Учебное пособие по чтению математических текстов на английском языке. Нижний Новгород: нгпу, 2011. 65с

CARTESIAN COORDINATES

Cartesian uniquely

coordinate n. conversely

coordinate v. arbitrarily

referred mutually

similarly rectangular

parallelepiped oblique

diagonal

In a plane the position of a point is defined by two coordinates, X and Y, referred to two straight lines OX, OY, the coordinate-axes.

To fix the position of a point in space we take three planes. These have a point O in common and intersect in pairs in three lines X'OX, Y'OY, Z'OZ. O is called the origin, the three lines the coordinate - axes, and the three planes the coordinate-planes.

Let P be any point. Through P draw PL parallel to XOX' cutting the plane YOZ in L, and similarly PM and PN parallel to the other axes. Let the plane MPN cut OX in L', and similarly obtain M' and N'. We obtain then a parallelepiped whose faces are parallel to the coordinate-planes, and edges parallel to the coordinate-axes, and OP is a diagonal. We then define the coor­dinates of the point P as the three lengths OL'=X, OM'=Y, ON'=Z

To every point P there corresponds uniquely a set of three numbers (X, Y, Z) and conversely to every set of three numbers, positive or negative, there corresponds a unique point.

CONVENTION OF SIGHS

Let the positive directions along the axes of X and Y be defined arbitrarily, say OX and OY; then in the plane XOY we may pass from OX to OY by a rotation through an angle XOY less than two right angles. Viewed from one side of the plane this rotation is clockwise, and from the other side it appears counter-clockwise. We define that side of the plane from which the rotation appears to be counter-clockwise as the positive side of the plane. Then the positive direction of the axis of Z is defined to be that which lies on the positive side of the plane XOY. This rela­tion then holds for each of the axes, viz., the positive direction of the axis of Y is on the positive side of the plane ZOX. This is called a right-handed system of Cartesian coordinates.

When the planes are mutually at right angles we call it a rectangular system, otherwise it is oblique.

I. ANSWER THE QUESTIONS:

1. 
   What are the coordinate-axes of the Cartesian system of coor­dinates?
   
2. 
   How many coordinates are necessary to define a point in a plane?
   
3. 
   How many coordinates are necessary to define a point in a space?
   
4. 
   What point is called the origin?
   
5. 
   What side of the plane is regarded positive?
   
6. 
   What system of coordinates is called rectangular?
   
7. 
   What system of coordinates is called oblique?
   

convention arbitrarily

definition commonly

direction definitely

intersection conversely

position mutually

relation similarly

rotation uniquely

III. TRANSLATE THE FOLLOWING SENTENCES:

1. 
   The straight lines in the same plane either intersect or are parallel.
   
2. 
   The position of a point P in a line may be defined to fixed points A and B instead of with respect to a single origin O.
   
3. 
   The angle is measured from OX to OP with the usual sign convention of trigonometry, similar convention is made for the angle, measured from OX to OP'.
   
4. 
   In the general Cartesian system the planes are not necessari­ly at right angles. The system will be defined by the angles between the coordinate axes, viz. YOZ= λ, ZOX= μ , XOY= ν .
   
5. 
   After the rotation of the axis OX counter-clockwise through angle 90° it will coincide with the axis OY.
   
6. 
   There is an advantage for certain problems in using oblique axes (XOY ≠ ½π) and many formulae are as easily obtained for oblique as for rectangular axes, but rectangular axes are often used in applications to coordinate geometry.
   
7. 
   Euclid posed the question: to find a rectangle such that, when a square is cut from it the remaining smaller rectangle has the same shape as the original.
   
8. 
   The rectangular axes are generally much more convenient in practice than are oblique (nonrectangular).
   

1. 
   To define the position of a point in a plane we take two (...) lines OX and OY.
   
2. 
   When the coordinate planes are mutually at right angles we
   have (...) system.
   
3. 
   The point of intersection of coordinate-axes is called (...).
   
4. 
   The faces of a cube are at (…. ….).
   
5. 
   Points in (…) can be represented by pairs of numbers.
   
6. 
   The (...) which represent a point are called the coordinates
   of that point.
   
7. 
   The sum of two algebraic numbers is an algebraic number, and
   the same (...) for the other operations of arithmetic.
   

1. 
   The position of a point is defined by two coordinates ...
   
2. 
   The position of a point is defined by three coordinates ...
   
3. 
   We take two axis to define ...
   
4. 
   We take three planes to define ...
   
5. 
   To every point in a plane there corresponds ...
   
6. 
   To every point in space there corresponds ...
   
7. 
   When the coordinate planes are at right angles, we have ...
   
8. 
   When the coordinate planes are not at right angles, we
   have …
   

координата, позиция, линия, параллелограм, фиксировать, диагональ, параллельный, параллелепипед, позитивный, система, негативный.

VII. DEFINE THE POSITION OF A POINT

1. 
   in a plane
   
2. 
   in space
   

Euclidean initial

pair argument

determines major

assigning measured

coordinatization liberty

coordinatized convenient
It is evident from your observation that every point in the Euclidean plane has an infinity of coordinate representations, but that each pair of coordinates of the form (± r, θ±2nπ) deter­mines one and only one point. The method of assigning to each point P of the Euclidean plane a distance r and an angle θ is known more formally as a polar coordinatization of the plane. The coor­dinates (r,θ) are called polar coordinates, and the coordinatized plane is called the polar plane. The initial ray is called the polar axis, the radius r (the segment OP) is called the radius vector, and the origin is called the pole. Finally, the angle θ will be referred to as the argument of the point P. You can become more familiar with this new coordinate system by com­paring it with a rectangular coordinate system. One of the major differences between them is the method by which a point P is lo­cated in the two systems. Recall that, by convention, the first coordinate in the ordered pair (X, Y) refers to the horizontal displacement of the point P from the origin, and the second coor­dinate gives the vertical displacement from the origin. If, how­ever, polar coordinates are used, then the first coordinate of the pair (r,θ) determines the distance to the point P, measured along the radius vector. If r is negative, then measured backward along the extended vector. The angle θ, generally measured in ra­dians, is the angle between the polar axis and the radius vector. When θ is positive, the angle is generated in a counter-clockwise direction, and in a clockwise direction when θ is negative. Also observe that when you construct a rectangular coordinate system you are at liberty to choose any convenient scales for both the horizontal and vertical distances. This is not the case when polar coordinates are used. You do not have a choice for the θ-scale, as there are exactly 2π radians in a complete rotation of the radius vector. Thus, no arbitrary unit angle exists for O'. You do, however, have freedom in choosing any convenient scale when you measure distance along the radius vector.

I. ANSWER THE QUESTIONS:

1. 
   What method is called a polar coordinatization of the plane?
   
2. 
   What coordinates are called polar?
   
3. 
   What do we call the initial ray?
   
4. 
   How can you become more familiar with the new coordinate system?
   
5. 
   What is the major difference between the two coordinate systems?
   
6. 
   What does the first coordinate in Cartesian system refer to?
   
7. 
   What does the first coordinate in polar system determine?
   
8. 
   How is the angle θ measured?
   
9. 
   Why has every point in a rectangular coordinate system an infinity of representations and only one in a polar system?
   

coordinatize displacement

symbolize argument

characterize assignment

familiarize measurement

factorize requirement

generalize movement

summarize statement

emphasize achievement

rationalize development

accomplishment

III. TRANSLATE THE FOLLOWING SENTENCES:

1. 
   The equation Rx+my+nz+pw=0 represents the condition that the
   point lies on the plane.
   
2. 
   Points whose coordinates satisfy an equation y=f(x) can be
   plotted by assigning value to x and calculating the corresponding value of y.
   
3. 
   An arbitrary point of the curve y=x³ has only one degree of
   freedom: it requires only one coordinate to determine the po­sition of a point on the curve.
   
4. 
   Cartesian coordinates are particularly convenient for the in­vestigation of problems in metrical geometry, i.e., problems in which distances are involved.
   
5. 
   The degree as a unit for measuring angles may be defined as the value of the angle formed by dividing a right angle into 90 equal parts.
   
6. 
   While a decimal representation is most convenient for practi­cal purposes, the well-known ratio 22/7 shows that rational approximations have their uses.
   
7. 
   With a given arbitrary segment AB as a unit, one could measure any segment CD which was an exact multiple of the unit.
   
8. 
   Every problem on changing from rectangular to polar form is a problem on the solution of a right triangle with two legs given.
   
9. 
   It is sometimes necessary to locate the maximum or minimum of
   a function f(x).
   
10. 
    It is customary to use the same scale on both axes and we will do so unless we state otherwise.
    
11. 
    There exists a number of system of coordinates, but the most convenient is a Cartesian one.
    
12. 
    The major difference between rectangular and oblique systems of coordinates lies in the fact that in a rectangular system the angle between coordinate planes is right.
    
13. 
    In algebra we deal with numbers and also with line segments and geometrical formations in general.
    
14. 
    When we compared segments we saw that they are equal.
    
15. 
    Since three mutually perpendicular planes meet in three mutually perpendicular lines, we may also consider the Cartesian coordinates of a particle as a displacement in the direction of these lines needed to move the particle from the point of intersection of the three lines to its actual position.
    

1. 
   A point in space (... ...) by three data, its three coordinates.
   
2. 
   When we (...) distances, we are at liberty to choose units of measurement.
   
3. 
   The way of (...) ordered pairs of real numbers to points in the plane is the basis of analytic geometry.
   
4. 
   If you are (...) with the rectangular system you know that the coordinate planes intersect at right angles.
   
5. 
   Usually the most (...) points to draw the graph of a straight line are those where the line crosses the two axes.
   
6. 
   The map of a country may be drawn on a (...) of 50 miles to an inch, or on any other convenient (...).
   
7. 
   A cylinder can be considered as a cone whose vertex is a point at (…).
   

1. 
   Every point in the Euclidean plane … .
   
2. 
   Each pair of coordinates determines … .
   
3. 
   The coordinates (r,θ) are called … .
   
4. 
   The initial ray is called ... .
   
5. 
   The angle θ will be referred to as ... .
   
6. 
   We can become more familiar with a polar coordinate system
   by comparing it ... .
   
7. 
   The first coordinate in the ordered pair (X,Y) refers to … .
   
8. 
   The first coordinate of the pair (r,θ) determines ... .
   
9. 
   If r is negative … .
   
10. 
    The angle θ is ... .
    
11. 
    When θ is positive … .
    

1. 
   Polar coordinates.
   
2. 
   The major difference between polar and Cartesian coordinates.
   

LOCUS OF A VARIABLE POINT

locus ordinate

aggregate satisfy

particular endeavour

relevant accurately

abscissa squared

associated emphasized

In analytical geometry we are concerned with two distinct problems. In the first, a condition is given, and it is our task to derive the equation of the locus or curve each point of which is to be in accordance with the given condition; and no point which is not in accordance with the given condition has coordi­nates which can satisfy the equation. In the second, the equation of the curve is given (or has been found) and it is then our endeavour to interpret this equation by investigating the properties of the curve and, if need be, to make a sketch of the curve or even to plot it accurately on squared-paper. It is, of cour­se, evident that we may have the composite problem involving (1) the derivation of the equation of the locus resulting from a given condition and (II) the discussion of the properties of the associated curve.

One important principle requires to be emphasized. When we have derived, or if we are given, the equation of the curve, this equation in x and y is then the analytical condition that a point P (x,y) should lie on the curve. In particular, if a point A(x1,y1) lies on the curve, then the values of x1 and y1 satisfy the equation of the curve; for example, it is easily seen that the point (2, 12) lies on the curve y = 3x² for, when x = 2 then 3x² =12, and hence y =12. Further, if A (x1,y1) does not lie on the curve then the values of x1, and y1 do not satisfy the equation.

I. ANSWER THE QUESTIONS:

1. 
   Where can the position of a point P be found if x and y are unspecified?
   
2. 
   When are the positions of P restricted?
   
3. 
   What is the locus of P?
   
4. 
   What relation is called the equation of the locus?
   
5. 
   Is a straight line included in the category of curves?
   
6. 
   What two distinct problems are we concerned with in analyti­cal geometry?
   
7. 
   If the equation of the curve is given what is our task?
   
8. 
   What parts does a composite problem consist of?
   
9. 
   What principle should be emphasized?
   

clarify algebraic

identify analytic

justify asymptotic

satisfy geometric

simplify elliptic

specify hyperbolic

III. TRANSLATE THE FOLLOWING SENTENCES:

1. 
   When the properties of a single parabola are to be investigated, it is best to use the equations y² = kx or y² = 4ax, or the parametric equation equivalent to them.
   
2. 
   The relation which holds between the coordinates x,y of the
   arbitrary point P on the locus must hold no matter which point of the locus is chosen as P.
   
3. 
   Most of us have a good intuitive understanding of the concept of area as a measure of size or extent, derived from the phy­sical plane.
   
4. 
   The problem of proving that a particular number is transcen­dental is difficult one.
   
5. 
   If we are concerned with triangles we shall show that a tri­angle can be dissected into three pieces that form a rectangle.
   
6. 
   The distance between parallel tangents to a circle is constant but it is not the only curve with this property.
   
7. 
   Among all curves of given circumference the circle has the greatest area.
   
8. 
   In space of three dimensions there are two fundamentally dif­ferent kinds of loci, of which the simplest examples are the plane and the straight line.
   
9. 
   We shall now consider a method of determining the n constants of integration from the initial conditions of the system.
   
10. 
    A straight line is specified uniquely if we are given one point on it and the angle which the line makes with OX.
    
11. 
    It should be emphasized, that a function is never determined before its domain is specified.
    
12. 
    If two lines have a common point, the coordinates of this point satisfy the equation of' the line and are found by sol­ving these equations.
    
13. 
    In particular if the coordinates can be separately expressed as rational algebraic functions of one parameter the curve is called a rational algebraic curve.
    
14. 
    Sometimes the solution is valid under the limited conditions imposed by our assumptions.
    
15. 
    The use of vector algebra is not restricted to the study of fluid mechanics.
    
16. 
    The study of vector quantities leads into what is known as vector analysis.
    

1. 
   An algebraic (...) is cut by an arbitrary plane in a finite number of points, this number is called the order of the (...).
   
2. 
   An important part of analytical geometry is the discussion
   of the (...) of special curves.
   
3. 
   Polar equations are rarely used in the general theory of al­gebraic (...), but an algebraic equation can always be expres­sed in polar form and this form is useful for (…) problems.
   
4. 
   The function Ø (t) can assume only integral values and there­fore cannot pass continuously from the (...) O to the (...) n.
   
5. 
   Sometimes it is almost impossible to describe physical phe­nomena with absolute mathematical (...).
   
6. 
   A point can be (...) as an intersection of two or more lines.
   

loci minor

circle coincide

parabola eccentricity

ellipses conjugate

hyperbola obtainable

degenerate

CONICS

As was mentioned before, the curve that results when a pla­ne intersects a cone is called a conic section. There are, how­ever, situations in which the plane and the cone may intersect in a single point (plane passing through the vertex of the cone), a line, two intersecting lines, or no intersection at all bet­ween plane and cone.

THE ELLIPSE

An ellipse is defined as the set of points the sum of whose distances from two fixed points is constant. The two fixed points denoted by F and F' are called the foci. The distance F'F is de­noted by 2c and the constant sum by 2a. A simple equation of an ellipse is obtained by placing the foci at (-c,o) and (c,o).

An arbitrary point is denoted by P (x,y), then the equation of the ellipse is

+ =1

The points V' (-a,o) and V (a,o) are called the vertices; the line segment V'V- the major axis, and the line segment from (o,-b) to (o,b)-the minor-axis. The origin is called the centre C of the ellipse.

If we permitted F' and F to coincide, C would be zero and the ellipse would be a circle radius 2a/2=a. Thus a circle is sometimes regarded as an ellipse of eccentricity zero.
THE PARABOLA

A parabola is the locus of a point whose distance from a line varies as the square of its distance from a perpendicular line. If the lines are taken as axes of coordinates, the equation of the parabola is x² = ky or

y² = kx.

All parabolas are similar, for if the value of K in y² = kx is changed to λK, this is equivalent to the substitution of λx for x and λy for y. It only changes the scale of the graph.

The parabola y² =kx is symmetrical about the line y = 0, which is called the axis of the parabola.
THE HYPERBOLA

A hyperbola is the locus of a point that remains in the plane and moves so that the difference of the distances from two fixed points is constant. The two fixed points, denoted by F and F' are called the foci.

The origin is called the centre of the hyperbola, the points V and V' the vertices, the line segment V'V the transverse axis, and the line segment from (o,-b) to (o,b) the conjugate axis. The equation is

– =1

I. ANSWER THE QUESTIONS:

1. 
   What is the degree of Cartesian equations of the following lo­ci: circles, parabolas, ellipses, and hyperbolas?
   
2. 
   Does the term in xy appear in the equations of conics?
   
3. 
   What curve results when a plane intersects a cone?
   
4. 
   When does the plane intersect the cone in a single point?
   
5. 
   How can we define an ellipse?
   
6. 
   How are the foci in the ellipse denoted?
   
7. 
   What points of the ellipse are called the vertices?
   
8. 
   In what case does the ellipse become a circle?
   
9. 
   What are the properties of a parabola as a locus?
   
10. 
    How can you prove that all parabolas are similar?
    
11. 
    What are the properties of a point of a hyperbola?
    
12. 
    What axes of the hyperbola do you know?
    

comparable arbitrary

measurable customary

movable necessary

notable ordinary

observable primary

variable stationary

III. TRANSLATE THE FOLLOWING SENTENCES:

1. 
   It is found that the only non-degenerate curves of order 2 are the ellipse, parabola, and hyperbola; the circle is re­garded as a special case of the ellipse.
   
2. 
   Square is a rectangle whose sides are all equal.
   
3. 
   A cone is a surface generated by a line which passes through a fixed point, the vertex, and through the points of a fixed curve.
   
4. 
   A cross section of a prism is a section that is perpendicular to the edges of the prism.
   
5. 
   A curve of order 2 is called a conic section or conic because these curves were first obtained as plane sections of circu­lar cones.
   
6. 
   If a point moves only on a line or curve it has one degree
   of freedom.
   
7. 
   Any meridian plane cuts the surface in conic congruent to the
   generating conic, and any plane perpendicular to this axis of revolution cuts it in a circle.
   
8. 
   The graph of an equation in two variables x and y is simply the set of all points (x,y) in the plane whose coordinates satisfy the given equation.
   
9. 
   We can regard the irrational number as the limit of a sequence of rationale, chosen so that their squares are clo­ser and closer to 2.
   
10. 
    The graphs show clearly how the values of the function increa­se as x varies.
    
11. 
    The Euclidean and hyperbolic geometries, which differ widely in the large, coincide so closely for relatively small figures that they are experimentally equivalent.
    
12. 
    Conics are of three main types and we shall regard their equations which particularly simple on account of the special choice of axes.
    

13. 
    A fundamental mathematical concept is that of one-one, or (1,1) correspondence between two sets of objects.
    
14. 
    The ordinary curves which occur in elementally geometry such as straight lines, circles and conies, have much more "regu­larity" than is implied by mere continuity.
    
15. 
    We will assume that the pole coincides with the origin and the polar axis coincides with the origin and the positive X-axis.
    
16. 
    The applications of conies to space mechanics are often simplified when a polar equation of a conic is employed.
    
17. 
    The projection of a circle is an ellipse, and any two perpen­dicular diameters of the circle project into conjugate diame­ters of the ellipse.
    

1. 
   A (...) is the intersection of a sphere and a plane.
   
2. 
   A surface may be (...) by the motion of a line or a plane.
   
3. 
   There is much in common in the generation of a cylinder and a (...).
   
4. 
   The curve represented by the general algebraic equation of (...) n is called a curve of order n.
   
5. 
   If we (...) the base by b and the altitude by a, the area of rectangle will be determined by the formula A=ab.
   
6. 
   The equation of the second order represents (…) (...) – that is, curves formed by the intersection of a plane with right circular (...).
   
7. 
   The line x = 0 does not meet the hyperbola, it is called the (...) axis.
   
8. 
   The graph of an equation of the second (...) in x,y is a conic section that is the section of a (...).
   
9. 
   The development of physical intuition is (...) as one of the important functions of engineering analysis.
   

When a plane intersecting a cone is perpendicular to its axis … (we have a circle in the section).

1. 
   When a plane intersecting a cone is not perpendicular to its axis …
   
2. 
   When a plane intersecting a cone is parallel to a generating line ...
   
3. 
   When a plane intersecting a cone is parallel to its axis …
   
4. 
   When a plane intersects a cone ...
   
5. 
   When F' and F coincide ...
   

1. 
   The curve represented by the general algebraic equation of degree n is called a curve of order n.
   
2. 
   The tangents are usually found by the method of repeated roots.
   
3. 
   We rarely used polar equations in the general theory of algebraic curves, but we applied them for particular problems.
   
4. 
   Created by the human mind to count the objects in various assemblages numbers have no reference to the individual characteristics of the objects counted.
   
5. 
   Although the method of undetermined coefficients cannot always be used, it is usually the simplest.
   

1. 
   the ellipse
   
2. 
   the parabola
   
3. 
   the hyperbola
   

THE FOLIUM OF DESCARTES

folium asimptote

straight neither

touches substitute

homogeneous

The general equation of a straight line is Ax + By + C =0, and if we substitute for y in (1) we get a cubic equation in x indicating that there are 3 points of intersection. However, if the line touches the curve, or goes through its double point, 2 of the roots will coincide. Still more special is an inflection­al tangent which both touches and crosses a curve, in this case 3 roots coincide at the point of contact. We see later that the Folium has such a tangent. We may lose 2 roots if our line, be­sides going through a double point, touches the curve there. It is general result that, for a curve through the origin, the tangent(s) there come from the homogeneous group of terms of lowest degree in its equation. Here this group consists of the right- hand side of (1). So the axes x=0, y=0 touch the curve at the origin.

Some lines lose points of intersection with a curve in quite a different way. Thus the line x + y + c = 0 when combined with (1), leads to a quadratic. The vanishing of the cubic term indicates that, as c varies, these lines all go through a point at infinity on the curve. One of them, with c=a, when combined with (1) leads to the result a³ =0, indicating that all 3 roots are at infinity.

This line, shown dotted in Figure 1, is an asymptote: it touches the curve at infinity. It is in fact a rather special asymptote, for the curve has an inflection at the infinite point of contact. This accounts for the loss of 3 roots rather than 2, and for the fact that the curve comes in from infinity on the same side of the line at either end.

I. ANSWER THE QUESTIONS:

1. 
   Who discovered the curve of Figure 5?
   
2. 
   What will we get if we substitute for y in (1)?
   
3. 
   When will two of the roots coincide?
   
4. 
   In what case do 3 roots coincide at the point of contact?
   
5. 
   At what point do the axes x=0, y=0 touch the curve?
   
6. 
   What does the vanishing of the cubic term indicate?
   
7. 
   What do we call the dotted line shown in Figure 5?
   
8. 
   Where does the asymptote touch the curve?
   

central eccentricity

exceptional familiarity

general infinity

horizontal perpendicularity

transversal polarity

vertical similarity

III. TRANSLATE THE FOLLOWING SENTENCES:

1. 
   Frequently, special problems involve simple equations because
   some of the terms of the general energy equation vanish.
   
2. 
   When mathematician presents the results of his analysis, he
   should state the underlying assumptions clearly and explicit­ly.
   
3. 
   Using physical laws we write equations expressing those re­lations which must hold for our particular problem.
   
4. 
   It is evident that the notation y = f(x) is most appropriate in the case in which y is defined by an explicit formula in x.
   
5. 
   It is natural to ask whether every continuous curve has a de­finite tangent at every point.
   
6. 
   An algebraic equation of the n-th degree has n roots.
   
7. 
   Physically, a point may be represented by a dot made on paper by a pencil.
   
8. 
   A number a² is read "a square", the figure 2 indicates that a must be taken twice as a factor.
   
9. 
   The root sign, or the radical sign indicates that a root of a number is sought.
   
10. 
    All terms of a dimensionally homogeneous equation have the same dimensions.
    
11. 
    Points of inflection are determined by using the second derivative instead of the first.
    
12. 
    The plus or minus sign is to be chosen in each of the above equations according as the circles are to be externally or internally tangent.
    
13. 
    It will be found that in general a cubic curve with no double point does not possess rational parametric equation.
    

1. 
   Descartes discovered ...
   
2. 
   He studied …
   
3. 
   Neither x nor y ...
   
4. 
   We get a cubic equation if ...
   
5. 
   Two of the roots will coincide if …
   
6. 
   Three roots coincide when ...
   
7. 
   Some lines lose ...
   
8. 
   The vanishing of the cubic term …
   

1. 
   A vector whose end points coincide is denoted by the symbol O.
   
2. 
   Any function that can be represented by a finite number of the
   five basic algebraic operations is called an algebraic function.
   
3. 
   The arguments, the justification of which will be dealt with
   later on, are essential for our purpose.
   
4. 
   Such forms as a parallelogram, a rectangle, a square, a trapezoid have already been discussed - they are very important, as any figure bounded by straight lines may be thought of as composed of rectangles and triangles.
   
5. 
   The body is projected away from a point P with a known velo­city and is acted upon by a force which is proportional to the distance x of the body from P and directed toward P.
   
6. 
   The learner is led to some of the principal elementary ideas of group theory and is given an opportunity of becoming familiar with them by using them in the analysis of several notions which are important in the later development.
   
7. 
   The process of successively getting rid of unknowns (elimination), which can be applied to simultaneous linear equations, is usually taught in elementary algebra just for the case of ordinary numbers (rational or real) as coefficients.