Download Transactions of the Literary and Historical Society of Quebec

THE THEORY OF PARALLEL
72
LINES.
THEORY OF PARALLEL LINES, BEING AN ATTEMPT TO DEMONSTRATE THE TWELFTH AXIOM OF EUCLID. BY THE REV. D. WILKIE.
ART. Ill— THE
READ BEFORE THE SOCIETY ON THE 21 ST
the truth of the
In the following paper,
axioms, and of the
first
—
I.
eleven
first
twenty-six propositions of the
hook of Euclid, and of these
DEFINITIONS.
JAN., 1SS2.
only,
When
is
first
to be admitted.
a line cuts
two other
lines,
the angles on the opposite sides of the cutting line, and at
its
opposite extremities, are called
II.
When
a line cuts two other
nate angles equal, these
two
Alternate
angles.
making the
lines,
lines are said to
alter-
be Parallel.
H
C
D
B
\
E
I
Thus,
gles
if
ACF
the line
and
CFD.
Theorem
1.
CFE
HI
cut the two lines
are called alternate
If a line cut two other
;
AB, DE,
and also
lines,
the an-
BCF
and
making one -pair
of the alternate angles equal, the remaining pair shall also
be equal.
Thus,
if
the angle
pair of alternate
equal.
ACF
angles,
CFE,
and CFD,
be equal to
the remaining
BCF
shall
also be
THE THEORY OF PARALLEL
LINES.
73
ACF, BCF, being equal to two right
angles (XIII. 1. Eucl.), and CFD, CFE, being also equal
to two right angles
the two ACF, BCF, are equal to
CFD, CFE (Ax. 1.). And since ACF is equal to CFE,
For, the angles
;
the remainder
Theorem
BCF is equal to the remainder CFD. — Q. E. D.
If a
2.
line cut
two parallel
secting the line which cuts them
lines,
ivill
have
any
line bi-
its alternate
angles equal.
M
A
O
Let
AB
any
N
B
AM be parallel
NB,
to
NM,
rallel lines, shall
For, the angle
AB
bisecting
have
and likewise the
consequence of the
its
alternate angles equal
in C,
and meeting the pa-
MAC being equal to CBN by supposition,
side
AC
BCN also equal (XV.l.Eucl.)
equal to
CB
by construction, the
(XXVI. 1. Eucl.)
the angle CNB, and
triangles are entirely equal
the angle
AMC
alternate angles.
NMS.
is
equal to
Hence,
Wherefore, any
parallel lines,
line
alternate angles also equal.
its
and the opposite angles ACM,
two
in
cutting them, and having
line, as
„
makes
its
.
also, the angle
MNO
is
Hence,
they are
equal to
AB, and meeting the
angles equal, as was to
line bisecting
alternate
be shown.
Cor.
—Hence,
a line passing through the point C, and
perpendicular to the one
* This was
first
line, is
perpendicular to both*.
proved by Dr. R. Simson ; but his proposition is limited to
and falling on the parallels at right angles, as i«J
lines bisecting the cutting line,
the Cor.
K
THE THEORY OF PARALLEL LINES.
74
Theorem
lines,
3.
— The perpendiculars
drawn from points in
distances
from
the line
to
one of two parallel
the other, ivhich are at equal
which
perpendicular to both,
is
are equal to one another.
Let
AC and DE
A
B
C
D
E
F
be parallel, and let
(by Cor. Prop. 2.)
points
A
BE be
perpendicular to both
is
the line which
;
and
the
let
and C, be at equal distances from B, the perpen-
diculars let
fall
from these points upon the other
parallel,
are equal to one another.
AE and EC be drawn, the triangles ABE, CBE
having AB equal to CB by supposition, BE common, and
the angle ABE equal CBE being both right angles; are
entirely equal.
(Eucl.
Consequently the line AE
4.)
equal to CE and the angle BEA equal to BEC.
Also, let AD be perpendicular to DF, and having made
For,
if
I.
EF
equal to
ED,
join
BED, and
being equal to the whole
BEA,
AED.
the part
der
the remainder
Hence, the triangles
first.)
equal to the angle at D, which
equal to
is
BEC
BEF
equal to
equal to the remain-
having the two sides
in the one,
and the angle between them
equal, (by the IV. of the
is
CEF
the part
AED, CEF
and the angle between them
sides
Then, the whole angle
CF.
equal to the two
in the other, are entirely
F
CF
Therefore the angle at
is
a right angle, and
AD.
Wherefore
AD and
CF, which are perpendicular
and drawn from the points
A
to
DF
x
and C at equal distances from
the point B, are equal to one another.
— Q. E. D.
THE THEORY OF PARALLEL
Theorem
making
If a line cut two lines,
4.
75
LINES.
angles unequal, every line bisecting that
the alternate
and meet-
line,
ing the other two, shall make unequal angles with these
two.
A
B
\F
E
Thus,
if
AF
the line
AFH, FAI,
the alternate angles
ing
cut the two lines AI,
unequal, every line bisect-
HF
AF, and meeting AI, and
HF, making
have
shall
alternate
its
angles unequal.
For, since the angle
FAI
is
AFE,
let
AB
equal to
AF
two
that
be bisected
is
in C,
:
sequently greater than
In the
lines
CEF.
— Hence
it
is
A I.
That
If a line cut
CBA,
That
— Q.
let
draw ECB, cutting the
ABC
(Eucl.
is,
is
equal to
CDA
is
16.)
I.
CEF
greater
and con-
the alternate angles
HF may be
AI and
line bisecting
shown
to
to
AF
have
its
E. D.
manifest that the line bisecting
and drawn perpendicular
with
it,
same manner, any
alternate angles unequal.
Cor.
let
AFE,
But the angle
than the interior opposite one
and meeting the
be parallel to HF, then,
the angle
(by the 2d of this paper.)
are unequal.
be
and through
B and E
parallels in
FAB
unequal to
HF,
will
make unequal
AF
angles
is,
two
lines,
qual, the line
which
to one of the
two
making the
bisects that line
lines,
alternate angles une-
and
is
perpendicular
makes oblique angles with the
other.
Theorem
o.
— If a
line
cut two
other lines,
alternate angles unequal, the lines
drawn from
making
the
the one line
THE THEORY OF PARALLEL
76
LINES.
perpendicular to the other, will continually decrease, the
nearer to the acute angle being always greater than the
more remote
on
:
the other hand, these perpendiculars will
continually increase on the side of the obtuse angle, the
nearer being always less than the one more remote.
U
j)
Thus,
if
BF
V
M
1
qual angles with them, so that
while
FBC
is
AC, DE, and make une-
cut the two lines
BFD may
be a right angle,
acute (by the 4th of this paper,) then,
from
if
L and K in the line AC, perpendiculars
DE at M and I, LM shall be less than BF,
any other points
be
let fall
and KI
less
upon
less
LM.
than
the triangle
FK to
FK
FBK,
(Eucl.
Much
greater than
right-angled at
FK,
drawing
LM
is
less than
GH, and
GH
it.
K, the
side
BF
For the same reason,
19.)
I.
more, then,
Further, let
I,
BF
the side
FK
is
greater
is
in the tri-
greater than
which was proved
be
to
greater than KI.
L be any point between B and K, then, Upon
perpendicular to
have the angle
16.)
is
be perpendicular to AC, then, in
angle FKI, right-angled at
KI.
BF
than any perpendicular beyond
For, supposing
than
Also,
BLF
DE,
the triangle
Therefore, in the triangle
BLF,
to the acute angle
is
FBL
son, in the triangle
must
obtuse because greater (by Eucl.
than the inward opposite angle
the obtuse angle
BFL
BFL,
FKL
which
FLM,
I.
is right.
the side BF, opposite to
greater than the side
(Eucl.
I.
19.)
the side
FL
FL
opposite
For the same
rea-
opposite the right-
THE THEORY OF PARALLEL
angle
FML, is
LM, opposite
greater than
77
LINES.
to the acute angle
LFM, which is acute because only a part of the right-angle
BFM. Much more must the line BF, which was proved to
be greater than FL, be greater than LM. For the like
LM
reasons,
is
Much more,
KI.
Thus,
it
MK,
greater than
then,
is
AC,
may
from points
DE, drawn from
continually decrease as these points
recede from the acute angle at B.
it
greater than
LM greater than KI.
appears, that perpendiculars to
points in the line
ner,
MK
and
And
in the
be proved that perpendiculars to
in the
AC,
same man-
DE, drawn
continually increase as these points
recede from the obtuse angle at B, as asserted.*
Theorem
hvo parallel
Let the
Every
6.
lines, is
lines
AC,
line
which
is
perpendicular to one of
ABC
perpendicular to the other.
F H
D
E
DF be parallel, having BE
ular to both (by the Cor.
our 2d
to
;)
perpendic-
any other
line
perpendicular to the one shall be perpendicular to the
other.
For,
if
BA
be taken equal to BC, and
drawn perpendicular
to
DF,
But
if
CF had its alternate
angles unequal, no other perpendicular as
to
DF, could be equal
equal to CF, the angle
—
be both
these perpendiculars shall be
equal (by the 3d of this paper. )
AC
AD, CF
to
BCF
CF.
is
AD, drawn from
Therefore since
equal to
AD is
CFH, which
is
a
*Note. This is the most important part of the theory, now attempted, as all
It implies, that, if
the rest of the demonstrations succeeding depend upon it.
two lines are not parallel, according to the definition given above, no two per~
pendiculars drawn from the one upon the other, can be found equal.
THE THEORY OF PARALLEL
78
Hence,
right angle.
FC
is
LINES.
same manner, any other perpendicular
shown to be so to AC.
the
Theorem
AC.
perpendicular to
to
And,
DF may be
Parallel lines are every where at the
7.
in
same
perpendicular distance.
Let
LM
and
OQ be
formed by some
L
M
N
O
P
Q
parallel, or
have the alternate angles
line cutting these equal,
any perpendiculars
NQ shall be equal.
For, bisect LN in M (Eucl.
10.) and join LP and PN.
Also, let PM be perpendicular to LM, and
will be so to OQ,
Then, the triangles LMP, MNP are entirely equal,
(6.)
(Eucl.
4.) and have the side LP equal to NP and the
to them, as
LO,
I.
it
I.
angles
Now
PLM, LPM,
since the
the part
to the
MLP
whole
equal to
MLO
is
MNP,
equal to
MNP, MPN.
MNQ
the remainder
and
(6.)
OLP
is
equal
QNP. In like manner the angle LPO is
angle NPQ.
Wherefore the triangles OLP
remainder
equal to the
and
respectively equal to
QNP,
having two angles equal, and the side
PN £ucl.
LO is lequal to
26.) are entirely equal.
LP
equal
Hence the
to
I.
lars cutting the
NQ, and thus all the perpendicutwo lines, may be shown to be equal.
Theorem
8.
side
the side
If a line cut two other
alternate angles unequal, the
two
lines,
lines will
duced, on the side of the acute angle.
making
the
meet y if pro-
THE THEORY OF PARALLEL
LINES.
79
THE THEORY OF PARALLEL
80
A, F, N, decrease by the equal
points,
EG.
LINES.
same manner,
In the
if
more
points
AE,
be taken on AB,
differences,
AF, the perpendiculars let fall
from these points upon CD, will decrease by differences
each equal to AE. By taking. AE, therefore, a sufficient
number of times, it must at last, either end at C, or beyond
But, by taking AF the same number of times
it at X.
at distances each equal to
—
upon AB,
it
must
at last end in the line
CD,
or on the other
has been just shown, that every line joining
side, since it
AC, and AB, is perpendicular
If, for example, four
parallel to CD.
the corresponding points in
AC, and
to
times
AE
therefore
should be found equal to
must end
precisely in
11. Cor.)
;
but
if
AC
some point of the
Hence
is
AB
CD
four times
line
CD,
AF
(Eucl.
I.
should be found to be more than four
times, but less than five times
would not meet
AC,
AE,
then, four times
AF
produced, but five times would pass
and CD, being both produced, must meet, and
it.
it
on the side of the acute angle, that they meet.
Quebec, December 27th, 1831.
In the preceding paper,
defect under
I
have endeavored to remove the
which the science of Geometry has hitherto
laboured from the imperfect elucidation of parallel
To
this
attempt
Length," inserted
I
alluded in
in the
my
lines.
paper on " Space and
second volume of the transactions
of this society.
I
have not had the pleasure of seeing the methods
posed by Legendre to remedy
this defect.
Professor Playfair, one of his methods
is
pro-
According to
founded on the
abstract doctrine of Functions, and therefore unsuitable to
THE THEORY OF PARALLEL
LINES.
81
be proposed to such as are entering on the study of Geometry.
The
other method
is
also stated
by the same Professor
and
intricate a nature to
to be of too abstruse, extensive
An
form part of an elementary system of that science.
ingenious demonstration of this property of Parallel lines,
was published
Mr. Ivory
in the Philosophical
but
;
it is
Magazine
As
also very complicated.
can judge, the preceding Theory
is
for 1822,
by
far as I
sufficiently simple to
be
put into the hands of the youngest students of Geometry.
The
more
5th proposition and the last might be rendered
easy of comprehension by dividing each of them into two,
a distribution of which they will readily admit.
In communicating this
it
Theory
I
beg further to
state, that
my
forms part of a system of Geometry which occupied
now nearly comfragment now presented, may
leisure hours for several years,
pleted.
Of that system, the
be considered as a
necessary to omit
fair
and which
is
specimen, except as far as
it
was
reference to the larger work, in an
all
extract requiring to be read independently by the lovers of
demonstrative science.
altered as to render
it
The
extract has only been so far
susceptible of separate perusal.
This system of Geometry, on which
have bestowed a
I
good deal of attention, possesses the following
which, of course,
1.
The
I
consider as so
many
peculiarities,
advantages.
usual treatises on this subject, presuppose the
use of the ruler and compass.
That which
presupposes only the use of the ruler.
use of the compass are in
The
derived from
it
I
speak
of,
properties and
first principles,
by the same synthetical reasoning, upon which the whole
science depends.
2.
The
science
is
definitions,
on which
it
is
entirely built, are all strictly
L
known that this
logical.
The terms
well
THE THEORY OF PARALLEL
82
which cannot be
logically defined, are explained,
principles they involve,
3.
No
involves
4.
assumed on
definition is attempted,
is
shown
LINES.
their proper grounds.
till
to be possible, or
and the
the property which
assumed
All the propositions are simple.
it
to be so.
It is at
no time
at-
tempted to prove two or more properties of a geometrical
figure in the
same demonstration.
But where a
series of
truths are intimately connected so as to require a connected
view
to be given of them, such a
scholia or notes
appended
view
is
given of them in
to the propositions.
There are other minor points of
difference
which need
not be here enumerated.
The
the strictest geometrical
reasoning with the most lucid
illustration of
object has been, to combine
which the subject admits.
Particular care
has been taken to render the introductory demonstrations
as simple and easy as possible.
trical
When
a habit of geome-
reasoning has once been formed, greater conciseness
of expression, and
be safely admitted.
more
difficult
forms of ratiocination,
may