First. A square, Fig. 120. On the ends _B A_ as centers, with the line
_A B_ as radius, describe arcs cutting at _C_; on _C_ describe arcs
cutting the others at _D E_; and on _D_ and _E_ cut these at _F G_. Draw
_A F_, _B G_ and join the intersections _H I_.
Second. A rectangle, Fig. 121. On the base _E F_ draw the perpendiculars
_E H_, _F G_, equal to the height of the rectangle, and join _G H_.
[Illustration: Fig. 122.]
EX. 17.--_To construct a parallelogram of which the sides and one of the
angles are given_, Fig. 122. Draw the side _D E_ equal to the given
length _A_, and set off the other side _D F_ equal to the other length
_B_, forming the given angle _C_. From _E_ with _D F_ as radius,
describe an arc, and from _F_, with the radius _D E_ cut the arc at _G_.
Draw _F G_, _E G_. Or, the remaining sides may be drawn as parallels to
_D E_, _D F_.
[Illustration: Fig. 123.]
EX. 18.--_To describe a circle about a triangle_, Fig. 123. Bisect two
sides _A B_, _A C_ of the triangle at _E F_, and from these points draw
perpendiculars cutting at _K_. On the center _K_, with the radius _K A_
draw the circle _A B C_.
[Illustration: Fig. 124.]
EX. 19.--_To describe a circle about a square, and to inscribe a square
in a circle_, Fig. 124.
First. To describe the circle. Draw the diagonals _A B_, _C D_ of the
square, cutting at _E_; on the center _E_ with the radius _E A_ describe
the circle.
Second. To inscribe the square. Draw the two diameters _A B_, _C D_ at
right angles and join the points _A B_, _C D_ to form the square.
_In the same way a circle may be described about a triangle._
[Illustration: Fig. 125.]
EX. 20.--_To inscribe a circle on a square, and to describe a square
about a circle_, Fig. 125.
First. To inscribe the circle. Draw the diagonals _A B_, _C D_ of the
square, cutting at _E_; draw the perpendicular _E F_ to one side, and
with the radius _E F_ describe the circle.
Second. To describe the square. Draw two diameters _A B_, _C D_ at right
angles, and produce them; bisect the angle _D E B_ at the center by the
diameter _F G_, and through _F_ and _G_ draw perpendiculars _A C_, _B
D_, and join the points _A D_ and _B C_ where they cut the diagonals to
complete the square.
[Illustration: Fig. 126.]
EX. 21.--_To inscribe a circle in a triangle_, Fig. 126. Bisect two of
the angles _A C_ of the triangle by lines cutting at _D_; from _D_ draw
a perpendicular _D E_ to any side, and with _D E_ as radius describe a
circle.
[Illustration: Fig. 127.]
EX. 22.--_To inscribe a pentagon in a circle_, Fig. 127. Draw two
diameters _A C_, _B D_ at right angles cutting at _O_; bisect _A O_ at
_E_, and from _B_ with radius _B E_ cut the circumference at _G H_ and
with the same radius step round the circle to _I_ and _K_; join the
points to form the pentagon.
[Illustration: Fig. 128.]
EX. 23.--_To construct a hexagon upon a given straight line_, Fig. 128.
From _A_ and _B_, the ends of the given line, describe arcs cutting at
_G_; from _G_ with the radius _G A_ describe a circle. With the same
radius set off the arcs _A C_, _C F_ and _B D_, _D E_; join the points
so found to form the hexagon.
[Illustration: Fig. 129.]
EX. 24.--_To inscribe a hexagon in a circle_, Fig. 129. Draw a diameter
_A C B_; from _A_ and _B_ as centers, with the radius of the circle _A
C_ cut the circumference at _D_, _E_, _F_, _G_, and draw _A D_, _D E_,
etc., to form the hexagon. The points _D E_, etc., may be found by
stepping the radius (with the dividers) six times round the circle.
[Illustration: Fig. 130.]
EX. 25.--_To describe an octagon on a given straight line_, Fig. 130.
Produce the given line _A B_ both ways and draw perpendiculars _A E_, _B
F_; bisect the external angles _A_ and _B_ by the lines _A H_, _B C_,
which make equal to _A B_. Draw _C D_ and _H G_ parallel to _A E_ and
equal to _A B_; from the center _G D_, with the radius _A B_, cut the
perpendiculars at _E F_, and draw _E F_ to complete the hexagon.
[Illustration: Fig. 131.]
EX. 26.--_To convert a square into an octagon_, Fig. 131.--Draw the
diagonals of the square cutting at _E_; from the corners _A_, _B_, _C_,
_D_, with _A E_ as radius, describe arcs cutting the sides at _G_, _H_,
etc., and join the points so found to complete the octagon.
[Illustration: Fig. 132.]
EX. 27.--_To inscribe an octagon in a circle_, Fig. 132. Draw two
diameters _A C_, _B D_, at right angles; bisect the arcs _A B_, _B C_,
at _E_, _F_, etc., to form the octagon.
[Illustration: Fig. 133.]
EX. 28.--_To describe an octagon about a circle_, Fig. 133. Describe a
square about the given circle _A B_, draw perpendiculars _H_ and _K_, to
the diagonals, touching the circle to form the octagon. Or, the points
_H_, _K_, etc., may be found by cutting the sides from the corners, by
lines parallel to the diagonals.
[Illustration: Fig. 134.]
EX. 29.--_To describe an ellipse when the length and breadth are given_,
Fig. 134. On the center _C_, with _A E_ as radius, cut the axis _A B_ at
_F_ and _G_, the foci, fix a couple of pins into the axis at _F_ and
_G_, and loop on a thread or cord upon them equal in length to the axis
_A B_, so as when stretched to reach the extremity _C_ of the conjugate
axis, as shown in dot-lining. Place a pencil or drawpoint inside the
cord, as at _H_, and guiding the pencil in this way, keeping the cord
equally in tension, carry the pencil round the pins _F_, _G_, and so
describe the ellipse.
NOTE.--The ellipse is an oval figure, like a circle in perspective.
The line that divides it equally in the direction of its great length
is the _transverse axis_, and the line which divides the opposite way
is the _conjugate axis_.
_Second Method._ Along the straight edge of a piece of stiff paper mark
off a distance _a c_ equal to _A C_, half the transverse axis; and from
the same point a distance _a b_ equal to _C D_, half the conjugate axis.
Place the slip so as to bring the point _b_ on the line _A B_ of the
transverse axis, and the point _c_ on the line _D E_; and set off on the
drawing the position of the point _a_. Shifting the slip, so that the
point travels on the transverse axis, and the point _c_ on the conjugate
axis, any number of points in the curve may be found, through which the
curve may be traced. See fig. 135.
[Illustration: Fig. 135.]
Trigonometry.
Trigonometry is that portion of geometry which has for its object the
measurement of triangles. When it treats of plane triangles, it is
called _Plane Trigonometry_; and as the engineer will continually meet
in his studies of higher mathematics _the terms_ used in plane
trigonometry, it is advantageous for him to become familiar with some of
the principles and definitions relating to this branch of mathematics.
The circumferences of all circles contain the same number of degrees,
but the greater the radius the greater is the absolute measures of a
degree. The circumference of a fly wheel or the circumference of the
earth have the same number of degrees; yet the same number of degrees in
each and every circumference is the measure of precisely the same angle.
The circumference of a circle is supposed to be divided into 360 degrees
or divisions, and as the total angularity about the center is equal to
four right angles, each right angle contains 90 degrees, or 90°, and
half a right angle contains 45°. Each degree is divided into 60 minutes,
or 60′; and for the sake of still further minuteness of measurement,
each minute is divided into 60″. In a whole circle there are, therefore,
360 × 60 × 60 = 1,296,000 seconds. The annexed diagram, fig. 136,
exemplifies the relative positions of the
Sine,
Co-sine,
Versed Sine,
Tangent,
Co-Tangent,
Secant and
Co-secant
of an angle.
[Illustration: Fig. 136.]
These may be defined thus: