Some Fundamental Topics in Analytic & Euclidean Geometry 1
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1 Lecture Notes – Math 119 Some Fundamental Topics in Analytic & Euclidean Geometry 1. Cartesian coordinates Analytic geometry, also called coordinate or Cartesian geometry, is the study of geometry using the principles of algebra. The algebra of the real numbers can be employed to yield results about geometry due to the Cantor – Dedekind axiom which asserts that there is a one to one correspondence between the real numbers and the points on a line. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. It is concerned with defining geometrical shapes in a numerical way and extracting numerical information from that representation. Some consider that the introduction of analytic geometry was the beginning of modern mathematics. History The Greek mathematician Apollonius of Perga, (262-190 BC) in On Determinate Section dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different than our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas or x-coordinates, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates or y-coordinates. He further developed relations between P = (2, –1 .5) x = 2 y = – 1.5
Some Fundamental Topics in Analytic & Euclidean Geometry 1
Microsoft Word - HO Notes 1 Analytic geometry2015Some Fundamental
Topics in Analytic & Euclidean Geometry
1. Cartesian coordinates Analytic geometry, also called coordinate
or Cartesian geometry, is the study of geometry using the
principles of algebra. The algebra of the real numbers can be
employed to yield results about geometry due to the Cantor –
Dedekind axiom which asserts that there is a one to one
correspondence between the real numbers and the points on a line.
Usually the Cartesian coordinate system is applied to manipulate
equations for planes, lines, curves, and circles, often in two and
sometimes in three dimensions of measurement. It is concerned with
defining geometrical shapes in a numerical way and extracting
numerical information from that representation. Some consider that
the introduction of analytic geometry was the beginning of modern
mathematics. History The Greek mathematician Apollonius of Perga,
(262-190 BC) in On Determinate Section dealt with problems in a
manner that may be called an analytic geometry of one dimension;
with the question of finding points on a line that were in a ratio
to the others. Apollonius in the Conics further developed a method
that is so similar to analytic geometry that his work is sometimes
thought to have anticipated the work of Descartes by some 1800
years. His application of reference lines, a diameter and a tangent
is essentially no different than our modern use of a coordinate
frame, where the distances measured along the diameter from the
point of tangency are the abscissas or x-coordinates, and the
segments parallel to the tangent and intercepted between the axis
and the curve are the ordinates or y-coordinates. He further
developed relations between
P = (2, –1 .5)
2
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2. Excerpts from Apollonius, Conics, Book I: FIRST DEFINITIONS 1.
If from a point a straight line is joined to the circumference of a
circle which is not in the same plane with the point, and the line
is produced in both directions, and if, with the point remaining
fixed, the straight line being rotated about the circumference of
the circle returns to the same place from which it began, then the
generated surface composed of the two surfaces lying vertically
opposite one another, each of which increases indefinitely as the
generating straight line is produced indefinitely, I call a conic
surface, and I call the fixed point the vertex, and the straight
line drawn from the vertex to the center of the circle I call the
axis. 2. And the figure contained by the circle and by the conic
surface between the vertex and the circumference of the circle I
call a cone, and the point which is also the vertex of the surface
I call the vertex of the cone, and the straight line drawn from the
vertex to the center of the circle I call the axis, and the circle
I call the base of the cone. 3. I call right cones those having
axes perpendicular to their bases, and I call oblique those not
having axes perpendicular to their bases. 4. Of any curved line
which is in one plane, I call that straight line the diameter
which, drawn from the curved line, bisects all straight lines drawn
to this curved line parallel to some straight line; and I call the
end of the diameter situated on the curved line the vertex of the
curved line, and I say that each of these parallels is drawn
ordinatewise to the diameter. 5. Likewise, of any two curved lines
lying in one plane, I call that straight line the transverse
diameter which cuts the two curved lines and bisects all the
straight lines drawn to either of the curved lines parallel to some
straight line; and I call the ends of the [transverse] diameter
situated on the curved lines the vertices of the curved lines; and
I call that straight line the upright diameter which, lying between
the two curved lines, bisects all the straight lines intercepted
between the curved lines and drawn parallel to some straight line;
and I say that each of
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the parallels is drawn ordinatewise to the [transverse or upright]
diameter. 6. The two straight lines, each of which, being a
diameter, bisects the straight lines parallel to the other, I call
the conjugate diameters of a curved line and of two curved lines.
7. And I call that straight line the axis of a curved line and of
two curved lines which being a diameter of the curved line or lines
cuts the parallel straight lines at right angles. 8. And I call
those straight lines the conjugate axes of a curved line and of two
curved lines which, being conjugate diameters, cut the straight
lines parallel to each other at right angles. PROPOSITION 1 The
straight lines drawn from the vertex of the conic surface to points
on the surface are on that surface.
Let there be a conic surface whose vertex is the point A, and let
there be taken some point B on the conic surface, and let a
straight line ACB be joined. I say that the straight line ACB is on
the conic surface. For if possible, let it not be, and let the
straight line DE be the line generating the surface, and EF be the
circle along which ED is moved. Then if, the point A remaining
fixed, the straight line DE is moved along the circumference of the
circle EF, it will also go through the point B (Def. 1), and two
straight lines will have the same ends. And this is absurd.
Therefore the straight line joined from A to B cannot not be on the
surface. Therefore it is on the surface. PORISM It is also evident
that, if a straight line is joined from the vertex to some point
among those within the surface, it will fall within the conic
surface; and if it is joined to some point among those without, it
will be outside the surface.
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PROPOSITION 2 If on either one of the two vertically opposite
surfaces two points are taken, and the straight line joining the
points, when produced, does not pass through the vertex, then it
will fall within the surface, and produced it will fall outside.
Let there be a conic surface whose vertex is the point A, and a
circle BC along whose circumference the generating straight line is
moved, and let two points D and E be taken on either one of the two
vertically opposite surfaces, and let the joining straight line DE,
when produced, not pass through the point A. I say that the
straight line DE will be within the surface, and produced will be
without. Let AE and AD be joined and produced. Then they will fall
on the circumference of the circle (I. 1). Let them fall to the
points B and C, and let BC be joined. Therefore the straight line
BC will be within the circle, and so too within the conic surface.
Then let a point F be taken at random on DE, and let the straight
line AF be joined and produced. Then it will fall on the straight
line BC; for the triangle BCA is in one plane (Eucl. XI. 2). Let it
fall to the point G. Since then the point G is within the conic
surface, therefore the straight line AG is also within the conic
surface (I. 1 porism), and so too the point F is within the conic
surface. Then likewise it will be shown that all the points on the
straight line DE are within the surface. Therefore the straight
line DE is within the surface. Then let DE be produced to H. I say
then it will fall outside the conic surface
For if possible, let there be some point H of it not outside the
conic surface, and let AH be joined and produced. Then it will fall
either on the circumference of the circle or within (I. 1 and
porism). And this is impossible, for it falls on BC produced, as
for example to the point K. Therefore the straight line EH is
outside the surface. Therefore the straight line DE is within the
conic surface, and produced is outside.
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Proposition 11 Deriving the Symptom of the Parabola
If a cone is cut by a plane through its axis, and also cut by
another plane cutting the base of the cone in a straight line
perpendicular to the base of the axial triangle, and if further the
diameter of the section is parallel to one side of the axial
triangle, then any straight line which is drawn from the section of
the cone to its diameter parallel to the common section of the
cutting plane and of the cone's base, will equal in square the
rectangle contained by the straight line cut off by it on the
diameter beginning from the section's vertex and by another
straight line which has the ratio to the straight line between the
angle of the cone and the vertex of the section that the square on
the base of the axial triangle has to the rectangle contained by
the remaining two sides of the triangle. And let such a section be
called a parabola (παραβολ).
Consider the cone with vertex A and a plane through the axis
intersecting the cone. This plane intersects the cone in the axial
triangle ABC, where BC is the diameter of the base circle of the
cone.
The parabola is like the section of a right-angled cone
(orthotome); if the cutting plane is orthogonal to a side of the
axial triangle, it must also be parallel to the other side of the
axial triangle (for right-angled cones). For Apollonius, the
parabola is generated by a plane cutting one side of the axial
triangle such that it is parallel to the other side. This works for
all cones in general, but we illustrate here with a right circular
cone.
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Consider an arbitrary ordinate (i.e. y value) ML constructed on the
axis at M. We wish to determine the relationship between ML and EM,
that is, the symptom of the conic. The ordinate ML is located in a
horizontal plane that cuts the cone in the circle with diameter PR.
In this horizontal plane construct the segments PL and LR, which
results in a right triangle inscribed in a semicircle. As we have
seen before, by similar triangles this implies
or
(1)
Next Apollonius constructs a segment EH perpendicular to EM such
that
(2)
How is this possible? All the lengths EA, BC, BA, and AC are known.
He is simply finding the point H that makes the ratio true. Now why
Apollonius does this is another story. Wait and see.
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Now consider some triangles in the axial plane, namely ABC, APR,
and EPM. These triangles are all similar, using the usual
properties of parallel lines (see previous derivations for
details). From the similarity we have
(3)
and
. (4)
. (5)
Also, in triangle APR, since EM is parallel to AR we know by
Elements Book VI, Prop. 2 that
. (6)
. (7)
.
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In this form we can understand Apollonius' use of the word parabola
for this section. He has proven that the square on ML is equal to a
rectangle applied (paraboli) to a line EH with a width equal to EM.
This is based on the idea of application of areas, a Greek
technique used to deal with multiplication and division of lengths
and areas.
Finally, if we let ML be y, EM be x, and EH be p, we have a
standard equation for a parabola,
y2 = p x .
Notice that all that was necessary to derive the symptom of the
parabola was a knowledge of similar triangles. We will see this
again as we derive the symptom of the next conic section, the
ellipse.
Proposition 13 Deriving the Symptom of the Ellipse
If a cone is cut by a plane through its axis, and also cut by
another plane on the one hand meeting both sides of the axial
triangle, and on the other extended neither parallel to the base
nor subcontrariwise, and if the plane the base of the cone is in,
and the cutting plane meet in a straight line perpendicular either
to the base of the axial triangle or to it produced, then any
straight line which is drawn from the section of the cone to the
diameter of the section parallel to the common section of the
planes, will equal in square some area applied to a straight line
to which the diameter of the section has the ratio that the square
on the straight line drawn from the cone's vertex to the triangle's
base parallel to the sections's diameter has to the rectangle
contained by the intercepts of this straight line (on the base)
from the sides of the triangle, an area having as breadth the
straight line cut off on the diameter beginning from the section's
vertex by this straight line from the section to the diameter, and
deficient (λλειπου) by a figure similar and similarly situated to
the rectangle contained by the diameter and parameter. And let such
a section be called an ellipse (λλειψις). We will still consider
the cone with vertex A and a plane through the axis intersecting
the cone. This plane intersects the cone in the axial triangle ABC,
where BC is the diameter of the base circle of the cone.
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To generate the ellipse the cutting plane must intersect both sides
of the axial triangle. Once again this works for all cones in
general, but we illustrate here with a right circular cone.
Consider an arbitrary ordinate (i.e. y value) ML constructed on the
axis at M. We wish to determine the relationship between ML and EM,
that is, the symptom of the conic. The ordinate ML is located in a
horizontal plane that cuts the cone in the circle with diameter PR.
In this horizontal plane construct the segments PL and LR, which
results in a right triangle inscribed in a semicircle. As we saw
before in the case of the parabola, by similar triangles this
implies
or
. (1)
Apollonius next constructs a segment EH, perpendicular to DE, such
that
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, (2)
where AK is parallel to ED. [As in the previous section, all the
lengths DE, AK, BK, and KC are known. He is simply finding the
point H that makes the ratio true.]
Notice in the figure that triangles DEH and DMX are similar. This
implies that
(3)
where the first equality is due to the similarity, and the second
one is due to the fact that MX = EO. Next, consider the cone
diagram carefully (a different viewpoint is below). The triangles
ABK, EBG, and EPM in the axial plane are all similar (parallel
lines), and also the triangles ACK, DCG, and DMR are all similar
(do you see why?). From these two sets of similar triangles we
have
(4)
and
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. . (5)
. (6)
Now compare (3) and (6) and notice that they imply PM . MR = EM .
EO. However, from (1) we have ML2 = PM . MR , so together we have
ML2 = EM . EO.
Taking another look at our ellipse, notice that triangles DEH and
XOH are similar. This implies
. (7)
Notice that EO = EH - HO, so ML2 = EM . EO becomes, using
(7),
. (8)
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In this form we can understand Apollonius' use of the word ellipse
for this section. He has proven that the square on ML is equal to a
rectangle applied to a line HE with a width equal to EM and
deficient (ellipsis) by a rectangle similar to one contained by ED
and HE.
.
Can this be written in the standard form of an ellipse? Complete
the square and see.
In the next section we derive the symptom of the hyperbola. The
derivation follows that of this section. You will notice very
little difference between the two until the very end. In
Apollonius's Conics, propositions concerning the ellipse and the
hyperbola are frequently proven together, due to their
similarities.
Proposition 12 Deriving the Symptom of the Hyperbola
If a cone is cut by a plane through its axis, and also cut by
another plane cutting the base of the cone in a straight line
perpendicular to the base of the axial triangle, and if the
diameter of the section produced meets one side of the axial
triangle beyond the vertex of the cone, then any straight line
which is drawn from the section to its diameter parallel to the
common section of the cutting plane and of the cone's base, will
equal in square some area applied to a straight line to which the
straight line added along the diameter of the section and
subtending the exterior angle of the triangle has a ratio that the
square on the straight line drawn from the cone's vertex to the
triangle's base parallel to the section's diameter has to the
rectangle contained by the sections of the base which this straight
line makes when drawn, this area having as breadth the straight
line cut off on the diameter beginning from the section's vertex by
this straight line from the section to the diameter and exceeding
(περβλλου) by a figure (εδος), similar and similarly situated to
the rectangle contained by the straight line subtending the
exterior angle of the triangle and by the parameter. And let such a
section be called an hyperbola (περβολ).
We will once again consider the cone with vertex A and a plane
through the axis intersecting the cone. This plane intersects the
cone in the axial triangle ABC, where BC is the diameter of the
base circle of the cone. To generate the hyperbola the cutting
plane must intersect only one side of the axial triangle but also
both "top" and "bottom" of the "double cone." As before this works
for all cones in general, but we illustrate here with a right
circular cone.
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Consider an arbitrary ordinate (i.e. y value) ML constructed on the
axis at M. We wish to determine the relationship between ML and EM,
that is, the symptom of the conic. The ordinate ML is located in a
horizontal plane that cuts the cone in the circle with diameter PR.
In this horizontal plane construct the segments PL and LR, which
results in a right triangle inscribed in a semicircle. As we saw
before in the case of the parabola, by similar triangles this
implies
or
. (1)
As with the ellipse, Apollonius next constructs a segment EH,
perpendicular to DE, such that
(2)
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where AK is parallel to ED. [As before, all the lengths DE, AK, BK,
and KC are known. He is simply finding the point H that makes the
ratio true.]
Notice in the figure that triangles DEH and DMX are similar. This
implies that
(3)
where the first equality is due to the similarity, and the second
one is due to the fact that MX = EO. Next, consider the cone
diagram carefully (a different viewpoint is below). The triangles
ABK, EBG, and EPM in the axial plane are all similar (parallel
lines), and also the triangles ACK, DCG, and DMR are all similar
(do you see why?). From these two sets of similar triangles we
have
(4)
and
. (5)
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. (6)
Now compare (3) and (6) and notice that they imply PM . MR = EM .
EO. However, from (1) we have ML2 = PM . MR, so together we have
ML2 = EM . EO.
Taking another look at our hyperbola, notice that triangles DEH and
XOH are similar. This implies
. (7)
Here is where the hyperbola derivation changes from that of the
ellipse. Notice now that
EO = EH + HO, so ML2 = EM . EO becomes, using (7),
. (8)
From this expression comes Apollonius' use of the word hyperbola
for this section. He has proven that the square on ML is equal to a
rectangle applied to a line HE with a width equal to EM and
exceeding (yperboli) by a rectangle similar to one contained by ED
and HE.
.
Can this be written in the standard form of a hyperbola? Again, it
is up to you to try it.
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Analytic geometry has traditionally been attributed to René
Descartes who made significant progress with the methods of
analytic geometry when in 1637 in the appendix entitled Geometry of
the titled Discourse on the Method of Rightly Conducting the Reason
in the Search for Truth in the Sciences, commonly referred to as
Discourse on Method. This work, written in his native French
tongue, and its philosophical principles, provided the foundation
for calculus in Europe.
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3. Congruence in geometry
Two geometrical objects are called congruent if they have the same
shape and size. This means that either object can be repositioned
so as to coincide precisely with the other object. More formally,
two sets of points are called congruent if, and only if, one can be
transformed into the other by an isometry, i.e., a combination of
translations, rotations and reflections. Congruent triangles Two
triangles are congruent if their corresponding sides are equal in
length and their corresponding angles are equal in size.
If triangle ABC is congruent to triangle DEF, the relationship can
be written mathematically as: ABC y DEF.
Sufficient evidence for congruence between two triangles in the
plane can be shown through the following comparisons:
• SAS (Side-Angle-Side): If two pairs of sides of two triangles are
equal in length, and the included angles are equal in measurement,
then the triangles are congruent.
• SSS (Side-Side-Side): If three pairs of sides of two triangles
are equal in length, then the triangles are congruent.
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• ASA (Angle-Side-Angle): If two pairs of angles of two triangles
are equal in measurement, and the included sides are equal in
length, then the triangles are congruent. The ASA Postulate was
contributed by Thales of Miletus .
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4. Similarity in geometry Two geometrical objects are called
similar if one is congruent to the result of a uniform scaling
(enlarging or shrinking) of the other. One can be obtained from the
other by uniformly "stretching", possibly with additional rotation,
i.e., both have the same shape, or additionally the mirror image is
taken, i.e., one has the same shape as the mirror image of the
other. For example, all circles are similar to each other, all
squares are similar to each other, and all parabolas are similar to
each other. On the other hand, ellipses are not all similar to each
other, nor are hyperbolas all similar to each other. Two triangles
are similar if and only if they have the same three angles, the
so-called "AAA" condition.
Similar triangles
If triangle ABC is similar to triangle DEF, then this relation can
be denoted as
ABC w DEF
In order for two triangles to be similar, it is sufficient for them
to have at least two angles that match. If this is true, then the
third angle will also match, since the three angles of a triangle
must add up to 180°. This condition guarantees that the side
lengths are locked in a common ratio, but can vary
proportionally.
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F
Suppose that triangle ABC is similar to triangle DEF in such a way
that the angle at vertex A is congruent with the angle at vertex D,
the angle at B is congruent with the angle at E, and the angle at C
is congruent with the angle at F. Then, once this is known, it is
possible to deduce proportionalities between corresponding sides of
the two triangles, such as the following:
EF DF
EF BC
DE AB == .
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5. Pythagorean Theorem The Pythagorean theorem:
In any right triangle, the area of the square whose side is the
hypotenuse (the side opposite the right angle) is equal to the sum
of the areas of the squares whose sides are the two legs (the two
sides that meet at a right angle). If we let c be the length of the
hypotenuse and a and b be the lengths of the other two sides, the
theorem can be expressed as the equation: 222 cba =+
or, solved for c : 22 bac += .
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Proof using similar triangles
Like most of the proofs of the Pythagorean theorem, this one is
based on the proportionality of the sides of two similar
triangles.
Let ABC represent a right triangle, with the right angle located at
C, as shown on the figure. We draw the altitude from point C, and
call H its intersection with the side AB. The new triangle ACH is
similar to our triangle ABC, because they both have a right angle
(by definition of the altitude), and they share the angle at A,
meaning that the third angle will be the same in both triangles as
well. By a similar reasoning, the triangle CBH is also similar to
ABC. The similarities lead to the two ratios..: As
BC = a, AC = b and AB = c .
so c a =
Summing these two equalities, we obtain
22 ba + = HBc × + AHc × = ) ( AHHBc +×
In other words, the Pythagorean theorem:
222 cba =+
6. Area in geometry
Area is a quantity expressing the size of a figure in the Euclidean
plane or on a 2-dimensional surface. Points and lines have zero
area, although there are space-filling curves. Depending on the
particular definition taken, a figure may have infinite area, for
example the entire Euclidean plane. In three dimensions, the analog
of area is called a volume.
How to define area Although area seems to be one of the basic
notions in geometry, it is not at all easy to define even in the
Euclidean plane. Most textbooks avoid defining an area, relying on
self-evidence. To make the concept of area meaningful one has to
define it, at the very least, on polygons in the Euclidean plane,
and it can be done using the following definition:
The area of a polygon in the Euclidean plane is a positive number
such that: 1. The area of the unit square is equal to one. 2.
Congruent polygons have equal areas. 3. Additivity: If a polygon is
a union of two polygons which do not have common
interior points, then its area is the sum of the areas of these
polygons. The area of an arbitrary square (i) If n is any positive
integer ( i.e. a “whole number” ) then it follows immediately from
the additivity property of area that the square with side length n
has area: A = 2n square units.
Proof
Observe that there are: 2 . . . nnn
n terms nnn =×=+++ 444 3444 21
unit square tiles which cover the square with side length n units.
∴ A = 2n × (1 2unit ) = 2n square units.
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(ii) If n is any positive integer then it also follows from the
additivity property that the square with side length
n 1 units must have area:
A =
Observe that there are 2n square tiles with side lengths
n 1 units which
⇒ A =
1 2n
square units .
( iii ) For any positive integers m and n , the square with side
lengths
n m units has area
A = 2
n m square units. We may obtain this result by additivity once
again.
Proof In this case, note that there are 2n square tiles of side
length
n m units
which cover the square having side length m units. ∴ 2n × A = 2m
2unit
⇒ A = 2
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( iv ) For any positive real number s the area of the square with
side length s is A = 2s square units. This follows from the well
known property that any positive number s (whether rational or
irrational) may be obtained as the limit of a sequence of numbers
of the
form i
n m
where im and in are positive integers for all i = 1 , 2 , 3, . . .
, Thus,
the difference:
n ms becomes infinitesimally small as i becomes larger and
larger. That is, there exists a sequence:
1
1
n m
→ .
So due to the continuity of geometrical area, it must be the case
that
A i =