Part 4

When a series of parallels are required perpendicular to a base line _A B_, they may be drawn as in fig. 107 through points in the base line set off at the required distances apart. This method is convenient also where a succession of parallels are required to a given line _C D_, for the perpendicular may be drawn to it, and any number of parallels may be drawn on the perpendicular.

EX. 6.--_To divide a line into a number of equal parts_, Fig. 108.

To divide the line _A B_ into, say, five parts. From _A_ and _B_ draw parallels _A C_, _B D_ on opposite sides; set off any convenient distance four times (one less than the given number), from _A_ on _A C_, and on _B_ on _B D_; join the first on _A C_ to the fourth on _B D_, and so on. The lines so drawn divide _A B_ as required.

_Second Method_, Fig. 109. Draw the line at _A C_, at an angle from _A_, set off, say, five equal parts; draw _B_ 5, and draw parallels to it from the other points of division in _A C_. These parallels divide _A B_ as required.

EX. 7.--_Upon a straight line to draw an angle equal to a given angle_, Fig. 110. Let _A_ be the given angle and _F G_ the line. With any radius from the points _A_ and _F_, describe arcs _D E_, _I H_, cutting the sides of the angle _A_ and the line _F G_.

Set off the arc _I H_, equal to _D E_ and draw _F H_. The angle _F_ is equal to _A_ as required.

EX. 8.--_To bisect an angle_, Fig. 111. Let _A C B_ be the angle; on the center _C_ cut the sides at _A B_. On _A_ and _B_ as centers describe arcs cutting at _D_ dividing the angle into two equal parts.

EX. 9.--_To find the center of a circle or of an arc of a circle._ Fig. 112. Draw the chord _A B_, bisect it by the perpendicular _C D_, bounded both ways by the circle; and bisect _C D_ for the center _G_.

EX. 10.--_Through two given points to describe an arc of a circle with a given radius_, Fig. 113. On the points _A_ and _B_ as centers, with the given radius, describe arcs cutting at _C_; and from _C_, with the same radius, describe an arc _A B_ as required.

Second, for a circle or an arc, Fig. 114. Select three points _A_, _B_, _C_ in the circumference, well apart; with the same radius describe arcs from these three points cutting each other, and draw two lines _D E_, _F G_, through their intersections according to Fig. 107. The point where they cut is the center of the circle or arc.

EX. 11.--_To describe a circle passing through three given points_, Fig. 114. Let _A_, _B_, _C_ be the given points and proceed as in last problem to find the center _O_, from which the circle may be described.

This problem is variously useful; in finding the diameter of a large fly-wheel, or any other object of large diameter when only a part of the circumference is accessible; in striking out arches when the span and rise are given, etc.

EX. 12.--_To draw a tangent to a circle from a given point in the circumference_, Fig. 115. From _A_ set off equal segments _A B_, _A D_, join _B D_ and draw _A E_, parallel to it, for the tangent.

EX. 13.--_To draw tangents to a circle from points without it_, Fig. 116. From _A_ with the radius _A C_ describe an arc _B C D_, and from _C_ with a radius equal to the diameter of the circle, cut the arc at _B D_, join _B C_, _C D_, cutting the circle at _E F_, and draw _A E_, _A F_, the tangents.

EX. 14.--_Between two inclined lines to draw a series of circles touching these lines and touching each other_, Fig. 117. Bisect the inclination of the given lines _A B_, _C D_ by the line _N O_. From a point _P_ in this line draw the perpendicular _P B_ to the line _A B_, and on _P_ describe the circle _B D_, touching the lines and cutting the center lines at _E_. From _E_ draw _E F_ perpendicular to the center line, cutting _A B_ at _F_, and from _F_ describe an arc _E G_, cutting _A B_ at _G_. Draw _G H_ parallel to _B P_, giving _H_, the center of the next circle, to be described with the radius _H E_, and so on for the next circle, _I N_.

EX. 15.--_To construct a triangle on a given base, the sides being given._

First. An equilateral triangle, Fig. 118. On the ends of a given base _A B_, with _A B_ as a radius describe arcs cutting at _C_, and draw _A C_, _C B_.

Second. Triangle of unequal sides, Fig. 119. On either end of the base _A D_, with the side _B_ as a radius describe an arc; and with the side _C_ as a radius, on the other end of the base as a center, describe arcs cutting the arc at _E_; join _A E_, _D E_.

This construction may be used for finding the position of a point _C_ or _E_ at given distances from the ends of a base, not necessarily to form a triangle.

EX. 16.--_To construct a square rectangle on a given straight line._

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_.

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_.

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_.

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._

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

These may be defined thus:

DEFINITIONS.

1. The _Complement_ of an arc is 90° minus the arc.

2. The _Supplement_ of an arc is 180° minus the arc.

3. The _Sine_ of an angle, or of an arc, is a line drawn from one end of an arc, perpendicular to a diameter drawn through the other end.

4. The _Cosine_ of an arc is the perpendicular distance from the center of the circle to the sine of the arc; or, it is the same in magnitude as the sine of the complement of the arc.

5. The _Tangent_ of an arc is a line touching the circle in one extremity of the arc, and continued from thence, to meet a line drawn through the center and the other extremity.

6. The _Cotangent_ of an arc is the tangent of the complement of the arc. The _Co_ is but a contraction of the word complement.

7. The _Secant_ of an arc is a line drawn from the center of the circle to the extremity of the tangent.

8. The _Cosecant_ of an arc is the secant of the complement.

9. The _Versed Sine_ of an arc is the distance from the extremity of the arc to the foot of the sine.

For the sake of brevity, these technical terms are contracted thus: for sine _A B_, we write _sin. A B_; for cosine _A B_, we write _cos. A B_; for tangent _A B_, we write _tan. A B_, etc.

Drawing Materials and Instruments.

Drawing tools or instruments are contrived solely for mechanical drawing; aside from this use they are perfectly worthless, hence the quality of these special utensils is a matter of the first consideration to the earnest student.

There are several degrees of excellence to be found in the make-up of drawing instruments and materials; it may be remarked with truth that “any kind are good enough, and the best none too good,” i. e., a learner in this delightful art should not stop at the lack of goodness or the low grade existing in his “tools,” but rather do the best work possible with the means at hand.

However, in order that acceptable work may be accomplished, fairly good instruments should be procured. The advice of some one experienced in the use and care of draughting tools should be sought before purchasing. A drawing board, a single sheet of paper and a pencil is the simplest “outfit” to be thought of; to this small beginning may be added, soon afterwards, an inexpensive pair of compasses, a T-square and a couple of triangles; a vast range of work can be executed with these few tools.

Nothing else will be needed to do fine work except, perhaps, one or two pairs of better compasses and a few sweeps or means of drawing irregular curves; all these had best be purchased separately; for in buying a “box of instruments,” it may contain some articles which are not desired, or that are of a wrong size, or even duplicates of those already possessed.

An outfit recommended by the author of “Reed’s Hand Book” is as follows.

Large compasses with movable leg.

A pair of dividers.

Bow pencil.

Bow pen.

Pencil leg for large compasses.

Pen leg for large compasses.

Drawing pen.

Louis Rouillon, B. S., Instructor of Drawing in Pratt Institute, New York, recommends the following:

Compasses, 5¹⁄₂ inches, with needle point; pen pencil and lengthening bar.

Drawing pen, 4¹⁄₂ inches.

T-square, 24-inch blade.

45-degree triangle, 9 inches.

30 and 60 degree triangle, 9 inches.

1 Scroll.

Dixon’s V. H. pencil.

12-inch boxwood scale, flat, graduated ¹⁄₁₆ inch the entire length.

Bottle of liquid India ink, four thumb-tacks, pen and ink eraser.

20 sheets drawing paper, 11 × 15 inches, and a drawing-board about 16 × 23 inches will also be necessary; students can usually make the board themselves for less money than it can be bought.

NOTE.--The purchasing of drawing tools is one of the most difficult points to settle that can present itself to a person about to buy a drawing outfit for the first time. Nothing can be so productive of distress to a person drawing as to have his tools getting out of order, joints one day too tight, next day too slack, points getting blunt or perhaps turning up altogether; if needle points, then the needles slip up, and drawing spoiled; in fact, the purchaser can be annoyed in numberless different ways.--W. H. THORN.

THE DRAWING BOARD.

A drawing-board should be made of well seasoned pine of a convenient size, say 23 × 16, which will take half a sheet of imperial paper, leaving ¹⁄₂-inch margin all around.

The working surface of the board--or its front side--should be perfectly smooth, but instead of being flat it should have a very slight camber, or rounding, breadthways, this latter feature in its construction being to prevent the possibility of a sheet of paper when stretched on its surface having any vacuity beneath it.

The _four edges_ of the board need not form an exact rectangle, as much valuable time is often wasted in the attempts to produce such a board; but it will answer every purpose of the draughtsman so long as the adjacent edges at the lower left-hand corner of it are at right angles, or square to each other.

An English authority recommends the use of two drawing-boards, 42 inches long and 30 inches wide, made of plain stuff, without cleets, 1¹⁄₄ inches thick--seasoned--with edges perfectly straight and at right angles to each other. _With two boards, one may be used for sketching and drawing details and the other for the finished drawing._

The board should be ³⁄₄ inch in thickness, and fitted at the back, at right angles to its longest side, with a couple of hardwood battens, about 2 inches wide and ³⁄₄ inch thick; the use of these battens being to keep the board from casting or winding and to allow of its expansion or contraction through changes of temperature. This latter purpose, however, is only effected by attaching the battens to the back of the board in the following manner: ... At the middle of the length of each batten--which should be one inch less than the width of the board--a stout, well-fitted wood screw is firmly inserted into it, and made to penetrate the board for about ¹⁄₂ inch, the head of the screw being made flush with the surface of the batten; on either side of the central screw, two others, about 3¹⁄₂ inches apart, are passed through oblong holes in the battens, and screwed into the body of the board until their heads are flush with the central one; fitted in this way the board itself can expand or contract lengthwise or crosswise, while its surface is prevented from warping or bending.

A further improvement in such a drawing board as above shown is made by cutting lengthwise along its ends a narrow groove and inserting an ebony or hardwood strip; this is cut or sawn apart at about every inch to admit of contraction; this strip serves as a guide to the stock of the drawing square, allowing an easy sliding movement.

To produce really good work in the shape of a mechanical drawing, one perfect straight edge only is required on a drawing board, and that the left one, which is always known as the _working-edge_; but for the convenience of being able to draw a long line across the board at right angles to its lower edge, this edge is made truly square with that on the left side of the board.

The details for building these drawing boards are given, because they are easy to be made by one who understands the use of a few wood-working tools; while the boards themselves are difficult of transportation--in case of the change of residence of their owners--quite unlike the instruments which are to accompany them.

Fig. 141 represents the board which has been described in the text, with provisions for the contraction and expansion; the very dark lines are intended to represent the ebony insertions--as described. Fig. 140 represents a plain pine board with dovetailed battens.

Fig. 142 represents the common means used to attach or secure slightly or temporarily the drawing paper to the drawing board; these are called thumb-tacks, and are usually forced through the paper into the wood by the hand, whence they are easily detached. These are made to have as slight a projection as may be, so as not to interfere with the free movement of the tee-square.

For mechanical drawing the invariable practice is to secure the paper on which the drawing is to be made to the drawing board by pinning it; this is effected by various kinds of _drawing pins_ or _thumb-tacks_.

The best kind for this purpose have a head as thin as possible without cutting at its edges, slightly concave on the under side next the paper, and only so much convex on its upper side as will give it sufficient thickness to enable the pin to be secured to it; better use four or more small pins along the edge of a sheet of paper, than use one clumsy, badly made pin at each end.

Fig. 143 and fig. 144 represent a pair of plain trestles or horses in common use for supporting large size drawing boards. This pattern is found frequently in the laying-out shop. Fig. 145 and fig. 146 represent _adjustable_ horses or trestles--these are designed, primarily, for office use. As will be seen by viewing the illustration, the upper part is supported by two hard-wood sliding pieces; these are provided with strong pins and numerous holes, and pass through the frame of the trestle, as shown, so that the upper portions can be arranged at any angle convenient to the draughtsman, as he lies over his work or stands by it.

Fig. 137 is introduced to exhibit the paper attached to the drawing board with the thumb-tacks, and with the T-square and set-squares arranged to commence work; the paper should not extend to the edges of the board; three, four or more tacks may be used on each edge of the sheet of paper, instead of two, as shown in illustration.

A most convenient--and except for its extreme lightness, which is not good in a drawing stand--a most admirable device is shown in fig. 138. The drawing table is simply a drawing board with folding legs; these are made from hard-wood, while the top is made of soft, seasoned pine, with square corners; while the device is strong and well braced, it can be folded and easily carried about--all as shown in the illustration.

THE TEE-SQUARE.

This is an instrument in the form of a letter T, as shown in the figures 149 and 150; the two parts are known as _stock_ and _blade_; the horizontal part of the letter (T) is the stock, and the vertical part the blade--hence the name, T-square; to form the square, the two parts are joined together in such a way as to make them exactly at right angles to each other; the stock, which is applied to the working edge of the drawing board, being about one-third the length of the blade, and about three times its thickness.