95

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the

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The initial velocities can never be pushed as far as That the reader may see with one glance the relaUse of the we have calculated for in this table ; but we mean it tion of those different quantities, we have given this last table. for a table of more extensive use than appears at first table, expressed in a figure (fig. 10.). The abscissa, Fig. 10. sight. Recollect, that while the proportion of the ve- or axis DA, is the scale of the initial velocities in feet 96

Relation of locity at the vertex to the terminal velocity remains the per second, measured in a scale of 400 equal parts in

the diffesame, the curves will be similar: therefore, if the initial an inch. The ordinates to the curve ACG

express rent quanvelocities are as the square-roots of the diameters of the yards of the range on a scale containing 800 yards in tilies in it. balls, they will describe similar curves, and the ranges

an inch. The ordinates to the curve A x y express (by
will be as the diameters of the balls.

the same scale) the height to which the ball rises in the
Therefore, to have the range of a 12 pound shot, if air.
projected at an elevation of 45, with the velocity 1500 ; The ordinate BC (drawn through the point of the
suppose the diameter of the 12-pounder to be d, and abscissa which corresponds to the initial velocity 2000)
that of the 24-pounder D; and let the velocities be v is divided on the points 4, 9, 12, 18, 24, 32, 42, in the
and V: Then say, di D=1500, to a fourth ratio of the diameters of cannon-shot of different weights;
proportional V. If the 24-pounder be projected with and the same ordinate is produced on the other side of
the velocity V, it will describe a curve similar to that the axis, till BO be equal to BA; and then BO is
described by the 12-pounder, having the initial velocity divided in the subduplicate ratio of the same diameters.
1500. Therefore find (by interpolation) the range of Lines are drawn from the point A, and from any point
the 24-pounder, having the initial velocity V. Call this D of the abscissa, to these divisions.
R. Then Did=R:r, the range of the 12-pounder We see distinctly by this figure how the effect of the
which was wanted, and whicli is nearly 3380 yards. initial velocity gradually diminishes, and that in very

We see by this table the immense difference between great velocities the range is very little increased by its
the motions through the air and in a void. We see augmentation. The dotted curve APQR, shows what
that the ranges through the air, instead of increasing in the ranges in vacuo would be.
the duplicate ratio of the initial velocities, really in- By this figure may the problems be solved. Thus,
crease slower than those velocities in all cases of mili- to find the range of the 12-pounder, with the initial
tary service; and in the most usual cascs, viz. from soo velocity 1500. Set off 1500 from B to F; draw FH
to 1600, they increase nearly as the square-roots of the parallel to the axis, meeting the line 12 A in H; draw
velocities.

the ordinate HK, draw HL parallel to the axis, meet-
A set of similar tables, made for different elevations, ing 24 B in L; draw the ordinate LM, cutting 12 B
would almost complete what can be done by theory, in N. MN is the range required.
and would be much more expeditious in their use than If curves, such as ACG, were laid down in the same
Mr Euler's Trajectories, computed with great labour manner for other elevations, all the problems might be
by his English translator.

solved with great dispatch, and with much more accu-
The same table may also serve for computing the racy than the theory by which the curves are drawn
ranges of bomb-shells. We have only to find what must can pretend to.
be the initial velocity of the 24 pound shot which cor-

Note, that fig. 10. as given on Plate CCCCXLII, is
responds to the proposed velocity of the shell. This one-half less than the scale according to which it is
must be deduced from the diameter and weight of the described; but the practical mathematician will find no
shell, by making the velocity of the 24-pounder such, difficulty in drawing the figure on the enlarged scale to
that the ratio of its weight to the resistance may be the correspond to the description.
same as in the shell.

PROJECTION OF THE SPHERE,

THE PROJECTION of the SPHERE is a perspective primitive circle. The pole of this circle is the pole Stereogratereographic Pro representation of the circles on the surface of the of projection, and the place of the eye is the projecting phie Pro

jection of ection of sphere; and is variously denominated according to the point.

the Spheres he Sphere. different positions of the eye and plane of projection. 2. The line of measures of any circle of the sphere is

There are three principal kinds of projection ; the that diameter of the primitive, produced indefinitely,
stereographic, the orthographic, and gnomic. In the which passes through the centre of the projected circle.

tereographic projection the eye is supposed to be placed
on the surface of the sphere ; in the orthographic it is

AXIOM.
supposed to be at an infinite distance; and in the gno-
mic projection the eye is placed at the centre of the

The projection, or representation of any point, is
sphere. Other kinds of projection are, the globular,

where the straight line drawn from it to the proMercator's, scenographic, &c. for which see the artieles

jecting point intersects the plane of projectiou.
GEOGRAPHY, NAVIGATION, PERSPECTIVE, &c.

SECTION I.
DEFINITIONS.
1. The plane npon which the circles of the sphere

Of the Stereographic Projection of the Sphere.
are described, is called the plane of projection, or the In the stereographic projection of the sphere, the

eye

sures.

a

Stereogra- eye is placed on the surface of the sphere in the pole of But if the circle MN (fig. 2.) be not parallel to the

Stereographic Pro- the great circle upon which the sphere is to be project- primitive circle BD, let the great circle ABCD, pas- phic to jection of ed. The projection of the bemisphere opposite to the sing through the projecting point, cut it at right angles jection of the Sphere.

eye falls within the primitive, to which the projection in the diameter MN, and the primitive in the diameter the Sphere is generally limited; it, however, may be extended to BD. Through M, in the plane of the great circle, let Fig

. 1. the other hemisphere, or that wherein the eye is placed, MF be drawn parallel to BD; let AM, AN be joined, the projection of which falls without the primitive. and meet BD in m, n. Then, because AB, AD are

As all circles in this projection are projected either quadrants, and BD, MF parallel, the arch AM is equal into circles or straight lines, which are easily described, to AF, and the angle AMF or Amn is equal to ANM, it is therefore more generally understood, and by many Hence the conic surface described by the revolution of preferred to the other projections.

AM about the circle MN is cut by the primitive in a

subcontrary position ; therefore the section is in this PROPOSITION I. THEOREM I.

case likewise a circle.
Every great circle which passes through the projecting

COROLLARIES.
point is projected into a straight line passing through
the centre of the primitive; and every arch of it,

1. The centres and poles of all circles parallel to the
reckoned from the other pole of the primitive, is pro- primitive bave their projection in its centre.
jected into its semitangent.

2. The centre and poles of every circle inclined to Plate Let ABCD (fig. 1.) be a great circle passing through the primitive have their projections in the line of meaccecxlii. A, C, the poles of the primitive, and intersecting it in fig. 1.

the line of common section BED, E being the centre 3. All projected great circles cut the primitive in
of the sphere. From A, the projecting point, let there two points diametrically opposite ; and every circle in
be drawn straight lines AP, AM, AN, AQ, to any the plane of projection, which passes through the extre-
namber of points P, M, N, Q, in the circle ABCD: mities of a diameter of the primitive, or through the
these lines will intersect BED, which is in the same projections of two points that are diametrically oppo-
plane with them. Let them meet it in the points p, m, site on the sphere, is the projection of some great
1, 9; then p, m, n, q, are the projections of P, M, N, circle.
Q : hence the whole circle ABCD is projected into the 4. A tangent to any circle of the sphere, which does
straight line BED, passing through the centre of the not pass through the projecting point, is projected into
primitive.

a tangent to that circle's projection ; also, the circular
Again, because the pole C is projected into E, and projections of tangent circles touch one another.
the point M into m; therefore the arch CM is project- 5. The extremities of the diameter, on the line of
ed into the straight line E m, which is the semitan- measures of any projected circle, are distant from the
gent of the arch CM to the radius AE. In like man- centre of the primitive by the semitangents of the least
ner, the arch CP is projected into its semitangent, E and greatest distances of the circle on the sphere, from

the pole opposite to the projecting point.

6. The extremities of the diameter, on the line of COROLLARIES.

measures of any projected great circle, are distant from 1. Each of the quadrants contiguous to the projects the centre of the primitive by the tangent and cotaning point is projected into an indefinite straight line, gent of half the great circle's inclination to the primiand each of those that are remote into a radius of the

tive. primitive.

7. The radius of any projected circle is equal to balf 2. Every small circle which passes through the pro

the sum, or half the difference of the semitangents of jecting point is projected into that straight line which is the least and greatest distances of the circle from the its common section with the primitive.

pole opposite to the projecting point, according as that 3. Every straight line in the plane of the primitive, pole is within or without the given circle. and produced indefinitely, is the projection of some circle on the sphere passing through the projecting

PROPOSITION III. THEOREM III. point.

An angle formed by two tangents at the same point 4. The projection of any point in the surface of the

in the surface of the sphere, is equal to the angle sphere, is distant from the centre of the primitive, by

formed by their projections.
the semitangent of the distance of that point from the
pole opposite to the projecting point.

Let FGI and GH (fig. 3.) be the two tangents, Fig. 3•

and A the projecting point; let the plane AGF cut PROPOSITION II. THEOREM II.

the sphere in the circle AGL, and the primitive in the Every circle on the sphere which does not pass through tion of the plane AGH with the primitive : then the

line BML. Also, let MN be the line of common secthe projecting point is projected into a circle.

angle FGH=LMN. If the plane FGH be parallel If the given circle be parallel to the primitive, then to the primitive BLD, the proposition is manifest. If a straight line drawn from the projecting point to any not, through any point K in AG produced, let the point in the circumference, and made to revolve about plane FKH, parallel to the primitive, be extended to the circle, will describe the surface of a cone ; which meet FGH in the line FH." Then, because the plane being cut by the plane of projection parallel to the base, AGF meets the two parallel planes BLD, FKH, the the section will be a circle. See Conic-Sections. lines of common section LM, FK are parallel ;

a