Encyclopaedia Britannica > Volume 15, PLA-RAM

Sea. It.
Orthogra¬
phic i‘ro-
PROJECTION of
Proposition VIII. Problem III.
jeftion of trough two given points in the plane of the primi-
the sphere. ^ °q defcribe the projedion of a great circle.
1. If the two given points and the centre of the
primitive be in the fame ilraight line, then a diameter
of the primitive being drawn through thele points will
be the projection of the great circle required.
2. When the two given points are not in the fame
ftraight line with the centre of the primitive ; and one
of them is in the circumference of the primitive.
Plate Let R} (fig. 34.) be the two given points, of
oCcccxxi. wlljch R {s in the circumference of the primitive. Draw
the diameter RCS, and GC^, FDH perpendicular to
it, meeting the primitive in G^F. Divide GC, £ C,
in A, B, in the fame proportion as FH is divided in D;
and defenbe the elhpfe whofe axes are RS, AB, and
centre C ; and it will be the projection required.
3. When the given points are within the primitive,
and not in the fame ftraight line with its centre.
Let D, E (fig. 35.) be the two given points;
through C the centre of the primitive draw the ftraight
lines IDi, KE/; draw DL perpendicular to I i, and
EO perpendicular to K i, meeting the primitive in L,
O. Through E, and towards the fame parts of C,
draw EP parallel to DC, and in magnitude a fourth
proportional to LD, DC, OE. Draw the diameter
CP meeting the primitive in R, S, and deferibe an
ellipfe through the points D and R or S, and it will
alfo pafs through E. This ellipfe will be the projedion
#f the propofed inclined circle.
Proposition IX. Problem IV.
To deferibe the projeftion of a lefs circle parallel to the
primitive, its diftauce from the pole of the primitive
being given.
From the pole of the primitive, with the fine of the
given diftance of the circle from its pole, deferibe a
circle, and it will be the proje&ion of the given lefs
circle.
Proposition X. Problem V.
About a given point as a projeded pole to deferibe the
projedion of an inclined circle, whofe diftance from
its pole is given.
Let P (fig. 36.) be the given projeded pole, through
which draw the diameter G,f, and draw the diameter
H/j perpendicular thereto. From P draw PL per¬
pendicular to GP meeting the circumference in L ;
through which draw the diameter L /. Make L P,
LK each equal to the chord of th# diftance of the lefs
circle from its pole, and join TK, which interfeds L /,
in From the points T, draw the lines FA,
QS, KB, perpendicular to Gg; and make OR, OS,
each equal to .QT, or QK. Then an ellipfe deferibed
through the points A, S, B, R will be the projedion
©f the propofed lefs circle.
Proposition XI. Problem VI.
To find the poles of a given projeded circle.
1. If the projeded circle be parallel to the primitive,
the centre of the primitive will be its pole.
Vol. XV. Part II.
THfcSpHEfcf. ^ 577
2. If the circle be perpendicular to the primitive, Orthogra-
then the extremities of a diameter of the primitive
drawn at right angles to the ftraight line reprclentingtiic sphere,
the projeded circle, will be the poles of that circle. ^
3. When the projeded circle is inclined to the pri¬
mitive.
Let ARBS (fig. 36, 37.) be the elliptical projedion
of any oblique circle ; through the centre of which,
and C the centre of the primitive, draw the line of mea-
fures CBA, meeting the ellipfe in B, - ; and the pri¬
mitive in G,g. Draw CH, BK, AT perpendicular
to G g, meeting the primitive in FI, K, d . Biled the
arch KT in L, and draw LP perpendicular to Gg ;
then P will be the projeded pole of the circle, of which
ARBS is the projedion. *
Proposition XII. Problem VII.
To meafure any portion of a projeded circle, and con-
verfely.
1. When the given projedion is that of a great cir¬
cle.
Let ADEB (fig. 38.) be the given great circle,
either perpendicular or inclined to the primitive, ot
which the portion DE is to be meafured, and let M m
be the line of meafures of the given circle. Through
the points D, E draw the lines EG, DF parallel to
M m ; and the arch EG of the primitive will be the
meafure of the arch DE of the great circle, and Con-
verfely.
2. When the projedion is that of a lefs cirde paral¬
lel to the primitive.
Let DE (fig. 39.) be the portion to be meafured, cc^at*
of the lefs circle DEH parallel to the primitive. From ^
the centre C draw the lines CD, CE, and produce them
to meet the primitive in the points B, F. Then the
intercepted portion BF of the primitive will be the
meafure of the given arch DE of the lefs circle DEH.
3. If the given lefs circle, of which an arch is to be
meafured, is perpendicular to the primitive.
Let ADEB (fig. 40.) be the lefs circle, of which,
the meafure of the arch DE is required. 1 hrough C,
the centre of the primitive, draw the line of meafures
M m, and from the interfedion O of the given right
circle and the line of meafures, wiih the radius OA, or
OB, deferibe the fcmicivcle AFGB ; through the points.
D, E draw the lines DF, EG parallel to the line ofc
meafures, and the arch EG will be the meafure of DE,
to the radius AO. In order to find a fimilar arch in the
circumference of the primitive, join OF, OG, and at
the centre C of the primitive, make the angle m CH
equal to FOG, and the arch m H to the radius C m
will be the meafure of the arch DE.
4. When the given projedion is of a lefs circle in¬
clined to the primitive.
Let RDS (fig. 41.) be the projedion of a lefs circle
inclined to the primitive, and DE a portion of that cir¬
cle to be meafured. Through O the centre of the pro¬
jeded circle, and C the centre of the primitive, draw
‘the line of meafures M m ; and from the centre O, with
the radius OR, or OS, deferibe the femicirclc RGFS ;
through the points D, E draw the lines DF, EG
parallel to the line of meafures, and EG will he the
meafure of the arch DE to the radius OR, or OS.
loin OF, OG, and make the angle mCH equal to
41 4 D VOG,