From these Propofitions, and their particular Examples, I fuppofe 'tis fufficiently evident how neceffary and useful the Art of Plain Trigonometry is in the Art of Gunnery, in Cafting Bombs, Fufees, and other Inftruments of Deftruction in Times of War, and in Cafes of Neceffity, fo as to effect the defigned Purposes with Certainty and Succefs. And this is all I intend, leaving what elfe concerns this terrible Affair to thofe who have more Knowledge therein than my self.

CHAP. VI.

Plain Trigonometry applied to Mechanics; in computing the Directions, Celerities, Momenta, and Forces of moving and striking Bodies.

I'

N order to understand the Do&trine of Motion, the following Definitions are neceffary to be premifed.

Definitions.

I. Motion is a continual and fucceffive Change of Place.

II. Celerity is an Affection of Motion, whereby a Body in Motion paffeth over a given Space in a given

- Time.

III. Reft is the Permanence, or Continuance of any Body in the fame Place.

Kk 2

IV. Place

IV. Place is the Space taken up by any Body; and is two-fold, abfolute or relative.

V. Abfolute Place is that Part of the immoveable Space which a Body takes up.

VI. Relative Place is the Situation of a Body in refpect of other Bodies, and can only be difcerned by our Senfes; and is changeable, while abfolute Place remains the fame; and e contra.

VII. Abfolute Motion is a Change of abfolute Place, and its Celerity is measured by abfolute Space. VIII. Relative Motion is the Change of relative Place, and its Celerity is measured by relative Space.

IX. Abfolute Reft is the Permanence of a Body in the fame abfolute Place.

X. Relative Reft is the Permanence of a Body in the fame relative Place.

XI. The Direction of Motion is a right Line, according to which the moving Body tends to any certain Point or Place.

XII. Equable Motion is that which is performed in every Part of Space paffed over, with equal or the fame Celerity.

XIII. Accelerated Motion is that whofe Celerity or Velocity continually increaseth.

XIV. Retarded Motion is that whofe Velocity is continually diminished.

XV. The Momentum, or Momenta, is the Quantity of Motion, or the Quantities of Motion in moving Bodies compared, which is compounded of the Quantity of Matter and Celerity of Motion, in any Body.

XVI. A Power is any Force impreffed, or ading on any Body, to change its State, either of Motion or Rest

XVII. Gravity is a Force tending downwards, or whereby Bodies tend towards the Center of the Earth. XVIII. A Centripetal Force, is that whereby any Body endeavours by Gravity to reach fome Point, as its proper Center, the contrary, is called the Centrifugal Force, receeding from the Center.

In the next Place 'twill be neceffary for the young Trigonometer to understand the Laws of Nature relating to Motion, or Bodies freely moving or falling; which Sir Isaac Newton has reduced to Three, in his Principia; and are as follows.

Law I.

Every Body will continue in a State of Reft, or will move uniformly in a right Line, except fo far as it is compelled to change its first State by Forces impreffed.

Law II.

The Change of Motion is always proportionable to the moving Forces impreffed, and is always made according to the Right Line, in which that Force is impreffed.

Law III

Action is always equal and contrary to Reaction; that is, the Actions of two Bodies on each other are always equal, and in contrary Directions.

These Laws of Motion, which all Bodies obferve, are illuftrated, proved and established, by the abovenamed great Author in the faid Book, and by many other Writers of mechanical and experimental Phi

losophy;

lofophy; which the Reader may confult at his Pleafure.

From thefe Axioms or Fundamental Laws of Nature, the following Corollary is deduced,

Corollary.

Every Body by conjoint Forces, will defcribe the Diagonal of a Parallelogram in the fame Time, wherein it would do the Sides by the Forces finglely. Suppofe a Body A, A

with the Force M = = 8,
were carried with an uni-
form Motion from A to
B in the Time T = 5.
and in the fame Moment C
it were urged by the
Force N3, from A
towards C; it would,
by thofe joint Forces,
be borne from A to D
in the fame Time. For

A

B

D

B

D

because the Force N acts according to the Line AC parallel to BD, it will nothing alter the Velocity of approaching to the Line BD produced by the first Force M, by Law II, it will therefore arrive to the Line BD in the fame Line T, whether the Force N acteth on it or not, and will in the End of that Time be found fomewhere in the Line BD. By the like Reafoning, at the End of the faid Time it will be found fomewhere in the Line CD; and therefore of Neceffity it will be in the Point where these two Lines meet, viz. in D, and the Force (thus compounded of the first two M and N) whereby 'tis carried in the Line AD, will be = M + N, or 89

9.8. For fince the Time is the fame, the two Forces

M

M, N, will be reprefented by the Sides AB = CD, and AC BD, in the firft Cafe of a Retangular Parallelogram; but √AB + BD = √AD = √ 9.8 the compound Force.

In the Second Cafe, where the Dirc&ion of the Forces are impreffed obliquely to each other, on the Body A; it will always hold, as the Sine of the greatest Acute Angle: is to the greater Force (or as the Sine of the leffer Acute Angle: to the leffer Force) :: fo the Sine of the Obtufe Angle: to the new compound Force fought. From licnce refults the lavention of that curious Art.

Of the Compofition and Refolution of Forces and Motion.

T

HE Example in the preceding Corollary, is fufficient to thew, how two direct or oblique Forces AB and AC, are to be reduced to one direct Force AD, and how the Quantity of that Force is to be estimated.

And if any fingle dire& Force AD, of a given Quantity, be propofed; 'tis cafy to fee how it may be refolved into two Forces AB and AC, that fhall act either in right or oblique Directions to each other; and to determine the Quantity of each.

Alfo, if a Body A be acted upon by three given Powers or Forces B, C, D, that are as the Sides of a Triangle made by Lines parallel to the Directions or the Powers, 'tis from hence evident, that Body will be at Reit.

Suppofe