the
county of Maine. Vid* Phihsoph. Collect*
‘ The
Jlying
B
F«etus.
FOE [ ^81
The philosophers of King Charles the second’s reign
were mightily busied about this art. The famous
( Bishop Wilkins was so confident of success in it, that
he says, he does not question but in future ages it will
be as usual to hear a man call for his wings, when he is
going a journey, as it is now to call for his boots.
Flying Bridge. See Bridge.
Flying Fish, a name given to several species of fish,
which, by means of long fins, can keep themselves out
of the water for some time. See Exocoetus, Ichthy¬
ology Index.
Flying Pinion, is part of a clock, having a fly or fan
whereby to gather air, and so bridle the rapidity of the
clock’s motion, when the weight descends in the
striking part.
FO, or Foe *, an idol of the Chinese. He was ori¬
ginally worshipped in the Indies, and transported from
thence into China, together with the fables with which
the Indian books were filled. He is said to have per¬
formed most wonderful things, which the Chinese have
described in several volumes, and represented by cuts.
Sect of Fo. See China, N° 104.
Fo-Kien. See Fokien.
FOAL, or Colt and Filly; the young of the horse
kind. The word colt, among dealers, is understood of
the male, ^ filly is of the female. See Colt.
FOCUS, in Geometry and Conic Sections, is applied
to certain points in the parabola, ellipsis, and hyper¬
bola, where the rays reflected from all parts of these
curves concur and meet. See Conic Sections.
Focus, in Optics, a point in which any number of
rays, after being reflected or refracted, meet.
FODDER, any kind of meat for horses or other cattle.
In some places, hay and straw, mingled together, is pe¬
culiarly denominated fodder.
Fodder, in the civil law, is used for a prerogative
that the prince has, to be provided with corn and other
meals for his horses, by the subjects, in his warlike ex¬
peditions.
Fodder, among miners,, a measure containing 22
hundred and a half weight y in London the fodder is
only 20 hundred weight.
FODDERING a ship. See Fothering.
FOENUGREEK. See Trigonella, Botany
Index.
FOENUS nauticum. Where money was lent to
a merchant to be employed in a beneficial trade, with
condition to be repaid with extraordinary, interest,
in case such voyage was safely performed, the agree¬
ment was sometimes called fiznus nauticinn, sometimes
usura maritime. But as this gave an opening for usuri¬
ous and gaming contracts, 19; Geo. II. c. 37. enacts,
that all money lent on bottomry, or at respondentia, on
vessels bound to or from the East Indies, shall be ex¬
pressly lent upon the ship or merchandise ; the lender
to have the benefit of salvage, &c. Blackst. Com. ii.
459. Mol. de lour. Mar. 361.
FOETOR, in Medicine, fetid effluvia arising from
the body or any part thereof.
FOETUS, the young of all viviparous animals
whilst in the womb, and of oviparous animals before
being hatched : the name is transferred by botanists to
the embryos of vegetables.
Strictly, the name is applied to the young after it
]
FOG
is perfectly formed ; previous to which it is usually
called Embryo. See Anatomy Index.
In the human foetus are several peculiarities not to
be found in the adult 5 some of them are as follow’.
1. The arteries of the navel string, which are continu¬
ations of the hypogastrics, are, after the birth, shri¬
velled up, and form the ligamenta umbilic. infer.
2. The veins of the navel string are formed by the
union of all the venous branches in the placenta, and
passing into the abdomen become the falciform ligament
of the liver. 3. The lungs, before being inflated with
air, are compact and heavy, but after one inspiration
they become light, and as it were spongy j and it may
be noted here, that the notion of the lungs sinking in
water before the child breathes, and of their swimming
after the reception of air, are no certain proofs that the
child bad or had not breathed, much less that it was
murdered : for the uninflated lungs become specifically
lighter than water as soon as any degree of putrefaction
takes place in them ; and this soon happens after the
death of the child ; besides, where the utmost care hath
been taken to preserve the child, it hath breathed once
or twice and then died. 4. The thymus gland is very
large in the foetus, but dwindles away in proportion as
years advance. 5. The foramen ovale in the heart of,
a foetus, is generally closed in an adult.
FOG, or Mist, a meteor, consisting of gross vapours,
floating near the surface of the earth.
Mists, according to Lord Bacon, are imperfect con¬
densations of the air, consisting of a large proportion
of the air, and a small one of the aqueous vapour $ and
these happen in the winter, about the change of the
weather from frost to thaw, or from thaw to frost; but
in the summer, and in the spring, from the expansion
of the dew.
If the vapours which are raised plentifully from the
earth and waters, either by the solar or subterraneous
heat, do at their first entrance into, the atmosphere
meet with cold enough to condense them to a consider¬
able degree, their specific gravity is by that means
increased, and so they will be stopped from ascending ;
and either return back in form of dew or of drizzling
rain, or remain suspended some time in the form of a
fog. Vapours may be seen on the high grounds as
well as the low, but more especially about marshy
places. . They are easily dissipated by the wind, as
also by the heat of the sun. They continue longest
in the lowest grounds, because these places contain
most moisture, and are least exposed to the action of
the wind.
Hence we may easily conceive, that fogs are only
low clouds, or clouds in the lowest region of the air ;
as clouds are no other than fogs raised on high. See
Cloud.
When fogs stink, then the vapours are mixed with 1
sulphureous exhalations, which smell so. Objects view¬
ed through fogs appear larger and more remote than
through the common air. Mr Boyle observes, that
upon the coast of Coromandel, and most maritime parts
of the East Indies, there are, notwithstanding the heat
of the climate, annual fogs so thick, as to-occasion peo¬
ple, of other nations who reside there, and even the more
tender sort of the natives, to keep their houses close shut up.
Fpga are commonly strongly electrified, as appears
from i
F O K
F()ff from Mr Cavallo’s observations upon them. See Elec-
TRICITY.
Fukien. FOGAGE, in the forest law, is rank grass not eaten
' 1 up in summer.
EOGO, or Fuego. See Fuego.
FOHI. See Fe ; and China, N° 7.
FOIBLE, a French term, frequently used also in
our language. It literally signifies weak; and in that
sense is applied to the body of animals and the parts
thereof, as foible re\ns, foible sight, &c. being derived
from the Italian fievole, of the Latin flebilis, to be “ la¬
mented, pitied.”
But it is chiefly used with ns substantively, to denote
* defect or flaw in a person or thing. Thus we say,
Every person has his foible ; and the great secret con¬
sists in hiding it artfully : Princes are gained by flatte¬
ry, that is their foible. The foible of young people is
pleasure 5 the foible of old men is avarice j the foible of
the great and learned is vanity ; the foible of women
and girls, coquetry, or an affectation of having gallants:
You should know the forte and the foible of a man be¬
fore you employ him : We should not let people per¬
ceive that we know their foible.
FOIL, in fencing denotes a blunt sword, or one that
has a button at the end covered with leather, used in
learning the art of fencing.
Foil, among glass-grinders, a sheet of tin, with
quicksilver, or the like, laid on the backside of a look¬
ing glass, to make it reflect. See Foliating.
Foil, among jewellers, a thin leaf of metal placed
under a precious stone, in order to make it look trans¬
parent, and give it an agreeable different colour, either
deep or pale: thus, if you want a stone to be of a pale
colour, put a foil of that colour under it •, or if you
would have it deep, lay a dark one under it.
These foils are made either of copper, gold, or gold
and silver together. The copper foils are commonly
known by the name of Nuremberg or German foils;
and are prepared as follows : Procure the thinnest cop¬
per plates you can get *, beat these plates gently upon
a well-polished anvil, with a polished hammer, as thin
as possible j and placing them between two iron plates
as thin as writing paper, heat them in the fire $ then
boil the foil in a pipkin, with equal quantities of tartar
and salt, constantly stirring them till by boiling they
become white 5 after which, taking them out and dry¬
ing them, give them another hammering, till they are
made fit for your purpose : however, care must be taken
not to give the foils too much heat, for fear of melting $
nor must they be too long boiled, for fear of attracting
too much salt.
The manner of polishing these foils is as follows :
Take a plate of the best copper, one foot long and about
five or six inches wide, polished to the greatest perfec¬
tion j bend this to a long convex, fasten it upon a half
roll, and fix it to a bench or table $ then take some chalk
washed as clean as possible, and filtered through a fine
linen cloth, till it be as fine as you can make it ; and
having laid some thereof on the roll, and wetted the
copper all over, lay your foils on it, and with a polish¬
ing stone and the chalk polish your foils till they are as
bright as a looking-glass; after which they must be
dried, and laid up secure from dust.
FOKIEN, a province of China in Asia, commodi-
ously situated for navigation and commerce, part of it
2
F O L
bordering on the sea, in which they catch large quan- Fokiea
titles of fish, which they send salted to other parts of Folard. fo
the empire. Its shores are very uneven, by reason of v—k"
the number and variety of its bays •, and there are many
forts built thereon to guard the coast. The air is hot,
but pure and wholesome.
The mountains are almost everywhere disposed into
a kind of amphitheatres, by the labour of the inhabi¬
tants, with terraces placed one above another. Tbs
fields are watered with rivers and springs, which issue
out of the mountains, and which the husbandmen
conduct in such a manner as to overflow the fields of
rice when they please, because it thrives best in watery
ground. They make use of pipes of bamboo for this
purpose.
They have all commodities in common with the rest
of China \ but more particularly musk, precious stones,
quicksilver, silk, hempen cloth, callico, iron, and all
sorts of utensils wrought to the greatest perfection.
From other countries they have cloves, cinnamon, pep¬
per, sandal wood, amber, coral, and many other things.
The capital city is Fou-tcheou Fou $ or, as others would
have it written, Fucherofu. But as for Fokien, which
most geographers make the capital, Grosier informs us
there is no such place.
FOLARD, Charles, an eminent Frenchman, fa¬
mous for his skill and knowledge in the military art,
was born at Avignon in 1669, of a noble family, but
not a rich one. He discovered an early turn for the
sciences, and a strong passion for arms ; which last was
so inflamed by reading Cgesar’s Commentaries, that he
enlisted at 16 years of age. His father got him off,
and shut him in a monastery : but he made his escape
in about two years after, and entered himself a second
time in quality of cadet. His inclination for military
affairs, and the great pains he took to accomplish him¬
self in that way, recommended him to notice ; and ha
was admitted into the friendship of the first-rate offi¬
cers. M. de Vendome, who commanded in Italy in
1702, made him his aid-de-camp, having conceived the
highest regard for him : and soon after sent him with
part of his forces into Lombardy. He was entirely
trusted by the commander of that army ; and no mea¬
sures were concerted, or steps taken, without consult¬
ing him. By pursuing his plans, many places were
taken, and advantages gained j and such, in short, were
his services, that he had a pension of 500 livres settled
upon him, and was honoured with the cross of St
Louis. He distinguished himself greatly, August 15.
1705, at the battle of Cassano •, where he received a
wound upon his left hand, which deprived him of the
use of it ever after. It was at this battle that he con¬
ceived the first idea of the system of columns, which
he afterwards prefixed to his Commentaries upon Poly¬
bius. The duke of Orleans sending De Vendome again
into Italy in 1706, Folard had orders to throw himself
into Modena to defend it against Eugene; where,
though he acquitted himself with his usual skill, he was
very near being assassinated. The description which
he has given of the conduct and character of the go¬
vernor of this town, may be found in his Treatise of
the Defence of Places, and deserves to be read. He
received a dangerous wound on the thigh at the battle
of Malplaquet, and was some time after made prisoner
by Piince Eugene. Being exchanged in 1711, he was
made
[ 782 ]
[
F O L
Fokrd nia{*e governor of Bourbonrg. In 1714, lie went to
[j Malta, to assist in defending that island against the
Foil-mote. rulks. Upon his return to France, he embarked for
^ Sweden, having a passionate desire to see Charles XII.
He acquned the esteem and confidence of that famous
general, who sent him to trance to negotiate the re¬
establishment of James II. upon the throne of Eng¬
land j but that project being dropped, he returned to
Sweden, followed Charles XII, in his expedition to
Norway, and served under him at the siege of Frede-
rickshall, where that prince was killed, Dec. 11. 1718.
Folard then returned to France; and made his last
campaign in 1719, under the duke of Berwick, in qua¬
lity of colonel. Ironi that time he applied himself in¬
tensely to the study of the military art as far as it
could be studied at home ; and built his theories upon
the foundation of his experience a*nd observations on
facts. He contracted an intimacy with Count Saxe,
who, as he then declared, would one day prove a very
great general. He was chosen a fellow of the Royal
Society of London in 1749; and, in 1751, made a
journey to Avignon, where he died in 1752, aged 83
years. He was the author of several works, the princi¬
pal of which are; 1. Commentaries upon Polybius, in
six volumes, 4to. 2. A Book of New Discoveries in
War. 3. A Treatise concerning the Defence of Places,
&c. in French. Those who would know more of this
eminent soldier, may consult a French piece, entitled,
Memoires pour servir a l' Histoire de M. le Chevalier de
Folard. Ratisbone, 1753, i2mo.
FOLC lands, (Sax.) copyhold lands so called in
the time of the Saxons, as charter-lands were called
hoc-lands, Kitch. 174. Falkland was terra vulgi or
popularis; the land of the vulgar people, who had no
certain estate therein, but held the same, under the
rents and services accustomed or agreed, at the will
only of their lord the thane ; and it was therefore not
put in writing, but accountedrusticum et ig-
nobile. Spelm. of Feuds, c. 5.
FOLCMOTE, or Folkmote, (Sax. Folcgemote,
i. e. conventus populi), is compounded of folk, populus,
and mote, or gemote, convenire ; and signified originally,
as Somner in his Saxon Dictionary informs us, a gene¬
ral assembly of the people, to consider of and order
matters of the commonwealth. And Sir Henry Spel-
man says, the folcmote was a sort of annual parliament
or convention of the bishops, thanes, aldermen, and
freemen, upon every May-day yearly ; where the lay¬
men were sworn to defend one another and the king,
and to preserve the laws of the kingdom ; and then con¬
sulted of the common safety. But Dr Brady infers
from the laws of the Saxon kings of England, that it
was an inferior court, held before the king’s reeve or
steward, every month, to do folk right, or compose
smaller differences, from whence there lay appeal to the
superior courts ; Gloss, p. 48. Squire seems to think
the folcmole not distinct from the shiremote, or common
general meeting of the county. See his Angl. Sax. Gov.
155. n.
Manwood mentions folcmote as a court holden in
London, wherein all the folk and people of the city did
complain of the mayor and aldermen, for misgovern-
ment within the said city; and this word is still in use
among the Londoners, and denotes celebrem ex tota ci-
vitate conventum. Stow's Survey. According to Kea-
783 ] F O L
net, the folcmote was a common council of al! the inha- FalcmoU
bitants of a city, town, or borough, convened often SI
by sound of bell, to the Mote Mall av Mouse; or it was Foliating,
applied to a larger congress of all the freemen within a w"v‘—*
county, called the shiremote, where formerly all knights
and military tenants did fealty to the king, and elected
the annual sheriff on the 1st of October; till this popu¬
lar election, to avoid tumults and riots, devolved to the
king’s nomination, anno 1315, 3 Edw. 1. After which
the city folkmote was swallowed up in a select commit¬
tee or common council, and the comity folkmote in the
sheriff’s tourn and assizes.
The viov& folkmote was also used for any kind of po¬
pular or public meetings; as of all the tenants at the
court Icet, or court baron, in which signification it was
of a less extent. Paroch. Antiq. 120.
I QLENGIO, 1 heofhilus, an Italian poet, was a
native of Mantua. He was known also by the title of
flerhn Coccaye, a name which he gave to a poem, and
which has been adopted ever since for all trifling per¬
formances of the same species, consisting of buffoonery,
puns, anagrams, wit without wisdom, and humour without
good sense. His poem was called The Macaroni, from an
Italiancakeof thesamename, whichissweet to the taste,
but lias not the least alimentary virtue, on the contrary
palls the appetite and cloys the stomach. These idle
poems, however, became the reigning taste in Italy and
in I ranee ; they give birth to macaroni academies, and
reaching England, to macaroni clubs; till, in the end,
every thing insipid, contemptible, and ridiculous, in the
character, dress, or behaviour, of both men and wo¬
men, is now summed up in the despicable appellation
of a macaroni. Folengio died in 1544.
FOLIA, among botanists, particularly signify the
leaves of plants ; those of flowers being expressed by the
word petals. See Botany.
FOLIAGE, aclusteror assemblage of flowers, leaves,
branches, &c.
Foliage, is particularly used for the representations
of such flowers, leaves, branches, rinds, &c. whether
natural or artificial, as are used for enrichments on ca¬
pitals, friezes, pediments, &c.
FOLIATING o/'Looking-glasses, the spreading
the plates over, after they are polished, with quicksil¬
ver, &c. in order to reflect the image. It is performed
thus: A thin blotting paper is spread on the table and
sprinkled with fine chalk; and then a fine lamina or
leaf of tin, called foil, is laid over the paper ; upon
this is poured mercury, which is to be distributed equal¬
ly over the leaf with a hare’s foot or cotton ; over thia
is laid a clean paper, and over that the glass plate,
which is pressed down with the right hand, and the pa¬
per gently drawn cut with the left; this being done,
the plate is covered with a thicker paper, and loaded
with a greater weight, that the superfluous mercury
may be driven out and the tin adhere more closely to
the glass. When it is dried, the weight is removed,
and the looking-glass is complete.
Some add an ounce of marcasite, melted by the fire ;
and, lest the mercury should evaporate in smoke, they
pour it into cold water; and when cooled, squeeze
through a cloth, or through leather.
Some add a quarter of an ounce of tin and lead to
the marcasite, that the glass may dry the sooner.
Foliating of Globe looking-glasses, is done as fol¬
lows :
F O L
[
Foliating lows: Take five ounces of quicksilver, and one ounce
H of bismuth ; of lead and tin, half an ounce each : first
Folkc«. pUt the iea(l and tin into fusion, then put in the bis-
mut|1 . and when you perceive that in fusion too, let it
stand till it is almost cold, and pour the quicksilver in¬
to it : after this, take the glass globe, which must be
very clean, and the inside free from dust: make a pa¬
per funnel, which put into the hole of the globe, as
near the glass as you can, so that the amalgam, when
you pour it in, may not splash and cause the glass to be
full of spots*, pour it in gently, and move it about,
so that the amalgam may touch everywhere : if you
find the amalgam begin to be curdly and fixed, then
hold it over a gentle fire, and it will easily How again ;
and if you find the amalgam too thin, add a little more
lead, tin, and bismuth to it. The finer and clearer
vour globe is, the better will the looking-glass be. .
" Dr Shaw observes, that this operation has consider¬
able advantages, as being performed in the cold ; and
that it is not attended with the danger of poisonous
fumes from arsenic, or other unwholesome matters usu¬
ally employed for this purpose : besides, how far it is ap¬
plicable to the more commodious foliating of the com¬
mon looking-glasses and other speculums, he thinks,
may deserve to be considered.
FOLIO, in merchants books, denotes a page, or ra¬
ther both the right and left hand pages, these being ex¬
pressed by the same figure, and corresponding to each
other. See Book-KEEPING.
Folio, among printers and booksellers, the largest
form of books, when each sheet is so printed that it may
be bound up in two leaves only.
FOLIS. See Follis.
FOLIUM, or Leaf, in Botang. See Leaf.
FOLKES, Martin, a philosopher and antiquarian
of considerable eminence, was born in Westminster
in the year 1690. A Mr Cappel, once professor of
Hebrew at Saumur, was his private tutor. When 17
years of age, he was sent to Clare-hall, Cambridge,
where he successfully applied himself to the study ot
philosophy and the mathematics; and when only twen¬
ty-three years of age he was chosen a fellow of the
Royal Society. His ingenious communications acquired
him so much applause, that he was frequently chosen
into its council. He was in habits of friendship with the
illustrious Newton, at that time president, and by his
influence was elected one of the vice-presidents in the
vear 1723. Mr Folkes became a candidate for the
chair on the death of Sir Isaac Newton j but the supe¬
rior interest of Sir Hans Sloane rendered his application
ineffectual. In 1733 and the two subsequent years,
his residence was for the most part in Italy, with the
view of improving himself in the knowledge of classical
antiquities. To ascertain the weight and value of an¬
cient coins, he carefully consulted the cabinets of the cu¬
rious ; and on his return home he presented to the Anti¬
quarian Society, of which he was a member, a dissertation
on this subject. He read before the same learned body,
a dissertation on the measurement of Trajan’s and Anto-
nine’s pillar, together with other remains of antiquity.
The fruits of his observations he presented to the Royal
Society j and, in particular, “ Remarks on the stand¬
ard measure preserved in the Capitol of Rome,” and
the model of an ancient globe in the Farnesian palace.
He visited Paris in 1739, where he was received with
3
784 ] F O L
great respect, and honoured with the company of the
most eminent literary characters in that metropolis. This
respect indeed he was entitled to by his unwearied appli¬
cation to many branches of knowledge which were both
curious and useful. His valuable work, entitled “ A
table of English silver coins, from the Norman Conquest
to the present time, with their weights, intrinsic values,
and some remarks upon the several pieces,” was printed
in the year 1745. Among the many honours conferred
upon Mr Folkes, he was created doctor of laws by both
universities, and chosen president of the Antiquarian
Society. He continued to furnish the Philosophical
Transactions with many learned papers, till his career
was stopped by a paralytic stroke, which terminated his
useful life in the year 1654. He was a man of very
extensive knowledge and great accuracy *, but the chief
benefit to science which resulted from his labours, was
his treatise on the intricate subject of coins, weights,
and measures. His cabinet and library were large and
valuable, and exposed to public sale after his death.
His private character was distinguished for politeness,
generosity, and friendship.
FOLKESTONE, a town of Kent, between Dover
and Hythe, 72 miles from London, appears to have
been a very ancient place, from the Roman coins and
British bricks often found in it. Stillingfleet and Tan¬
ner take it for the Lapis Tituli of Nennius. It was
burnt by Earl Godwin, and by the French in the reign
of Edward III. It had five churches, now reduced to
one. It is a member of the town and port of Dover;
and has a weekly market and an annual fair. It is
chiefly noted for the multitude of fishing boats that be¬
long to its harbour, which are employed in the season
in catching mackerel for London $ to which they are
carried by the mackerel boats of London and Barking.
About Michaelmas, the Folkestone barks, with others
for Sussex, go away to the Suftolk and Norfolk coasts
to catch herrings for the merchants of Yarmouth and
Leostofl’.—Folkestone gives the title of Viscount to
William Henry Bouverie, whose grandfather, Jacob,
was so created in 1747. It has been observed of some
hills in this neighbourhood, that they have visibly sunk
and grown lower within memory.
FOLKLAND, and Folkmote. See Folcland.
FOLL1CULUS, (from follis, “ a bag,”) a species
of seed-vessel first mentioned by Liumseus in his Deli-
ncatin Plantce, generally consisting of one valve, which
opens from bottom to top on one side, and has no suture
for fastening or attaching the seeds within it.
FOLLICULI are likewise defined by the same au¬
thor to be small glandular vessels distended with air,
which appear on the surface of some plants ; as at the
foot of water-milfoil, and on the leaves of aldrovanda.
In the former the leaves in question are roundish, and
furnished with an appearance like two horns $ in the
latter, pot-shaped, and semicircular.
FOLLIS, or Folis, anciently signified a little bag
or purse ; whence it came to be used for a sum ot mo¬
ney,.and very difl'erent sums were called by that name :
thus the scholiast on the Basilics mentions a follis of
copper which was worth but the 24th part of the nu-
liarensis ; the glossse nomicee, quoted by Gronovius and
others, one of 125 miliarenses, and another of 250 de¬
narii, which was the ancient sestertium j and three dif¬
ferent sums of eight, four, and two pounds of gold, were
each
PON [ 785 ]
each called follis. According to the account of the dious place.
scholiast, the ounce oi silver, which contained five mi-
liarenses of 60 in the pound, was worth 120 folles of
copper. The glossographer, describing a foil is of 250
denarii, says it was equal to 312 pounds 6 ounces of
copper \ and as the denarius ot that age was the 8th
part of an ounce, an ounce of silver must have been
worth 120 ounces of copper j and therefore the scho¬
liast’s follis was an ounce of copper, and equal to the
glossographer’s nummus. But as Constantine’s copper
money weighed a quarter of a Roman ounce, the scho¬
liast’s follis and the glossographer’s nummus contained
lour of them, as the ancient nummus contained four
asses.
FOLLY, according to Mr Locke, consists in the
drawing of false conclusions from just principles ; by
which it is distinguished from madness, which draws
just conclusions from false principles.
But this seems too confined a definition ; /o//y, in its
most general acceptation, denoting a weakness of intel¬
lect or apprehension, or some partial absurdity in senti¬
ment or conduct.
FOMAHAUT, in Astronomy, a star of the first
magnitude in the constellation Aquarius.
FOMENTATION, in Medicine, is a fluid exter¬
nally applied, usually as warm as the patient can bear
it, and in the following manner. Two flannel cloths
are dipped into the heated liquor, one of which is
wrung as dry as the necessary speed will admit, then
immediately applied to the part affected : it lies on un¬
til the heat begins to go off, and the other is in readi¬
ness to apply at the instant in which the first is remov¬
ed : thus these flannels are alternately applied, so as to
keep the aflected part constantly supplied with them
warm. This is continued 15 minutes or half an hour,
and repeated as occasion may require.
Every intention of relaxing and soothing by fomen¬
tations may be answered as well by warm water alone
as when the whole tribe of emollients are boiled in it;
but when discutients or antiseptics are required, such
ingredients must be called in as are adapted to that
end.
The degree of heat should never exceed that of pro¬
ducing a pleasant sensation ; great heat produces effects
very opposite to that intended by the use of fomenta¬
tion.
FONG YANG, a city of China, in the province of
KiANQ-Nang. It is situated on a mountain, which hangs
over the Yellow river, and encloses with its walls seve¬
ral fertile little hills. Its jurisdiction is very extensive:
for it comprehends 18 cities; five of which are of the
second, and 13 of the third class. As this was the
birth-place of the emperor Hong-vou, chief of the pre¬
ceding dynasty, this prince formed a design of rendering
it a famous and magnificent city, in order to make it
the seat of empire. After having expelled the Western
Tartars, who had taken possession of China, he transfer¬
red his court hither, and named the city Fong yang;
that is to say, “ The Place of the Eagle’s Splendour.”
His intention, as we have said, was to beautify and en¬
large it ; but the inequality of the ground, the scarcity
of fresh water, and above all the vicinity of his father’s
tomb, made him change his design. By the unanimous
advice of his principal officers, this prince established
his court at Nan-king, a more beautiful and commo-
Vol. VIII. Part II. t
B
Fontaine.
F O N
When he had formed this resolution, a Fong-Tang
stop was put to the intended works : the imperial pa¬
lace which was to have been enclosed by a triple wall,
the walls of the city to which a circumference of nine
leagues were assigned, and the canals that were begun,
all were abandoned ; and nothing was finished, but three
monuments that still remain. The extent and magnifi¬
cence of these sufficiently show what the beauty of this
city would have been, had the emperor pursued his ori¬
ginal design. The first is the tomb of the father of
Hong-vou, to decorate which no ex pence was spared ;
it is called Hoan-lin, or the Royal Tomb. The second
is a tower built in the middle of the citv, which is of
an oblong form, and 100 feet high. The third is a
magnificent temple erected to the god Fo. At first it
was only a pagod, to which Hong-vou retired after ha¬
ying lost his parents, and where he was admitted as an
infeiior domestic ; but having soon become weary of
this kind of life, he enlisted with the chief of a band of
banditti, who had revolted from the Tartars. As he
was bold and enterprising, the general made choice of
him for his son-in-law: soon after he was declared his
successor by the unanimous voice of the troops. The
new chief seeing himself at the head of a large party,
had the presumption to carry his views to the throne.
The lartars, informed of the progress of his arms, sent
a numerous army into the field ; but he surprised and
attacked them with so much impetuosity, that they were
obliged to fly ; and, though they several times returned
to the charge, they were still defeated, and at length
driven entirely out of China. As soon as he mounted
the throne, he caused the superb temple which we have
mentioned to be raised out of gratitude to the Bonzes,
who had received him in his distress, and assigned them
a revenue sufficient for the maintenance of 300 persons,
under a chief of their own sect, whom he constituted a
mandarin, with power of governing them, independent
of the officers of the city. This pagod was supported
as long as the preceding dynasty lasted ; but that of
the Eastern Tartars, which succeeded, suffered it to fall
to ruin.
Fong-CZioui, the name of a ridiculous superstition
among the Chinese. See China, N° 105.
FONT, among ecclesiastical writers, a large bason
in which water is kept for the baptizing of infants or
other persons.
Font, in the art of printing, denotes a complete as¬
sortment of letters, accents, &c. used in printing. See
Fount.
FONTAINE, John, a celebrated French poet,
and one of the first-rate geniuses of his age, was born
at Chateau-Thierri in Champagne, the 8th of July
1621, of a good family. At the age of 19 he en¬
tered amongst the Oratorians, but quitted that order
18 months after. He was 22 years of age before he
knew his own talents for poetry ; but hearing an ode
of Malherbe read, upon the assassination of Henry IV.
he was so taken with admiration of it, that the poeti¬
cal fire, which had before lain dormant within him,
seemed to be enkindled from that of the other great
poet. He applied himself to read, to meditate, to re¬
peat, in fine to imitate, the works of Malherbe. The
first essays of his pen he confined to one of his rela¬
tions who made him read the best Latin authors, Ho¬
race, Virgil, Terence, Quintilian, &c. and then the
J G best
F O N
FonUsne,
[786
best composition in French and Italian. He applied
himself likewise to the study of the Greek authors, par¬
ticularly Plato and Plutarch. Some time afterwards
his parents made him marry a daughter of a lieutenant-
general, a relation of the great Racine. This young
lady, besides her very great beauty, was remarkable
for the delicacy of her wit, and Fontaine never com¬
posed any work without consulting her. But as her
temper was none of the best, to avoid dissension, he se¬
parated himself from her company as often as he well
could. The famous duchess ol Bouillon, niece to Car¬
dinal Mazarine, being exiled to Chateau-1 hierri, took
particular notice of Fontaine. Upon her recal, he fol¬
lowed her to Paris-, where by the interest ot one
his relations, he got a pension settled upon him. He
met with great friends and protectors amongst the most
distinguished persons of the court, but Madame dela Sa-
bHere was the most particular. She took him to live
at her house, and it was then that 1‘ ontaine, divested
of domestic concerns, led a life conformable to his
disposition, and cultivated an acquaintance with all the
great men of the age. It was his custom, after he was
fixed at Paris, to go every year, during the month of
September, to his native place of Chateau-1 hierri, and
pay a visit to his wife, carrying with him Racine, Des-
preaux, Chapellp, or some other celebrated writers.
When he has sometimes gone thither alone by himself,
he has come away without remembering even to call
upon her 5 but seldom omitted selling some part of his
lands, by which means he squandered away a consider¬
able fortune. After the death of Madame de la Sa-
bliere, he was invited into England, particularly by
Madame Mazarine, and by St Evremond, who promi¬
sed him all the sweets and comforts of life -, but the
difficulty of learning the English language, and the
liberality of the duke of Burgundy, prevented his
voyage.
About the end of the year 1692 he fell dangerously
ill : and, as is customary upon these occasions in the
Romish church, be made a general confession of his
whole life to P. Poguet, an Oratorian -, and, before he
received the sacrament, he sent for the gentlemen of
the French Academy, and in their presence declared
his sincere compunction for having composed his Tales ;
a work he could not reflect upon without the greatest
repentance and detestation j promising that it it should
please God to restore his health, he would employ his
talents only in writing upon matters of morality or
piety. He survived this illness two years, living in
the most exemplary and edifying manner, and died the
13th of March 1695, being 74 years of age.. When
they stripped his body, they found next his skin a hair
shirt which gave room for the following expression
of the younger Racine:
Et V Auteur de Jaconde est or re d'un Cilice.
Fontaine’s character is remarkable for a simplicity, can¬
dour, and probity seldom to be met with. He was of
an obliging disposition ; cultivating a real friendship
with his brother poets and authors ; and what is very
rare, beloved and esteemed by them all. His conversa¬
tion was neither gay nor brilliant, especially when he
was not amongst his intimate friends. One day being
invited to a dinner at a farmer general’s, he ate a great
deal, but did not speak. Rising up from table very
] F O N
early, under pretext of going to the academy, one of Fontame
the company represented to him that it was not yet a 11
proper time: “ Well (says he), if it is not, I will stay Font^!iei-
a little longer.” He had one son by his wife in the
year 1660. At the age of 14, he put him into the
hands of M. de Harley, the first president, recommend¬
ing to him his education and fortune. It is said, that
having been a long time without seeing him, he hap¬
pened to meet him one day visiting, without recollect¬
ing him again, and mentioned to the company that he
thought that young man had a good deal of wit and
understanding. When they told him it was his own son,
he answered in the most tranquil manner, “ Ha ! truly
I am glad on’t.” An indifference, or rather an ab¬
sence of mind, influenced his whole conduct, and ren¬
dered him often insensible to the inclemency of the wea¬
ther. Madame de Bouillon going one morning to Ver¬
sailles, saw him, abstracted in thought, sitting in an ar*
hour returning at night, she found him in the same
place, and in the same attitude, although it was very
cold, and had rained almost the whole day. He carried
this simplicity so far, that he was scarcely sensible of
the bad effects some of bis writings might occasion,
particularly his Tales. In a great sickness, his confes¬
sor exhorting him to prayer and alms deeds : “ As for
alms deeds (replied Fontaine), I am not able, having
nothing to give ; but they are about publishing a new
edition of my Tales, and the bookseller owes me a hun¬
dred copies j you shall have them to sell, and distribute
their amount amongst the poor.” Another time P. Po¬
guet exhorting him to repent of his faults, “ If he has
committed any (cried the nurse), I am sure it is more
from ignorance than malice, for he has as much simpli¬
city as an infant.” One time having composed a tale,
wherein he made a profane application of those words
of the Gospel, “ Lord, five talents thou didst deliver to
me,” he dedicated it, by a most ingenious prologue, to
the celebrated Arnauld, telling him, it was to show to
posterity the great esteem he had for the learned doctor.
He was not sensible of the indecency of the dedication,
and the profane application of the text, till Boileau and
Racine represented it to him. He addressed another,
by a dedication in the same manner, to the archbishop
of Paris. His Fables are an immortal work, exceeding
every thing in that kind, both ancient and modern, in
the opinion of the learned. People of taste, the often-
er they read them, will find continually new beauties
and charms, not to be met with elsewhere. The des¬
cendants of this great poet were long exempted in France
from all taxes and impositions > a privilege which the
intendants of Soisson thought it an honour to confirm
to them.
FONTAINBLEAU, a town in the Isle of France,
and in the Gatinois, remarkable for its fine palace,
which has been the place where the kings of France
used to lodge when they went a hunting. It was first
embellished by Francis I. and every successive king has
added something to it; so that it may now be called the
finest pleasure house in the world. It stands in the
midst of a forest, consisting of 26,424 arpents of land,
each containing 100 square perches, and each perch 18
feet. F. Long. 2. 23* N. Lat. 40. 22.
FONTAINES, Peter Francis, a French critic,
was horn of a good family at Rouen in 1685. At 15,
he entered into the society of the Jesuits j and at 30,
quitted
run
fso
F O N C 787 ] F O N
Fontaines quitted It, for the sake of returning to the world. He
jj was a priest, and had a cure in Normandy : but left it,
Fontarabia.and was, as a man of wit and letters, some time with
the cardinal d’Auvergne. Having excited some at¬
tention at Paris by certain critical productions, the
Abbe Bignon in 1724 committed to him the Journal
des Spavans. He acquitted himself well in this depart¬
ment, and was peaceably enjoying the applauses of the
public, when his enemies, whom by critical strictures
in his Journal he had made such, formed an accusation
against him of a most abominable crime, and procured
him to be imprisoned. By the credit of powerful
friends, he was set at liberty in 15 days ; the magistrate
of the police took upon himself the trouble of justifying
him in a letter to the Abbe Bignon and this letter hav¬
ing been read amidst his fellow labourers in the Journal,
he was unanimously re-established in his former credit.
This happened in 1725. But with whatever repute
he might acquit himself in this Journal, frequent dis¬
gusts made him frequently abandon it. He laboured
meanwhile in some new periodical works, from which
he derived his greatest fame. In 1731, he began one
under the title of Nouvelliste du Parnasse, ou Reflections
sur les Ouvrages nouveaux : but only proceeded to two
volumes ; the work having been suppressed by autho¬
rity, from the incessant complaints of authors ridiculed
therein. About three years after, in 1735, he ob¬
tained a new privilege for a periodical production, en¬
titled, Observations sur les Ecrits Modernes ; which, af¬
ter continuing to 33 volumes, was suppressed again
in 1743* Yet the year following, 1744> published
another weekly paper, called, Jugemens sur les Ouv¬
rages nouveaux, and proceeded to 11 volumes : the two
last being done by other hands. In 1745, he was attack¬
ed with a disorder in the breast, which ended in a drop¬
sy that proved fatal in five weeks. “ He was (says
M. Freron) born a sentimental person ; a philosopher
in conduct as well as in principle ; exempt from ambi¬
tion ; and of a noble firm spirit, which would not sub¬
mit to sue for preferments or titles. In common con¬
versation he appeared only a common man : but when
subjects of literature, or any thing out of the ordinary
way, were agitated, he discovered great force of ima¬
gination and wit.” Besides the periodical works men¬
tioned above, he was the author of many others : his
biographer gives us no less than 17 articles ; many of
them critical, some historical, and some translations
from English writers, chiefly from Pope, Swift,
Fielding, &c. The Abbe de la Porte, published, in
1757, 1?Esprit de VAbbe des Fontaines, in 4 vols.
I 2mo.
FONTANA, Felix, a celebrated Italian physiolo¬
gist. See Supplement.
Fontana, Gregory, an eminent Italian mathemati¬
cian. See Supplement.
FONTANELLA, in Anatomy, imports the qua¬
drangular aperture found betwixt the os frontis and
ossa sincipitis, in children just born ; which is also call¬
ed fans pulsatilis.
FONTARABIA, a sea port town of Spain, in Bis¬
cay, and in the territory of Guipuscoa, seated on a
peninsula on the sea shore, and on the river Bidassoa.
It is small, but well fortified both by nature and art}
and has a good harbour, though dry at low water. It
is built in the form of an amphitheatre, on the decli¬
vity of a hill, and surrounded on the land side by the Fontiuabia.
lofty Pyrenean mountains. It is a very important Konttnelle.
place, being accounted the key of Spain on that side.'■*"—v——■
W. Long. 1. 43. N. Lat. 43.
FONTENELLE, Bernard le Bovier de, was a
man of letters, born at Rouen in 1657, the most uni¬
versal genius of the age of Louis XIV. in the estima¬
tion of Voltaire. He received his education in the
college of Jesuits at Rouen, where the quickness of his
parts became conspicuous at a very early period. He
was capable of writing Latin verses when only 13,
which u'ere deemed worthy of being published. He
studied the law at the desire of bis father} but as be
lost the very first cause in which be was employed as
an advocate, he became disgusted with his profession,
and devoted himselyentirely to literature and philosophy.
He composed a considerable part of the operas of Psyche
and Beilerophon, which were printed under the name
of his uncle Thomas Corneille. He wrote a tragedy
called Aspar ; but as it did not succeed, he consigned
the manuscript to the flames, and never afterwards at¬
tempted that species of composition. His Dialogues of
the dead were published in the year 1683, which were
well received, as a specimen of elegant composition,
combining morality with the charms of literature. His
“ Lettres du Chevalier d’Her,” published in l68^
without his name, discovered much wit and ingenuity,
hut at the same time no small share of affectation. His
“ Entretiens sur la Pluralite des Mondes,” has been
regarded as one of his ablest performances, com¬
bining science and philosophy with vivacity and hu¬
mour, a talent which may he said to belong almost ex¬
clusively to the French. It was perused by all, and
translated into several foreign languages.
In I) is “ Flistory of Oracles,” he supported the opi¬
nion that oracles were forgeries, in opposition to those
who contended that they were supernatural operations of
evil spirits, put to silence by the appearing of Christ,
and of consequence he exposed himself to clerical ani¬
madversion. His “ Pastoral Poems” appeared in the
year 1688, with a discourse on the nature of the eclogue,
which were very much admired for their delicacy
of sentiment, as was also his opera of “ Thetis and
Peleus }” but his “./Eneas and Lavinia” was not so suc¬
cessful. In the year 1699, Fontenelle was chosen
secretary of the Academy of Sciences, which office he
Ireld during the long period of 42 years. He publish¬
ed a volume annually of the history of that; learned
body, filled with analyses of memoirs, and eulogiums on
deceased members.
As a poet, he did not rise above elegance and in¬
genuity } as a man of science, he rather excelled in
throwing light on the inventions of others, than in dis¬
covering any new truth himself, and as a general writer,
he united solid sense with the delicacy and refinement
of a man of wit. He studied his own happiness as much
as most men, but he never sacrificed to the promotion
of it, the duties of a man of honour and virtue. He
had many friends, and towards the close oflife, scarce¬
ly a single enemy. He was never married, and for
a man of letters he acquired considerable affluence.
Although of a delicate constitution, he reached the
great age of 90 without any complaint but dulness of
hearing. He died on the 9th of January 1757, being
almost a hundred years of age. When asked by a certain
5 G 3 person
F O O [ ?88 ] F O O
Fon'.enelle person how he could pass so easily through the world,
A he replied, “ hy virtue of these two axioms j All is
Food. possible, and every one is in the right.”
' PONTE NOY, a town or village of the Nether¬
lands, in the province of Hainault, and on the bor¬
ders of Flanders j remarkable for a battle tought be¬
tween the allies and the French on the first of May
1745. The French were commanded by Mareschal
Saxe, and the allies by the duke of Cumberland. On
account of the superior numbers of the French army,
and the superior generalship of their commander, the
allies were defeated with great slaughter. The British
troops behaved with great intrepidity, as their enemies
themselves acknowledged. It has been said, that the
battle was lost through the cowardice of the Dutch,
who failed in their attack on the village of Fontenoy,
on which the event of the day depended. E. Long. 2. 20.
N. Lat. 50. 3S' '
FoNTENOY, a village of France, in the duchy of
Burgundy, remarkable for a bloody battle fought
there in 841, between the Germans and the French,
in which were killed above 100,000 men ; and the
Germans were defeated. E. Long. 3. 48. N. Lat.
47. 28.
FONTEVRAUD, or Frontevaux, Order of, in
ecclesiastical history, a religious order instituted about
the latter end of the nth century, and taken under
the protection of the holy see by Pope Pascal II. in
1106, confirmed by a bull'in II13, and invested by his
successors with very extraordinary privileges. The chief
of this order is a female, who is appointed to inspect
both the monks and nuns. The order was divided into
four provinces, which were those of France, Aquitaine,
Auvergne, and Bretagne, in each of which they had
formerly several priories.
FONTICULUS, or Fontanella, in Surgery, an
issue, seton, or small ulcer, made in several parts of the
body, in order to excite irritation, or to produce the
discharge of matter.
FONTINALIA, or Fontanalia, in antiquity, a
religious feast held among the Romans in honour of the
deities who presided over fountains or springs. Yarro
observes, that it was the custom to visit the wells on
those days, and to cast crowns into fountains. Scali-
ger, in his conjectures on Varro, takes this not to be
a feast of fountains in general, as Festus insinuates,
but of the fountain which had a temple at Rome, near
the Porta Capena, called also Poi'ta Fontinalis: he adds,
that it is of this fountain Cicero speaks in his second
book De Legibus. The Fontinalia were held on the
13th of October.
FONTINALIS, Water-moss, a genus of plants
belonging to the cryptogamia class, and to the order of
nrusci. See Botany Index.
FOOD, in the most extensive signification of the
word implies whatever aliments are taken into the
body, whether solid or fluid j but in common language,
it is generally used to signify only the solid part of our
aliment.
We are told, that in the first ages men lived upon
acorns, berries, and such fruits as the earth spontane¬
ously produces j then they proceeded to eat the flesh
of wild animals taken in hunting : But their numbers
decreasing and mankind multiplying, necessity taught
them the art of cultivating the ground, to sow corn,
&c. By and by they began to assign to each other, by Food,
general consent, portions of land to produce them their w—y-w
supply of vegetables; after this, reason suggested the
expedient of domesticating certain animals, both to as¬
sist them in their labours and supply them with food.
Hogs were the first animals of the domestic kind that
appeared upon their tables ; they held it to be ungrate¬
ful to devour the beasts that assisted them in their la¬
bours.—When they began to make a free use of do¬
mestic animals, they roasted them only : boiling was a
refinement in cookery which for ages they were stran¬
gers to *, and fish living in an element men were unused
to, were not eaten, till they grew somewhat civilixed.
Menelaus complains, in the Odyssey, that they had
been constrained to feed upon them.
The most remarkable distinction of foods, in a me¬
dical view, is into those which are already assimilated
into the animal nature, and such as are not. Of the
first kind are animal substances in general j which if
not entirely similar, are nearly so, to our nature. The
second comprehends vegetables, which are much more
difficultly assimilated. But as the nourishment of all
animals, even those which live on other animals, can
be traced originally to the vegetable kingdom, it 18
plain, that the principle of all nourishment is in vege¬
tables.
Though there is perhaps no vegetable which Aots Cullen cn
not afford nourishment to some species of animals or i/5* Wat.
other-, yet, with regard to mankind, a very consider-^- nt
able distinction is to be made. Those vegetables which
are of a mild, bland, agreeable taste, are proper nou¬
rishment ; while those of an acrid, bitter, and nauseous
nature, are improper. We use, indeed, several acrid
substances as food j but the mild, the bland, and agree¬
able, are in the largest proportion in almost every ve¬
getable. Such as are very acrid, and at the same time
of an aromatic nature, are not used as food, but as
spices or condiments, which answer the purposes of
medicines rather than any thing else. Sometimes, in¬
deed, acrid and bitter vegetables seem to be admitted
as food. Thus celeri and endive are used in common
food, though both are substances of considerable acri¬
mony j but it must be observed, that when we use
them, they are previously blanched, which almost to¬
tally destroys their acrimony. Or if we employ other
acrid substances, we generally, in a great measure, de¬
prive them of their acrimony by boiling. In different
countries, the same plants grow with different degrees
of acrimony. Thus garlic here seldom enters our
food; but in the southern countries, where the plants
grow more mild, they are frequently used for that pur¬
pose. The plant which furnishes cassada, being very
acrimonious, and even poisonous, in its recent state,
affords an instance of the necessity of preparing acrid
substances even in the hot countries: and there are
other plants, such as arum root, which are so exceed¬
ingly acrimonious in their natural state, that they caa-
not be swallowed with safety ; yet, when deprived of
that acrimony, will afford good nourishment.
The most remarkable properties of different vegeta¬
ble, substances as food, are taken notice of under their
different names : here we shall only compare vegetable
foods in general with those of the animal kind.
I. In the Stomachy they differ remarkably, in that
the vegetables always have a tendency to acidity, while
animal
b'ooJ.
F O O [ ^89
animal food of all kinds rather tends to alkalescency
-' and putrefaction. Some animal foods, indeed, turn ma¬
nifestly acid before they putrify ; and it has been assert¬
ed, that some degree of acescency takes place in every
kind of animal tood before digestion. This acescency
of animal food, however, never comes to any morbid
degree, but the disease is always on the side of putres-
cency. The acescency of vegetables is more frequent,
and ought to be more attended to, than the alkalescen¬
cy of animal food ; which last, even in weak stomachs,
is seldom felt j while acescency greatly affects both the
stomach and system.
With regard to their difference of solution :—Heavi¬
ness, as it is called, is seldom felt from vegetables, ex¬
cept from tough farinaceous paste, or the most viscid
substances j while the heaviness of animal food is more
frequently noticed, especially when'in any great quan¬
tity. Difficulty of solution does not depend so much
on firmness of texture (as a man, from fish of all kinds,
is more oppressed than from firmer substances), as on
viscidity ; and hence it is more frequent in animal food,
especially in the younger animals.
With regard to mixture :—There is no instance of
difficult mixture in vegetables, except in vegetable oils $
while animal foods, from both viscidity and oiliness,
especially the fatter meats, are refractory in this re¬
spect. Perhaps the difference of animal and vegetable
foods might be referred to this head of mixture. For
vegetable food continues long in the stomach, giving
little stimulus: Now the system is affected in propor¬
tion to the extent of this stimulus, which is incompa¬
rably greater from the animal viscid oily food, than
from the vegetable, firmer, and more aqueous. How¬
ever, there are certain applications to the stomach,
which have a tendency to bring on the cold fit of fe¬
ver, independent of stimulus, merely by their refrige¬
ration : and this oftener arises from vegetables; as we
see, in those hot countries where intermittents prevail,
they are oftener induced from a surfeit of vegetable
than of animal food. A proof of this is, that when
one is recovering of an intermittent, there is nothing
more apt to cause a relapse than cold food, especially
if taken on those days when the fit should return, and
particularly acescent, fermentable vegetables, as salads,
melons,, cucumbers, See. acido-dulces, &c. which, ac¬
cording to Dr Cullen, are the most frequent causes of
epidemics; therefore, when an intermittent is to be
avoided, we shun vegetable diet, and give animal foods,
although their stimulus be greater.
' II. In the Intestines. When the putrescency of ani¬
mal food has gone too far, it produces an active stimu¬
lus, causing diarrhoea, dysentery, &.c. But these ef¬
fects are but rare ; whereas from vegetable food and
its acid, which, united with the bile, proves a pretty
strong stimulus, they more frequently occur j hut luckily
are of less consequence, if the refrigeration is not very
great. In the autumnal season, when there is a ten¬
dency to dysentery, if it is observed that eating of fruits
brings it on, it is rather to be ascribed to their cooling
than stimulating the intestines.
As to i-mo/.—Wherever neither putrefaction nor a-
cidity has gone a gr'**^ length, animal food keeps the
belly more regular. Vegetable food gives a greater
proportion of succulent matter; and, when exsiccated by
the stomach and intestines, is more apt to stagnate, and
T F o o
produce slow belly and costiveness, than animal stimu¬
lating food ; which, before it comes to the great guts, '
where stoppage is made, has obtained a putrefactive
tendency, and gives a proper stimulus : and thus
those who are costive from the use of vegetables j when
they have recourse to animal lood are in this respect
better.
HI. In the blood-vessels. They both give a blood of
the same kind, but of dillerent quality. Animal food
gives it in great quantity, being in great part, as the
expression is, convertible in succum et sanguinem, and
ot easy digestion j whereas vegetable is more watery,
and contains a portion of unconquerable saline matter,
which causes it to be thrown out of the body by some
excretion. Animal food allords a more dense stimula¬
ting elastic blood than vegetable j stretching and caus¬
ing a great resistance in the solids, and again exciting
their stronger action. It has been supposed that ace¬
scency of vegetable food is carried into the blood-ves¬
sels, and there exerts its effects j but the tendency of
animal fluids is so strong to alkalescency, that the exis¬
tence of an acid acrimony in the blood seems very im¬
probable. Animal food alone will soon produce, an
alkalescent acrimony ; and if a person who lives entire¬
ly on vegetables were to take no food for a few days,
his acrimony would be alkalescent.
IV. We are, next to take notice of the quantity of
nutriment these different foods afford. Nutriment is of
two kinds : the first repairs the waste of the solid fibres;
the other supplies certain fluids, the chief of wjiich i»
oil. Now, as animal food is easier converted, and also
retained longer in the system, and as it contains a.,
greater proportion of oil, it will afford both kinds of
nutriment more copiously than vegetables..
V. Lastly, As to the different degrees of perspirability
of these foods. This is not yet properly determined.
Sanctorius constantly speaks of mutton as the most
perspirable of all food, and of vegetables as checking
perspiration. This is a consequence of the different
stimulus those foods give to the stomach, so that per¬
sons who live on vegetables have not their perspiration
so suddenly excited.. In time of digestion, perspira¬
tion is stopped from whatever food, much more so from
cooling vegetables. Another reason why vegetables
are less perspirable is, because their aqceo-saline juices
determine them to go off by urine, while the more per¬
fectly mixed animal food is more equally diffused over
the system, and so goes off by perspiration. Hence
Sanctorius’s accounts may be understood ; for vegetable
aliment is not longer retained in the body, but mostly
takes the course of the kidneys. Both are equally per¬
spirable in this respect, viz. that a person living on ei¬
ther returns once a day to his usual weight; and if we
consider the little nourishment of vegetables, and the
great tendency of animal food to corpulency, vve must
allow that vegetable is more quickly perspired than a-
nimal food.
As to the question, Whether man was originally de¬
signed for animal or vegetable food, see the article
Carnivojious.
With regard to the effects of these foods on men, it
must be observed, that there are no persons who live
entirely on vegetables. The Pythagoreans themselves
ate milk ; and those who do so mostly, as these Py¬
thagoreans, are weakly, sickly, and meagre, laboi}j>
ing.
Food.
F O O
I 79° ]
F O O
Food under a constant diarrhoea anil several other ihs-
- gases. None of the hardy, robust, live on these *, but
chiefly such as gain a livelihood by the exertion of their
mental faculties, as (in the East Indies) factors and
brokers ; and this method of life is now confined to
the hot climates, where vegetable diet, without incon¬
venience, may be carried to great excess. Though it
be granted, therefore, that man is intended to live on
these different foods promiscuously, yet the vegetable
should be in very great proportion. Thus the Lap¬
landers are said to live entirely on animal food : but
this is contradicted by the best accounts ; for Linnaeus
says, that besides milk, which they take sour, to ob¬
viate the bad effects of animal food, they use also cala-
menyanthes, and many other plants, copiously. . So
there is no instance of any nation living entirely either
on vegetable or animal food, though there are indeed
some who live particularly on one or other in the great¬
est proportion. In the cold countries, e. g. the inha¬
bitants live chiefly on animal food, on account of the
rigour of the season, their smaller perspiration, and
little tendency to putrefaction.
Of more importance, however, is the following than
the former question, viz. In what proportion animal and
vegetable food ought to be mixed ?
I. Animal food certainly gives most strength to the
system. It is a known aphorism of Sanctorius, that
pondus addit robur; which may be explained from the
impletion of the blood-vessels, and giving a proper de¬
gree of tension for. the performance of strong oscilla¬
tions. Now animal food not only goes a greater way
in supplying fluid, but also gives the fluid more
dense and elastic. The art of giving the utmost
strength to the system is best understood by those who
breed fighting cocks. These people raise the cocks to
a certain weight, which must bear a certain proportion
to the other parts of the system, and which at the same
time is so nicely proportioned, as that, on losing a few
ounces of it, their strength is very considerably im¬
paired. Dr Robinson of Dublin has observed, that
the force and weight of the system ought to be deter¬
mined by the largeness of the heart, and its proportion
to the system: for a large heart will give large blood¬
vessels, while at the same time the viscera are less, par¬
ticularly the liver} which last being increased in size,
a greater quantity of fluid is determined into the cel¬
lular texture, and less into the sanguineous system.
Hence we see how animal food gives strength, by fill¬
ing the sanguiferous vessels. What pains we now
bestow on cocks, the ancients did on the athletse, by
proper nourishment bringing them to a great degree of
strength and agility. It is said that men were at first
fed on figs, a proof of which we have from their nutri¬
tious quality : however, in this respect they were soon
found to fall short of animal food ; and thus we see,
that men, in some measure, will work in proportion to
the quality of their food. The English labour more
than the Scots ; and wherever men are exposed to hard
labour, their food should be animal.—Animal food,
although it gives strength, yet loads the body j and Hip¬
pocrates long ago observed, that the athletic habit,
by a small increase, was exposed to the greatest ha¬
zards. Hence it is only proper for bodily labours,
and entirely improper for mental exercises $ for who¬
ever would keep his mind acute and penetrating,
will exceed rather on the side of vegetable food. Even
the body is oppressed with animal food ; a full meal al-
ways produces dullness, laziness, and yawning and
hence the feeding of gamesters, whose mind must be
ready to take advantage, is always performed by
avoiding a large quantity of animal food. Farther,
With regard to the strength of the body, animal food,
in the first stage of life, is hardly necessary to give
strength : in manhood, when we are exposed to active
scenes, it is more allowable j and even in the decline
of life, some proportion of it is necessary to keep the
body in vigour. There are some diseases which come
on in the decay of life, at least are aggravated by it ;
among these the most remarkable is the gout. This,
when it is in the system, and does not appear with in¬
flammation in the extremities, has pernicious effects
there, attacking the lungs, stomach, head, &c. Now,
to determine this to the extremities, a large proportion
of animal food is necessary, especially as the person is
commonly incapable of much exercise.
Animal food, although it gives strength, is yet of
many hazards to the system, as it produces plethora and
all its consequences. As a stimulus to the stomach and
to the whole system, it excites fever, urges the circula¬
tion, and promotes the perspiration. The system,
however, by the repetition of these stimuli, is soon worn
out; and a man who has early used the athletic diet, is
either early carried off by inflammatory diseases, or, if
he takes exercise sufficient to render that diet saluta-
such an accumulation is made of putrescent fluids,
F ood.
ry. . .
as in his after life lays a foundation for the most inve¬
terate chronic distempers. Therefore it is to be ques¬
tioned, whether we should desire this high degree of bo¬
dily strength, with all the inconveniences and dangers
attending it. Those who are chiefly employed in mental
researches, and not exposed to too much bodily labour,
should always avoid an excess of animal food. There
is a disease which seems to require animal food, viz.
the hysteric or hypochondriac $ and which appears to be
very much a-kin to the gout, affecting the alimentary
canal. All people affected with this disease are much
disposed to acescency : which sometimes goes so far,
that no other vegetable but bread can be taken in,
without occasioning the worst consequences. Here then
we are obliged to prescribe an animal diet, even to those
of very weak organs j for it generally obviates the
symptoms. However, several instances of scurvy in ex¬
cess have been produced by a long-continued use of this
diet, which it is always unlucky to be obliged to pre¬
scribe •, and when it is absolutely necessary to prescribe,
it should be joined with as much of the vegetable as
possible, and when a cure is performed we should gra¬
dually recur to that again.
2. Next, let us consider the vegetable diet. The chief
inconveniency of this is difficulty of assimilation; which,
however, in the vigorous and exercised, will not be li¬
able to occur. In warm climates, the assimilation of ve¬
getable aliment is more easy, so that there it may be
more used, and when joined to exercise gives a pretty
tolerable degree of (strength and vigour; and though
the general rule be in favour of animal diet, for giving
strength, yet there are many instances of its being re¬
markably produced from vegetable. Vegetable diet
has this advantage, that it whets the appetite, and
that we can hardly suffer from a full meal of it. Besides
the
F O O [ 791 ] F O O
the disorders it is liable to produce in the primce vice,
and its falling short to give strength, there seem to be
no bad consequences it can produce to the blood ves¬
sels; for there is no instance where its peculiar acri¬
mony was ever carried there, and it is certainly less
putrefiable than animal food ; nor, without the utmost
indolence, and a sharp appetite, does it give plethora,
or any of its consequences : so that we cannot here but
conclude, that a large proportion of vegetable food is
useful for the generality of mankind.
There is no error in this country more dangerous,
or more common, than the neglect of bread : for it is
the safest of vegetable aliments, and the best corrector
of animal food ; and, by a large proportion of this
alone, its bad consequences, when used in a hypochon¬
driac state, have been obviated. The French appar-
rently have as much animal food on their tables as the
British ; and yet, by a greater use of bread, and the
dried acid fruits, its bad effects are prevented ; and
therefore bread should be particularly used by the
English, as they are so voracious of animal food. Ve¬
getable food is not only necessary to secure health, but
long life: and, as we have said, in infancy and youth
we should be confined to it mostly: in manhood, and
decay of life, use animal food ; and near the end, ve¬
getable again.
There is another question much agitated, viz. What
are the effects of variety in food? Is it necessary and
allowable, or universally hurtful ? Variety of a certain
kind seems necessary ; as vegetable and animal foods
have their mutual advantages, tending to correct each
other. Another variety, which is very proper, is that
of liquid and solid food, which should be so managed
as to temper each other; and liquid food, especially of
the vegetable kind, is too ready to pass off before it is
properly assimilated, while solid food makes a long
stay. But this does not properly belong to the question,
whether variety of the same kind is necessary or pro¬
per, as in animal foods, beef, fish, fowl, &c. It doth
not appear that there is any inconvenience arising from
this mixture, or difficulty of assimilation, provided a
moderate quantity be taken. When any inconvenience
does arise, it probably proceeds from this, that one of
the particular substances in the mixture, when taken
by itself, would produce the same effects; and indeed it
would appear that this effect is not heightened by the
mixture, but properly obviated by it. There are few
exceptions to this, if any, e. g. taking a large propor¬
tion of acescent substances with milk. The coldness,
&c. acidity, flatulency, &o. may appear; and it is pos¬
sible that the coagulum, from the acescency of the
vegetables, being somewhat stronger induced, may give
occasion to too long retention in the stomach, and to
acidity in too great degree. Again, the mixture offish
and milk often occasions inconveniences. The theory
of this is difficult, though, from universal consent, it
must certainly be just. Can we suppose that fish gives
occasion to such a coagulum as runnet ? If it does so,
it may produce bad effects. Besides, fishes approach
somewhat to vegetables, in giving little stimulus ; and
are accused of the same bad effects as these, viz. bring¬
ing on the cold fit of fever. ^ ,
Thus much may be said for variety. But it also
has its disadvantages, provoking to gluttony ; this,
and the art of cookery, making men take in more than
they properly can digest : and hence, perhaps very
justly, physicians have universally almost preferred sim¬
plicity of diet ; for, in spite of rules, man’s eating will
only be measured by his appetite, and satiety is sooner
produced by one than by many substances. But this
is so far from being an argument against variety, that
it is one for it, as the only way of avoiding a full meal
of animal food, and its bad effects, is by presenting a
quantity of vegetables. Another mean of preventing
the bad effects of animal food, is to take a large pro¬
portion of liquid ; and hence the bad effects of animal
food are less felt in Scotland, on account of their drink¬
ing much with it, and using broths, which are at once
excellent correctors of animal food and preventives of
gluttony.
With regard to the difference between ANIMAL
FOODS, properly so called, the first regards their solubi¬
lity, depending on a lax or firm texture of their different
kinds.
I. Solubility of animal food seems to deserve less
attention than is commonly imagined; for there are
many instances of persons of a weak stomach, incapable
of breaking down the texture of vegetables, or even of
dissolving a light pudding, to whom hung beef, or a
piece of ham, was very grateful and easily digested.
None of the theories given for the solution of animal
food in the human stomach seem to have explained
the process sufficiently. Long ago has been discarded
the supposition of an active corrosive menstruum there ;
and also the doctrine of trituration, for which, indeed
there seems no mechanism in the human body; and,
till lately, physicians commonly agreed with Boerhaave
in supposing nothing more to be necessary than a wa¬
tery menstruum, moderate heat, and frequent agita¬
tion. This will account for solution in some cases, but
not entirely. Let us try to imitate it out of the body
with the same circumstances, and in ten times the time
in which the food is dissolved in the stomach we shall
not be able to bring about the same changes. Take
the coagulated white of an egg, which almost every
body can easily digest, and yet no artifice shall be able
to dissolve it. Hence, then, we are led to seek another
cause for solution, viz. fermentation ; a notion, indeed,
formerly embraced, but on the introduction of me¬
chanical philosophy, industriously banished, with every
other supposition of that process taking place at all in
the animal economy.
Many of the ancients imagined this fermentation to
be putrefactive. But this we deny, as an acid is pro¬
duced ; though hence the fermentation might be rec¬
koned the vinous; which, however, seems always to be
morbid. Neither indeed is the fermentation purely
acetous, but modified by putrescence ; for Pringle has
observed, that animal matters raise and even expede the
acetous process. The fermentation, then, in the sto¬
mach is of a mixed nature, between the acetous and the
putrefactive, mutually modifying each other; though,
indeed, in the intestines, somewhat of the putrefactive
seems to take place, as may be observed from the state
of the feces broke down, and from the little disposition
of such substances to be so, which are not liable to the
putrefactive process, as the firmer parts of vegetables,
&c. Upon this view solution seems to be extremely
easy, and those substances to be most easily broke down
which
F O O
Faod. which are most subject to putrefaction. See Anato-
—v—' MY, and Gastric Juice.
But solution also depends on other circumstances, and
hence requires a more particular regard.
1. There is a difference of solubility with respect to
the manducation of animal food, for which bread is ex¬
tremely necessary, in order to keep the more slippery
parts in the mouth till they be properly comminuted.
From want of proper manducation persons are subject
to eructations $ and this more frequently from the
firm vegetable foods, as apples, almonds, &c. than from
the animal, though, indeed, even from animal food,
very tendinous, or swallowed in unbroken masses, such
sometimes occur. Manducation is so much connected
with solution, that some, from imperfectly performing
that, are obliged to belch up their food, remanducate
it, and swallow it again before the stomach can dis¬
solve it, or proper nourishment be extracted. Another
proof of our regard to solubility, is our rejecting the
firmer parts of animal food, as bull beef, and generally
carnivorous animals.
2. Its effects with regard to solubility seem also to
be the foundation of our choice between fat and lean,
young and old meats. In the lean although perhaps
a single fibre might be sufficiently tender, yet these,
when collected \n fasciculi, are very firm and compact,
and of difficult solution $ whereas in the fat there is a
greater number of vessels, a greater quantity of juice,
more interposition of cellular substance, and conse¬
quently more solubility. Again, in young animals,
there is probably the same number of fibres as in the
older, but these more connected : whereas, in the old¬
er, the growth depending on the separation of these,
and the increase of vessels and cellular substance, the
texture is less firm and more soluble j which qualities,
with regard to the stomach, are at that time too increas¬
ed, by the increased alkalescency of the animal. To
this also may be referred our choice of castrated ani¬
mals, viz. on account of their disposition to fatten af¬
ter the operation.
3. It is with a view to the solubility, that we make
a choice between meats recently killed, and tho?e
which have been kept for some time. As soon as
meat is killed, the putrefactive process begins $ which
commonly we allow to proceed for a little, as that
process is the most effectual breaker down of animal
matters, and a great assistance to solution. The length
of time during which meat ought to be kept, is pro¬
portioned to the meat’s tendency to undergo the pu¬
trid fermentation, and the degree of those circum¬
stances which favour it: Thus, in the torrid zone,
where meat cannot be kept above four or five hours,
it is used much more x’ecent than in these northern cli¬
mates.
4. Boiled or roasted meats create a difference of so¬
lution. By boiling we extract the juices interposed
between the fibres, approximate them more to each
other, and render them of more difficult solubility;
which is increased too by the extraction of the juices,
which are much more alkalescent than the fibres : but
when we want to avoid the stimulus of alkalescent
food, and the quick solution, as in some cases of dis¬
ease, the roasted isiiot to be chosen. Of roasted meats
it may be asked, which are more proper, those which
are most or least roasted ? That which is least done is
3
F O O
certainly the most soluble : even raw meats are more Food,
soluble than dressed, as Dr Cullen was informed by a
person who from necessity was obliged, for some time,
to eat such. But at the same time that meats little
done are very soluble, they are very alkalescent ; so
that, wherever we want to avoid alkalescency in the
prbyice vice, the most roasted meats should be chosen.
Those who throw away the broths of boiled meat do
very improperly j for, besides their supplying a fluid,
from their greater alkalescency they increase the solu¬
bility of the meat. Here we shall observe, that pure
blood has been thought insoluble. Undoubtedly it is
very nutritious j and though out of the body, like the
white of eggs, it seems very insoluble, yet, like that
too, in the body it is commonly easily digested. Mo¬
ses very properly forbade it the Israelites, as in warm
countries it is highly alkalescent ; and even here,
when it was used in great quantity, the scurvy was
more frequent : but to a moderate use of it, in these
climates, no such objection takes place.
5. Solubility is varied from another source, viz. vis¬
cidity of the juice of aliment. Young animals, then,
appear more soluble than old, not only on account of
the compaction and firmness of texture in the latter,
but also their greater viscidity of juice. And nothing
is more common, than to be longer oppressed from a
full meal of veal, than from the same quantity of beef,
&c. Upon account, too, of their greater viscidity of
juice, are the tendinous and ligamentous parts of ani¬
mals longer retained than the purely muscular, as well
as on account of their firmness of texture. Even fishes,
whose muscular parts are exceedingly tender, are, on
account of their gluey viscosity, longer in solution in
the stomach. And eggs, too, which are exceedingly
nourishing, have the same effect, and cannot be taken
in great quantity : For the stomach is peculiarly sen¬
sible to gelatinous substances •, and by this means has
nature perhaps taught us, as it were by a sort of in¬
stinct, to limit ourselves in the quantity of such nutri¬
tive substances.
6. With regard to solution, we must take in the oils
of animal food; which, when tolerably pure, are the
least putrescent part of it, and, by diminishing the co¬
hesion of the fibres, render them more soluble. On
this last account is the lean of fat meat more easily dis¬
solved than other lean. But when the meat is expo¬
sed to much heat, this oil is separated, leaving the so¬
lid parts less easily soluble, and becoming itself empy-
reumatic, rancescent, and of difficult mixture in the stoi
mach. Fried meats, from the reasons now given, and
baked meats, for the same, as well as for the tenacity
of the paste, are preparations which diminish the solu¬
bility of the food. From what has been said, the pre¬
paration of food by fattening it, and keeping it for
some time after being killed, although it may administer
to gluttony, will yet, it must be confessed, increase the
solution of the food.
II. The second difference of animal food is with re¬
gard to Alkalescency.
Of this we have taken a little notice aheady under
the head of Solubility.
1. From their too great alkalescency we commonly
avoid the carnivorous animals, and the ferae; and
choose rather the granivorous. Some birds, indeed,
which live on insects, are admitted into cur food ; but
on
[ 792 1
Food.
( F 0 0 [ 793 ]
no man, without nausea^ can live upon these alone for tious.'
any length of time. Fishes, too, are an exception to
this rule, living almost universally on each other. But
in these the alkalescency does not proceed so far; whe¬
ther from the viscidity of their juice, their want of heat,
or some peculiarity in their economy, is not easy to
determine.
2. Alkalescency is determined by difference of age.
The older animals are always more alkalescent than the
young, from their continual progress to putrefaction.
Homberg always found in his endeavours to extract an
acid from human blood, that more was obtained from
the young than from the old animals.
3. A third circumstance which varies the alkalescen¬
cy of the food, is the wildness or tameness of the ani¬
mal ; and this again seems to depend on its exercise.
Dr Cullen knew a gentleman who was fond of cats for
food; but he always used to feed them on vegetable
food, and kept them from exercise; and in the same
manner did the Romans rear up their rats, when in¬
tended for food. In the same way the flesh of the
partridge and the hen seems to be much the same ;
only, from its being more on the wing, the one is more
alkalescent than the other. Again, tame animals are
commonly used without their blood; whereas the wild
are commonly killed in their blood, and upon that ac¬
count, as well as their greater exercise, are more alka¬
lescent.
4. The alkalescency of food may be determined from
the quantity of volatile salt it affords. The older the
meat is, it is found to give the greater proportion of
volatile salt.
5. The alkalescency of aliment may also, in some
measure, be determined from its colour, the younger
animals being whiter and less alkalescent. We also
take a mark from the colour of the gravy poured out,
according to the redness of the juices judging of the
animal’s alkalescency.
6. The relish of food is found to depend much on its
alkalescency, as does also the stimulus it gives and the
fever it produces in the system. These effects are also
complicated with the viscidity of the food, by which
means it is longer detained in the stomach, and the want
of alkalescency supplied.
Having mentioned animal food as differing in solu¬
bility and alkalescency, which often go together in the
same subject, we come to the third difference, viz.
III. Quantity of Nutriment. Which is either ab¬
solute or relative: absolute with respect to the quanti¬
ty it really contains, sufficient powers being given to
extract it; relative, with respect to the assimilatory
powers of those who use it. The absolute nutriment
is of some consequence : but the relative, in the robust
and healthy, and except in cases of extraordinary
weakness, may, without much inconvenience, be dis¬
regarded. In another case is the quantity of nourish¬
ment relative, viz. with regard to its perspirability;
for if the food is soon carried off by the excretions, it
is the same thing as if it contained a less proportion of
nourishment. For, giving more fluid, that which is
longer retained affords most; and, for the repair of the
solids, that retention also is of advantage. Now, gela¬
tinous substances are long retained ; and, besides, are
themselves animal substances dissolved; so that, both
absolutely and relatively, such substances are nutri-
Vol. VIII. Part II. f
F O O
Of this kind are eggs, shell fish, &c. In
adults, though it is disputed whether their solids need
any repair, yet at any rate, at this period, fluid is
more required ; for this purpose the alkalescent foods
are most proper, being most easily dissolved. They are,
at the same time, the most perspirable ; on one hand
that alkalescency leading to disease, while on the other
their perspirability obviates it. Adults, therefore, as
writers justly observe, are better nourished on the al¬
kalescent ; the young and growing, on gelatinous
foods. All this leads to a comparison of young and
old meats ; the first being more gelatinous, and the last
more alkalescent. This, however, by experience, is
not yet properly ascertained. Mr Geoffrey is the only
person who has been taken up with the analysis of
foods. 'SttMemoires del'Academic, 1731 & 1732.
His attempt was certainly laudable, and in some re¬
spects usefully performed ; but, in general, his experi¬
ments were not sufficiently repeated, nor are indeed
sufficiently accurate. He has not been on his guard
against the various circumstances which affect meats;
the cow-kind liking a moist succulent herbage, which
is not to be got in warm climates; while the sheep are
fond of dry food, and thrive best there. Again, some
of his experiments seem contradictory. He says, that
veal gives more solution than beef, while lamb gives less
than mutton, which is much to be doubted. If both
he and Sanctorius had examined English beef, the re¬
sult probably would have been very different as to its
perspirability, &c. Besides, Mr Geoffrey has only ana¬
lyzed beef and veal when raw; has made no proper
circumstantial comparisons between quadrupeds and
birds ; and has examined these last along with their
bones, and not their muscles, &c. by themselves, as
he ought to have done, &c. If a set of experiments
of this kind were properly and accurately performed,
they might be of great use; but at present, for the
purpose of determining our present object, we must
have recourse to our alkalescency, solubility, &c.
IV. The fourth difference of animal food is, The
Nature of the Fluids they afford. The whole of this
will be understood from what has been said on alkales¬
cency ; the fluid produced being more or less dense
and stimulating, in proportion as that prevails.
V. The fifth difference of animal foods is with re¬
spect to their
Perspirability. The sum of what can be said on
this matter is this, that such foods as promote an ac¬
cumulation of fluid in our vessels and dispose to ple¬
thora, are the least perspirable, and commonly give
most strength; that the more alkalescent foods are the
most perspirable, though the viscid and less alkalescent
may attain the same property by long retention in the
system. The authors on perspirability have determin¬
ed the perspiration of foods as imperfectly as Mr Geof-
froy has done the solubility, and in a few cases only.
We must not lay hold on what Sanctorius has said on
the perspirability of mutton, because he has not exa¬
mined in the same way other meats in their perfect
state; far less on what Keil says of oysters, as he him¬
self was a valetudinarian, and consequently an unfit
subject for such experiments, and probably of a pecu¬
liar temperament.
As to the effects of Food on the Mind, we have
already hinted at them above. It is plain, that deli-
5 H eacy
Food.
F O O [ 794 ]
Food c ity of feeling, liveliness of imagination, quickness of circumference.
f[ apprehension, and acuteness of judgment, more fre-
Foosht. quently accompany a weak state of the body.. Irue
' it is, indeed, that the same state is liable to timidity,
fluctuation and doubt J while the strong have that
steadiness of judgment, and firmness of purpose, which
are proper for the higher and more active scenes ot
life. The most valuable state of the mind, however,
appears to reside in somewhat less firmness and vigour
of body. Vegetable aliment, as never over-distending
the vessels or- loading the system, never interrupts the
stronger motions of the mind j while the heat,.fulness,
and weight, of animal food, are an enemy to its vigo¬
rous efforts. Temperance, then, does not so much con¬
sist in the quantity, for that always will be regulated
by our appetite, as in the quality, viz,, a large propor¬
tion of vegetable aliment.
A considerable change has now taken place in the
articles made use of as food by the ancients, by substi¬
tuting, instead of what were then used, particularly of
the vegetable kind, a number of more bland, agxeeable,
and nutritive juices. The acorns and nuts of the pri¬
mitive times have given way to a variety of sweeter fa¬
rinaceous seeds and roots. To the malvaceous tribe of
plants, so much used by the Greeks and Romans, has
succeeded the more grateful spinach-, and to the blite,
the garden orach. The rough borage is supplanted by
the acescent sorrel} and asparagus has banished a num¬
ber of roots recorded by the Roman writers under the
name of bulbs; bnt Linnaeus is of opinion, that the
parsnip has undeservedly usurped the place of the skir-
ret. The bean of the ancients, improperly so called, be¬
ing the roots as well as other parts ot the nymphxa ne-
lumbo, or Indian water-lily, is superseded by the kidney
bean. The garden rocket, eaten with and as an antidote
against the chilling qualities of the lettuce, is banished
by the more agreeable cress and tarragon \ the apium.
by the meliorated celery ; the pompion, and others of
the cucurbitaceous tribe, by the melon } and the sumach
berries, by the fragrant nutmeg. The silphium, or
succus Cyrenaicus, which the Romans purchased from
Persia and India at a great price, and is thought by
some to have been the asafoetida of the present time, is
no longer used in preference to the alliaceous tribe.
To turn from the vegetable to some of the animal
substitutes, we may mention the carp among fishes as
having excluded a great number held in high estima¬
tion among the Romans •, the change of oil for butter j
of honey for sugar j of mulsa, or liquors made of wine,
water, and honey, for the wines of modern times ; and
that of the ancient zythus for the present improved
malt liquors j not to mention also the Calhda of the
Roman taverns, analogous to our tea and coffee. See
Food and Dietetics, in the Supplement.
Foov of P/ants. See Agriculture Index.
FOOL, according to Mr Locke, is a person who
makes false conclusions from right principles-, where¬
as a madman, on the contrary, draws right conclusions
from wrong principles. See Folly.
Fool-Stones. See Orchis, Botany Lidex.
FOOSITT, an island in the Red sea situated, ac¬
cording to the observations of Mr Bruce, in N. Lat.
JB0 59' dS”- -^t ^ described by him as about five miles
in length from north to south, though only nine in
F O O
It is low and sandy in the southern
part, hut the north rises in a black bill of inconsider¬
able height. It is covered with a kind of bent grass,
which never arises at any great length by reason of
want of rain and the constant browzing of the goats.
There are great appearances of the black hill having
once been a volcano ; and near the north cape the
ground sounds hollow like the Solfaterra in. Italy.
There are a vast number of beautiful fish met with up¬
on the coasts, but few fit for eating j and onr traveller
observed, that the most beautiful were the most noxious
when eaten j none indeed being salutary food except¬
ing those which resembled the fish of the northern seas.
There are many beautiful shell-fish, as the concha ve¬
neris, of several colours and sizes j sea urchins, &c.
Sponges are likewise found all along the coast. There
are also pearls, but neither large nor of a good water j
in consequence of which they sell at no great price.
They are produced by a species of bivalve shells. Se¬
veral large shells, from the fish named bissery are met
with upon stones of ten or twelve tons weight along
the coast. They are turned upon their faces and sunk
into the stones, as into a paste, the stone being raised
all about them in such a manner as to cover the edge
of the shell j “ a proof (says Mr Bruce) that this stone
must some time lately have been soft or liquefied . for
had it been long ago, the sun and air would have worn
the surface of the shell *, but it seems perfectly entire,
and is set in that hard brown rock as the stone of a
ring is in a golden chasing.”—The water in this island
is very good. f
The inhabitants of Foosht are poor fishermen ot a
swarthy colour *, going naked, excepting only a rag
about their waist. They have no bread but what they
procure in exchange for the fish they catch. What
they barter in this manner is called seajan.^ But be¬
sides this they catch another species, which is flat, with
a long tail, and the skin made use of for shagreen, of
which the handles of knives and swords are made.
There is a small town on the island, consisting of about
30 huts, built with faggots of bent grass or sparturo,
supported by a few sticks, and thatched with grass of
the same kind of which they are built.
FOOT, a part of the body of most animals whereon
they stand, walk, &c. See Anatomy.
Foot, in the Latin and Greek poetry, a metre or
measure, composed of a certain number of long and
short syllables. j n c 1 ' L
These feet are commonly reckoned 28 : of which
some are simple, as consisting of two or three syllables,
and therefore called dissijllubic or trisyllabic feet; others
are compound, consisting of four syllables, and are there¬
fore called tetrasyllable feet.
The dissyllabic feet are four in number, viz. the
pyrrichius, spondeus, iambus, and trocheus. See rYR-
rhichius, &c. . ,
The trisyllabic feet are eight in number, viz. the
dactylus, anapsestus, tribrachys, molossus, amphibra¬
chys, amphimacer, bacchius, and antibacchius. ee
Dactyl* &c.
The tetrasyllable are in number 10, viz. the pro-
celeusmaticus, dispondeus, choriamhus, antispastus, di-
ambus, dichoreus, ionicos a majore, ionicus a mmore,
epitritus primus, epitritus secundus, epitritus tertius,
s epifntus
Pooiht,
Foot.
[ 795 ] F O O
Poot epltritus quartus, paeon primus, paeon secundus, paeon
Foote tertius, and paeon quartus. See Proceleusmaticus,
&c.
Foot is also a long measure consisting of 12 inches.
Geometricians divide the foot into 10 digits, and the
digit into 10 lines.
FoOT-Ha/t, the name of a disorder peculiar to sheep.
It is occasioned by an insect, which when it comes to a
certain maturity, resembles a worm of two, three, or
four inches in length. See Farriery Index.
FooT-Square, is the same measure both in breadth
and length, containing 144 square or superficial
inches.
Cubic or Solid Foot, is the same measure in all the
three dimensions, length, breadth, and depth or thick¬
ness, containing 1728 cubic inches.
„ Foot of a Horse, in the manege, the extremity of
the leg, from the coronet to the lower part of the
hoof.
Foot Level, among artificers, an instrument that
serves as a foot rule, a square, and a level. See Level,
Rule, and Square.
FOOTE, Samuel, Esq. the modern Aristopha¬
nes, was born at Truro, in Cornwall $ and was descend¬
ed from a very ancient family. His father was member
of parliament for Tiverton, in Devonshire, and enjoyed
the post of commissioner of the prize office and fine-
contract. His mother was heiress of the Dinely and
Goodere families. In consequence of a fatal misunder¬
standing between her two brothers, Sir John Dinely
Goodere, Bart, and Samuel Goodere, Esq. captain of
his majesty’s ship the Ruby, which ended in the death
of both, a considerable part of the Goodere estate,
which was better that 5000I. per annum, descended to
Mr Foote.
He was educated at Worcester college, Oxford,
which owed its foundation to Sir Thornes Cookes
Windford, Bart, a second cousin of our author’s. On
leaving the university, he commenced student of law in
the Temple *, but as the dryness of this study did not
suit the liveliness of his genius, he soon relinquised it.
He married a young lady of a good family and some
fortune j but their tempers not agreeing, a perfect
harmony did not long subsist between them. He now
launched into all the fashionable foibles of the age, gam¬
ing not excepted, and in a few years spent his whole
fortune. His necessities led him to the stage, and he
made his first appearance in the character of Othello.
He next performed Fondlewife with much more ap¬
plause $ and this, indeed, W'as ever after one of his ca¬
pital parts. He attempted Lord Foppington likewise,
but prudently gave it up. But as Mr Foote was never
a capital actor in the plays of others, his salary was
very unequal to his gay and extravagant turn j and he
contracted debts which forced him to take refuge with¬
in the verge of the court. On this occasion, he reliev¬
ed his necessities by the following stratagem. Sir
Er s D« 1—1 had long been bis intimate friend, and
had dissipated his fortune by similar extravagance.
Lady N-ss-u P—let, who wms likewise an intimate ac¬
quaintance of Foote’s, and who was exceedingly rich,
was fortunately at that time bent upon a matrimonial
scheme. Foote strongly recommended to her to con¬
sult upon this momentous affair the conjuror in the Old
Bailey, whom be represented as a. man of surprising
skill and penetration. He employed an acquaintance
of his own to personate the conjurer ; who depicted
Sir 1 r s D—1—] at full length 5 described the time
when, the place where, and the dress in which she
would see him. The lady was so struck with the coin¬
cidence of every circumstance, that she married D 1 1
in a few days. For this service Sir Francis settled an
annuity upon Foote; and this enabled him once more
to emerge from obscurity.
In 1747 he opened the little theatre in the Ilay-
maiket, taking upon himself the double character of
author and performer ; and appeared in a dramatic piece
of his own composing, called the Diversions of the
Morning. 1 he piece consisted of nothing more than
the exhibition of several characters well known in real
life ; whose manner of conversation and expression this
author very happily hit off in the diction of his drama,
and still more happily represented on the stage, by an
exact and most amazing imitation, not only of the man¬
ner and tone of voice, but even of the very persons, of
those whom he intended to take off. In this perform¬
ance, a certain phy'sician, Dr L — n, well known for
the oddity and singularity of his appearance and con¬
versation, and the celebrated Chevalier Taylor, who
Was at that time in the height of his popularity, were
made objects of Foote’s ridicule ; the latter, indeed
very deservedly : and, in the concluding part of his
speech, under the character of a theatrical director,
Mr I note took oft, with great humour and accuracy,
the several styles of acting of every principal performer
on the English stage. The performance at first met
with some opposition from the civil magistrates of
Westminster, under the sanction of the act of parlia¬
ment for limiting the number of playhouses, as well
as from the jealousy of one of the managers of Drury-
lane playhouse: but the author being patronized by'
many of the principal nobility, and other persons of
distinction, this opposition was overruled: and having
altered the title of his performance, Mr Foote proceed¬
ed, without further molestation, to give Tea in a Morn¬
ing to his friends, and represented it through a run of
40 mornings to crowded and splendid audiences.—
The ensuing season he produced another piece of the
same kind, which he called An Auction of Pictures. In
this performance he introduced several new and popu¬
lar characters; particularly Sir Thomas de Veil, then
the acting justice of peace for Westminster, Mr Cock
the celebrated auctioneer, and the equally famous Ora¬
tor Henley. This piece also had a very great run.—
His Knights, which was the production of the ensuing
season, was a performance of somewhat more dramatic
regularity: but still, although his plot and characters
seemed less immediately personal, it was apparent that
he kept some particular real persons strongly in his eye
in the performance ; and the town took upon them¬
selves to fix them where the resemblance appeared to
be the most striking. Thus Mr Foote continued from
time to time to select, for the entertainment of the
public, such characters, as well general as indivi¬
dual, as seemed most likely to engage their attention.
His dramatic pieces, exclusive of the interlude called
Piety in Pattens, are as follows : Taste, the Knights,
The Author, The Englishman in Paris, The English¬
man returned from Paris, The Mayor of Garrat, The
Liar, The Patron, The Minor, The Orators, The
5 H 2 Commissary,
F O O
Foote. Commissary, The Devil upon two Sticks, Ihe Lame
——y- Lover, The Maid of Bath, Ihe Nabob, the Cozeners,
The Capuchin, the Bankrupt, and an unfinished co¬
medy called The Slanderer. All these works are only
to be ranked among the petites pieces of the theatre.
In the execution they are somewhat loose, negligent,
and unfinished •, the plots are often irregular, and the
catastrophes not always conclusive } but with all these
deficiencies, they contain more strength of character,
more strokes of keen satire, and more touches of tem¬
porary humour, than are to be found in the writings of
any other modern dramatist. Even the language spo¬
ken by his characters, incorrect as it may sometimes
seem, will on a closer examination he found entirely
dramatical j as it abounds with those natural minutiae
of expression which frequently form the very basis of
character, and which render it the truest mirror of the
conversation of the times in which he wrote.
In the year 1^766, being on a party of pleasure with
the late duke of York, Lord Mexborough, and Sir
Francis Delaval, Mr Foote had the misfortune to
break his leg, by a fall from his horse $ in consequence
of which he was compelled to undergo an amputation.
This accident so sensibly affected the duke, that he
made a point of obtaining for Mr Foote a patent for
life; whereby he was allowed to perform, at the little
theatre in the Haymarket, from the 15th of May to
the 15th of September every year.
He now became a greater favourite of the town
than ever : his very laughable pieces, with his more
laughable performance, constantly filled his house; and
his receipts were sometimes almost incredible. Par¬
simony was never a vice to be ascribed to Mr Foote j
his hospitality and generosity were ever conspicuous j he
was visited by the first nobility, and he was sometimes
honoured even by royal guests.
The attack made upon his character by one of his
domestics, whom he had dismissed for misbehaviour, is
too well known to be particularized here. Suffice it
to say, he was honourably acquitted of that charge :
but it is believed by some, that the shock which he re¬
ceived from it accelerated his death *, others pretend,
that his literary altercation with a certain then duchess,
or rather her agents, much affected him, and that from
that time his health declined. It is probable, however,
that his natural volatility of spirits could scarcely fail
to support him against all impressions from either of
these quarters.
Mr Foote, finding his health decline, entered into
an agreement with Mr Colman, for his patent of the
theatre j according to which, he was to receive from
Mr Colman, x6ool. per annum, besides a stipulated sum
whenever he chose to perform. Mr Foote made his
appearance two or three times in some oi the most ad¬
mired characters ; but being suddenly aflected with a
paralytic stroke one night whilst upon the stage, he
was compelled to retire. He was advised to bathe $
and accordingly retired to Brighthelmstone, where he
apparently recovered his former health and spirits, and
was what is called \\\e fiddle of the company who resort¬
ed to that agreeable place of amusement. A few weeks
before his death, he returned to London ; but, by the
advice of his physicians, set out with an intention to
spend the winter at Paris and in the south of France.
He had got no farther than Dover, when he was sud-
F O R
denly attacked by another stroke of the palsy, which Foote
in a few hours terminated his existence. He died on ||
the 21st of October 1777, in the 56th year of his age, ^orcc*
and was privately interred in the cloisters of Westmin- ~L" ^
ster abbey.
FOP, probably derived from the vappa of Horace,
applied in the first satire of his first book to the wild
and extravagant Neevius, is used among us to denote a
person who cultivates a regard to adventitious ornament
and beauty to excess.
FORAMEN, in Anatomy, a name given to several
apertures or perforations in divers parts of the body J
as, 1. The external and internal foramina of the cra¬
nium or skull. 2. The foramina in the upper and lower
jaw. 3. Foramen lachrymale. 4. Foramen membranfe
tympani.
Foramen Ovale, an oval aperture or passage through
the heart of a foetus, which closes up after birth. It
arises from the coronal vein, near the right auricle, and
passes directly into the left auricle of the heart, serving
for the circulation of the blood in the foetus, till such*
time as the infant breathes, and the lungs are open :
in this the foetus differs from the adult $ although
almost all anatomists, Mr Cheselden excepted, assure
us, that the foramen ovale has sometimes been found in
adults. See Foetus.
FORBES, Duncan, Esq. of Culloden, lord presi¬
dent of the court of session in Scotland, was born in
the year 1685. In his early life, he was brought up
in a family remarkable for hospitality j which, perhaps,
led him afterwards to a freer indulgence in social plea¬
sures. His natural disposition inclined him to the ar¬
my : but, as he soon discovered a superior genius, by
the advice of his friends he applied himself to letters.
He directed his studies particularly to the civil law $ in
which he made a quick progress, and in 1709 was ad¬
mitted an advocate. From 1722 to I737> l'e repre¬
sented in parliament the boroughs oflnverness, Fortrose,
Nairn, and Forres. In 1725, he was made king’s ad¬
vocate; and in 1737 lord president of the courtof session.
In the rebellions which broke out in Scotland in 1715
and 1745 he espoused the roval cause ; but with so much
prudence and moderation did he conduct himself at this
delicate conjuncture, that not a whisper was at any time
heard to his prejudice. The glory he acquired in ad¬
vancing the prosperity of his country, and in contribu¬
ting to re-establish peace and order, was the only reward .
of his services. He had even impaired, and almost ruin¬
ed, his private fortune in the cause of the public : but
government did not make him the smallest recompense.
The minister, with a meanness for which it is difficult
to account, desired to have a state of his disbursements.
Shocked at the incivility and rudeness of this treatment,
he left the minister without making any reply. Through¬
out the whole course of his life he had a lively sense of
religion, without the least taint of superstition ; and his
charity was extended to every sect and denomination oi
religionists indiscriminately. He was well versed in the
Hebrew language ; and wrote in a flowing and orato-
rial style, concerning religion natural and revealed,
some important discoveries in theology and philosophy,
and concerning the sources of incredulity. He died in
1747, in the (>2d year of his age ; and his works have
since been published in two volumes octavo.
FORCE, in Philosophy, denotes the cause of the
change
[ 796 ]
F O It [
Force change in the state of a body, when, being at rest, it
—v-——' begins to move, or has a motion which is either not
uniform or not direct. While a body remains in the
same state, either of rest or of uniform and rectilinear
motion, the cause of its remaining in such a state is in
the nature of the body, and it cannot be said that any
extrinsic force has acted on it. Ibis internal cause or
principle is called inertia.
Mechanical forces may be reduced to two sorts : one
of a body at rest, the other of a body in motion.
The force of a body at rest, is that which we con¬
ceive to be in a body lying still on a table, or banging
by a rope, or supported by a spring, &c. and this is
called by the names of pressure, tension, force, or vis
mortua, solicitatio, conatus movendi, conamen, &.c.
To this class also of forces we must refer centripetal
and centrifugal forces, though they reside in a body in
motion j because these forces are homogeneous to
weights, pressures, or- tensions of any kind.
The force of a body in motion is a power residing
in that body so long as it continues its motion by
means of which it is able to remove obstacles lying
in its way •, to lessen, destroy, or overcome the force
of any other moving body which meets it in an oppo¬
site direction j or to surmount any dead pressure or re¬
sistance, as tensity, gravity, friction, &c. for some
time j but which will be lessened or destroyed by such
resistance as lessens or destroys the motion of the bo¬
dy. This is called moving force, vis matrix, and by
some late writers vis viva, to distinguish it from the
vis mortua spoken of before ; and by these appellations,
however different, the same thing is understood by all
mathematicians •, namely, that power of displacing,
of withstanding opposite moving forces, or of overcom¬
ing any dead resistance, which resides in a moving
1
FOR
body, and which, in whole or in part, continues to
accompany it, so long as the body moves. See Me-.
CHANICS.
We have several curious as well as useful observa¬
tions in Desaguliers’s Experimental Philosophy, con¬
cerning the comparative forces of men and horses, and
the best way of applying them. A horse draws with
the greatest advantage when the line of direction is le¬
vel with his breast; in such a situation, he is able to
draw 200lb. eight hours a-day, walking about two
miles and a half an hour. And if the same horse is
made to draw 2401b. he can work but six hours a-day,
and cannot go quite so fast. On a carriage, indeed,
where friction alone is to be overcome, a middling
horse will draw looolb. But the best way to try a
horse’s force, is by making him draw up out of a well,
over a single pulley or roller j and, in such a case, one
horse with another will draw 200lb. as already ob¬
served.
Five men are found to he equal in strength to one
horse, and can, with as much ease, push round the ho¬
rizontal beam of a mill, in a walk 40 feet wide ;
whereas three men will do it in a walk only 19 feet
wide.
The worst way of applying the force of a horse, is
to make him carry or draw up hill j for if the hill be
steep, three men will do more than a horse, each man
climbing up faster with a burden of loolb. weight,
than a horse that is loaded with 300lh. ", a difference
which is owing to the position ol the parts of the hu¬
man body being better adapted to climb than those of
a horse. ||
On the other hand, the he^t way of applying the Forcible,
force of a horse, is in a horizontal direction, wherein a v—
man can exert least force j thus a man weighing 1401b.
and drawing a boat along by means of a rope coming
over his shoulders, cannot draw above 271b. or exert
above one-seventh part of the force of a horse employed
to the same purpose.
The very best and most effectual posture in a man,
is that of rowing, in which he not only acts with more
muscles at once for overcoming the resistance, than in
any other position ; but as he pulls backward, the
weight of his body assists by way of lever. See Desa-
guiiers, Exp. Phil. vol. i. p. 241. where we have seve¬
ral other observations relative to force acquired by cer¬
tain positions of the body, from which that author ac¬
counts for most feats of strength and activity. See also
a Memoire on this subject by M. de la Hire, in Mem.
Koy. Acad. Sc. 1629 j or in Desagnliers, Exp. &c.
267, &c. who has published a translation of part of it,
with remarks.
Citizen Regnierhas invented an instrument for ascer¬
taining the relative strength of men and animals, for an
account of which, see Dynamometer ; and for a fuller
description of the apparatus, the reader may consult the
original paper on the subject in Jour, de PEcole Pohj-
ter/i. vol. ii. or the translation in Phil. Mag. vol. i.
Force, in Law, signifies any unlawful violence of¬
fered to things or persons, and is divided into simple
and compound. Simple force is what is so committed,
that it has no other crime attending it 5 as where a
person by force enters on another’s possession, without
committing any other unlawful act. Compoundforce,
is where some other violence is committed, with such
an act which of itself alone is criminal ; as if one en¬
ters by force into another’s house, and there kills a per¬
son, or ravishes a woman. There is likewise a force
implied in law, as in every trespass, rescue, or dissei¬
sin, and an actual force with weapons, number of per¬
sons, &c.—Any person may lawfully enter a tavern,
inn, or victualling-house 5 so may a landlord his te¬
nant’s house to view repairs, &c. But if, in these
cases, the person that enters commits any violence or
force the law will intend that he entered for that pur¬
pose.
FORCEPS, in Surgery, &c. a pair of scissars for-
cutting off, or dividing, the fleshy membranous parts of
the body, as occasion requires. See SURGERY.
FORCER, in Mechanics, is properly a piston, with¬
out a valve. For by drawing up such a piston, the air
is drawn up, and the water follows $ then pushing the
piston down again, the water, being prevented from de¬
scending by the lower valve, is forced up to any height
above, by means of a side branch between the two.
FORCIBLE ENTRY, is a violent and actual entry
into houses or lands •, and a forcible detainer, is where
one by violence withholds the possession of lands, &e.
so that the person who has a right of entry is barred,
or hindered, therefrom.
At common law, any person that had a right to en¬
ter into lands, &c. might retain possession of it by
force. But this liberty being abused, to the breach
of the peace, it. was therefore found necessary that the
same should be restrained: Though, at this day, he
who
FOR
Forcible, who is wrongfully dispossessed ol goods may by force
——y—■»' retake them. By statute no person shall make an en¬
try on any lands or tenements, except where it is given
by law, and in a peaceable manner, even though they
have title of entry, on pain of imprisonment j and where
a forcible entry is committed, justices of the peace are
authorized to view the place, and inquire of the force
bv a jury, summoned by the sheriff ot the county ; and
they may cause the tenements, &c. to be restored, and
imprison the offenders till they pay a fine. likewise a
writ of forcible entry lies, where a person seized of
freehold, is by force put out thereof.
Forcible Marriage, of a woman of estate, is felony.
For by the statute 3 H. VII. c. 2. it is enacted, “ That
if anv persons shall take away any woman having lands
or goods, or that is heir-apparent to her ancestor, by
force and against her will, and marnj or defile her j the
takers, procurers, abettors, and receivers, of the woman
taken away against her will, and knowing the same,
shall be deemed principal felons $ but as to procurers
and accessories before the fact, they are to be excluded
the benefit of clergy, by 39 Elizabeth, c. 9. The in¬
dictment on the statute Hen. VII. is expressly to set
forth, that the woman taken away had lands or goods,
or was heir apparent; and also that she wras married or
defiled, because no other case is within the statute j and
it ought to allege that the taking was for lucre. It is
no excuse that the woman at first was taken away with
her consent *, for if she afterwards refuse to continue
with the offender, and be forced against her will, she
may from that time properly be said to be taken against
her will j and it is not material whether a woman so
taken away be at last married or defiled with her own
consent or not, if she was under force at the time the
offenders being in both cases equally within the words
of the act.
Those persons who, after the fact, receive the of¬
fender, are but accessories after the offence, according
to the rules of common law > and those that are only
privy to the damage, but not parties to the forcible
taking away, are not within the act H. P. C. 119.
A man may be indicted for taking away a woman by
force in another country 5 for the continuing of the
force in any country, amounts to a forcible taking
there. Ibid. Taking away any woman child under
the age of 16 years and unmarried, out of the custody
and without the consent of the father or guardian, &c.
the offender shall suffer fine and imprisonment j and if
the woman agrees to any contract of matrimony wdth
F O K
such person, she shall forfeit her estate during life, to Forcible
the next of kin, to whom the inheritance should de- j)
scend, &c. stat. 4 & 5 P. &. M. c. 8. This is a force Fordwicb,
against the parents •, and an information will lie for se-
ducing a young man or woman from their parents, a-
gainst their consents, in order to marry them, &c. See
Marriage.
FORCING, in Gardening, a method of producing
ripe fruits from trees before their natural season. See
Gardening Index.
Forcing, in the wine trade, a term used by the wine
coopers, for the fining down wines, and rendering them
fit for immediate draught. The principal inconvenience
of the common way of fining down the white wines
with isinglass, and the red with whites of eggs, is the
slowness of the operation ; these ingredients not per¬
forming their office in less than a week, or sometimes a
fortnight, according as the weather proves favourable,
cloudy or clear, windy or calm : this appears to be
matter of constant observation. But the wine mer¬
chant frequently requires a method that shall, with
certainty, make the wines fit for tasting in a few hours.
A method of this kind there is, but it is kept in a few
hands a valuable secret. Perhaps it depends upon a
prudent use of a tartarised spirit of wine, and the
common forcing, as occasion is, along with gypsum,
as the principal *, all which are to be well stirred
about in the wine, for half an hour before it is suf¬
fered to rest.
FORDOUN, John of, the father of Scottish his¬
tory, flourished in the reign of Alexander III. to¬
wards the end of the 13th century. But of his life
there is nothing known with certainty, though there
was not a monastery that possessed not copies of his
work. The first five books of the history which bears
his name were written by him : the rest wrere fabrica¬
ted from materials left by him, and from new collec¬
tions by different persons. There is a manuscript in
vellum of Fordoun’s History, in the library of the uni¬
versity of Edinburgh.
FORDWICH, a town of Kent, called in Dooms¬
day Book “ the little borough of Fordwich,” is a mem¬
ber of the port of Sandwich, and was anciently incor¬
porated by the style of the barons of the town of Ford¬
wich, but more lately by the name of the mayor,
jurats, and commonalty, who enjoy the same privileges
as the cinque ports. This place is famous for excellent
trouts in its river Stour.
[ 798 ]
END OF THE EIGHTH VOLOME.
DIRECTIONS for placing the PLATES of
Part I.
Plate CC. CCI. to face
ecu.
CCIIL—CCVI.
CCVII.—CCIX.
ccx.
Part II.
CCXL—CCXVI.
CCXVII.
CCXVIII.
CCXIX. ccxx.
Vol. VIII.
Page 10
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238
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