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ON THE MEASUREMENT OF ALTITUDES BY THE
BAROMETER AND THERMOMETER.

391. FROM the principles laid down in the scholium to prop. 76, concerning the measuring of altitudes by the barometer, and the foregoing descriptions of the barometer and thermometer, we may now collect together the precepts for the practice of such measurements, which are as follow:

First. Observe the height of the barometer at the bottom of any height, or depth, intended to be measured; with the temperature of the quicksilver, by means of a thermometer attached to the barometer, and also the temperature of the air in the shade by a detached thermometer.

Secondly. Let the same thing be done also at the top of the said height or depth, and at the same time, or as near the same time as may be. And let those altitudes of barometer be reduced to the same temperature, if it be thought necessary, by correcting either the one or the other, that is, augment the height of the mercury in the colder temperature, or diminish that in the warmer, by its part for every degree of difference of the two.

Thirdly. Take the difference of the common logarithms of the two heights of the barometer, corrected as above if necessary, cutting off 3 figures next the right hand for decimals, when the log-tables go to 7 figures, or cut off only 2 figures when the tables go to 6 places, and so on; or in general remove the decimal point 4 places more towards the right hand, those on the left hand being fathoms in whole numbers.

Fourthly. Correct the number last found for the difference of temperature of the air, as follows; Take half the sum of the two temperatures, for the mean one: and for every degree which this differs from the temperature 31°, take so many times the part of the fathoms above found, and add them if the mean temperature be above 31o, but subtract them if the mean temperature be below 31°; and the sum or difference will be the true altitude in fathoms: or, being multiplied by 6, it will be the altitude in feet.

45

392. Example 1. Let the state of the barometers and thermometers be as follows; to find the altitude, viz.

Barom.

Thermom. attach. i detach. Lower29.68 67 57 Upper25.28 43

42

Ans. the alt. is
7191 fathoms.

393. Exam,

THE RESISTANCE OF FLUIDS, &c.

393. Exam. 2. To find the altitude, when the state of the barometers and thermometers is as follows, viz.

Barom.

Thermom. attach. detach.

31

35

236

Lower29.45 38
Upper26.82 41

ON THE RESISTANCE OF FLUIDS, WITH THEIR
FORCES AND ACTIONS ON BODIES.

PROPOSITION LXXVII.

394. If any Body Move through a Fluid at Rest, or the Fluid
Move against the Body at Rest; the Force or Resistance of
the Fluid against the Body, will be as the Square of the Ve-
locity and the Density of the Fluid. That is, Roc dv2.

Ans. the alt. is
409 fathoms,
or 2458 feet.

T3

J

FOR, the force or resistance is as the quantity of matter or particles struck, and the velocity with which they are struck. But the quantity or number of particles struck in any time, are as the velocity and the density of the fluid. Therefore the resistance, or force of the fluid,, is as the density and square of the velocity.

395. Corol. 1. The resistance to any plane, is also more or less, as the plane is greater or less; and therefore the resistance on any plane, is as the area of the plane a, the denș sity of the medium, and the square of the velocity. That is, R¤ adv2.

396. Corol. 2. If the motion be not perpendicular, but oblique to the plane, or to the face of the body; then the resistance, in the direction of motion, will be diminished in the triplicate ratio of radius to the sine of the angle of inclination of the plane to the direction of the motion, or as the cube of radius to the cube of the sine of that angle. So that R∞ adv3s3, putting 1= radius, and s sine of the angle of inclination CAB.

1

1

THE RESISTANCE OF FLUIDS, &c.

237

ticle, striking the plane obliquely in the direction ca, is also
diminished as AB to BC, or as 1 tos; therefore the resistance,
which is perpendicular to the face of the plane by art 52, is
as 12 to se. But again, this resistance in the direction perpen-
dicular to the face of the plane, is to that in the direction ac,
by art 51, as AB to Be, or as 1 to 8. Consequently, on all these
accounts, the resistance to the plane when moving perpendi-
cular to its face, is to that when moving obliquely, as 13 to s3,
or 1 to $3. That is, the resistance in the direction of the
motion, is diminished as 1 to s3, or in the triplicate ratio of
radius to the sine of inclination.

PROPOSITION LXXVIII.

397. The Real Resistance to a Plane, by a Fluid acting in a
Direction perpendicular to its Face, is equal to the Weight
of a Column of the Fluid, whose Base is the Plane, and Al-
titude equal to that which is due to the Velocity of the Motion,
or through which a Heavy Body must fall to acquire that Ve-
locity.

THE resistance to the plane moving through a fluid, is the
same as the force of the fluid in motion with the same veloci-
ty, on the plane at rest. But the force of the fluid in motion,
is equal to the weight or pressure which generates that motion;
and this is equal to the weight or pressure of a column of the
fluid, whose base is the area of the plane, and its altitude that
which is due to the velocity.

398. Corol. 1. If a denote the area of the plane, v the
velocity, a the density or specific gravity of the fluid, and
g= 1611⁄2 feet, or 193 inches. Then the altitude due to the
v2 anv2
velocity v being
therefore a Xnx
will be the
4g'
whole resistance, or motive force R.

22

4g. 4g

399. Corol. 2. If the direction of motion be not perpendicular to the face of the plane, but oblique to it, in any angle, whose sine is s. Then the resistance to the plane will be and2 §3

4g

400. Corol. 3. Also, if w denote the weight of the body, whose plane face a is resisted by the absolute force R; then anv2 $3

the retarding force f, or will be

R
W

4gw

401. Corol. 4. And if the body be a cylinder, whose face

or

238

THE RESISTANCE OF FLUIDS, &c.

;

=

=

or end is a, and radius r, moving in the direction of its axis
because then s = 1, and a = pr3, where p 3.1416; then
pnv3r3
will be the resisting force R, and pnv2 2
4gw

the retarding

4g force f.

402. Corol. 5. This is the value of the resistance when the end of the cylinder is a plane perpendicular to its axis, or to the direction of motion. But were its face an elliptic section, or a conical surface, or any other figure every where equally inclined to the axis, or direction of motion, the sine or inclination being s; then, the number of particles of the fluid striking the face being still the same, but the force of each opposed to the direction of motion, diminished in the duplicate ratio of radius to the sine of inclination, the resisting force & would be pnr2 va sa

4g

PROPOSITION LXXIX.

403. The Resistance to a Sphere moving through a Fluid, is
but Half the Resistance to its Great Circle, or to the End of a
Cylinder of the same Diameter, moving with an Equal Velo-
city.

B

LET AFEB be half the sphere, moving
in the direction CEG. Describe the para-
boloid AIEKB on the same base. Let any
particle of the medium meet the semicir-
cle in F, to which draw the tangent FG,
the radius Fc, and the ordinate FIH. Then
the force of any particle on the surface at
F, is to its force on the base at H, as the
square of the sine of the angle &, or its
equal the angle FCH, to the square of radius, that is, as
HF2 to CF3. Therefore the force of all the particles, or the
whole fluid, on the whole surface, is to its force on the circle
of the base, as all the HF2 to as many times cr2. But cr2
is = CA2
= CA2 AC: CB, and HF2 = AH. HB by the nature of the
circle : also, AH. HB: AC. CB : : HI: CE by the nature of the
parabola: consequently the force on the spherical surface, is.
to the force on its circular base, as all the HI's to as many ce's,
that is, as the content of the paraboloid to the content of its
circumscribed cylinder, namely as 1 to 2.

!

404. Corol. Hence, the resistance to the sphere is R pn v 2 pa

8g

A

H

being the half of that of a cylinder of the same
diameter.

SPECIFIC GRAVITY.

239

diameter. For example, a 9lb iron ball, whose diameter is 4 inches, when moving through the air with a velocity of 1600 feet per second, would meet a resistance which is equal to a weight of 132 lb, over and above the pressure of the atmosphere, for want of the counterpoise behind the wall.

PRACTICAL EXERCISES CONCERNING SPECIFIC

GRAVITY.

The Specific Gravities of Bodies are their relative weights contained under the same given magnitude; as a cubic foot, or a cubic inch, &c.

The specific gravities of several sorts of matter, are expressed by the numbers annexed to their names in the Table of Specific Gravities, at page 211; from which the numbers are to be taken, when wanted.

Note. The several sorts of wood are supposed to be dry. Also, as a cubic foot of water weighs just 1000 ounces avoirdupois, the numbers in the table express, not only the specific gravities of the several bodies, but also the weight of a cubic foot of each in avoirdupois ounces; and hence, by proportion, the weight of any other quantity, or the quantity of any other weight, may be known, as in the following problems.

PROBLEM I.

To find the Magnitude of any Body, from its Weight.

As the tabular specific gravity of the body,
Is to its weight in avoirdupois ounces,

So is one cubic foot, or 1728 cubic inches,
To its content in feet, or inches, respectively.

EXAMPLES.

EXAM. 1. Required the content of an irregular block of common stone, which weighs 1 cwt or 112lb.

Ans. 12284 cubic inches. EXAM. 2. How many cubic inches of gunpowder are there in llb weight? Ans. 291 cubic inches nearly. feet are there in a ton weight Ans. 38138 cubic feet.

EXAM. 3. How many cubic of dry oak?

PROBLEM