As Radius

10.000000Ɔ

1s to the Sine of the Inclination 30°= 9.6989700

So the Velocity in the Perpen-}

dicular

100 2,0000000

To the Velocity on the given } 50 = 1.6989700

Plane

That is just half the Velocity in the Perpendicular.

Thus, if the Velocity on the Plane, and the Inclination be given, the Velocity in the Perpendicular may be found.

Alfo, if both the Velocities are given, the Inclination of the Plane is thus cafily found.

Theorem III. exemplified.

Suppofe a Body defcend thro' an inclined Plane, whofe Inclination is 35° 50', in 16 Seconds of Time, in bow many Seconds will it fall from the Perpendicular Height of that Plane? Say thus;

As Radius

10.0000000

Is to the Sine of the Incli-} 35° 50′ = 9.7674746

nation

So is the Time of the De-7

fcent in the Plane

De-} 16°

To the Time of Defcent

in the Perpendicular

1.2041200

9′′ 1 = 0.971 5946

And thus may the two other Cafes of this Theo

rem be folved.

Theorem

Theorem IV. exemplified.

Suppofe the Inclinations of the two contiguous Planes AB and AD, were ABC 32° 14'; and ABC= 43° 25′; and it were required to find the Weight of the Body F, which on the Plane AD hall fuftain or move the Body E on the Plane AB, weighing 465 Pounds? Say.

As the Sine of the

Inclination

} ABC

32° 14' 9.7270272.

Is to the Sine of the ABC =
Inclination
So is the Weight of the Body E-465= 2.6674529

To the Weight of the } F = 360 = 2.5572345

Body required

And thus for any other of the Cafes of this Theo

rem.

Theorem V. exemplified.

This is but as it were a fecond Cafe of the laft; and therefore, to fee the Difference between them, let the Example there be here again refolved. Saying,

As the Co-fine of

the Inclination

Is to the Co-fine of

the Inclination

BAC = 57° 45′ = 9.9273103

DAC = 46.35 = 9.8611608

So is the Weight of the Body E-465= 2.6674529

To the Weight of the

Body

} F = 400 = 2.6013034

Whence it appears, that in this Cafe 40 Pounds more is neceffary, than in the Cafe of the laft Theorem, to fufpend the Body E in Equilibrio, or to move it. And therefore the Pulley A ought to to be placed, both in the fingle and double inclined Planes, that the Line coming from the Body over it may be parallel to the Plane.

Theorem VI. exemplified.

=

Suppose the Arch DB 40°, and the Arch EB = 100°; and that the Pendulum defcends from D to B, and thereby dces acquire 15 Degrees of Velocity in B; quere, how many Degrees of Velocity it will have in B, by falling thro' the Arch EB? Say thus;

So are the Degrees of Velocity acquired by defcending thro'

To the Degrees of Ve

locity acquired by de- EB=3301.5262954 fcending thro'

That is, it will now be above twice as fwift in B, as before.

To

To reprefent the Times, Velocities, and Spaces palled thro' by falling Bodies, by the Parts of Right-angled Triangle.

third Minute, &c. Laftly, The Triangular Area ADd, will reprefent the Space paffed thro' in the first Minute, AEe the Space paffed thro' in the fecond Minute, AFf the Space paffed thro' in the third Minute, &c. And thus if AC be fix Minutes, the Side CB will be the Velocity, and the Area ACB the Space paffed over at the End of the fix Minutes.

From hence 'tis evident, that the Velocity of a falling Body is always proportionate to the Time; for as the Time AD: is to the Time AC: fo is the Velocity or Celerity Dd to the Celerity CB; and therefore the Motion of falling Bodies is a Motion equally accelerated, or equally increased in equat Times.

Hence alfo the Spaces gone thro' from the Beginning of the Fall are as the Squares of the Times; for the Area ADd, the Space gone thro' in one Minute, is to the Area ACB, the Space gone thro' at the End of fix Minutes: As the Square of the Time AD is to the Square of the Time AC; Hence

the

the Spaces increase according to the odd Numbers, 1, 3, 5, 7, 9, 11, &c. All this is evident from the Nature of a Triangle itself.

Plain Trigonometry applied to Surveying, or Measuring of Land; and alfo of other Plain Superficies.

I

N Surveying Land, the Inftruments in common Ufe are the Semicircle, Plain Table, and Theodolite for taking the Angles, and a Protractor, or Plain Scale, and Compaffes for Plotting or Delineating the Dimenfion of the Field on Paper, in order to reduce it to Triangles, and thereby to find the true fuperficial Content or Area in Acres, Roods, and Poles.

I fhall here exhibit fome of the best and most usual Methods for taking the Plot of a Field, and then fhew how to find its Area; which may be as fufficient for the ingenious young Artift, as fome larger Tra&t wherein abundance is often faid to little Purpofe; fince a Word to the Wife is enough, and there is no making a Silken Purfe of a Sow's Ear when a Perfon has faid all he can.