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When a Plane cuts a Cone parallel to one of its fides, it makes a Parabola ; when it cuts the Cone parallel to its Bale, it makes a Circle. See Conicks.
The Sphere is wholly explained by Planes, imagin'd to cut the Celeftial Luminaries, and to fill the Areas or Cir- cumferences of their Orbits. See Sphere.
Altronomers fhew, that the Plans of the Moon's Orbit is inclined to the Plane of the Earth's Orbit, or the Ecliptic, by an Angle of about 5 Deg. and piffes thro' the Center of the Earth. See Orbit.
The Interferon of this Plane with that of the Ecliptic, has a proper Motion of 3' 11" each Day, from Eaft to Weft ; fo that the Nodes anfwer fucceffively to all the De- grees of the Ecliptic, and make a Revolution round the ^ Earth in about 19 Years. See Node.
The Planes of the Orbits of the other Planets, like that
Of the Ecliptic, p.ifs thro' the Center of the Sun. The
Plane of the Orbit of Saturn, is inclined to the Ecliptic by 2° 33' 30", and cuts it, at prefent, in the 22d Degree of Cancer and Capricorn. See Inclination ; fee alfo Moon and Planet.
The Centre of the Earth, then, being in the Plane of the Moon's Orbit, the Circular Section of that Plan in the Moon's Disk, is reprefented to us in Form of a Right Line paffing thro' the Center of the Moon.— This Line is inclined to the Plane of the Ecliptic by 5 when the Moon is in her Nodes : But,.this Inclination diminifhes as that Planet recedes from the Nodes ; and at three Degrees diftance, the Section of the Moon's Orbit in its Disk, becomes parallel to the Plane of the Ecliptic. The fame Appearances attend the primary - Planets, with regard to the Sun.
But the Cafe is very different in the Planets feen from
one another, efpecially from the Earth -The Planes of
their Orbits only pifs thro' the Center of the Earth when they are in their Nodes : In every Other Situation, the Plane is rais'd above the Orbit of the Planet, either to the North or the South. And the Circular Section of the Plane of the Orbit on its Disk, or in the Orbit of one of its Satellites, does not appear a Right Line, but an Ellipfis, broader or narrower as the Earth is more or lefs elevated above the Plane of the Orbit of the Planet.
P L a N E, in Mechanicks A Horizontal Plane, is a Plane
level or parallel to the Horizon. See Horizon.
The determining how far any given Plane, Sec. deviates from a Horizontal one, makes the whole Bufmefs of Level- ling. See Levelling.
Inclined Plane, in Mechanicks, is a Plane which makes an oblique Angle with an Horizontal Plane. See Oblique.
The Doftrine of the Motion of Bodies on Inclined Planes makes a very confiderable Article in Mechanicks ; the Sub- ftance whereof is as follows:
Laws of the Defcent of Bodies on an Inclined Plane.
If a Body be placed on an inclined Plane, its relative Gra- vity will be to its abfolute Gravity, as the Length of the Plane, e. gr. AC [Tab. Mechanicks, Fig. 58.) to Its Height AB. See Gravity.
Hence, i° fince the Ball D only gravitates on the inclined Plane, with its relative Gravity, the Weight L, applied in a Direction, parallel to the Length of the Plane, will re- tain or fufpend it, provided its Weight be to that of the Ball, as the Altitude of the Plane BA is to its Length AC.
2 C If the Length of the Plane C A be taken for the whole Sine ; A B will be the Sine of the Angle of Inclination A C B. —The abfolute Gravity of the Body, therefore, is to its re- fpective Gravity applied on the inclined Plane ; and there- fore, alfo, the Weight Dto the Weight L afting according to the Direction D A which fuftains it ; as the whole Sine to the Sine of the Angle of Inclination.
3 Hence the refpeftive Gravities of the fame Body on different inclined Planes, are to each other as the Sines of the Angle of Inclination.
4 The greater .therefore the refpeftive Gravity is, the greater is the Angle of Inclination.
5 As, therefore, in a vertical Plane, where the Inclina- tion is greateft, viz. perpendicular, the refpective Gravity degenerates into abfolute ; fo in a horizontal Plane, where there is no Inclination, the refpective Gravity vanifhes.
II. To find the Sine of the Angle of Inclination of a Plane, on which a given Power will be able to fuflain a given Weight._Say, as the given Weight, is to the give* Power, fo is the whole Sine to the Sine of the Angle of Inclination of the Plane. Thus, fuppofe a Weight of 1000 be to be fuftained by a Force of 50; the Angle of Inclination will be found 2 Q 52'.
III. If the Weight L defcend according to the perpendi- cular Directon A B, and raife up the Weight D in a Dire- ction parallel to the inclined Plane ; the Height of the Afcent of D will be to that of the Defcent of L, as the Sine of the Angle of Inclination C, to the whole Sine.
Hence, 1 ° The Height of the Defcent C D of the Weiid* L is to the Height of Afcent D H of the Weight D y recipro eally as the Weight D to the equivalent Weight L.
2° Since then CDL = DHD, and the Actions of the equiponderating Bodies D and L are equal; the Moments of the Weights D and L are in a Ratio compounded of their Maffes, and Altitudes, thro' which they afcend or defcend in a Plane, either inclined or perpendicular.
3 Q The Powers that raife Weights thro' Altitudes recipro- cally proportional to them, are equal This Des Cartes
affumes as a Principle whereby to demonftrate the Powers of Machines. Hence we fee why a loaden Waggon is drawn with more Difficulty on an inclined than an horizontal Plane - as being prefs'd with a Part of the Weight which is to the whole Weight in a Ratio of the Altitude to the Length.
IV. Weights E and F, equiponderating upon inclined Planes A C and C B of the fame Height C D, are to each other as the Lengths of the Planes AC and CB.
S. Stevinus gives a very pretty Demembration of this Theo- rem, which, for its Eafinels and Ingenuity, we fhall here add.— Put a Chain, whofe Parts do all exactly weigh in Pro- portion to their Length, over a Triangle, G I H : (Fig. 59.) lis .evident the Parts GK and KH do balance each other. If then I H did not balance G I, the preponderating Part would prevail; and there would arife a perpetual Motion of the Chain about G I H •, but this being abfurd, it follows, that the Parts of the Chain IH and GI-, and confequently all other Bodies which are as the Lengths of the Planes I H and I, G will balance each other.
V. A heavy Body defcends on an inclined Plane, with a Motion uniformly accelerated. See Acceleration.
Hence, 1° The Spaces of Defcent are in a duplicate Ratio of the Sines, and likewife of the Velocities; and therefore in equal times increafe according to the unequal Numbers i, 3, $,7,9, &c.
2 The Space pafs'd over by a heavy Body defending on an inclined Plane, is fubduple of that which it would pafs over in the fame Time, with the Velocity it has acquired at' the End of its Fall.
3° Heavy Bodies, therefore, defcend by the fame Laws ori inclined Planes, as in perpendicular Planes. Hence it was, that Gallileo, to find the Laws of perpendicular Defcent, made his Experiments on inclined Planes, in regard of the Motions being flower in the latter than the former ; as in the following Theorem.
VI. The Velocity of a heavy Body defending on an in- clined Plane, at the End of any given Time ; is to the Ve- locity which it would acquire in falling perpendicularly, in the lame Time ; as the Height of the inclined Plane is to its Length.
VII. The Space pafs'd over by a heavy Body on an inclined Plane A D, (Fig. 60.) Is to the Space A B, it would pafs over in the fame time in a perpendicular Plane : As its Ve- locity on the inclined Plane is to its Velocity in the perpen- dicular Defcent, at the End of any given time.
Hence, i c The Space pafs'd over in the inclined Plane, is to the Space it would defcend in the fame time in the per- pendicular Plane, as the Altitude of the Plane A B to its Length A C ; and therefore as the Sine of the Angle of In- clination to the whole Sine.
2° If, then, from the Right Angle B, a Perpendicular be let fall to AC; AC : AB :: AB : AD. So that in the lame time wherein the Body would fall perpendicularly from A to B ; in an inclined Plane it will defcend from A to D.
3° The Space, therefore, of perpendicular Defcent being given in the Altitude of the Plane A B ; by letting fall a Per- pendicular from B to A C, we have the Space AD to be pafs'd over in the fame time on the inclined Plane.
4 In like manner, the Space AD, pafs'd over on tbez'a- clmed Plane, being given ; we have the Space A B, thro' which it would defcend perpendicularly in the fame time, by raifing a Perpendicular meeting the Side of the Plane in B.
5° Hence in the Semicircle ADEFB, the Body will de- icend thro' all the Planes A D, A E, A F, A C, in the fame time ; viz. in that time wherein it would fall thro' the Diameter AB, fuppofing that perpendicular to the horizon- tal Plane L M.
VIII. The Space AD, pafs'd over in an inclined Plane A C, being given ; to determine the Space which would be pafsd over in any other inclined Plane in the fame time.
From the Point D erect a Perpendicular D B, meeting the Altitude A B in B ; then will A B be the Space, thro' which the Body would fall perpendicularly in that time. Wherefore if from B a Perpendicular BE be let fall to the Plane A F ; A E will be the Space in the inclined Plane which the Body will pafs over, in the fame time wherein it falls perpendicularly from A to B ; and confequently A D will be the Space in the other inclined Plane A C, which it pafe thro' in the fame time.
Hencu;
