PL A
(82$)
P L A
Hencei fince A B is to AD, as the whole Sine to the Sine of the Angle of Inclination C. And A B is to A E as the whole Sine to the Sine of the Angle of Inclination F; the Spaces AD and AE, which the Body will pafs over in the fame Time oh different inclined Planes, are as the Sines of the Angles of Inclination, C and F, and reciprocally as the refpective Gravities on the fame Planes. And confequently, alfo, reciprocally as the Lengths of Planes equally high AC and A F. — Whence the Problem may berefolved vari- ous Ways by Calculation, p.,
IX. The Velocities acquired in the fame time on different inclined Planes, are as the Spaces pafs'd over in the fame Time. — Hence, alfo, they are as the Sines of the Angles of Inclination C and F ; reciprocally as the refpective Gravities on the fame Plums ; and reciprocally as the Lengths of e- qually high Planes, A C and A F.
X. A Body defcending on an inclined Plane A C, when it arrives at the horizontal Line C B, has acquired the lame Velocity which it wouid have acquired in a perpendicular Deferent A B, to the fame horizontal Line C F.
Hence, i° A heavy Body defcending thro' different in- clined Planes, AC, A G, A F, has acquired the fame Velocity when it arrives at the fame horizontal Line C F.
Hence alfo a Body continuing its Defcent thro' feveral contiguous inclined Planes acquires the fame Velocity which it would acquire in defcending perpendicularly to the fame horizontal Plane.
XI. The Time of Defcent along an inclined Plane A C, is to the Time of perpendicular Defcent thro' A B, as the Length of the Plane AC, to its Altitude A B : But Times of Defcents thro' different inclined Planes equally high A C and AG, are as the Lengths of the Planes.
XII. If the Diameter of a Circle A B, (Fig. 60J be paral- lel to the horizontal Line LM;a Body will defcend from any Point of the Periphery D, E, or C to B, along an in- clined Plane DC, E B, and C B, in the fame Time wherein it will deicend thro' the Diameter A B. Hence,
XIII. The Defcents of a Body thro' a Semicycloid DEF, (Kg. 61.) and thro' any Arch thereof B A K, are always equidiurnal, or perform'd in the fame Time; on which Principle is built the Doctrine of Pendulums vibrating in a Cycloid. See Cycloid and Pendulum.
Laws of the Afcent of Bodies on Inclined' Planes.
T. If aBodyafcend in a Medium void of Refiftance, in any Direction, whether perpendicular, or along an inclined Plane ; its Motion will be uniformly retarded. See Retar- dation.
Hence, 1° A Body afcending either perpendicularly or obliquely, in fuch a Medium paffes over a Space which is fubduple of that it would pafs over in the ftme Time on a horizontal Plane, with an uniform Celerity equal to that it has at the Beginning of its Motion.
2° Such Spaces, therefore, perform'd in equal Times, de- creafe in a retrograde Order, as the uneven Numbers 7, 5, 3, 1: and therefore the Afcent is fo much impeded; confe- quently, when the imprefs'd Force is exhaufted, the Body will defcend again by the Force of Gravity.
3 They are therefore, inverfcly, as the Spaces defcrib'd in the fame Times by a Body defcending thro' the lame Al- titude.— For, fuppofe the Time divided into four Parts; In the firft Moment, the Body A defcends thro' the Space 1, and B afcends thro' 7 ; in the fecond, A defcends thro' 3, B afcends thro' 5, &c.
4 Hence, a Body rifing with an imprefs'd Force, afcends to that Altitude, from which it muft fall to acquire that Velocity in failing, wherewith it afcended.
5 Hence, by" falling it acquires a Force to rife again to the Height whence it fell. See Pendulum.
II. The Time wherein a Body afcends to a given Altitude, being given ; to determine the Spaces pafs'd over each Mo- ment.— -Suppofe the fame Body to defcend from the fame Altitude in the fune time; and find the Spaces pafs'd over each Moment. (See Motion.) Thefe, taken inverfely, are the fame with the Spaces of Afcent required.
Suppofe, v.g- a Body projected perpendicularly, to afcend thro' a Space of 240 Feet in 4 Seconds; and the Spaces of Afcent perform'd in the feveral Times required? If, now, the Body had defcended, the Defcent in the firft Minute had been 15 Feet, in the fecond 45, in the third 75, in the fourth 105, &c. The Defcent therefore will be in the firft Moment 105, in the fecond 75, &c.
III. If a Body defcend either perpendicularly thro U ft, {Fig. 61.) or in any other Surface FED, and with the Velocity it has there acquired, again afcend along anot j>" Surface D C, at Points equally high, e. gr. at G and H, and Q. and D, it will have the fame Force and the fame Velocity.
Hence, if a Body defcend along any Surface, FED, and again afcend along another fimilar and equal Surface DGC; 'tis the fame as if it pafs'd over the feveral Parts of the fame Line twice.
Whence, the Times of Afcent and Defcent thro' equal Spaces are equal.
On this Principle is founded the Conflructiou and Ufe of Pendulums. See Pendulum and Oscillation,
Plane of Gravity, or Gravitation, is a PLme fuppofed to pafs thro' the Center of Gravity of the Body, and in the Direction of its Tendency ; that is, perpendicular to the Horizon. See Gravity and Gravitation.
Plane of Rtfletlion, in Catoptrics, is a Plane which paffes through the Point of Reflection ; and is perpendicular to the Plane of the Glafs, or reflecting Body. See Re- flection.
Plane of RcfraBion is a Plane drawn thro' the incident and refracted Ray. See R_ r.r- R A c r 1 N.
PerjpeUive P l a N e, is a plain pellucid Surface, ordinari- ly perpendicular to the Horizon, and placed between the Spectator's Eye and the Object he views; thro' which the optic Rays, emitted from the feveral Points of the Object,
- fuppofed to pafs to the Eye, and in their Paffage to leave
rks that reprefent them on the faid Plane. See Per-
are
Marks that repr
SPECTIVE.
Such is the Plane HI ; (Tab. PerfpeSive Fig. I.) Tome call it the Table, becaufe the Draught, or Perfpective of the Object, is fuppofed to be thereon ; others, the Section, from its cutting the vifual Rays; and others, the Glafs, from its fuppofed Tranfparencv.
Geometrical P l a M e, in Perfpective, is a Plane parallel to the Horizon, whereon the Object to be delineated Is fuppofed to be placed.
Such is the Plane L M. (Fig. I. Tab. PerfpeUive)— This Plane is ufually at right Angles with the perjpeclive Plane.
Horizontal Plane, in Perfpective, is a Plane palling thro' the Spectator's Eye, parallel to the Horizon, cutting the Perfpective Plane v/hen that is perpendicular to the Geo- metrical one, at right Angles.
Vertical Vt. km, in Perfpective, a plane puffing thro' the Spectator's Eye, perpendicular to the Geometrical Plane; and ufually parallel to the Perfpective Plane. See Vertical.
Objeftive Plane, in Perfpective, is any Plane fituate in the horizontal Plane, whole Reprefentation in Perfpe- ctive is required. See Object.
Plane of the Horopter, in Optics, is a Plane that paf- fes thro' the Horopter, A B, (Tab. Optics Fig. 67.) and is perpendicular to a Plane paffing thro' the Optic Axes IGH. See Horopter.
Pianh/ the Projection, in the Stereographic Projection of the Sphere, is the fame with the perfpeftive Plane, which fee. See alfo Projection, &c.
Pl a n e of a Dial, or Dial Plan e, the Surface where- on a Dial is drawn. See Di al.
We have Horizontal, Vertical, Inclining, Declining, Re- clining, Deinclining, Direct, &c. Dial Planes. See In- clining, Declining, Reclining, Direct, ere.
Plane Glafs, Mirror, &c. See P l a i n Glajs, Mir- ror, Ike.
Plane, in Joinery, &c. an F.dge-Tnflrument, ufed to pare or fhave Woods fmootb, even, c>c.
It coufifts of a Piece of Wood, very fmoot'i at bottom, fervlng as a Stock, or Shaft ; in the middle whereof is an Aperture, thro' which paffes a Steel Edge, or Chiffel, obliquely placed, and very (harp, which takes off the Ine- qualities of the Wood it is Aid along.
The Plane acquires various Names according; to its various Forms, Sizes and Ufes : as, i° The Fore-Plane, which is very long, and is that commonly firft ufed. The Edge of its Iron is not ground (freight, but rifes with a Convex- Arch in the middle, to bear being fet the ranker ; its Ufe being to take off the greater Irregularities of the Stuff, and to prepare it for the Smoothing Plane.
2° The Smoothing Plane is fhortand final!, its Iron, fine; it takes off the greater Irregularities left by the Fore* Plane, and prepares the Wood for the Jointer.
3 The Jointer is the longeft of all ; its Edge very fine, not (landing out above a Hair's Breadth ; it comes after the Smoothing-Plane, and is chiefly intended to fhoot the Edge of a Board perfectly ftreight for jointing finooth Tables, d-c. 4 The Strike-Block is like the Jointer, but fhorter; its Ufe, to fhoot fhort Joints, &c.
5° Rabbet-Plane, ufed to cut the upper Edge of a Board, (trait or fquare, down into the Stuff; fo as the Edge of an- other, cut after the fame manner, may join in with it on the Square; it is alfo ufed to finite Fafcia's in Mouldings. Its Iron is full as broad as its Stock, that the Angle may cutftraif, and it delivers its Shavings at the Sides, not, like the others, at the Top.
6" The Plow, a narrow Rabbet-Plane, with the Addi- tion of two Staves, whereon are Shoulders, and on the Shoulders a Fence.— Its Ufe is to plow a narrow fquare Groove on the Edge of a Board, Ore.
7 Moulding-Planes; of thefe there are various Kinds, 10 A accommo-
