PAR 
ParA-LIAX, annual, the change of (he 
apparent place of a heavenly body, which 
is caused by being viewed from the earth in 
dilferent parts of its orbit round the sun. 
The annual parallax of all the planets is 
found very considerable, but that ot the 
fixed stars is imperceptible. 
Parallax, in levelling, denotes the 
angle contained between the line of the 
true level, and that of the apparent level. 
PARALLEL. The subject of parallel 
lines, says Playfair, is one of the most dif- 
ficult in the Elements of Geometry. It has 
accordingly been treated in a great variety 
of different ways, of which, pefhaps, there 
is none which can be said to have given en- 
tire satisfaction. The difficulty consists in 
converting the twenty-seventh and twenty- 
eighth of Euclid, or in demonstrating, that 
parallel straight lines (or such as do riot 
meet one another) when they meet a third 
line, make the alternate angles with it equal, 
or which comes to the same, are equally in- 
clined to it, and make the exterior angle 
equal to the interior and opposite. In or<ler 
to demonstrate this proposition, Euclid as- 
sumed it as an axiom, that if a straight line 
meet two straight lines, so as to make the 
interior angles on the same side of it less 
than two right angles, these straight lines 
being continually produced, will at length 
meet on the side on which the angles are 
that are less than two right angles. This 
proposition, however, is not self-evident ; 
and ought the less to be received without 
proof, that the converse of it is a proposi- 
tion that confessedly requires to be demon- 
strated. In order to remedy this defect, 
three sorts of methods have been adopted — 
a new definition of parallel lines ; a new 
manner of reasoning on the properties of 
straight lines without any new axiom ; and, 
the introduction of a new axiom less excep- 
tionable than Euclid’s. Playfair adopts the 
latter plan; but we do not perceive that his 
axiom is by any means self-evident upon 
Euclid’s definition which he retains, viz. 
Parallel straight lines are such as are in tlie 
same plane, and which being produced ever 
so far both ways do not meet. A more in- 
telligible, and we think an equally rigid, de- 
monstration of the property of parallels, 
may be obtained without any axiom, by 
means of a new definition. It may at first 
sight be thought that the objection urged by 
Playfair against the definition in T. Simp- 
son’s first edition, must equally hold against 
ours ; but we think that if his objection 
really hold good against that definition, 
(though ve confess we cannot feel the force 
PAR 
of it), it is obviated by distinguishing as 
ought to be done between the distance and 
the measure of that distance. 
We must of course suppose our readers 
acquainted with the propositions in Euclid 
preceding the twenty-seventh ; bnt to save 
the necessity of reference we shall give an 
enunciation of those which we shall have to 
employ in our demonstration, in the form in 
which w'e employ them. 1. (Prop. 16.) If 
one side of a triangle be produced, the out- 
ward angle is greater than either of the in- 
ward opposite angles. 2. (Prop. 19.) The 
greater angle of every triangle has the 
greater side opposite to it. 3. (Prop. 4.) If 
two triangles have two sides of the one 
respectively equal to two sides of the other, 
and have the included angles equal, the 
other angles will be respectively equal, viz, 
those to which the equal sides are opposite. 
4. (Prop. 15.) If two straight lines cut 
each other, tlie vertical or opposite angles 
will be equal. 5. (Prop. 13.) If a straight 
line meet another, the sum of the adjacent 
angles is equal to the sum of two right 
angles. 
6. Definition. Parallel straight lines are 
those whose least distances from each other 
are every where equal. 
7. Theorem I. The perpendicular drawn 
to a straight line from any point, is the least 
line that can be drawn from that point to 
the given line. 
Let C D, (Plate XII. Miscell. fig. 2) be 
a straight line drawn from C perpendicular 
to A B ; and let C E be any other straight 
line from C to A B ; then is C D less than 
C E. For the angle ODE equals angle C D A 
by construction; and CD A is greater than 
C E D (1) ; therefore C D E is greater than 
C E D. Hence (2) C D is less tlian C E. 
8. Cor. 1. Hence the perpendicular from 
any point to a straight line is the true 
measure of the least distance of that point 
from that line. 
9. Cor. 2. Hence (6) the perpendiculars 
to one of two parallel straight lines, from 
any points in the other, are every where 
equal to each otlier. 
10. Cor. 3. Hence tw'o parallel straight 
lines, however far they may be produced, 
can never meet. 
11. Theorem II. If a line meeting two 
parallel straight lines be perpendicular to 
one of them, it is also perpendicular to the 
other. 
If A B, (fig. 3) be parallel to C D, and 
E F meet them so as to be perpendicular to 
A B, it will also be perpendicular to C D. 
If not, draw E G perpendicular to C D and 
