PRO 
tlie Ime L be a third propertional to E P 
and AP ; and it will be (by 17 El. 6) 
L X E P = A P 9 , but A P g. : A G g. 
::EP:B1M : :LX EP:LxBM; 
whence since it is L X E P ~ A P g. it 
will be L X B M z= A G g. In like man- 
ner it will be L X C N — A H g, &c. 
But because it is B G : A G; : (E P : AP : : 
by hypothesis) A P : L ; it w ill be L X 
BG = AGxAP = AGx ag+ag 
X GP = AGg. + A G X GP. But it 
has been shown that, it is L X B M = AG g, 
wherefore it will be L X B G — L X B M 
;= A G X G P, that is, L X M G = A G X 
G P. By the same way of reasoning it 
will be LxNH— AH xHP, ike. 
Wherefore the rectangle under M G and ly, 
will be equal to the square of A G, which is 
the property of the parabola; and so thd 
curve A M N O P K, wherein the projectile 
is moved, will be a parabola. 
Cor: 1. Hence the right line L is the la- 
tiis rectum oi' parameter of the parabola, 
that belongs to its axis. 
Cor. 2. Let A H z= H P, and it will be 
L X C N = A H g. = L X N H, whence 
it will be N H =; C N ; and consequently 
the right line A F being the line of di- 
rection of the projectile, will be a tangent 
to the parabola. 
Cor. 3. If a heavy body be projected 
dow'nwards, in a direction oblique to the 
horizon ; the path of the projectile will be a 
parabola. 
Theorem 3. The impetus of a projected 
body in different parts of the parabola, 
are as the portions of the tangents intercept- 
ed between two right lines parallel to the 
axis ; that is, the impetus of the body pro- 
jqcted, in the points A and B (fig. 9) to 
which A D, and BE are tangents, will be 
as CD, and E B, the portions of the tan- 
gents intercepted between two right lines 
C B, and DE parallel to the axis. 
AVe have here treated the path of a pro- 
jected body as an exact parabola, though 
from the resistance of the air, the line of a 
projectile is not exactly parabolical, but 
rather a kind of hyperbola ; which, if con- 
sidered and applied to practice, would 
render the computations tar more operose, 
and the very small difference (as experi- 
ence shows in heavy shot) would, in a 
great measure, lessen the elegancy of the 
demonstrations given by accounting for it ; 
since the common rules are sufficiently 
exact, and easy for practice. 
PKOJECITON, in mechanics, the act of 
communicating motion to a body, from 
PRO 
thence called projectile. In perspective, 
projection is the appearance or leprescnta- 
tion of an object on the perspective plane. 
The projection of the sphere is either ortho- 
graphic, or stereographic. The former, or 
orthographic projection, supposes the eye 
placed at an infinite distance ; whereas, in 
the stereographic projection, it is supposed 
to be only 90 degrees distant from the pri- 
mitive circle, or placed in its pole, and 
thence viewing the circles on the sphere. 
The primitive circle is that great circle 
which limits or bounds the representation or 
projection ; and the place of the eye is 
called the projecting point. 
PROLATE, in geometry, an epithet ap- 
plied to a spheroid produced by tlie revo- 
lution of a semi-ellipses about its larger 
diameter. 
PROLEGOMENA, in philology, certain 
preparatory observations or discourses pre- 
fixed to a book, &c. containing something 
necessary for the reader to be apprized of, 
to enable him the better to understand the 
book, or to enter deeper into the science, 
&c. 
PROMISE, in law, is where, upon a 
valuable consideration, persons bind them- 
selves by words to do or perform such a 
thing agreed on : it is in the nature of a 
verbal covenant, and wants only the solem- 
nity of writing and sealing to make it abso- 
lutely the same. Yet for the breach of it 
the remedy is different ; for instead of an 
action of covenant, there lies only an action 
upon the case, the damages whereof are to 
be estimated and determined by the jury. 
If there is no consideration it is void, and 
it is called a nude compact, or in Latin a 
nudum pactum. 
PROMISSORY note. See Bills of 
Exchange. 
PRONOUN, pronomen, in grammar, a 
declinable part of speech, which being put 
instead of a noun, points out some person, 
or thing. See Grammar. 
PRONUNCIATION, in grammar, the 
manner of articulating or sounding the 
words of a language. Pronunciation makes 
much the most difficult part of a written 
grammar ; in regard that a book expressing 
itself to the eyes, in a matter that wholly 
concerns the ears, seems next akin to that 
of teaching the blind to distinguish colours; 
hence it is that there is no part so defective 
in grammar as that of the pronunciation, as 
the writer has frequently no term whereby 
to give the reader n idea of the sound lie 
would express; f want of a proper term, 
