_LINES AND CURVES._
Although the beginner will find that a study of geometry is not
essential to the production of such elementary examples of mechanical
drawing as are given in this book, yet as more difficult examples are
essayed he will find such a study to be of great advantage and
assistance. Meantime the following explanation of simple geometrical
terms is all that is necessary to an understanding of the examples
given.
The shortest distance between two points is termed the radius; and, in
the case of a circle, means the distance from the centre to the
perimeter measured in a straight line.
[Illustration: Fig. 38.]
[Illustration: Fig. 39.]
[Illustration: Fig. 40.]
Dotted lines, thus, <----->, mean the direction and the points at which a
dimension is taken or marked. Dotted lines, thus,-----, simply connect
the same parts or lines in different views of the object. Thus in
Figure 38 are a side and an end view of a rivet, and the dotted lines
show that the circles on the end view correspond to the circle of the
diameters of the head and of the stem, and therefore represent their
diameters while showing that both are round. A straight line is in
geometry termed a right line.
A line at a right angle to another is said to be perpendicular to it;
thus, in Figures 39, 40, and 41, lines A are in each case perpendicular
to line B, or line B is in each case perpendicular to line A.
A point is a position or location supposed to have no size, and in cases
where necessary is indicated by a dot.
[Illustration: Fig. 41.]
[Illustration: Fig. 42.]
[Illustration: Fig. 43.]
Parallel lines are those equidistant one from the other throughout their
length, as in Figure 42. Lines maybe parallel though not straight; thus,
in Figure 43, the lines are parallel.
[Illustration: Fig. 44.]
[Illustration: Fig. 45.]
[Illustration: Fig. 46.]
A line is said to be _produced_ when it is extended beyond its natural
limits: thus, in Figure 44, lines A and B are _produced_ in the point C.
A line is bisected when the centre of its length is marked: thus, line A
in Figure 45 is bisected, at or in, as it is termed, _e_.
The line bounding a circle is termed its circumference or periphery and
sometimes the perimeter.
A part of this circumference is termed an arc of a circle or an arc;
thus Figure 46 represents an arc. When this arc has breadth it is termed
a segment; thus Figures 47 and 48 are segments of a circle. A straight
line cutting off an arc is termed the chord of the arc; thus, in Figure
48, line A is the chord of the arc.
[Illustration: Fig. 47.]
[Illustration: Fig. 48.]
[Illustration: Fig. 49.]
[Illustration: Fig. 50.]
[Illustration: Fig. 51.]
A quadrant of a circle is one quarter of the same, being bounded on two
of its sides by two radial lines, as in Figure 49.
When the area of a circle that is enclosed within two radial lines is
either less or more than one quarter of the whole area of the circle the
figure is termed a sector; thus, in Figure 50, A and B are both sectors
of a circle.
A straight line touching the perimeter of a circle is said to be tangent
to that circle, and the point at which it touches is that to which it is
tangent; thus, in Figure 51, line A is tangent to the circle at point B.
The half of a circle is termed a semicircle; thus, in Figure 52, A B and
C are each a semicircle.
[Illustration: Fig. 52.]
[Illustration: Fig. 53.]
The point from which a circle or arc of a circle is drawn is termed its
centre. The line representing the centre of a cylinder is termed its
axis; thus, in Figure 53, dot _d_ represents the centre of the circle,
and line _b b_ the axial line of the cylinder.
To draw a circle that shall pass through any three given points: Let A B
and C in Figure 54 be the points through which the circumference of a
circle is to pass. Draw line D connecting A to C, and line E connecting
B to C. Bisect D in F and E in G. From F as a centre draw the semicircle
O, and from G as a centre draw the semicircle P; these two semicircles
meeting the two ends of the respective lines D E. From B as a centre
draw arc H, and from C the arc I, bisecting P in J. From A as a centre
draw arc K, and from C the arc L, bisecting the semicircle O in M. Draw
a line passing through M and F, and a line passing through J and Q, and
where these two lines intersect, as at Q, is the centre of a circle R
that will pass through all three of the points A B and C.
[Illustration: Fig. 54.]
[Illustration: Fig. 55.]
To find the centre from which an arc of a circle has been struck: Let A
A in Figure 55 be the arc whose centre is to be found. From the extreme
ends of the arc bisect it in B. From end A draw the arc C, and from B
the arc D. Then from the end A draw arc G, and from B the arc F. Draw
line H passing through the two points of intersections of arcs C D, and
line I passing through the two points of intersection of F G, and where
H and I meet, as at J, is the centre from which the arc was drawn.
A degree of a circle is the 1/360 part of its circumference. The whole
circumference is supposed to be divided into 360 equal divisions, which
are called the degrees of a circle; but, as one-half of the circle is
simply a repetition of the other half, it is not necessary for
mechanical purposes to deal with more than one-half, as is done in
Figure 56. As the whole circle contains 360 degrees, half of it will
contain one-half of that number, or 180; a quarter will contain 90, and
an eighth will contain 45 degrees. In the protractors (as the
instruments having the degrees of a circle marked on them are termed)
made for sale the edges of the half-circle are marked off into degrees
and half-degrees; but it is sufficient for the purpose of this
explanation to divide off one quarter by lines 10 degrees apart, and the
other by lines 5 degrees apart. The diameter of the circle obviously
makes no difference in the number of decrees contained in any portion of
it. Thus, in the quarter from 0 to 90, there are 90 degrees, as marked;
but suppose the diameter of the circle were that of inner circle _d_,
and one-quarter of it would still contain 90 degrees.
[Illustration: Fig. 56.]
So, likewise, the degrees of one line to another are not always taken
from one point, as from the point O, but from any one line to another.
Thus the line marked 120 is 60 degrees from line 180, or line 90 is 60
degrees from line 150. Similarly in the other quarter of the circle 60
degrees are marked. This may be explained further by stating that the
point O or zero may be situated at the point from which the degrees of
angle are to be taken. Here it may be remarked that, to save writing the
word "degrees," it is usual to place on the right and above the figures
a small deg., as is done in Figure 56, the 60 deg. meaning sixty degrees, the
deg., of course, standing for degrees.
[Illustration: Fig. 57.]
Suppose, then, we are given two lines, as _a_ and _b_ in Figure 57, and
are required to find their angle one to the other. Then, if we have a
protractor, we may apply it to the lines and see how many degrees of
angle they contain. This word "contain" means how many degrees of angle
there are between the lines, which, in the absence of a protractor, we
may find by prolonging the lines until they meet in a point as at _c_.
From this point as a centre we draw a circle D, passing through both
lines _a_, _b_. All we now have to do is to find what part, or how much
of the circumference, of the circle is enclosed within the two lines. In
the example we find it is the one-twelfth part; hence the lines are 30
degrees apart, for, as the whole circle contains 360, then one-twelfth
must contain 30, because 360/12 = 30.
[Illustration: Fig. 58.]
If we have three lines, as lines A B and C in Figure 58, we may find
their angles one to the other by projecting or prolonging the lines
until they meet as at points D, E, and F, and use these points as the
centres wherefrom to mark circles as G, H, and I. Then, from circle H,
we may, by dividing it, obtain the angle of A to B or of B to A. By
dividing circle I we may obtain the angle of A to C or of C to A, and by
dividing circle G we may obtain the angle of B to C or of C to B.
[Illustration: Fig. 59.]
It may happen, and, indeed, generally will do so, that the first attempt
will not succeed, because the distance between the lines measured, or
the arc of the circle, will not divide the circle without having the
last division either too long or too short, in which case the circle may
be divided as follows: The compasses set to its radius, or half its
diameter, will divide the circle into 6 equal divisions, and each of
these divisions will contain 60 degrees of angle, because 360 (the
number of degrees in the whole circle) /6 (the number of divisions) =
60, the number of degrees in each division. We may, therefore, subdivide
as many of the divisions as are necessary for the two lines whose
degrees of angle are to be found. Thus, in Figure 59, are two lines, C,
D, and it is required to find their angle one to the other. The circle
is divided into six divisions, marked respectively from 1 to 6, the
division being made from the intersection of line C with the circle. As
both lines fall within less than a division, we subdivide that division
as by arcs _a_, _b_, which divide it into three equal divisions, of
which the lines occupy one division. Hence, it is clear that they are at
an angle of 20 degrees, because twenty is one-third of sixty. When the
number of degrees of angle between two lines is less than 90, the lines
are said to form an acute angle one to the other, but when they are at
more than 90 degrees of angle they are said to form an obtuse angle.
Thus, in Figure 60, A and C are at an acute angle, while B and C are at
an obtuse angle. F and G form an acute angle one to the other, as also
do G and B, while H and A are at an obtuse angle. Between I and J there
are 90 degrees of angle; hence they form neither an acute nor an obtuse
angle, but what is termed a right-angle, or an angle of 90 degrees. E
and B are at an obtuse angle. Thus it will be perceived that it is the
amount of inclination of one line to another that determines its angle,
irrespective of the positions of the lines, with respect to the circle.
[Illustration: Fig. 60.]
TRIANGLES.
A right-angled triangle is one in which two of the sides are at a right
angle one to the other. Figure 61 represents a right-angled triangle, A
and B forming a right angle. The side opposite, as C, is called the
hypothenuse. The other sides, A and B, are called respectively the base
and the perpendicular.
[Illustration: Fig. 61.]
[Illustration: Fig. 62.]
[Illustration: Fig. 63.]
[Illustration: Fig. 64.]
An acute-angled triangle has all its angles acute, as in Figure 63.
An obtuse-angled triangle has one obtuse angle, as A, Figure 62.
When all the sides of a triangle are equal in length and the angles are
all equal, as in Figure 63, it is termed an equilateral triangle, and
either of its sides may be called the base. When two only of the sides
and two only of the angles are equal, as in Figure 64, it is termed an
isosceles triangle, and the side that is unequal, as A in the figure, is
termed the base.
[Illustration: Fig. 65.]
[Illustration: Fig. 66.]
When all the sides and angles are unequal, as in Figure 65, it is termed
a scalene triangle, and either of its sides may be called the base.
The angle opposite the base of a triangle is called the vertex.
[Illustration: Fig. 67.]
[Illustration: Fig. 68.]
A figure that is bounded by four straight lines is termed a quadrangle,
quadrilateral or tetragon. When opposite sides of the figure are
parallel to each other it is termed a parallelogram, no matter what the
angle of the adjoining lines in the figure may be. When all the angles
are right angles, as in Figure 66, the figure is called a rectangle. If
the sides of a rectangle are of equal length, as in Figure 67, the
figure is called a square. If two of the parallel sides of a rectangle
are longer than the other two sides, as in Figure 66, it is called an
oblong. If the length of the sides of a parallelogram are all equal and
the angles are not right angles, as in Figure 68, it is called a rhomb,
rhombus or diamond. If two of the parallel sides of a parallelogram are
longer than the other two, and the angles are not right angles, as in
Figure 69, it is called a rhomboid. If two of the parallel sides of a
quadrilateral are of unequal lengths and the angles of the other two
sides are not equal, as in Figure 70, it is termed a trapezoid.
[Illustration: Fig. 69.]
[Illustration: Fig. 70.]
[Illustration: Fig. 71.]
If none of the sides of a quadrangle are parallel, as in Figure 71, it
is termed a trapezium.
THE CONSTRUCTION OF POLYGONS.
[Illustration: Fig. 71_a_.]
[Illustration: Fig. 72.]
The term polygon is applied to figures having flat sides equidistant
from a common centre. From this centre a circle may be struck that will
touch all the corners of the sides of the polygon, or the point of each
side that is central in the length of the side. In drawing a polygon,
one of these circles is used upon which to divide the figure into the
requisite number of divisions for the sides. When the dimension of the
polygon across its corners is given, the circle drawn to that dimension
circumscribes the polygon, because the circle is without or outside of
the polygon and touches it at its corners only. When the dimension
across the flats of the polygon is given, or when the dimension given is
that of a circle that can be inscribed or marked within the polygon,
touching its sides but not passing through them, then the polygon
circumscribes or envelops the circle, and the circle is inscribed or
marked within the polygon. Thus, in Figure 71 _a_, the circle is
inscribed within the polygon, while in Figure 72 the polygon is
circumscribed by the circle; the first is therefore a circumscribed and
the second an inscribed polygon. A regular polygon is one the sides of
which are all of an equal length.
NAMES OF REGULAR POLYGONS.
A figure of 3 sides is called a Trigon.
" 4 " " Tetragon.
polygon 5 " " Pentagon.
" 6 " " Hexagon.
" 7 " " Heptagon.
" 8 " " Octagon.
" 9 " " Enneagon or Nonagon.
[Illustration: Fig. 73.]
[Illustration: Fig. 74.]
The angles of regular polygons are designated by their degrees of angle,
"at the centre" and "at the circumference." By the angle at the centre
is meant the angle of a side to a radial line; thus in Figure 73 is a
hexagon, and at C is a radial line; thus the angle of the side D to C is
60 degrees. Or if at the two ends of a side, as A, two radial lines be
drawn, as B, C, then the angles of these two lines, one to the other,
will be the "angle at the centre." The angle at the circumference is the
angle of one side to its next neighbor; thus the angle at the
circumference in a hexagon is 120 degrees, as shown in the figure for
the sides E, F. It is obvious that as all the sides are of equal length,
they are all at the same angle both to the centre and to one another. In
Figure 74 is a trigon, the angles at its centre being 120, and the angle
at the circumference being 60, as marked.
The angles of regular polygons:
Trigon, at the centre, 120 deg., at the circumference, 60 deg..
Tetragon, " 90 deg., " " 90 deg..
Pentagon, " 72 deg., " " 108 deg..
Hexagon, " 60 deg., " " 120 deg..
Octagon, " 45 deg., " " 135 deg..
Enneagon, " 40 deg., " " 140 deg..
Decagon, " 36 deg., " " 144 deg..
Dodecagon, " 30 deg., " " 150 deg..
THE ELLIPSE.
An ellipse is a figure bounded by a continuous curve, whose nature will
be shown presently.
The dimensions of an ellipse are taken at its extreme length and
narrowest width, and they are designated in three ways, as by the length
and breadth, by the major and minor axis (the major axis meaning the
length, and the minor the breadth of the figure), and the conjugate and
transverse diameters, the transverse meaning the shortest, and the
conjugate the longest diameter of the figure.
In this book the terms major and minor axis will be used to designate
the dimensions.
The minor and major axes are at a right angle one to the other, and
their point of intersection is termed the axis of the ellipse.
In an ellipse there are two points situated upon the line representing
the major axis, and which are termed the foci when both are spoken of,
and a focus when one only is referred to, foci simply being the plural
of focus. These foci are equidistant from the centre of the ellipse,
which is formed as follows: Two pins are driven in on the major axis to
represent the foci A and B, Figure 75, and around these pins a loop of
fine twine is passed; a pencil point, C, is then placed in the loop and
pulled outwards, to take up the slack of the twine. The pencil is held
vertical and moved around, tracing an ellipse as shown.
[Illustration: Fig. 75.]
Now it is obvious, from this method of construction, that there will be
at every point in the pencil's path a length of twine from the final
point to each of the foci, and a length from one foci to the other, and
the length of twine in the loop remaining constant, it is demonstrated
that if in a true ellipse we take any number of points in its curve, and
for each point add together its distance to each focus, and to this add
the distance apart of the foci, the total sum obtained will be the same
for each point taken.
[Illustration: Fig. 76.]
[Illustration: Fig. 77.]
In Figures 76 and 77 are a series of ellipses marked with pins and a
piece of twine, as already described. The corresponding ellipses, as A
in both figures, were marked with the same loop, the difference in the
two forms being due to the difference in distance apart of the foci.
Again, the same loop was used for ellipses B in both figures, as also
for C and D. From these figures we perceive that--
1st. With a given width or distance apart of foci, the larger the
dimensions are the nearer the form of the figure will approach to that
of a circle.
2d. The nearer the foci are together in an ellipse, having any given
dimensions, the nearer the form of the figure will approach that of a
circle.
3d. That the proportion of length to width in an ellipse is determined
by the distance apart of the foci.
4th. That the area enclosed within an ellipse of a given circumference
is greater in proportion as the distance apart of the foci is
diminished; and,
5th. That an ellipse may be given any required proportion of width to
length by locating the foci at the requisite distance apart.
The form of a true ellipse may be very nearly approached by means of the
arcs of circles, if the centres from which those arcs are struck are
located in the most desirable positions for the form of ellipse to be
drawn.
[Illustration: Fig. 78.]
Thus in Figure 78 are three ellipses whose forms were pencilled in by
means of pins and a loop of twine, as already described, but which were
inked in by finding four arcs of circles of a radius that would most
closely approach the pencilled line; _a b_ are the foci of all three
ellipses A, B, and C; the centre for the end curves of _a_ are at _c_
and _d_, and those for its side arcs are at _e_ and _f_. For B the end
centres are at _g_ and _h_, and the side centres at _i_ and _j_. For C
the end centres are at _k_, _l_, and the side centres at _m_ and _n_.
It will be noted that, first, all the centres for the end curves fall on
the line of the length or major axis, while all those for the sides fall
on the line of width or the minor axis; and, second, that as the
dimensions of the ellipses increase, the centres for the arcs fall
nearer to the axis of the ellipse. Now in proportion as a greater number
of arcs of circles are employed to form the figure, the nearer it will
approach the form of a true ellipse; but in practice it is not usual to
employ more than eight, while it is obvious that not less than four can
be used. When four are used they will always fall somewhere on the lines
on the major and minor axis; but if eight are used, two will fall on the
line of the major axis, two on the line of the minor axis, and the
remaining four elsewhere.
[Illustration: Fig. 79.]
In Figure 79 is a construction wherein four arcs are used. Draw the line
_a b_, the major axis, and at a right angle to it the line _c d_, the
minor axis of the figure. Now find the difference between the length of
half the two axes as shown below the figure, the length of line _f_
(from _g_ to _i_) representing half the length of the figure (as from
_a_ to _e_), and the length or radius from _g_ to _h_ equalling that
from _e_ to _d_; hence from _h_ to _i_ is the difference between half
the major and half the minor axis. With the radius (_h i_), mark from
_e_ as a centre the arcs _j k_, and join _j k_ by line _l_. Take half
the length of line _l_ and from _j_ as a centre mark a line on _a_ to
the arc _m_. Now the radius of _m_ from _e_ will be the radius of all
the centres from which to draw the figure; hence we may draw in the
circle _m_ and draw line _s_, cutting the circle. Then draw line _o_,
passing through _m_, and giving the centre _p_. From _p_ we draw the
line _q_, cutting the intersection of the circle with line _a_ and
giving the centre _r_. From _r_ we draw line _s_, meeting the circle and
the line _c, d_, giving us the centre _t_. From _t_ we draw line _u_,
passing through the centre _m_. These four lines _o_, _q_, _s_, _u_ are
prolonged past the centres, because they define what part of the curve
is to be drawn from each centre: thus from centre _m_ the curve from _v_
to _w_ is drawn, from centre _t_ the curve from _w_ to _x_ is drawn.
From centre _r_ the curve from _x_ to _y_ is drawn, and from centre _p_
the curve from _y_ to _v_ is drawn. It is to be noted, however, that
after the point _m_ is found, the remaining lines may be drawn very
quickly, because the line _o_ from _m_ to _p_ may be drawn with the
triangle of 45 degrees resting on the square blade. The triangle may be
turned over, set to point _p_ and line _q_ drawn, and by turning the
triangle again the line _s_ may be drawn from point _r_; finally the
triangle may be again turned over and line _u_ drawn, which renders the
drawing of the circle _m_ unnecessary.
To draw an elliptical figure whose proportion of width to breadth shall
remain the same, whatever the length of the major axis may be: Take any
square figure and bisect it by the line A in Figure 80. Draw, in each
half of the square, the diagonals E F, G H. From P as a centre with the
radius P R draw the arc S E R. With the same radius draw from O as a
centre the arc T D V. With radius L C draw arc R C V, and from K as a
centre draw arc S B T.
[Illustration: Fig. 80.]
[Illustration: Fig. 81.]
A very near approach to the true form of a true ellipse may be drawn by
the construction given in Figure 81, in which A A and B B are centre
lines passing through the major and minor axis of the ellipse, of which
_a_ is the axis or centre, _b c_ is the major axis, and _a e_ half the
minor axis. Draw the rectangle _b f g c_, and then the diagonal line _b
e_; at a right angle to _b e_ draw line _f h_, cutting B B at _i_. With
radius _a e_ and from _a_ as a centre draw the dotted arc _e j_, giving
the point _j_ on line B B. From centre _k_, which is on the line B B and
central between _b_ and _j_, draw the semicircle _b m j_, cutting A A at
_l_. Draw the radius of the semicircle _b m j_, cutting it at _m_, and
cutting _f g_ at _n_. With the radius _m n_ mark on A A at and from _a_
as a centre the point _o_. With radius _h o_ and from centre _h_ draw
the arc _p o q_. With radius _a l_ and from _b_ and _c_ as centres, draw
arcs cutting _p o q_ at the points _p q_. Draw the lines _h p r_ and _h
q s_ and also the lines _p i t_ and _q v w_. From _h_ as a centre draw
that part of the ellipse lying between _r_ and _s_, with radius _p r_;
from _p_ as a centre draw that part of the ellipse lying between _r_ and
_t_, with radius _q s_, and from _q_ as a centre draw the ellipse from
_s_ to _w_, with radius _i t_; and from _i_ as a centre draw the ellipse
from _t_ to _b_ and with radius _v w_, and from _v_ as a centre draw the
ellipse from _w_ to _c_, and one-half of the ellipse will be drawn. It
will be seen that the whole construction has been performed to find the
centres _h_, _p_, _q_, _i_ and _v_, and that while _v_ and _i_ may be
used to carry the curve around on the other side of the ellipse, new
centres must be provided for _h_ _p_ and _q_, these new centres
corresponding in position to _h_ _p_ _q_. Divesting the drawing of all
the lines except those determining its dimensions and the centres from
which the ellipse is struck, we have in Figure 82 the same ellipse drawn
half as large. The centres _v_, _p_, _q_, _h_ correspond to the same
centres in Figure 81, while _v'_, _p'_, _q'_, _h'_ are in corresponding
positions to draw in the other half of the ellipse. The length of curve
drawn from each centre is denoted by the dotted lines radiating from
that centre; thus, from _h_ the part from _r_ to _s_ is drawn; from _h'_
that part from _r'_ to _s'_. At the ends the respective centres _v_ are
used for the parts from _w_ to _w'_ and from _t_ to _t'_ respectively.
[Illustration: Fig. 82.]
[Illustration: Fig. 83.]
The most correct method of drawing an ellipse is by means of an
instrument termed a trammel, which is shown in Figure 83. It consists of
a cross frame in which are two grooves, represented by the broad black
lines, one of which is at a right angle to the other. In these grooves
are closely fitted two sliding blocks, carrying pivots E F, which may be
fastened to the sliding blocks, while leaving them free to slide in the
grooves at any adjusted distance apart. These blocks carry an arm or rod
having a tracing point (as pen or pencil) at G. When this arm is swept
around by the operator, the blocks slide in the grooves and the
pen-point describes an ellipse whose proportion of width to length is
determined by the distance apart of the sliding blocks, and whose
dimensions are determined by the distance of the pen-point from the
sliding block. To set the instrument, draw lines representing the major
and minor axes of the required ellipse, and set off on these lines
(equidistant from their intersection), to mark the required length and
width of ellipse. Place the trammel so that the centre of its slots is
directly over the point or centre from which the axes are marked (which
may be done by setting the centres of the slots true to the lines
passing through the axis) and set the pivots as follows: Place the
pencil-point G so that it coincides with one of the points as C, and
place the pivot E so that it comes directly at the point of intersection
of the two slots, and fasten it there. Then turn the arm so that the
pencil-point G coincides with one of the points of the minor axis as D,
the arm lying parallel to B D, and place the pivot F over the centre of
the trammel and fasten it there, and the setting is complete.
[Illustration: Fig. 84.]
To draw a parabola mechanically: In Figure 84 C D is the width and H J
the height of the curve. Bisect H D in K. Draw the diagonal line J K
and draw K E, cutting K at a right angle to J K, and produce it in E.
With the radius H E, and from J as a centre, mark point F, which will be
the focus of the curve. At any convenient distance above J fasten a
straight-edge A B, setting it parallel to the base C D of the parabola.
Place a square S with its back against the straight-edge, setting the
edge O N coincident with the line J H. Place a pin in the focus F, and
tie to it one end of a piece of twine. Place a tracing-point at J, pass
the twine around the tracing-point, bringing down along the square-blade
and fasten it at N, with the tracing-point kept against the edge of the
square and the twine kept taut; slide the square along the
straight-edge, and the tracing-point will mark the half J C of the
parabola. Turn the square over and repeat the operation to trace the
other half J D. This method corresponds to the method of drawing an
ellipse by the twine and pins, as already described.
[Illustration: Fig. 85.]
To draw a parabola by lines: Bisect the width A B in Figure 85, and
divide each half into any convenient number of equal divisions; and
through these points of division draw vertical lines, as 1, 2, 3, etc.
(in each half). Divide the height A D at one end and B E at the other
into as many equal divisions as the half of A B is divided into. From
the points of divisions 1, 2, 3, etc., on lines A D and B E, draw lines
pointing to C, and where these lines intersect the corresponding
vertical lines are points through which the curve may be drawn. Thus on
the side A D of the curve, the intersection of the two lines marked 1 is
a point in the curve; the intersection of the two lines marked 2 is
another point in the curve, and so on.
TO DRAW A HEART CAM.
[Illustration: Fig. 86.]
Draw the line A B, Figure 86, equal to the length of stroke required.
Divide it into any number of equal parts, and from C as a centre draw
circles through the points of division. Draw the outer circle and divide
its circumference into twice as many equal divisions as the line A B was
divided into. Draw radial lines from each point of division on the
circle, and the points of intersection of the radial lines with the
circles are points for the outline of the cam, and through these points
a curved line may be drawn giving the shape of the cam. It is obvious
that the greater the number of divisions on A B, the more points and the
more perfect the curve may be drawn.