Page:Cyclopaedia, Chambers - Volume 2.djvu/510

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treffion of 52 Places, where the difference is 3, and the firft Term 5 j 51 being multiply'd by 3, produces 155, to which adding 5, the Sum 158 is the latt Term required;

4. . If the Trogrejjion begin with o, the Sum of all the Terms is equal to half the Producl of the laft Term mul- tjply'd by the Number of Terms.

Whence it follows, that the Sum of a Progrejjion begin- ning from o, is fubduple the Sum of fo many Terms, all equal to the greateft.

5». In an Arithmetical Progreffion, as the difference of the Sum of the firft and laft Term from double the Sum of the Progrejjion, is to the difference of the firit Term from the laft j fo is the Sum of the firft and laft Terms to the Progrejfional difference.

Geometrical Progression, isa-Series of Quantities in- creafing or decreasing in the fame Ratio, or Proportion ; or a Series of Quantities continually proportional. See Pro- portion.

Thus r, 2, 4, 8, 16, 32, 64, ti?c. make a Geometrical Trogrejjion j or 7:9, 243, 81, 27, 9, 3, r.

i u. In every Geometrical Progrejjion, theProducl of the extreme Terms is equal to the Product of two interme- diate Terms equidiitant from the Extremes; as al fo, if the Number of Terms be uneven, to the Square of the middle Term.

For Example ;

3, 6", 12, 24, 48, 90"

12, 6, 3

288, 288, 288

2 . If the difference of the firft and laft Term of a Geo- metrical Progrejjion be divided by a Number lcfs than the Denominator of the Ratio, i. e. than the Quotient of a greater Term divided by a lefs ; the Quotient will be the bum of all the Terms except the greateft : Hence, by ad- ding the greateft Term, we have the Sum of the whole Trogrejjion.

Thus, in a Trogrejjion of 5 Terms, beginning with 3 ; and the Denominator being likewife 3, the greateft Term will be 243. If then the difference of the firft and laft Term 240, be divided by 2, a number lefs by 1 than the Denominator j the Quotient 120 added 10243, g' ves 3^3» the Sum of the Progrejfion.

Hence, 3°, the firft, or leaft term of a Trogrejjion, is to the Sum of the Trogrejjion., as the Denominator leffen'd by i, to its Power likewife leffen'd by 1 5 the Exponent of which Power is equal to the number of Terms.

Thus, fuppofing the firft Term 1, the Denominator 2, and the Number of Terms 8; the Sum will be 255.

4 Q. Hence, alfo, the difference between the laft Term and the Sum is to the difference between the firft Term and the Sum, as Unity to the Denominator : Wherefore, if the difference between the firft Term and the Sum, be di- vided by the difference between the Sum and the laft Term, the Quotient is the Denominator.

PROHIBITED Goods, in Commerce, fuch Commodi- ties as are not allow'd to be either imported or exported. See Contraband.

PROH1BITIO de vafto diretla Tarti, is a Writ judicial, directed to the Tenant, prohibiting him from making Wafte upon th«Land in controverfy, during the Suit. It is fome- times directed to the Sheriff.

PROHIBITION, the Act of forbidding, or inhibiting any thing. 'Tis the Prohibition of the Law that makes ti.e Sin. A Teftator frequently bequeaths things with a Tro- hibition to alienate.

Prohibition, in Law, is a Writ iffued to forbid any Court, either Spiritual or Secular, to proceed in a Caufe there depending ; upon fuggeftion, that the Cognizance thereof belongeth not to that Court.

It is now ufually taken for that Writ which lieth for one, who is impleaded in the Court Chriftian for a Caufe belonging to the Temporal Jurifdi£tion,or the Cognizance of ; he King's Court j whereby, as well the Party and his Council, as the Judge himfelf, and the Regifter, are forbid to proceed any farther in that Caufe.

PROJECTILE, or Project, in Mechanics, a heavy Body put in Motion by an external Force imprefs'd thereon 5 or, more fully, a ProjeBile is a heavy Body, which being put into a violent Motion, is difmifs'd from the Agent, and left topurfueits Courfe. See Motion.

Such, e. gr. is a Stone thrown out of the Hand or a Sling, an Arrow from a Bow, a Bullet from a Gun, &c. See Projection.

+ be Caufe of the Continuation of the Motion of Projectiles f or what it is determines 'em to perfift in Motion after the firft Caufe ceafes to aft, has puzzled the Philofophers. See Motion.

, The 'Peripatetics account for it from the Air, which being violently agitated by the Motion of the projecting Caufe,

rSf.) th f Hantl an ^ Sling, and forced to follow the Fro- jcBile while accelerated therein, does, upon the diirm&od of the ProjeBile, prefs after it, and protrude it forward ; to prevent a Vacuum. See "Vacuum.

The Moderns account for the Motion of Projectiles on a much mere rational and eafy Principle 5 it being, in effeft, a natural Confequence from one of the great Laws of Na- ture, viz. That all Bodies being indifferent as to Motipn or Reft, will neceffarily continue the State they are put into, except fo far as they are hindred, and forced to chanW it by fome new Caufe. See Nature.

Thus a Project put in Motion, mutt continue to move eternally on in the fame right Line, and with the fame Ve- locity; were it to meet with no Refiftance from the Medium nor had any force ef Gravity to encounter. _ The Doctrine of the Motion of Projectiles is the Founda- tion of all Gunnery. See Gunnery.

The Laws thereof are as follow.

Laws of the Motion of Projectiles.

t°. If a heavy Body be projected perpendicularly, it will continue to afcend or defcend perpendicularly : Becaufe,both the ProjeBing, and the Gravitating Force are found in the fame Line or Direction.

2« If a heavy Body be pro'.eBed horizontally, it will in its Motion, defcribe a Parabola 5 the Medium 'being fup! pofed void of Refiftance.

For, the Body is equably impell'd by the impreffed Force, according to the right Line A R, (Tab. Mecha- nics, Fig. 45.) and by the Force of Gravity according to t e right Line A C, perpendicular thereto. While, then, the Body by the Action of the impreffed Force is arrived inQj by the Force of Gravity it will be arrived in Q^M j and, therefore will be found in M. But the Morion in the direction A R will itill be uniform ; (fee Motion) and, therefore the Sp ices QjV and q A are as the Times j and the Spaces Q_M and q m, are likewife as the Squares of the Times. Therefore, A Q_" : A q 1 : : Q_M : qm. That is, P M :

f m ; : APj/.

The Courfe, or Path, therefore, of a heavy Body pro- jected horizontally A M m, is a Parabola. See Parabola.

Two hundred Years ago, the Philofophers took the Line defcribed by a Body projected horizontally, e.gr. a Bullet out of a Cannon, while the force of the Powder exceeded the Weight of the Bullet confiderably, to be a right Line 5 afier which it became a Curve.

N.Tartaglia was the firrt who perceiv'd the Miftake, and maintain'd the Path of the Bullet to be a crooked Line, throughout its whole Extent 5 but it was Galileo who firlt determined the precife Curve the Bullet defcribed 5 and fhew'd the Path of the Bullet, projected horizontally from an Eminence, to be a Parabola ; the Vertex whereof is the Point where the Bullet quits the Cannon.

3 . If a heavy Body be projected obliquely, either up- wards, or downwards, in a Medium void of Refiftance ; it will likewife defcribe a Parabola, in a Medium uniformly refilling.

Cor. Hence, 1. the Parameter of the Diameter of the Parabola AS (Fig. 47.) isathird Proportional to theSpace thro' which the Body defcends in any given Time, and the Celerity, which is defined by the Space pafs'd over in the fame time ; i.e. to A P and A Q. 2. Since the Space de- fcribed by a Body falling perpendicularly in one Minute, is 15 rV Paris feet in a Second ; the Parameter of the Dia- meter of the Parabola to he defcribed is found, if the Square of the Spice pafs'd over by the 'Projectile with the imprefs'd Force in a Second, be divided by 15 r». 5. If, then, the Velocity o£ the Projectiles be the fame, the Spaces defcribed in the fame time by the Force imprefs'd, are equal j confequently the Parameter of the Parabola's pafs'd over by the compound Motion, is the fame. 4. If from the Parameter of the Diameter be fubftrac"ted quadruple the Altitude of A P, the Remainder is the Parameter of the Axis j the fourth part whereof is the diftance of the Vertex of the Axis from the Focus of the Parabola. Hence the Celerity of the TrojeBile being given, the Parabola defcribed by the Projectile may be laid down on Paper. 5. The Line of Direction of the TrojeBile A R is a Tangent to the Parabola in A.

Sir If Ne-wt07i fhews, in his Principia, that the Line a Projectile defcribes, approaches nearer to an Hyperbola than a Parabola.

4 Q. A Projectile in equal times defcribes Portions of its parabolic Path, as A M, km, which are fubtended by equal Spaces of the Horizon A T, T t. i.e. in equal times it paffes over equal horizontal Spaces.

f. The Quantity, or Amplitude of the Path A B, i. e. the Range of the TrojeBile is to the Parameter of the Dia- meter A S, as the Sine of the Angle of Elevation R A B to its Secant.

Hence, 1. the Semiparameter ia to the Amplitude of the Path A B, as the whole Sine to the Sine of double the

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