PRO 
Projection stercographic of the sphere, is that 
In which the surface and circles of the sphere 
are drawn upon the plane of a , threat circle, the 
eve being in the pole of that circle. 
Projection gnomonical of the sphere, is that in 
which the surface of the sphere is drawn upon 
a plane withoutside of it, commonly touching 
it, the eye being at the centre of the sphere. 
Latvs of the orthographic projection. — 1 , The 
rays coming from the eye, being at an infinite 
distance, and making the projection, are parallel 
to each other, and perpendicular to the plane of 
projection. 
A A right line perpendicular to the plane of 
projection, is projected into a point where that 
line meets the said plane. (Fig. 8.) 
3. A right line, as AB, or CD, not perpendi- 
cular, but either parallel or oblique to the plane 
of the projection, is projected into a right line, 
as EF or GH, and is always comprehended be- 
tween the extreme perpendiculars AE and BF, 
or CG and DH. 
4. The projection of the right line AB is the 
greatest, when AB is parallel to the plane of the 
projection. 
5. Hence it is evident, that a line parallel to 
the plane of the projection, is projected into a 
right line equal to itself ; but a line that is ob- 
lique to the plane of projection, is projected 
into one that is less than itself. (Fig. 9.) 
G. A plane surface, as ACBD, perpendicular 
to the plane of the projection, is projected into 
the right line, as AB, in which it cuts that plane. 
Hence it is evident, that the circle ACBD per- 
pendicular to the plane of projection, passing 
through its centre, is projected into that diame* 
ter AB in which it cuts the plane of the projec- 
tion. Also any arch as CV is projected into Oo, 
equal to ca, the right sine of that arch ; and the 
eomplemental arc rB is projected into oB, the 
versed sine of the same arc dB. 
_ 7. A circle parallel to the plane of the projec- 
tion, is projected into a circle equal to itself, 
having its centre the same with the centre of the 
projection, and its radius equal to the cosine of 
its distance from the plane. And a circle ob- 
lique to the plane of the projection, is projected 
into an ellipsis, whose greater axis is equal to 
the diameter of the circle, and its less axis equal 
to double the cosine of the obliquity of the cir- 
cle to a radius equal to half the greater axis. 
Properties oj the stereographic projection . — 1. In 
this projection a right circle, or one perpendi- 
cular to the plane of projection, and passing- 
through the eye, is projected into a line of halG 
tangents. 
2. The projections of all other circles, not 
passing through the projecting point, whether 
parallel or oblique, are projected into circles. 
Figs. 10, 1 1, and 3 2. 
I hus, let ACEDB represent a sphere, cut by 
a plane RS, passing through the centre I, per- 
pendicular to the diameter EH, drawn from E 
the place of the eye ; and let the section of the 
sphere by the plane RS be the circle CFDL, 
whose poles are H and E. Suppose now AGB 
is a circle on the sphere to be projected, whose 
pole most remote from the eye is P ; and the 
visual rays from the circle HGB meeting in E, 
form the cone AG BE, of which the triangle 
AEB is a section through the vertex E, and dia- 
meter of the base AB ; then will the figure aglf 
which is the projection of the circle AGB, be it- 
self a circle. Hence, the middle of the projected 
diameter is the centre of the projected circle, 
whether it is a great circle or a small one : also 
the poles and centres of all circles parallel to 
the pkne of projection, fall in the centre of the 
projection : and all oblique great circles cut the 
primitive circle in two points diametrically op- 
posite. 
3. The projected diameter of any circle sub- 
tends an angle at the eye equal to the distance 
«f that circle from its nearest pole, taken on the 
VOL. II. 
PRO 
sphere ; and tnat angle is bisected bv a right 
line joining the eye and that pole. Thus, let the 
plane RS (rig. 13) cut the sphere HFEC through 
its centre 1 ; and let ABC be any oblique great 
circle, whose diameter AC is projected into ac\ 
and KOL any small circle parallel to ABC, 
whose diameter KL is projected in LI. The dis- 
tances of those circles from their pole P, bein°- 
the arcs AHP, KFIP, and the angles aEc, LFJ, 
are the angles at the eye, subtended by their 
projected diameters, nc and LI. Then is the 
angle aEc measured by the arc AHP, and the 
angle LEI measured by the arc KHP, and those 
angles are bisected by EP. 
3. Any point of a sphere is projected at such 
a distance from the centre of projection, as is 
equal to the tangent of half the arc intercepted 
between that point and the pole opposite to the 
eye, the semidiameter of the sphere being radius. 
1 hus, let C^EB (fig. 14) be a great circleof the 
sphere, whose centre is c ; GH the plane of pro- 
jection cutting the diameter of the sphere in b 
and B ; also E and C the poles of the section by 
that plane; and a the projection of A. Then ca 
is equal to the tangent of half the arc AC, as is 
evident by drawing CF = the tangent of half 
that arc, and joining cF. 
4. The angle made by two. projected circles, is 
equal to the angle which these circlesinake on the 
sphere. For let IACE (fig. 15) and ABL be two 
circles on a sphere intersecting in A ; E the pro- 
jecting point ; and RS the plane of projection, 
in which the point A is projected in a, in the 
line IC, the diameter of the circle ACE, Also 
let DH and FA be tangents to the circles ACE 
and ABL. Then will the projected angle d.f 
be equal to the spherical angle BAC. 
5. 1 lie distance between the poles of the pri- 
mitive circle and an oblique circle, is equal to 
the tangent of half the inclination of those cir- 
cles; and the distance of their centres is equal 
to the tangent ol their inclination, the semidi- 
ameter of the primitive being radius. For let AC 
(fig. 16) be the diameter of a circle, whose poles 
are P and Q, and inclined to the plane of pro- 
jection in the angle A1F ; and let a, c,p, be the 
projections of the points A, C, P; also let HaE 
be the projected oblique circle, whose centre is 
? Now when the plane of projection becomes 
the primitive circle, whose pole is I, then is 
Ip = tangent of half the angle AIF, or of half 
the arch AF : and I q — tangent of AF, or of the 
angle FH*i — AIF. 
_ 6- If through any given point in the primi- 
tive circle, an oblique circle is described, 
then the centres of all other oblique circles 
passing through that point, will be in a right 
line drawn through the centre of the first ob- 
lique circle, and perpendicular to a line passing 
through that centre, the given point, and the 
centre of the primitive circle. Thus, let GACE 
(fig. 17) be the primitive circle, and ADEI a great 
circle described through D, its centre being B, 
UK is a right line drawn through B, perpendi- 
cular to a right line Cl passing through D and B 
and the centre of the primitive circle. Then 
the centres of all other great circles, as FDG, 
passing through D, will fall in the line HK. 
7. Equal arcs of any two great circles of the 
sphere will be intercepted between two other 
circles drawn on the sphere through the remotest 
poles of those great circles. For let PBEA.(fig. 18) 
be a sphere, on which AGB and CFD are” two 
great circles, whose remotest poles are FI and P- 
and through these poles let the great circle 
PBEC and the small circle PGE be drawn, cut- 
ting the great circles AGB and CFD in the points 
B, G, D, F. Then are the intercepted arcs BG 
and DF equal to one another. 
8. If lines are drawn from the projected pole 
of any great circle, cutting the peripheries of 
the projected circle and plane of projection, 
the intercepted arcs of those peripheries art- 
equal; that is, the arc BG = df. 
3 !» 
? R Cf SO 5 
9. The radius of any lesser circle, whose plane 
is perpendicular to that of the primitive circle, 
is equal to the tangent of that lesser circle's 
distance from its pole; and the secant of that 
distance is equal to the distance of the centres 
of the primitive and lesser circle. For let P (rig. 
19) be the pole and AB the diameter of a lesser 
circle, its plane being perpendicular to that of 
the primitive circle, whose centre is C: then J 
being the centre of the projected lesser circle, da 
is equal to the tangent of the arc PA, and dC =z 
the secant of PA. 
PROJECTURE. See Architecture. 
PROLAPSUS. See Surgery. 
PROLATE, n geometry, an epithet ap- 
plied to a spheroid produced by the revolu- 
tion ot a semi-ellipsis about its larger dia- 
meter. 
PROMISE, is where, upon a valuable 
consideration, persons bind themselves by 
words to do or perform such a thing agreed 
on : it is m the nature of a verbal covenant, 
and wants only the solemnity of writing and 
sealing to make it absolutely the same. Yet 
tor the breach of it, the remedy is different ; 
for instead of an action of covenant, there 
lies only an action upon the case, the tla- 
mages whereof are to be estimated and de- 
termined by thejurv. 
PROMISSORY NOTE. See Bills of 
Exchange. 
PRONOUN, in grammar, a declinable 
part of speech, which being put instead of a 
noun, points out some person or thing. 
PROOF, the shewing or making plain the 
truth of any matter alleged ; either in giving 
evidence to a jury on a trial, or else on in- 
terrogatories or by copies of records, or 
exemplifications of them. See Evidence. 
Proof of art ili try and small arms, is a 
trial whether they stand the quantity of pow- 
der allotted for {hat purpose. The rule of 
the board of ordnance is, that all guns under 
24-pounders are loaded w ith powder as much 
as their shot weighs ; that is, a brass 24-- 
pounder with 21 lb. a brass 32-poimdcr w ith 
26 lb. 12 oz. and a 42-pounder with 311b. 
8 o z. the iron 24-pounder with 1 8 lb. the 
32-pounder with 21 lb. 8 oz. and the 42- 
pounder with 25 lb. 
i he brass light field-pieces are proved 
with powder that Weighs half as much as 
their shot, except the 24-pounder, which is 
loaded with 10 lb. only. 
Government allows 1 1 bullets of lead in 
the pound for the proof of muskets ; and 14 5, 
or twenty-nine in two, for service; seventeen 
in the pound for the proof of carbines, and 
twenty for service ; twenty-eight in the pound 
for the proof of pistols,' and thirty-four for 
service. 
A' hen guns of a new metal, or of lighter 
construction, are proved ; then, besides the 
common proof, they are fired 200 or 300 
times, as quick as they can be loaded w ith 
the common charge given in actual service. 
Guv light six'pounders were fired 300 times 
in three hours and twenty-seven minute-,, 
loaded with 1 lb. 4 oz., without receiving 
any damage. 
Proof of ponder, is in order to try its 
goodness and strength. See Gunpowder. 
Proof of cannon, is made to ascertain 
their being well cast, their having no cavities 
in the metal, and in a word, their being m 
to resist the effort of their charge of p<nvder 
In making this proof, tire piece is laid upg* 
