Page 466

LO C
(
AB(m):. BE (n):: AP(x); -PF_          . And confe-
quentlyGMorPM-PF-FG=y- --r, and
C G or.- D G-D C      --s.    But from the Nature of
the Parabola G ~M  - C G x C 1., which Equation will
become that of the general Formula, by putting the Li-
teral Values of thofe Lines.
Again; if thro' the fixed Point A you draw the inde-
finite right Line A Q(Fig. 9.) parallel to P M, and you
take A B = m, and draw B E -n parallel to A P, and
thro' the determinate Points A, E, the Line A E -e
and if in A P you take A D _ r, and draw the indefinite
{trait Line D G parallel to A E, and take D C = s; this
being done, if with the Diameter C G, whofe Ordinates
areparallel to A P. and Paraneter the Line CH -P,
you defcribe a Parabola C M; the Portion of this Para-
bola contain'd in the Angle B A P, Ihall be the Locus of
this fecond Equation or Formula,
nn f   o             2.1?  r
xx m YX+smYy2zrxx+-Prr =o.
mm~~~
-epy +ps.
For if the Line M Qbe drawn from any Point M, therein,
parallel to A P5 then will A B (m) : A E (e): : A Q_ or
P M(y):AFor D G          Y. AndAB(rm);BE(n)::
A Q(y): QF =      Y. And therefore G M   or QM-
QF-FG          x-! - r; and CG orD G-DC
m
-my5     And fo by the common Property of the Pa-
rabola, you will have the aforegoing fecond Equation or
Formula.
So likewife may be found general Equations or For-
mula's to the other Conic Seaions.
Now if it be required to draw the Parabola, which
we find to be the Locus of this propofed Equation y y-
2. ay -  b x + c c = o; compare every Term   of the
firit Formula with the Terms of the Equation, becaufe
y y in both is without Fraaions; and then will m
becaufe the Reaangle xy not being in the propofed
Equation, the faid Reaangle may be efleem'd as multi-
plied by o; whence n _ o, and m = e; becaufe the Line
A E falling in A B, that is, in A P in the Conflruffion of
the Formula, the Points B, E, do coincide. Therefore
deftroying all the Terms adfeaed with -nin the Formula,
m
and fubflituting m fore, we fhall get yy -  z ry -p x
+  r r + p s = o. Again, by comparing the correfpon-
dent Terms -   r y and-2 ay, as alfo -_p x and- b x,
we have r = a, and p - 1; and comparing the Terms
wherein are neither of the unknown Quantities x, y, we
get r r +jp s = c c, and fublituting a and b for r and p,
c c-a a
then wills =     r-, which is a negative Exprefflon
when a is greater than c, as is here fuppofed. There is
no need of comparing the ftrll Terms y y and y y, becaufe
they are the very fame. Now the Values of nr , s
being thus found, the fought Locus may be con{lru&edby
means of the Conflruffion of the Formula, after the
following manner.
Becaufe B E (n)= o, (Fig. io.) the Points B, E, do co-
incide, and the Line A E falls in A P; therefore thro'
the fixed Point A draw the Line A D (r) _ a parallel to
P M, and draw D G parallel to A P, in which take
DC   = O      - s ; then with C G, as a Diameter,
whofe Ordinates are right Lines parallel to P M, and Pa-
rameter the Line C H (p ) = b, defcribe a Parabola : I
fay the two Portions 0 M M, R M S, thereof, contain'd in
the Angle P A 0, form'd by the Line A P, and the Line
AO drawn parallel to PM, will be the Locus of the
given Equation, as is eafily proved. If in a given Equa-
tion, whofe Locus is a Parabola, x x is without a Frac-
tion, then the Terms of the fecond Formula mufl be com-
pared with thofe of the given Equation.
Thus much for the Method of Conflruaing the LocS
of Equations, which are Conic Seafions. If, now, an
Equation whofe Locus is a Conic Seffion be given; and
the particular Seffion whereof it is the Locus, be re-
quired:
S6)                   L O G
All the Ternis of the given Equation being brought
over to one fide, fo that the other be equal to o, there
will be two Cafes.
Cafe I. When the Reffangle x y is not in the given E-
quation. i. If eitheryy or xx be in the fame Equation,
the Locus will be a Parabola. 2 - If both x x and yy are
in the Equation with the fame Signs, the Locus will be
an Ellipfis or a Circle.  If x x and yy have different
Signs, the Locus will be an Hyperbola: or the oppofite
Sedfions; regarding their Diameters.
Cafe a. When the Reriangle x y is in the given Equa-
tion. r. If neither of the Squares x x or yy, or only one
of them, be in the fame, the Locus of it will be an Hy-
perbola between the Afymtotes.  2. If yy and xx be
therein, having different Signs, the Locus will be an
Hyperbola, regarding its Diameters.  3. If both the
Squares x x and y y are in the Equation, having the fame
Signs, you mull free the Square yy from Fradions, and
then the Locus will be a Parabola, when the Square of
T the Fraffion multiplying x y, is equal to the Fradtion
multiplying x x ; an Ellipfis or Circle, when the fame is
lefs ; and an Hyperbola, or the oppofiteSedions, regard-
ing their Diameters, when greater.
LOCUSTJE, the Beards and pendulous Seeds of
Oats, and of the Graminna Panicuelata; to which the Bo-
tanifts gave this Name, from their Figure, which fome-
thing refembles that of a Locufs.
LODESMAN, or Locman, a Pilot eflablifh'd for con-
dufing Vefels in and out of Harbours, up and down
navigable Rivers. See Pilot.
LODGMENT, in Military Affairs, is Sometimes an
Incampment made by an Army; but oftener, a Retrench-
ment dug for a Cover or Shelter, when the Counterfcarp,
or fome other Polo is gain'd. It is alfo taken for the
Place where the Soldiers quarter among the Burghers, in
Huts, Barracks, or Tents. Lodgment of an Attack, is a
Work caft up by the Befiegers, during their Approaches
in a dangerous Poll, where it is abfolutely neceffhry to
fecure themfelves againfl the Enemies Fire; as in a
Covert-Way, in a Breach, in the Bottom of the Moat, ,c.
This Lodgment confills of all the Materials that are capa-
ble to make refiflance, viz. Barrels, and Gabions of
Earth, Pallifadoes, Wool-packs, Mantelets, Faggots, &.c.
LOG, a Sea-Term fignifying a piece of Board or
Timber 7 or 8 Inches long, and of a triangular figure, on
board a Ship; into one end whereof, a convenient quan-
tity of Lead is call, to make it fwim upright in the
Water.
Log-Line is a little Cord or Line faflen'd to one end of
the Log, and wound round a Reel fix'd for that purpofe
in the Gallery of the Ship. This Line, from the diftance
of about ten Fathom off the Log, has certain Knots or
Divifions, which ought to be at leall 50 foot from each
other: tho 'tis the common prafice at Sea, not to have
them above 42 feet afunder. The Ufe of the Log and
Line is to keep account, and make an eflimate of the
Ship's Way: which is done by obferving the Length of
Line unwound in half a Minute's time, told by a Half-
Minute Glafs; for, fo many Knots as run out in that
time, fo many Miles the Ship fails in an Hour. Thus if
there be four Knots veer'd out in half a Minute, the Ship
is computed to run four Miles an hour. To heave the
Log, as they call it, they let it down into the Water
letting it run till it comes without the Eddy of the Ship's
Wake; when one holding the Half-Minute Glafs, turns
it up juil as the firfl Knot turns off the Reel (tho fome
turn the Glafs as foon as the Log touches the Water)
as foon as the Glafs is out, the Reel is llopt, and the
Knots run off are told, and their Parts eflimated.
The Log is a very precarious Way of computing, and
mull always be correaed by Experience and Good Senfe,
there being a great deal of Incertainty both in the Heav-
ing of it, in the Courfe of the Currents, and in the
Strength of the Wind, which feldom keeps the fame Te-
nor or two Hours together, which is the Interval be-
tween the Times of ufing the Log in lhort Voyages, tho
in longer ones they heave it every hour. Yet is this  a
much more exacf Way of Computing than any other   id
ufe; much preferable certainly to that of the Spaniards
and Portuguefe, who guefs at the Ship's Way by the run-
ning of the Froth or Water by the Ship's fide; or to that
of the Dutcb, who ufe to heave a Chip over-board, and to
number the Paces they walk on the Deck while the Chip
fwims between any two Marks or Bolt-heads on the fide.
Log-Board is a Table divided into four or five Columns,
whereon are mark'd the Reckonings of every Day, from
whence they are enter'd into the Log.Book, or Traverfe-
Book, ruled and column'd jull as the Log-Board is:
Whence it may be tranfcribed into the Journals, and how
much the Ship gains in her Courfe estimated daily.  In
the firfe Column of the Log-Board are fhewn the Hours of
the Day from i to   i. In the fecond is fhewn  the Rhumb,
or