LOC
(4*0
LOG
A B 00 : B E («) : : A P 0) : P F = — . And confe-
- ■ J m
quentl'y GMorPM-l'F-FG=> >', and
C G or- D G — D C = — — i, But from the Nature of m
theParabola G M* = C G x C H, which Equation will become that of the general Formula, by putting the Li- teral Values of thole Lines.
Again ; if thro' the fixed Point A you draw the inde- finite right Line A Q_(fig-?0 parallel to P M, and you take AB = », and draw BE = « parallel to A P, and thro' the determinate Points A, E, the Line A E = e ; and if in A P you take AD=', and draw the indefinite flrait Line D G parallel to A E, and take DC = i: this being done, if with the Diameter C G, whofe Ordinates are parallel to A P, and Parameter the Line CH=;, you defcribe a Parabola C M i the Portion of this Para- bola contain'd in the Angle B A P, fliall be the Locus of this fecond Equation or Formula,
2 it , tin . 2 » r ,
xx i x A v y — a r x •+■ y-rf-rr — o.
■m ■> ~ mm J ■> ' m
— ef ,
m
For if the Line M Q^be drawn from any Point M, therein, parallel to A P ; then will A B(» : A E (e) : : A Q_ or
PM(;):AForDG = ^. And A B(»0 ; B E (») : : A 0,00 : QF = — ■ And therefore G M or Q_M —
and C G or D G — D C
Q.F-FG = I - y - m
e y — ~S — *• 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 Sections.
Now if it be required ro draw the Parabola, which we find to be the Locks of this propofed Equation y y — nj- l»-f «e=5«j compare every Term of the firll Formula with the Terms of the Equation, becaufe
y y in both is without Fractions 5 and then will — =
m '
becaufe the Rectangle xy not being in the propofed Equation, the faid Reclangle may be efteem'd as multi- plied by ; whence n = 0, and m = e ; becaufe the Line A E falling in A B, that is, in A P in the Conftrucfion of the Formula, the Points B, E, do coincide. Therefore
deftroying all the Terms adfefled with —in the Formula
m and fubftituting m for e, we fliall get yy — 2 r y ■ — t> x -■ 1- r -f- f s = 0. Again, by comparing the correfpon- dent Terms — %ry and — ■ 2 ay, as alfo — j> x and — bx we have >■ = a, and f = lj and comparing the Terms wherein are neither of the unknown Quantities x, y we get r r -\-f s = c c, and fubftituting a and b for r and p,
then will s
which
a negative Expreffion
when a is greater than c, as is here fuppofed. There is no need of comparing the firft Terms y y and yy, becaufe they are the very fame. Now the Values of lt',r,p,s, being thus found, the fought Locus may be conftructed by means of the Conftrudlion of the Formula, after the following manner.
Becaufe B E (»)=»> C%- 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 = — 7 J 5 then with C G, as a Diameter,
whofe Ordinates are right Lines parallel to P M, and Pa- rameter the Line CH(/i)=J, defcribe a Parabola ■ I fay the two Portions OMM, RMS, thereof, contain'd in the Angle P A O, 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 rhe fecond Formula muft be com- pared with rhofe of the given Equation.
Thus much for the Method of Conftrufling the Loci of Equations, which are Conic Sections. If, now an Equation whofe Locus is a Conic Section be given ; and the particular Seftion whereof it is the Locus, be re- quired ;
All the Terms of the given Equation being bronoht over to one fide, fo that the other be equal to o there will be two Cafes.
Cafe 1. When the Reftangle x y is not in the given E- quation. 1 . If either yy or x x 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 Sections j regarding their Diameters. _ Cafe 2. When the Rectangle x y is in the given Equa- tion. 1. It 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. H yy and xx be therein, having different Signs, the Locus will be an Hyperbola, regarding its Diameters. 5. If both the Squares x x and y y are in the Equation, having the fame Signs, you muft free the Square yy from Fractions, and then the Locus will be a Parabola, when the Square of i the Fraflion multiplying x y, is equal to the Fraction multiplying uiffl Ellipfis or Circle, when the fame is lefs ; and an Hyperbola, or the oppofite Sections, regard ing their Diamerers, when greater.
LOCUSTjE, the Beards and pendulous Seeds of Oats, and of the Gramina Taniculata ; to which the Bo- tanifts gave this Name, from their Figure, which fome- thing refembles rhat of a Locutt.
LODESMAN, or Locman, a Pilot eftablifh'd for con- duBmg Veffels in and out of Harbours, up and down navigable Rivers. See Pilot.
LODGMENT, in Military Affairs, is fometimes an Incampment made by an Army ; but oftener, a Retrench- ment dug for a Cover or Shelter, when the Counterfcarp or fome other Poll is gain'd. It is alfo taken for the Place where the Soldiers quarter among the Burghers, in Fluts, Barracks, or Tents. Lodgment of an Attack* is a Work call: up by the Befiegers, during their Approaches in a dangerous Poft, where it is abfolutely neceffary to fecure themfelves againft the Enemies Fire ; as in a Covert- Way, in a Breach, in the Bottom of the Moat, igc. This Lodgment confifts of all the Materials that are capa- ble to make refiftance, w. 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 caft, to make it fwim upright in the Water.
Log-Line is a little Cord or Line faften'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 leaft 50 foot from each other : tho tis the common praflice at Sea not to have them above 42 feet afunder. The Ufe of the Log and Line is to keep account, and make an eftimate 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- Mmute Glafs ; for, fo many Knots as run out in that time, fo many Miles the Ship fails in an Hour. Thus'if rhere 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 Shin's Wake ; when one holding the Half-Minute Glafs, turns itupjuft as the firft Knot turns off the Reel (tho fome turn the Olais as foon as the Log touches the Water) as foon as the Glafs is out, the Reel is ftopt and the Knots run off are told, and their Parts eftimated
The Log is a very precarious Way of computing, and muft always be correSed by Experience and Good Senfe there being a great deal of Incertainty both in the Heavl ing of it ,n the Courfe of the Currents, and in the Strength of the Wind, which feldom keeps the fame Te nor for two Hours together, which is the Interval be- tween the Times of ufing rhe Log in /hort Voyages, tho in longer ones rhey heave it every hour. Yet is his 1 much more exact Way of Computing than any other in ufe ; much preferable cerrainly to that of the : Spaniard, and T.rtugueje, who guefs at the Ship's Way by the run- ning of the Froth or Water by the Ship's fide ; or to that ot the Butch who ule to heave a Chip over-board and to number rhe I aces they walk on the Deck while the Chin fwims between any two Marks or Bolr-hcads on the fide
Log-Board ,s a Table divided into four or five Column's whereon are mark'd the Reckonings of every Day from whence rhey are enter 'd into the Log. Boo/:, or Traverfe Book, ruled and column'd juft as the Zog-JSoard is- Whence it may be tranfenbed into the Journals, and how much the Ship ga ,„s ,n her Courfe eftimated daily. In the firft Column of the Log-Board are (hewn the Hours of the Day from 1 to ,. In thefecond is Ihewn the Rhumb
