:

1

This equation shows us that we shall have y≈0 as often as

t

n. hyp. log. – becomes equal to any complete number of se-
3·1416, and N = any num-
ber in the series 1, 2, 3, 4, 5, &c. we can have y=0 only when

micircumferences: if therefore x=3·1416, and N

which shows the relation between the number of vibrations N
and the time T in which they are performed.

Hence it is manifest, that the times or durations of the se-
veral successive vibrations constitute a series in geometrical
progression.

DESCRIPTIVE GEOMETRY.

CHAPTER I.

Containing the First Principles of Descriptive Geometry,
with Illustrations.

DESCRIPTIVE GEOMETRY is the art of determining by constructions performed on one plane the various points of lines and surfaces which are in different planes. The principle on which this art is founded, consists in projecting the points of any line or surface on two given planes at right angles to each other. These two planes are usually denominated the horizontal and vertical planes, or the fundamental or primitive planes, or the planes of projection. In the constructions the vertical plane is supposed to have revolved about the line of their common intersection, and to be coincident with the horizontal plane; and it is by means of this coincidence that both the projections on the horizontal and vertical planes are effected by constructions performed on the horizontal plane.

D

་

C

G

A

B

To illustrate this, let ABCD be the horizontal plane, and
EFGH the vertical plane at right angles to it, and meeting it in

VOL. II,

72

the

562

DESCRIPTIVE GEOMETRY.

Р

the straight line of their common section EF. Suppose r to
be any point in space, from which on the plane ABCD let fall the
perpendicular PP', meeting that plane in r; and on the vertical
plane EFGH, let fall from P the perpendicular PP" meeting it in
p"; then and r' are the projections of the point P on the
primitive planes and it is obvious that the projections of any
other point besides P cannot be coincident with both the points
p' and p', and therefore, when the points p' and p" are given,
there is but one point P of which they are the projections.
From the point r' draw in the horizontal plane the straight
line P'k at right angles to the common section EF, and join P’K.
It is obvious that P'K is at right angles to the plane EFGH, and
by supposition PP" is at right angles to the same plane, conse-
quently PP", P'K are parallels, and therefore in one plane ; and
since the angle PPK is a right angle, therefore PPP" is also a
right angle and because PPK is a right angle, it follows that
P'KP" is likewise a right angle; thus it appears that the plane
figure PP KP is a rectangle, and the two distances P'K, P'K, are
equal to the two projecting perpendiculars PP" and PP'. Those
perpendiculars PP and PP', or their equals P'K, PK, are called
the ordinates of the point P.

:

Suppose now, after the points p' and p" are determined, that the plane EFGH revolves about its intersection EF from its position at right angles to ABCD, until it coincides with the horizon

tal plane during this revolution the straight line KP' in its
motion continues at right angles to the common section EF;
and when the vertical plane EFGH, has coincided with the

horizontal

་

horizontal plane ABCD, the line PK of the former plane evidently falls in the continuation of PK; so that PK, KP", make one straight line at right angles to EF, and lying in the horizontal plane; the distances P'K, P ́K, being the ordinates or co-ordinates to r, the point in space.

The

The point in space is said to be given, when the two perpendiculars or ordinates, P'K and P′′K, are given in magnitude and position; and a point sought, P is said to be found when the two ordinates P'K and PK have been found. various positions of the projections P' and P" corresponding to the different situations of the point in space, should be clearly conceived by the learner; on this account the following varieties of position deserve attention: and it is particularly to be noted. that the horizontal projection of the point. in space is marked with one accent, and the vertical projection with two accents. by means of which the several points of the horizontal and vertical planes will be easily distinguished,

If the point P which is to be projected, be in the ground line or common intersection EF of the fundamental planes, its projections P', P", must evidently coincide with the point itself as in No. 1.

If the point P be in one of the fundamental planes but not in the other, let it first be in the horizontal plane at r', as in N. 2, No 3. In each of which the vertical projection P falls on the ground line in No. 2, the point is before

the

*

the vertical plane, and in N°. 3. the point P is behind the vertical plane. Next let the point P be in the vertical but not in the horizontal plane, as at p" No. 4, No. 5. In each of these cases the horizontal projection is manifestly on the ground line at r'. In No. 4, the point P or P" is in the vertical plane directly above the point r of the ground line, and by the revolution of the vertical plane EFGH into a horizontal position, the point p" falls behind the ground line EF. In N°. 5, the point p or p" is directly below the point r' of the ground line in the continuation of the vertical plane EFGH below the horizontal plane ABCD; and by the same revolution of EFGH as before, the point p" of the vertical plane immediately below p', is brought up to the horizontal plane; so that in this last case the point p" is before the ground line: and therefore the points p" and p" of N°. 4 and N°. 5, fall on opposite sides of EF on the horizontal plane by the revolution of the vertical plane. When the point of space r is in neither of the primitive planes, there are four different situations in which it may be found, that require to be particularly distinguished from one another.

D

C

E

F

A

P

B

1. When the point r is above the horizontal plane and be fore the vertical plane. In this case the horizontal projection p' falls before the ground line EF, and the vertical projection falls behind EF; the horizontal and vertical ordinates being KP' and KP". To conceive distinctly the place of the point P, take in Er the distance KL equal to KP", and on the horizontal plane complete the rectangle KLMг. Imagine now that the rectangle KLMP'revolves about its fixed side Kr from a horizontal to a vertical position by the ascent of the rectangle above the horizontal plane; and when the rectangle