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  THE
  DRAUGHTSMAN’S HANDBOOK
  OF
  PLAN AND MAP DRAWING.


[Illustration: PLATE 1.

  =PLAN= SHEWING PRINCIPAL CHARACTERS OF WORK USED IN MAPPING.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]




  THE
  DRAUGHTSMAN’S HANDBOOK
  OF
  PLAN AND MAP DRAWING,
  INCLUDING INSTRUCTIONS FOR THE PREPARATION OF
  ENGINEERING, ARCHITECTURAL, AND MECHANICAL
  DRAWINGS.

  ~With Numerous Illustrations and Coloured Examples.~

  BY
  GEORGE G. ANDRÉ, C.E., M.S.E.

  [Illustration]

  LONDON:
  E. & F. N. SPON, 48, CHARING CROSS.

  NEW YORK:
  446, BROOME STREET.
  1874.




PREFACE.


The main purpose of the present work is to be a handy book of reference
for draughtsmen engaged chiefly in Topographical Drawings. But to have
limited its use to one class of draughtsmen, and especially to the
skilled members of that class, would have necessitated the discovery of
more cogent reasons for its publication than its author has yet been
able to adduce. Works of such a character exist already, and though
their imperfections are numerous, they fulfil their purpose in a fairly
satisfactory manner. But had the field been clear in this direction, it
is so restricted in extent that to have entered upon it would have been
to undertake a labour that promised little fruit, for such a work could
be only of small utility even to those for whom it was specially
intended. It was, therefore, determined to make the present handbook
_generally_ useful by giving it a much wider scope. And hence, if the
intention has been efficiently carried out, it may claim a place in
every drawing office, be it that of the Topographer, the Hydrographer,
the Surveyor, the Military, Civil, or Mechanical Engineer, or the
Architect. Whether or not this degree of success has been achieved, is
not for the author to judge. But should he have failed to reach the high
mark at which he has aimed, he hopes, with some degree of confidence,
that he has at least succeeded in producing a book which the experienced
draughtsman will find valuable as a book of reference, and which the
pupil may constantly consult with profit. A want has long been felt by
draughtsmen for some work of this kind to which they might refer their
pupils in the office, and it may not be presumptive to suppose that the
present work has supplied that want. To render it convenient for this
twofold purpose, it has been divided into two parts. In the first part
the principles and practices of the art have been clearly but briefly
explained and illustrated; while in the second part, the application of
the principles previously learned has been treated of, and such
information given as relates directly to the duties of the practitioner.

Of course, in a work of the present character, originality in the matter
is neither to be expected nor desired; enough if the manner shows some
novelty, and is such as to add value to the matter.

Although the preparation of maps and plans has received the chief share
of attention, engineering, architectural, and mechanical drawings have
been largely treated of. Projection, orthographic, isometric and
perspective, has been altogether omitted as beyond the scope of the
work; but Colouring and Shading have been fully considered and profusely
illustrated.

The Plates appended as examples for reference are numerous and varied in
character; they have been specially prepared by B. Alexander, to whom
the author offers his warmest thanks for the truly admirable manner in
which he has executed the work entrusted to him.

  16, CRAVEN STREET, CHARING CROSS,
  _September 7th, 1874_.




CONTENTS.


  PART I.--THE ESSENTIAL ELEMENTS.

  SECTION I.--THE DRAWING OFFICE AND ITS FURNISHINGS.

                                                           PAGE

  The Drawing Office                                          1
  Instruments                                                 2
  Materials                                                   5
  Precautions and Remarks                                     9


  SECTION II.--GEOMETRICAL PROBLEMS.                         15


  SECTION III.--LINES, DOTS, AND THEIR COMBINATIONS.

  Straight and Curved Lines                                  27
  Lines of uneven thickness                                  30
  The Broken Line                                            30
  The Dotted Line                                            31
  Combinations of Straight, Broken, and Dotted Lines         31
  The Wavy Line                                              33
  Grass-land                                                 34
  Swamps and Marshy Ground                                   35
  Sand and Gravel                                            35
  Woodland                                                   36
  Uncultivated Land                                          37
  Contour Lines                                              37


  SECTION IV.--COLOURS.

  Flat-tints                                                 40
  Conventional Colours                                       44
  Water                                                      45
  Grass-land                                                 45
  Marsh                                                      45
  Sand and Gravel                                            46
  Mud                                                        46
  Woodland                                                   46
  Cultivated Land                                            47
  Uncultivated Land                                          47
  Buildings                                                  47
  Roads and Streets                                          47
  Fences                                                     47


  SECTION V.--SHADING.

  Application of Shade Lines                                 48
  Cylindrical Surfaces                                       50
  Shading Lines                                              50
  Shading Lines on Cylindrical Surfaces                      51
  Shading Lines in Topographical Drawings                    52
  The Vertical System of Shading                             57
  Shading in Colours                                         63
  Hill Slopes                                                63
  Cylindrical Surfaces in Mechanical Drawings                64


  PART II.--APPLICATIONS.


  SECTION I.--LETTERING, BORDERING, AND NORTH POINTS.

  Lettering                                                  66
  Borders                                                    69
  North Points                                               69


  SECTION II.--SCALES.

  Scales of Distances                                        70
  Scales of Construction                                     74


  SECTION III.--PLOTTING.

  Reference Lines and Points                                 78
  Plotted Points                                             78
  To Plot Reference Lines and Points                         78
  To Plot Traverse Reference Lines                           84
  To Plot Detail                                             89
  To Plot Contours                                           90
  To Plot Sounded Points in Submerged Districts              90
  Errors and Error-sheets                                    91
  To Plot Vertical Sections                                  92
  To lay down Gradients                                      95
  To Plot a Section from a Contour Map                       96


  SECTION IV.--CIVIL ENGINEERS’ AND SURVEYORS’ PLANS.

  Standing Orders of Parliament                              98
      „      „          „       Documents required           99
      „      „          „       Plans                       100
      „      „          „       Book of Reference           101
      „      „          „       Sections                    101
  Working Sections                                          103

  _Regulations of Local Government Board_:--
    Boundary Maps                                           104
    Maps for Division into Wards                            104
    Plans of Proposed Works                                 105
    General Plan                                            105
    Detailed Plan                                           106
  Mining Plans                                              106
  Estate and Town Plans                                     107


  SECTION V.--MAP DRAWING.

  Single Stroke Streams                                     109
  Double Line Streams and Rivers                            110
  Colouring Streams or Rivers                               110
  Islands and Sand-banks, Sandy and  Pebbly Beds of Rivers  110
  Roads and Pathways                                        111
  Mountain Passes                                           111
  Fords and Ferries, Toll-gates                             111
  Telegraph Lines and Stations                              112
  Railways, Stations, and Termini                           112
  Size of Cities, Towns, and Villages                       112
  Sketching, Shading, and Copying Hills                     113
  Field Sketching                                           114
  Examination of Maps in the Field                          118


  SECTION VI.--MECHANICAL AND ARCHITECTURAL DRAWINGS.       121


  SECTION VII.--COPYING AND REDUCING.

  Drawing from Copy                                         127
  Copying by Tracing                                        128
  Copying by Transfer                                       129
  Reducing and Enlarging                                    130
  The Pantograph                                            131
  The Eidograph                                             136
  Drawings for Lithographers and Engravers                  141

  TRIGONOMETRICAL FORMULÆ                                   142

  INCLINED MEASURE                                          143

  CURVATURE AND REFRACTION                                  143

  INDEX                                                     144




LIST OF ILLUSTRATIONS.


  --------------------------------------+-------------+---------------
                                        |    Page.    |     Plate.
                                        +-------------+---------------
  ALPHABETS, examples of                |      ..     |   4, 5, 6
  Angle, to bisect                      |      16     |      ..
  Angles, to construct                  |    16, 17   |      ..
  Arch, equilateral                     |      23     |      ..
  ----, horse-shoe                      |      24     |      ..
  ----, lancet                          |      24     |      ..
  ----, obtuse                          |      24     |      ..
  ----, ogee                            |      25     |      ..
  ----, semi-elliptical                 |      23     |      ..
  ----, Tudor                           |      24     |      ..
  Architectural drawings, colouring of. |      ..     |      24
                                        |             |
  BORDERS                               |      ..     | 1, 3, 8, 9, 13
  Boundaries, parish, &c.               |      ..     |     3, 15
                                        |             |
  CANAL LOCKS                           |      ..     |     1, 11
  Chart, example of                     |      ..     |      18
  Cinquefoil, Gothic                    |      26     |      ..
  Circle, to describe through           |      17     |      ..
    given points.                       |             |
  ----, to find the centre of           |      18     |      ..
  Cliffs                                |      ..     |   1, 11, 14
  Colouring architectural drawings.     |      ..     |      24
  ---- maps and plans                   |      ..     |   1, 3, 13,
                                        |             |  17, 28, 33
  ---- mechanical drawings              |      ..     |  22, 23, 27
  Copse                                 |      ..     |     1, 10
  Corners                               |      ..     | 1, 3, 8, 9, 13
  Cylinders shaded                      |    51, 52   |      ..
  Cyma recta                            |      25     |      ..
  ---- reversa                          |      25     |      ..
                                        |             |
  DOCKS                                 |      ..     |    1, 11
  Drawings, architectural, colouring of.|      ..     |      24
  ----, isometrical                     |      ..     |      27
  ----, mechanical, colouring of.       |      ..     |  22, 23, 27
                                        |             |
  EIDOGRAPH                             |      ..     |      26
  Ellipse, to draw                      |      22     |      ..
  Equilateral triangle, to construct.   |      16     |      ..
                                        |             |
  FLOURISHES                            |      ..     |      25
  Fortifications, plans                 |      ..     |      32
  ----, sections                        |      ..     |      31
                                        |             |
  GEOLOGICAL MAPS                       |      ..     |      28
  ---- sections, coloured               |      ..     |    20, 21
  Grass                                 |      34     |     1, 17
  Gravel                                |      35     |       1
                                        |             |
  HARBOURS                              |      ..     |      11
  Hexagon, to describe                  |      21     |      ..
  Hill shading                          | 53, 55, 56, |      ..
                                        | 58, 61, 62, |
                                        |      63     |
  Hills                                 |      ..     | 1, 12, 14, 17
  ---- in colour                        |      ..     |    12, 14
                                        |             |
  ISOMETRICAL DRAWINGS                  |      ..     |      27
                                        |             |
  LAKES                                 |      ..     | 1, 3, 11, 17
  Land, cultivated                      |      32     |     1, 13
  ----, uncultivated                    |      37     |      ..
  Lettering, examples of                |      ..     | 4, 5, 6, 7, 8,
                                        |             |      25
  Line, to divide into equal parts.     |      15     |      ..
  Lines, broken                         |      30     |      ..
  ----, contour                         |      37     |      ..
  ----, dotted                          |      31     |      ..
  ----, section                         |      29     |      ..
  ----, shade                           |      49     |      ..
  ----, to bisect                       |      15     |      ..
                                        |             |
  MAPS, geological                      |      ..     |      28
  ----, Ordnance, example of.           |      ..     |      18
  ---- and plans, colouring of.         |      ..     |   1, 3, 13,
                                        |             |  17, 28, 33
  Marsh                                 |      35     |   1, 10, 11
  Mechanical drawings, colouring of.    |      ..     |  22, 23, 27
  Mining plans                          |      ..     |      33
                                        |             |
  NORTH POINTS                          |      ..     |       9
                                        |             |
  OVAL, to construct                    |      18     |      ..
                                        |             |
  PANTOGRAPH                            |      ..     |      26
  Parabola, to draw                     |      21     |      ..
  Pentagon, to describe                 |      20     |      ..
  Perpendicular, to erect               |      15     |      ..
  Plans, estate                         |      ..     |     3, 17
  ----, fortifications                  |      ..     |      32
  ----, mining                          |      ..     |      33
  ----, office                          |      ..     |       2
  ----, parliamentary                   |      ..     |    13, 19
  ----, reducing or enlarging           |      ..     |      26
  ----, town improvements               |      ..     |      13
  ---- and maps, colouring of.          |      ..     |  1, 3, 13, 17,
                                        |             |     28, 33
  Plotting, examples of                 | 82, 85, 86, |      ..
                                        |   88, 93    |
                                        |             |
  QUARRIES                              |      ..     |       1
  Quatrefoil, Gothic                    |      26     |      ..
                                        |             |
  RADII OF CIRCLE, to draw              |      18     |      ..
  Railways                              |      ..     |     1, 3
  Rectangles, similar, to construct     |      20     |      ..
  Rivers                                |      ..     | 1, 11, 12, 17
  ----, outlines of                     |      30     |      ..
  Roads                                 |      ..     | 1, 3, 12, 17
  Rocks                                 |      ..     |     1, 11
  Roofs                                 |      30     |      ..
                                        |             |
  SAND                                  |      35     |       1
  ---- banks                            |      ..     |     1, 11
  Scales                                |  71, 75, 76 | 2, 3, 8, 9, 13
  Section plotting, example of          |      93     |      ..
  Sections, fortifications              |      ..     |      31
  ----, parliamentary                   |      ..     |    19, 21
  ---- of strata, examples of colouring.|      ..     |    20, 21
  Shade, scales of, for hills           |    53, 58   |      ..
  Signs, various, used in Indian and    |      ..     |    29, 30
    Colonial maps.                      |             |
  ----, ----, used in maps, plans, &c.  |      ..     |    15, 16
  ----, ----, used in military maps and |      ..     |    31, 32
    fortifications.                     |             |
  Soundings                             |      ..     |    11, 18
  Square, to construct                  |      19     |      ..
  ----, multiple of, to construct.      |      19     |      ..
  Squares, proportional, to construct.  |    19, 20   |      ..
  Swamps and marshy ground.             |      35     |     1, 10
                                        |             |
  TANGENT, to draw                      |      18     |      ..
  Titles, examples of                   |      ..     |    3, 7, 8
  Towns                                 |      ..     |  1, 3, 11, 13
  Traverse plotting, example of.        |  85, 86, 88 |      ..
  Trees                                 |      36     |  1, 3, 10, 13,
                                        |             |      17
  Trefoil, Gothic                       |      25     |      ..
  Triangles, to construct               |    16, 17   |      ..
                                        |             |
  VILLAGES                              |      ..     |       1
                                        |             |
  WATER, flowing                        |      33     |      11
  ----, standing                        |      29     |     1, 11
  ---- in section                       |      30     |      ..
  Wood-graining                         |      32     |      ..
  Wood in section                       |      32     |      ..
  Woods                                 |      36     | 1, 3, 10, 17
  --------------------------------------+-------------+---------------




PLAN AND MAP DRAWING.




PART I.--THE ESSENTIAL ELEMENTS.


SECTION I.--THE DRAWING OFFICE AND ITS FURNISHINGS.

There are few occupations so dependent for their correct performance
upon minute matters of detail as that of the draughtsman. Things
apparently the most trivial are sufficient to render inaccurate or to
mar the appearance of the otherwise most carefully and skilfully
executed design, and as the value of a drawing depends wholly upon its
accuracy and its appearance, it is obvious that such matters of detail,
however trivial they may be in themselves, demand careful attention. We
have, therefore, deemed it desirable to preface our remarks on Plan and
Map Drawing with a brief description of the instruments and materials
required, and of the mode of using them which experience has shown to be
the best.


_The Drawing Office._--The first essentials of a room for drawing in
are--that it shall be quite free from damp and be well lighted. The
position of the windows is a matter of some importance, and though
persons have largely to accommodate themselves to circumstances in this
respect, it is desirable to know what are the most suitable conditions,
in order that they may be complied with as far as circumstances permit.
Skylights are unsuitable, because the light entering from above is
liable to be intercepted by the body, and especially by the hands of the
draughtsman; besides which, the light from a skylight is seldom
sufficient. For the same reasons, a window placed very high in the room
is objectionable. When possible, a western aspect is to be preferred, as
the light from this direction is less variable and lasts later in the
day than from other directions. Blinds of some kind are necessary to
modify the light when the sun shines directly upon the window. Gaslights
should be situate about 3 feet above the drawing table, and there should
be two burners, placed not less than 2 feet apart, as otherwise the
hands and the instruments will cast shadows which will prevent fine
lines and points from being seen.

The drawing table should be placed under the window; it should have a
breadth of about 2 feet 6 inches, and its height should be 3 feet 8
inches at the back and 3 feet 6 inches at the front. The front edge
should be rounded over.

Dusters and means for washing the hands must also be provided, as it is
requisite to frequently dust the paper and the instruments, and to keep
the hands perfectly clean.


_Instruments._--All drawing instruments should be of the best
workmanship, for it is impossible to obtain accuracy with imperfect
tools, and they must be kept in order by careful handling. For all kinds
of drawing, compasses of three sizes are required, the ordinary compass,
the bows, and the spring bows. The best compasses are those which are
sector-jointed. The points should be kept sufficiently sharp not to slip
on the paper, but not so sharp as to readily penetrate it. It is also
important that the points be thin and round, as otherwise, when several
arcs have to be struck from the same centre, the compass leg is apt to
make a large hole, which is utterly destructive of accuracy. The pencil
leg should be kept exactly equal in length to the steel leg, for true
circles cannot be drawn when one leg is shorter than the other. In
removing the movable leg, care should be taken to draw it straight out,
as nothing spoils the instrument so soon as wrenching the leg from side
to side. In using the compasses, the instrument should be held lightly
between the thumb and the forefinger only. It should not be pressed upon
the paper; but it should rest equally upon both points. If the weight of
the hand be thrown upon the instrument the points will penetrate the
paper. Care should also be taken to bend the joints so as to keep both
legs perpendicular to the paper. If attention be not given to this
matter, the steel leg will make a large hole in the paper, and the ink
leg will make a ragged line, because only one of the nibs will touch.

Next in importance to the compass, and of more frequent use, is the
drawing pen. The draughtsman should possess at least two of these
instruments, one for fine, and another for medium lines. When the proper
opening of the nibs for fine lines has once been obtained, it is
desirable not to change it; the pen can always be cleaned by passing a
piece of drawing paper between the nibs. The cleaning of the pen should
be carefully attended to; it should never be put away without having
every particle of dried ink removed from it; and frequently, while in
use, it should be wiped out to remove the dust, which is constantly
settling in it, as well as the particles of lead that are taken up from
the paper. The ink is supplied by breathing between the nibs and dipping
them in the liquid, or by means of a camel’s hair brush. When the latter
method is adopted, care should be taken to protect the brush from the
dust floating in the atmosphere of the room.

After considerable wear, the drawing pen will require setting. The
operation of setting requires some judgment and considerable practice,
and is one of those mechanical niceties which it is difficult to
describe. Generally it will be found advantageous to have the pen set by
an instrument maker. As, however, this resource is not always at hand,
it is desirable that the draughtsman should be able to set his own pen.
The following description of the operation given by W. Binns, in his
admirable work on Projection, is the best we have seen. “The nibs must
be precisely of the same length, rounded in two directions, and as sharp
as it is possible to make them without producing to the touch a
sensation of cutting, and without scratching the surface of the paper
when drawing a line, which is generally the case when one nib is longer
than the other. This irregularity may be detected by placing
alternately the sides of the pen at an acute angle with the forefinger,
and slipping the edge of the nail over the point, when the difference in
length will be at once perceived; and it may be reduced by drawing a few
lines, as it were, on a turkey stone, with the pen applied to the edge
of a set square in the same manner as if drawing lines upon paper, but
with this difference, that during the longitudinal motion of the pen the
handle must be turned over in a circular manner, so as to give a rounded
form to the point of the pen. If the pen be now held with the point
directed towards the eye, and gently moved about so as to catch the
angle of reflexion, a bright speck on one or both nibs will be observed,
which must be reduced by rubbing the pen to and fro upon the stone,
giving at the same time a slight rotary motion to the handle, which must
be held at an angle of about 20° with the face of the stone: the point
of the pen being examined from time to time, and the process of reducing
the bright specks continued until the point is as fine as can be used
without cutting or scratching the paper. If at this stage the two nibs
are of the same length, a perfectly solid and fine line can be drawn.
The beginner, however, must not be disappointed if sixty minutes are
thus expended before he can produce a satisfactory result; whereas two
minutes in the hands of a practitioner would suffice.”

The instrument most frequently in the hands of the draughtsman is the
lead pencil. These are required of various degrees of hardness, but for
lines that are to be ruled an H H is the best. The most suitable
qualities of lead are those which are the most easily rubbed out; these
qualities are sometimes gritty, but this defect is more than compensated
by the facility with which a line may be removed from the paper. There
is some art in cutting a pencil properly. If the point is intended for
sketching, it should be cut equally from all sides, so as to produce a
perfectly acute cone. But for line-drawing a flat or chisel point should
always be used. This point is much stronger, and will last much longer
than the cone point. To produce the chisel point, first cut the pencil
from two sides only with a long slope, and afterwards cut the other
sides away only just sufficiently to round the first edge a little.
This side wood is needed both to afford a support to the lead, and to
show in what direction the point stands. To avoid breaking the lead, the
knife should be held at an acute angle with it. A point cut in this
manner may be kept in order for some time by rubbing it upon a fine file
or upon a piece of glass-paper or fine sandstone.

Of the other instruments used in drawing, nothing need be said in this
work, as their use presents no difficulties. It may, however, be well to
remark that no straight-edge employed for ruling lines should be less
than a fourteenth of an inch thick, for if the edge be very thin, it
will be impossible to prevent the ink from escaping from the pen on to
it.


_Materials._--The drawing papers known as Whatman’s are the best
prepared of any obtainable, and they are almost universally employed. Of
these there are two kinds, the smooth and the rough; the former is
technically called _not_ paper, and is the more suitable for mechanical
and architectural drawings; the rough is more effective for perspectives
and Gothic elevations. A third kind is known as the _hot-pressed_, but
as it does not take colour so well as the not and the rough, it is not
often used. The various sizes are indicated by their names, which are
the following:--

  Antiquarian       53    × 31    inches.
  Atlas             34    × 26       „
  Columbier         34¹⁄₂ × 23¹⁄₂    „
  Demy              20    × 15       „
  Double Elephant   40    × 26³⁄₄    „
  Elephant          28    × 23       „
  Emperor           68    × 48       „
  Imperial          30    × 22       „
  Medium            22³⁄₄ × 17       „
  Royal             24    × 19¹⁄₄    „
  Super Royal       27¹⁄₂ × 19¹⁄₄    „

The sizes considered best, and almost universally used for engineering
and architectural drawings, are the elephant, the double elephant, and
the imperial. If smaller sizes are required, the half or quarter sheet
is used. Antiquarian has generally a good surface to draw upon, and it
is preferred by some architects. The atlas is also a very good paper.
Besides the foregoing, there is the machine-made or cartridge paper,
which is very commonly employed for detail drawings. It has not so good
a surface as the other kinds, nor is it so white; its chief advantage is
found in its dimensions, it being made uniformly 53 inches wide and
continuous. Hence the exact length required may be obtained. For large
plans and competition drawings, either cartridge or emperor paper is
used.

Paper that is to receive an elaborate drawing must be stretched and
glued to the board. This operation is one requiring a little skill and
some practice to perform successfully. The following is the best manner
of proceeding. The sheet to be strained is laid face upward upon the
board, and a wet sponge is passed rapidly along the margins, and then
across the face, including the margins, until the whole surface is
sufficiently and uniformly wetted. The object of wetting the margins
first is to prevent cockling by allowing them a longer time to expand in
than the middle of the paper. The sheet must now be left for about ten
minutes, or until the wet gloss has entirely disappeared. The process of
glueing to the board is as follows. A straight-edge is laid along one
end of the sheet, and about ³⁄₈ of an inch of the margin is turned up
against it, and glued by means of a brush. The margin is then turned
down and rubbed quickly with a knife-handle or, still better, a
paper-knife. The opposite end of the sheet is next pulled outwards and
glued in the same way, and the same method is afterwards applied to the
top and bottom margins. Some draughtsmen prefer to glue down the
_adjoining_ edges, but generally it will be found that laying down
successively _opposite_ edges will give more satisfactory results. The
contraction of the paper in drying should leave the face quite flat and
solid. During the process of drying, it is important to keep the board
in a perfectly horizontal position, as otherwise the water will
gravitate towards the lower side and soften the glue, and as the sheet
will dry unequally, the lower edge will break away.

The thinner the glue used the better, and for this reason the best
quality should be obtained, and care should be taken to keep the water
supplied that is lost by evaporation. When it becomes necessary to
replenish the glue-pot, the cake should be soaked in cold water for at
least eight hours.

The removal of a drawing from the board presents no difficulty. A pencil
line is drawn along the margin at a sufficient distance from the edge to
be clear of the glue, and a pen-knife is guided along this line by a
straight-edge not used for drawing.

As duplicates of drawings, especially if they be working drawings, are
usually tracings, tracing paper is an important material in every
drawing office. It is too well known to need a description. It is sold
in various sizes, and of various prices, but the most usual sizes are 30
× 20 inches, and 40 × 30 inches, the price of the former being 3_d._ and
that of the latter 6_d._ a sheet. It may also be purchased in continuous
lengths of 24 yards, 42 inches wide, for about 8_s._, or if extra stout,
16_s._ A much less expensive mode of obtaining tracing paper is to make
it one’s self. Common silk or _tissue_ paper may be purchased in
quantities at less than a halfpenny a sheet of the ordinary size. This
may be prepared by placing a single sheet at a time flat upon a board or
other smooth horizontal surface, and applying a mixture of boiled
linseed oil and turpentine. This mixture should be composed of one part
of oil to five of turpentine, and it should be applied with a small
sponge. One coating is sufficient, and it should not be put on too
thickly. Each sheet as prepared should be hung over a string stretched
across the room to dry, and when all the clear oily marks have entirely
disappeared, it will be ready for use. Five gills of turpentine and one
of oil is enough for two quires of double-crown tissue paper. That
tracing paper is best which is toughest, most transparent, and most free
from greasiness. The continuous papers are more economical than those in
sheets, because just the quantity required can always be taken from the
roll. For durability, tracing cloth is to be recommended; it is sold in
continuous lengths of 24 yards, and it may be had from 18 inches to 41
inches in width. That known as “Sager’s vellum cloth” is of excellent
quality, both for transparency and strength.

Some kinds of drawings, such as specifications for Letters Patent, plans
upon deeds, &c., have frequently to be made upon parchment. Special
kinds of parchment can be obtained for these purposes. There is a kind
made which is quite transparent, and which can be purchased cut to the
Patent Office regulation size. As parchment has always a more or less
greasy surface, before commencing to ink or to colour, it should be
pounced over with pouncet of finely-powdered French chalk. Besides this
precaution, it will be necessary to add a little ox-gall to the ink or
colour.

Blacklead and carbonic paper are used to transfer a drawing. The former
is prepared by rubbing thin paper over with a soft block of Cumberland
lead; the latter by painting one side of the paper with lamp-black
ground to perfect fineness in slow drying oil. Carbonic paper is used
for coarser work than blacklead paper. Both may be purchased, properly
prepared, at a trifling cost. The drawing to be copied is laid over the
sheet of paper which is to receive the copy, with a sheet of the
blacklead or carbonic paper interposed, and a tracer is passed with a
light pressure over the lines. This method is mostly used to reproduce a
drawing from a tracing, to obtain a finished copy from a rough draught
that has become soiled and marked in designing, or to avoid errors or
small alterations in the first drawing.

A very convenient kind of paper for small working drawings, or for
sketching to scale, is that known as sectional paper. This is paper
ruled into small squares to a given scale with pale ink. The spaces in
ordinary use are ¹⁄₁₀, ¹⁄₈, ¹⁄₆, ¹⁄₅, and ¹⁄₄ inch. Thicker lines are
drawn either to mark off the inches or to count the spaces in tens. With
this paper, the scale may be dispensed with, as the eye is capable of
subdividing the spaces with sufficient accuracy for practical purposes.
Sectional paper is much used for sections of railway cuttings and
embankments, as it affords a ready means of calculating the contents. It
is also made up into sketching books and architects’ pocket-books, for
which purposes it is peculiarly convenient.

Indian ink is used for all kinds of geometrical drawings. Being free
from acid, it does not corrode the steel points of the instruments, and
it preserves its colour unchanged. It is difficult to get the genuine
ink, but even that, as it is imported from China, varies considerably in
quality. For line-drawing, that is the best quality which will wash up
least when other colours are passed over it. This quality is ascertained
in the trade, though not with absolute certainty, by breaking off a
small portion. If it be of the right quality, it will show a very bright
and almost prismatic-coloured fracture.

The ink is prepared for use by rubbing it with water on a slab or in a
saucer. The saucer should be quite smooth inside, so as not to abrade
the ink. When mixed to the requisite thickness, which may be ascertained
by drawing a line with a common pen, it should be covered to protect it
from the particles of dust floating about the room. Ink should be rubbed
up perfectly black, for pale ink makes the boldest drawing look weak.
But after it has become black, any further mixing will only injure it by
rendering it viscid. It is best to use it immediately after it is mixed,
for if re-dissolved, it becomes cloudy and irregular in tone. The
addition of a little ox-gall will make it flow more freely from the pen.

For erasing Cumberland lead-pencil marks, native or bottle indiarubber
is sufficient; but for other kinds of pencils, fine vulcanized
indiarubber is better. This, besides being a more powerful eraser,
possesses the important quality of keeping clean, as it frets away with
the friction of rubbing, and thus presents a continually renewed
surface. Vulcanized rubber is also very useful for cleaning off
drawings.


_Precautions and Remarks._--It is essential to the good appearance of a
drawing that the paper be preserved perfectly clean. The hands
especially should be kept as much as possible from resting on it, as the
perspiration makes it greasy, and when once it has acquired this defect,
clear, sharp lines become impossible. A sheet of clean paper should be
constantly interposed between the draughtsman’s hands and the drawing
upon which he is working. Brown or printed paper is unfit for this
purpose, as the former is either greasy or tarry, and the latter is apt
to soil from the printed matter. White paper can be had of large size,
or, if necessary, several sheets may be pasted together.

To prevent risk of smearing the lines when inking in, it is well to
begin at the top of the drawing and to work downwards, also from the
right to the left for vertical lines. The ink slab or saucer should be
kept on one side and never in front of the drawing. Should a drawing get
a grease spot, it may be removed by the application of a hot smoothing
iron to a piece of clean blotting-paper laid over the spot, but not
sufficiently to be coloured over.

Great care should be taken to correctly place the centre lines of a
drawing; these lines should be drawn very fine and distinct. In working
drawings the centre lines are of great importance, as the dimensions are
always measured from them; in such cases it is customary to draw them in
red or blue colour. In all cases where a plane figure is symmetrical
with respect to a given line, whether the line exists in the figure or
may be considered as existing in it, that line should be drawn first,
and such a line is known as a centre line.

The centres of all arcs should be marked for the ink compasses at the
time the arc is struck by the pencil, by placing a small hand-drawn
circle around it. It is also necessary to mark distinctly by short
intersecting straight lines the exact points at which the arc begins and
ends. When a number of concentric circles have to be struck, the smaller
ones should be struck first, as it is more difficult when the hole in
the paper becomes enlarged to draw a small circle than a large one.

Whenever it is practicable, lines should be drawn _from_ a given point
rather than _to_ it; and if there are several points in one of which two
or more lines meet, the lines should be drawn from that one to the
others; thus, for example, radii should be drawn from the centre to the
points in the circumference of a circle. When a point has to be
determined by the intersection of circular arcs or straight lines, these
should not meet at an angle less than 30°. In dividing a line into a
number of parts, instead of setting off the part repeatedly along the
line, it is better to set off a convenient multiple of the given part,
and subdivide it; that is, to work from the whole to the parts, rather
than from the parts to the whole. This is an important principle in
surveying as well as in plan drawing, and in the construction of scales
it ought always to be observed.

Ink lines should never be erased with a knife, nor should an ink-eraser
be used, especially if the drawing is to be coloured. A needle point
will take out a short line in a way that leaves little trace of the
error. A very good means of taking out a line is furnished by a piece of
Oakey’s No. 1 glass-paper folded several times until it presents a round
edge; the application of this leaves the surface of the paper in a much
better condition for drawing upon than it is left in by the knife. When
the drawing is to be coloured, it is best to wash out a wrong line with
a small hard brush, and to slightly sponge over the place through a hole
of the requisite size cut in a scrap of drawing paper, to save the other
parts of the drawing. When a line has been drawn a little beyond the
point at which it should terminate, it will generally be found better to
avoid erasure by laying a little Chinese white over the line with a fine
sable-brush. Sometimes, when erasures are unavoidable upon a drawing
that is to be coloured, it will be found expedient to take the surface
off the whole of the paper with glass-paper, the colour will then flow
equally.

In copying from a tracing, it is well to put a sheet of drawing paper
underneath the tracing, for it not only shows up the lines more
distinctly, but it prevents the dividers from tearing the drawing while
taking off measurements.

Before commencing a drawing, a cutting-off line should be drawn all
round the sheet clear of the glued portion. The portions outside of this
line are useful to try the drawing pen upon before drawing a line, or
for trying a tint when colouring. Care should be taken not to leave too
narrow a margin, for nothing detracts more from the appearance of a good
drawing. For a drawing occupying a space of 1 foot or 15 inches square
over all, there should be a margin of at least 5 inches all round, with
the border line from 1¹⁄₂ to 2 inches from the cut-off line. Other
sizes should be in proportion. This rule is given by Maxton in his
‘Engineering Drawing,’ who also has the following remarks on cutting off
and preserving drawings. “The opposite side should never be cut first,
for if so cut, upon nearly completing the cutting of the third side the
paper undergoes contraction, and the fourth side pulling against it, is
apt to snap off the remaining inch or so, and generally in towards the
sheet, seldom in the margin on the outside of the cutting-off line. The
sheet should be cut off all round, taking care, by applying the
knife-blade under the edge of the sheet, that it is free from the board
before proceeding to cut off the side or end adjoining. When the sheet
has been removed, the strips of drawing paper left on the board should
be simply sponged over two or three times, and they will peel off
easily.

“For preserving a rolled drawing, a common substitute for string, and
one less likely to crease the drawing, is made as follows:--Take a strip
of drawing paper from 1¹⁄₂ to 2 inches wide and an inch longer than the
circumference of the rolled drawing. About half an inch from each end
make incisions, at one end in the middle and one-third of the breadth
across, and at the other end at the sides, each one-third of the breadth
across. Fold in these sides, so that they may pass through the incision
in the opposite end of the strip; on being opened again after they have
passed through, the whole will form a hoop, which, when slipped over the
drawing, will keep it secure.”

As cartridge paper is not always suitable, it sometimes becomes
necessary to join the smaller sizes end to end. To do this neatly the
edges should be cut straight, and a straight-edge laid upon the paper,
allowing ³⁄₈ inch to project beneath it. This portion of the paper
should then be rubbed down with sand or glass-paper until the outer edge
is quite thin. The edges of both sheets to be joined must be treated in
this way, and covered with a thin coating of gum. Having placed these
edges in contact, a strip of paper 1¹⁄₂ or 2 inches wide should be laid
upon the joint, and well rubbed with the handle of a paper-knife. If the
paper thus joined has afterwards to be stretched on a board, it should
be done while the joint is damp. In sponging the paper, care must be
taken not to go over the joint.

In joining sheets of tracing paper, the joint should never be made more
than ¹⁄₄ inch broad. The gum used for this purpose should be very thin,
and a strip of drawing paper should be placed upon each side of the
joint until it is quite dry. It is a good plan to roll the joined sheet
upon a roller with the joint in a line with the roller and the strips
infolded over the joint. When left to dry in this position, the joint
will be perfectly smooth.

Drawings have frequently to be mounted on stretchers, and the operation
of mounting is one requiring some care and practice. Generally it will
be found more convenient to purchase the stretcher ready made complete;
but when this is not done, care must be taken to have the frame made of
sufficient strength to resist the tension of the paper when dry. The
sides and the ends of a stretcher, 8 or 9 feet long, should be 4 inches
broad and not less than ⁷⁄₈ inch thick, and for any length above 18
inches there should be one or more bars across. A frame 6 feet long
should have two cross-bars dividing the length into three equal parts,
and they should be of such a thickness as not to come up flush with the
sides and ends by about ¹⁄₈ inch. The inner edges on the face of the
latter should be rounded down to the level of the cross-bars, and the
same degree of rounding should be given to the edges of the cross-bars
themselves. This is necessary to prevent the edges from showing a soiled
mark on the paper. When the frame has been thus prepared, the linen or
calico should be spread out on some flat surface and the frame laid upon
it face downwards. The ends of the linen should then be pulled over and
nailed to the back; next, the middle of the sides should be pulled over
and fixed in the same way. The intermediate spaces are afterwards tacked
down by placing a tack alternately on opposite sides, care being taken
to pull the linen tight and smooth before inserting the tack. It is a
good plan to fold the edge, as the double thickness will hold the
tacking better than if single.

To mount the paper on the stretcher, it should be laid face downwards
upon a clean flat surface, which will be all the better if covered with
a clean cloth, and sponged with clean water. When the water has soaked
in, apply with a flat brush some cold flour paste, and, if necessary,
remove all knots or particles of gritty matter, as these would prevent
the paper from lying close to the linen. The addition of a little alum
to the paste improves its adhesive property, and also tends to make the
drawing less stiff when dry. When a good coating of paste has been well
distributed over the paper, place the stretcher upon the paper and rub
the back of the linen well; then turn the stretcher over and rub down
the edges of the paper. Air-bubbles between the linen and the paper may
be got rid of by puncturing the spot with a fine needle and rubbing it
down. Paper thus mounted may be drawn upon nearly as well as when
stretched on a board. To give an edge for the [T]-square, a strip of
wood with parallel edges may be temporarily nailed on.

Some drawings, such as large plans of estates, have frequently to be
varnished. This operation requires some skill, and can be satisfactorily
accomplished only by a practised hand. The process generally adopted is
to stretch the drawing upon a frame, and to give it three or four coats
of isinglass size with a flat broad brush, taking care to well cover it
each time, and to allow it time to dry between each coat. The best
varnish is Canada balsam, diluted in oil of turpentine. This requires to
be put on evenly in a flowing coat with a fine flat brush, and to be
left in a warm room free from dust until it is thoroughly dry. The
drawing must be in a perfectly horizontal position while the size and
the varnish are being applied. In drawings to be varnished, thick lines,
such as shade lines, and chalky colours should never be put on before
sizing, as they are apt to blot during the process.

Should a fir drawing-board get accidentally dented, an application of
water to the part will, within certain limits, bring it up to its proper
level.


SECTION II.--GEOMETRICAL PROBLEMS.


_To bisect a given Straight Line._--Let A B (Fig. 1) be the given line.
From A and B, with any radius greater than ¹⁄₂ A B, draw arcs cutting
each other in C and D; then the line joining C D will bisect the line
A B as at E.

[Illustration: FIG. 1.]

[Illustration: FIG. 2.]


_To erect a Perpendicular to a given Straight Line._--Let it be required
to erect a line perpendicular to the point B (Fig. 2) in the line A B.
From any point C above the line, with radius B C, describe an arc as
A B D; join A C and produce the line until it cuts the arc in D, and
join D B; then will D B be perpendicular to A B.

[Illustration: FIG. 3.]


_To divide a Line into any number of equal parts._--Let it be required
to divide the line A B (Fig. 3) into five equal parts. From B, at any
angle, draw B C, and on the line B C lay off five equal parts, 1, 2, 3,
4, 5; then take a set square E, and make one of the sides containing the
right angle coincide with the points A and 5, and to the other side
apply a straight-edge D; then by passing the set square along the edge
of the straight-edge and drawing lines from the points 4, 3, 2, 1,
through the line A B, we shall have the line A B divided into five equal
parts through the points 1′, 2′, 3′, 4′.


_To draw a Line making, with another line, a given Angle._--Let it be
required to draw a line making with the line A B (Fig. 4) an angle of
35°. From a scale of _chords_, which will be found on most scales
supplied with a set of instruments, take off 60°; from the point A, with
this distance for radius, describe an arc C D; lay off on this arc the
distance of 35° taken from the same scale of chords; from A draw a line
through this point. Then will the line A E make with the line A B an
angle of 35°. The same result may be more readily arrived at by means of
a protractor. If the centre point of the protractor be placed on the
point A and its base made to coincide with the line A B, we can from its
circumference prick off the distance of 35°, and a line drawn from A
through the point thus found will make, with the line A B, the required
angle of 35°.

[Illustration: FIG. 4.]

[Illustration: FIG. 5.]


_To bisect an Angle._--Let B A C (Fig. 5) be the angle which it is
required to bisect. From A, with any radius, describe an arc cutting the
lines A B and A C in D and E; from D and E, with the same or any other
radius, describe arcs cutting each other in F, and from A draw a line
through F; this line will bisect the angle as required.

[Illustration: FIG. 6.]

[Illustration: FIG. 7.]


_To construct an Equilateral Triangle on a given base._--Let A B (Fig.
6) be the given base. From A and B, with radius A B, describe arcs
cutting each other in C; join A C and C B, which will complete the
required triangle.


_To construct a Triangle, the lengths of the Sides being given._--Let it
be required to construct a triangle whose sides shall be equal
respectively to 6, 5, and 4. Lay down the base A B (Fig. 7), making it
equal to 6 divisions of the scale; from A with radius equal to 4
divisions, and from B with radius of 5 divisions of the scale describe
arcs cutting each other in C; join A C and C B, which will complete the
required triangle.

[Illustration: FIG. 8.]


_To construct an Angle equal to a given angle._--It is required to draw
a line making with the line D E (Fig. 8) an angle equal to that
contained by the lines B A C. From A, with any radius, draw an arc F G,
and from D, with the same radius, draw the arc H I, and make H I equal
F G; then a line drawn from D through I will make, with the line D E, an
angle equal to the angle B A C.

[Illustration: FIG. 9.]


_To construct a Triangle, the length of the base and the angles at the
base being given._--It is required to construct a triangle whose base
shall equal 1 inch, and the angles at the base be 30° and 45°
respectively. Having made the base A B (Fig. 9) of the required length,
make the angles at A and B of the required magnitude in the manner
already described (see Fig. 4), and the continuation of these lines
meeting in the point G will complete the construction of the required
triangle.

[Illustration: FIG. 10.]


_To describe a Circle which shall pass through three given points._--Let
A B C (Fig. 10) be the points through which it is required to draw the
circle. From each of these points, with any radius, describe arcs
cutting each other in D and E; join the points D and E, and the point
where these lines intersect will be the centre from which to describe
the circle which will pass through the points A B C as required.

[Illustration: FIG. 11.]


_To draw a Tangent to a circle._--I. Let B (Fig. 11) be the point from
which it is required to draw the tangent. Draw the radius O B, and at B
erect a perpendicular (see Fig. 2); then will the line B D be a tangent
to the circle. II. It is required to draw a tangent from the point E in
the same circle. Draw the radius O E extending beyond the circumference
to F, and make E G equal to E F. From F and G, with any radius, describe
arcs cutting each other in H and I; then a line drawn through these
points will be a tangent to the circumference at E.

[Illustration: FIG. 12.]


_To find the Centre of a circle._--From any point in the circumference,
as B, (Fig. 12), describe an arc cutting the circumference in A and C,
and from A and C, with the same radius, describe arcs cutting the first
arc in two points; through the points of intersection draw lines to the
interior of the circle, and the point O where these lines intersect will
be the centre of the circle.

[Illustration: FIG. 13.]


_To draw lines which shall be Radii of a circle, the centre being
inaccessible._--Having laid off on the circumference of the arc, the
distances apart of the radii, as A, B, C, &c. (Fig. 13), from each of
these points, with radius greater than a division, describe arcs cutting
each other as at _a_, _b_, _c_, &c., join A _a_, B _b_, C _c_, &c., and
the lines so drawn will be radii of the circle as required.

[Illustration: FIG. 14.]


_To construct an Oval, the width being given._--Draw the line A B (Fig.
14) equal to the width, and bisect A B by C D (see Fig. 1). From the
point of intersection E, with radius E A or E B, describe the circle
A C B F, and from A and B through F, draw the lines A G and B H. From
A, with radius A B, describe the arc B G, and from B, with the same
radius, describe the arc A H; also from F, with radius F G or F H,
describe the arc G D H, which will complete the required oval.

[Illustration: FIG. 15.]


_To construct a Square on a given line._--Let A B (Fig. 15) be the given
line. At A erect a perpendicular (see Fig. 2), and from A, with radius
A B, describe an arc cutting the perpendicular in C; also from B and C,
with the same radius, describe arcs cutting each other in D; join C D
and B D, which will complete the required square.

[Illustration: FIG. 16.]


_To construct a square that shall be a Multiple of any given
square._--Let A B C D (Fig. 16) be the given square, and let it be
required to construct a square that shall contain 2, 3, 4, &c., times
its surface. Draw the diagonal B C, then the square described on B C
will be double the square A B C D. Lay off D E, equal to B C, and draw
C E; then the square described on C E will be three times the square
A B C D. In the same manner lay off D F, equal to C E, and the square
described on C F will be four times the square A B C D; and so for any
multiple of the square A B C D.

[Illustration: FIG. 17.]


_To construct a square that shall be equal to ¹⁄₂, ¹⁄₄, &c., of any
given square._--Let A B C D (Fig. 17) be the given square. On A B, as a
diameter, describe the semicircle A G B, and erect at the centre E the
perpendicular E G. Draw G B, which will be the side of a square equal to
one-half of A B C D. Lay off B F, equal to one-fourth of A B, and erect
the perpendicular F H; then the square described upon H B will be equal
to one-fourth of A B C D. In the same manner a square may be constructed
equal to _any_ part of A B C D.

[Illustration: FIG. 18.]


_To construct a square that shall be in any Proportion to a given
square._--Let A B C D (Fig. 18) be the given square. It is required to
construct a square which shall be to A B C D as 2 is to 5. Upon the side
A B as a diameter describe the semicircle A F B, and divide the line A B
into five equal parts. At the second point of division erect the
perpendicular E F and join A F; the square described upon A F will be to
the given square A B C D as 2 is to 5.

[Illustration: FIG. 19.]


_To construct, upon a given base, a Rectangle, which shall be similar to
a given rectangle._--Let A E F G (Fig. 19) be the given rectangle. It is
required to construct upon the base A B, one that shall be similar to
A E F G. Produce A E and lay off the given base from A to B; draw the
diagonal A G and produce it indefinitely. Erect a perpendicular to A B
at B, and from the point D where it intersects the diagonal produced,
draw D C perpendicular to A F produced. Then A B C D will be similar to
A E F G. All rectangles having their diagonals in the same line are
similar.

[Illustration: FIG. 20.]


_To describe a regular Pentagon on a given line._--Let A B (Fig. 20) be
the given line. Bisect A B at C, draw C F perpendicular to A B, and make
C D equal to A B. Draw A D and produce it indefinitely; make D E equal
to half A B. From A as a centre, with A E as a radius, describe an arc
cutting the perpendicular C D in F; and from A F and B as centres, with
radius A B, describe arcs cutting each other in G and H; join A G, B H,
F G and F H; then A G F H B will be the pentagon required.

[Illustration: FIG. 21.]


_To describe a regular Hexagon._--With a radius equal to the length of
one side of the required hexagon, describe a circle (Fig. 21), and set
off the same radius round the circumference of the circle, which will be
thus divided into six equal parts. Join the points thus found, and the
required hexagon will be completed as A B C D E F.

[Illustration: FIG. 22.]


_To draw a Parabola, the base and height being given._--Let C A (Fig.
22) equal half the base, and C D the height. From the point D draw D E
parallel and equal to A C, and from the point A draw A E parallel and
equal to C D. Divide D E and A E similarly, making the end E of A E
correspond to the end D of E D. Through 1, 2, &c., in DE draw 1, 1; 2,
2, &c., parallel to D C. Join D to the several points 1′, 2′, &c., in
A E. The parabola will pass through the points of intersections of these
lines with the verticals drawn from D E to C A.

[Illustration: FIG. 23.]

[Illustration: FIG. 24.]

[Illustration: FIG. 25 _a_.]

[Illustration: FIG. 25 _b_.]


_To draw an Ellipse._--I. By means of a piece of string and pins. Place
the diameters A B and C D (Fig. 23) at right angles to each other, and
set off from C half the major axis at E and F; then will E and F be the
two foci in the ellipse. Fix a pin at E and another at F; take an
endless string equal in length to the three sides of the triangle E F C
and pass it round the pins, stretch the string with a pencil G, which
will then describe the required ellipse. II. From the centre O (Fig. 24)
describe a circle of the diameter of the minor axis of the required
ellipse. From the same centre, describe another circle with a diameter
equal to its major axis. Divide the inner circle into any number of
equal parts as 1, 2, &c., and through these points draw radii cutting
the outer circle in 4, 3, &c. From 1, 2, &c., draw horizontals, and from
3, 4, &c., draw perpendiculars cutting each other in E F, &c.; the curve
traced from C through the points C E F A, &c., will complete the curve
of the required ellipse. III. Let A B (Fig. 25 _a_) be the major and C D
the minor axis of the required ellipse. On any convenient part of the
paper draw two lines F G, F H (Fig. 25 _b_) at any angle with each
other. From F with the distance E C or E D, the semi-axis minor,
describe an arc cutting the lines F G, F H, in I and K; and from F with
the distance E A or E B, the semi-axis major, describe the arc L M. Join
I M, and from L and K draw lines parallel to I M, cutting F G, F H, in N
and O. From A and B (Fig. 25 _a_) set off the distance F N (Fig. 25 _b_)
in points N′, and from these points as centres, with F N as radius,
describe an arc of about 15° on each side of the major axis. From C and
D (Fig. 25 _a_) set off on the minor axial line the distance FO (Fig.
25 _b_) in points O′, and from these points as centres, with radius FO,
describe arcs of about 15° on each side of the axis C D. To obtain any
number of intermediate points take a slip of paper (Fig. 25 _a_) and
mark upon one edge, with a sharp-pointed pencil, 1, 3, equal to the
semi-axis major, and 2, 3, equal to the semi-axis minor. If the slip of
paper be now applied to the figure and moved over it in such a manner
that the point 2 is always in contact with the major axis, and the point
1 in contact with the minor axis, the outer point 3 will describe a
perfect ellipse, any number of points in which can be marked off as the
“trammel” is moved into successive positions.

For this last method, which in practice is by far the best, we are
indebted to Binns’ ‘Orthographic Projection.’

[Illustration: FIG. 26.]


_To construct a Semi-Elliptical Arch._--The span A B (Fig. 26) and rise
C D being given, divide C A and C B into any number of equal parts.
Through the point D, draw E F parallel to A B, and from the points A and
B erect the perpendiculars A E and B F. Divide A E and B F similarly to
C A and C B. Produce C D and make C G equal C D. From D draw lines to
the points 1, 2, 3, &c., in the lines A E and B F; also from G draw
lines through the points 1, 2, 3, &c., in the line A B, and produce
these lines until they cut those drawn from D to the corresponding
numbers in A E and B F. Through the points thus obtained draw the curve
of the ellipse.

[Illustration: FIG. 27.]


_To draw the Gothic Equilateral Arch._--From the points A and B (Fig.
27), with radius A B equal to the span, describe the arcs B C and A C.
By joining C to A and B we obtain an equilateral triangle from which
this arch derives its name.

[Illustration: FIG. 28.]


_To draw the Gothic Lancet Arch._--In this arch, the centres E and D
(Fig. 28) from which the arcs are struck, are situate outside of and in
a line with the points of springing A and B; thus it is constructed on
an acute-angled triangle, as will be seen by joining C to A and B.

[Illustration: FIG. 29.]


_To draw the Gothic Obtuse Arch._--This arch, called sometimes the
Drop-Arch, is constructed on an obtuse-angled triangle; the centres E
and D (Fig. 29) being situate below and within the points of springing A
and B.

[Illustration: FIG. 30.]


_To draw the Gothic Tudor Arch._--On the line of springing A B (Fig.
30), take any two points as F and G, so that A F is equal to G B. Draw
F E and G D cutting each other on the bisecting line through C; from F
and G, with radius F A or G B, describe the short arcs, and from E and
D, with radius E C or D C, describe the arcs meeting in C.

[Illustration: FIG. 31.]


_To draw the Moorish Horse-Shoe Arch._--The centres E and D (Fig. 31)
from which the arcs forming this arch are struck, are situate above and
within the points of springing A and B. One of the most graceful forms
of this arch is obtained when the height of the points E and D above the
line of springing and their distance from the bisecting line through C
are equal to one-third of the span A B.

[Illustration: FIG. 32.]


_To draw the Gothic Ogee Arch._--The most pleasing form of this arch is
that constructed on an equilateral triangle, in the following manner.
Having drawn the equilateral triangle A B C (Fig. 32), draw F G parallel
to A B. Bisect the sides A C and C B and produce the bisecting lines to
F G and H, which will complete the triangle F G H similar and equal to
the triangle A B C. From H, with radius H A or H B, describe the arcs A
E and B D, and from F and G, with the same radius, describe the arcs E C
and C D.

[Illustration: FIG. 33.]

[Illustration: FIG. 34.]


_To draw the Roman Cyma Recta and Cyma Reversa._--Join A B (Fig. 33) and
bisect A B in C. From the points C and B, with the distance B C,
describe arcs cutting each other in E; and from A and C, with the same
radius, describe arcs cutting each other in D; from D, with the same
radius, describe the arc A C, and from E describe the arc C B. The
projection of the upper end of the curve over the under, as F B, is
generally equal to the height, A F, of the moulding. The same
description applies to the Cyma Reversa (Fig. 34) letter for letter.

[Illustration: FIG. 35.]


_To draw the Gothic Trefoil._--Having drawn the equilateral triangle
A B C (Fig. 35), bisect the angles and produce the bisecting lines
D E F which will bisect the sides of the triangle in G H I. From A B and
C as centres, with radius A H or A I, equal to half the side of the
triangle, describe the arcs K L M, and those concentric with them, and
from the centre O of the triangle describe the outer circles and
concentric arcs, which will complete the figure.

[Illustration: FIG. 36.]


_To draw the Gothic Quatrefoil._--Draw the square A B C D (Fig. 36);
bisect the sides of the square at I K L M and produce the bisecting
lines to E F G H. From the angles A B C D of the square as centres, with
radius A I or A M equal to half the side of the square, describe the
arcs P N R S, and draw the outer concentric arcs. The circles,
completing the figure, are drawn from the centre O of the square.

[Illustration: FIG. 37.]


_To construct the Gothic Cinquefoil._--Having drawn the regular pentagon
A B C D E (Fig. 37), bisect the angles and produce the bisecting lines
to F G H I K, which will cut the sides of the pentagon in _a_, _b_, _c_,
_d_, _e_. From A B C D and E as centres, with radius A _a_ or A _b_,
equal to one-half of the side of the pentagon, describe the arcs
L M N P R, and draw the outer concentric arcs and those concentric with
them. The circles are drawn from the centre O of the pentagon, as in the
preceding example.


SECTION III.--LINES, DOTS, AND THEIR COMBINATIONS.

All kinds of drawings are made up of lines and dots; these are the
constituent parts, the materials which the draughtsman has to employ. It
is therefore essential that he should make himself acquainted with their
various forms and uses, and familiar with those means of producing them
which experience has shown to be the best, before commencing the study
of the principles by which the representation of an object is
delineated. And moreover, it is desirable that he should acquire a
familiarity with the operations required in the delineation of isolated
objects, previously to making any attempt to place them in combination
for the purpose of producing a complete drawing. The student will,
therefore, do well to study carefully and to practise diligently the
forms and examples given in this Section.


_Straight and Curved Lines._--All straight lines, however short, should
be ruled, whether they be drawn with the pencil or the pen. Pencil
lines, which are intended to serve merely as guides to the pen, should
be drawn lightly, as otherwise it will be difficult to rub them out
without injuring the ink. They should also be drawn a little beyond the
point at which the line is required to terminate, because the
intersection of the lines at that point makes it more distinctly
visible, and there is, consequently, less danger of passing beyond that
point or of stopping short of it when inking in. It is very important
not to stop short of the required length when ruling a straight line
with a pen, for it is extremely difficult to lengthen the line
subsequently without leaving the join visible. An accurate line cannot
be drawn unless the point of the pencil or the pen be kept close up to
the rule, and to do this the top should be inclined a little outward.
Before inking in a line that has been drawn in pencil, the indiarubber
should be passed lightly over it, to remove the particles of lead
adhering to the paper, for if these particles are allowed to remain,
they get between the nibs of the pen and prevent the ink from flowing
freely. The chief difficulties in ruling a straight line with the pen
are, to keep it of a regular thickness throughout, and, when numerous
parallel lines have to be drawn, to keep them at equal distances apart.
To draw an even line, a first requisite is that the pen be in good
condition. Frequently it will be found when drawing fine lines that the
pen ceases to mark before the end of the line is reached, and as we have
already said, it is very difficult to join a line without leaving
visible traces of the operation. To remedy this defect, the pen must be
reset as described in Section I. If a very hard pencil has been used, or
if the pencil has been pressed heavily upon the paper, the pencil line
will lie in a groove in the paper, and as the nib of the pen will not
touch the bottom of this groove, the line drawn will be ragged. Another
cause of unevenness is unduly pressing the pen against the rule; this
pressure closes the nibs, and besides producing an irregularity in the
thickness of the line, is very apt to cause a blot by forcing out the
ink, which adheres to the rule when brought into contact with it. To
prevent this, care should be taken to press the pen very lightly against
the edge of the rule. A pen is manufactured by Stanley, of Holborn,
London, which has the back nib much stiffer than the other, so that all
danger of defect from this cause is removed by the construction of the
instrument. To ensure a good line, the pen should rest lightly upon the
paper, and the handle of the pen should make the same angle with the
paper from the beginning to the end of the line. A considerable amount
of practice is required to accomplish this, and to acquire the habit,
the same attention should be given to the pencil as to the pen. The
ability to draw a number of parallel lines at equal distances apart
without measuring requires considerable training of the eye, and this
training can be obtained from practice alone. This ability must be
acquired before anything further is attempted, and the student who
spends a good deal of time in its acquisition may have the satisfaction
of knowing that while he is going through this somewhat monotonous
practice, besides exercising himself in drawing accurate lines, he is
acquiring a correctness of eye and a power of hand that will be of
incalculable service to him later.

The straight line, besides being used for the outlines of regular
objects, is employed conventionally for various purposes. When it is
required to show an object in section, the part in section is covered
with straight and parallel lines drawn at an angle of 45° and at equal
distances apart, as in Fig. 38. To represent standing water, as ponds
and lakes, horizontal straight lines are drawn parallel to each other
and at equal distances apart over the surface, as shown in Fig. 39.

[Illustration: FIG. 38.]

[Illustration: FIG. 39.]

Curved lines, when arcs of circles, are drawn by the compasses. Other
curves are drawn by hand through points previously found. To draw the
curve correctly through these points, unless they be very numerous, a
knowledge of the nature of the curve is necessary, which the draughtsman
should in all cases endeavour to obtain. When the curved line is long,
it is usually inked in with the drawing pen, with the aid of an
instrument called the French curve, or cardboard moulds cut for the
purpose; but for short lines an ordinary fine-pointed steel-pen point,
or better, a good quill is used. In general, all lines drawn by hand
should be drawn _towards the body_, as a better command of the pen can
be obtained in that direction than in any other. In inking in curves by
this means, the draughtsman should proceed continuously along the
pencil-drawn line by partly repeated touches with the pen point, so held
that the divided points of the pen may follow partly in the same track.
Each touch should be made about one-thirtieth of an inch in length, and
it should begin and end fine. Each succeeding touch must begin half its
length back, so that the line is advanced by one-sixtieth of an inch. In
map drawing all irregular lines are drawn in this way. Tracing maps
will afford the student excellent practice in this mode of using the
pen.

[Illustration: FIG. 40.]


_Lines of uneven thickness._--Though generally a line is required to be
of even thickness throughout, cases sometimes occur in which a variation
in the thickness may be made to express some feature or quality of the
landscape. The usual application of this kind of line is to mark the
outline of rivers, lakes, and ponds, as shown in Fig. 40. The drawing of
such a line presents no difficulty; the increased thickness is produced
by going over those parts of the line again with the pen. Care must,
however, be taken not to make a sudden increase in the breadth of the
line, but to begin and end imperceptibly.

[Illustration: FIG. 41.]

[Illustration: FIG. 42.]

[Illustration: FIG. 43.]


_The Broken Line._--The broken line, shown in Fig. 41, is of frequent
occurrence in all kinds of drawings. In architectural and engineering
drawings it is usually employed in roofs, as in Fig. 42, and for water
in sections, as in Fig. 43. It is also used in combination with other
lines for various purposes. In drawing a succession of broken lines,
care must be taken not to allow the break in one line to be immediately
over that in another. The effect may be varied considerably by
increasing or diminishing the extent of the break. As in section lining,
the lines should be at regular intervals apart, and be all of the same
degree of fineness. Broken lines are sometimes used upon the face of
stone buildings, instead of marking in the joints and etching or
colouring. In such a case the breaks are long, and the lines widely
spaced.

[Illustration: FIG. 44.]


_The Dotted Line._--Of still more frequent occurrence is the dotted
line. There are two kinds of dotted lines, distinguished by the shape of
the dot, and known as the _long_ and the _round_ dotted line. These are
shown in Fig. 44, as well as a combination of the two.

The round dotted line is of very general application. In architectural
and mechanical drawings, it is used to distinguish hidden parts, and to
mark the path of a moving piece in a machine. In plans, it is used to
show the position of proposed works, to denote the walks through
pleasure grounds and gardens, to indicate lines chained over in
surveying, and frequently for other purposes, at the pleasure of the
draughtsman. The long dotted line is employed to mark the boundaries of
a township, the navigable channel of a river or creek, and in
large-scale maps to show farm and bridle roads, footpaths, and the
divisions of land among different tenants. The combination of the long
and round dotted lines is used for the boundaries of a parish. Another
combination of two round and one long dots, or sometimes of three round
and one long, is used to denote proposed railways, canals, roads, and
other similar works.

To draw a good dotted line requires some care. The difficulty lies in
keeping the dots at equal distances apart, and in making them equal in
size; and unless both these conditions are fulfilled, the line will not
present a pleasing appearance. To obviate this difficulty, an instrument
is sold by mathematical instrument makers, called the dotting or wheel
pen. But it requires very great care in using, as otherwise it
frequently happens that the ink escapes from it and spoils the drawing.
For this reason, its use has been generally abandoned by draughtsmen.
But if the instrument were better constructed and carefully handled, it
might be made to do good service.

[Illustration: FIG. 45.]

[Illustration: FIG. 46.]

[Illustration: FIG. 47.]

[Illustration: FIG. 48.]

[Illustration: FIG. 49.]


_Combinations of Straight, Broken, and Dotted Lines._--Combinations of
the foregoing lines are used for various purposes. Some draughtsmen
employ alternate, full, and dotted lines, to denote wood in section, as
in Figs. 45 and 46; when wood is used in combination with iron or other
metal, this is a very good way of distinguishing it. Wood-graining,
though not made up of straight, broken, or dotted lines, yet partakes
somewhat of the nature of all three kinds, and may therefore be
introduced here. Oak-graining is shown in Fig. 47, and fir-graining in
Fig. 48. The former is executed with the drawing pen, and requires some
care and practice; the latter is most readily done with a common pen or
a crow-quill. End wood is grained as shown in Fig. 49. The spring bows
are very suitable for drawing in the circles, as a certain degree of
turn to the nut will open the ink leg to the required distance after
drawing each circle. A few broken wavy lines, called _shakes_, radiating
from the centre, produce a good effect. When several pieces of end wood
come together, the centres in each should not be in the same relative
position.

[Illustration: FIG. 50.]

Cultivated land is represented by alternate broken and dotted lines,
suggesting furrows, as shown in Fig. 50. For the sake of variety, these
lines are put in in sets, and in different directions, one set being
usually parallel to one side of the enclosure. The lines are first ruled
in continuously with the pencil, and the broken and dotted lines
afterwards drawn in over them by hand. The portions of the broken lines
must in this case be short, and the breaks still shorter. The dots must
be fine and close together; they are made by touching the paper with
the point of the pen, and immediately lifting it off without _dragging_
it over the paper. All round dots must be made in this way.


_The Wavy Line._--The wavy line is very important in topographical
drawings, as it is employed to represent running water, and frequently
large bodies of standing water to which motion is communicated by the
wind, as lakes and the sea. These rippled lines are intended to
represent the ripples in the water, a purpose which they fulfil in a
very pleasing manner. They must, however, be well executed, or the
pleasing effect will not be produced. The operation of drawing these
lines is usually regarded by the draughtsman as a tedious and an
uninteresting one. But such ought not to be the case, for there is ample
scope in it for the exercise of the taste and the judgment, and in
proportion to the taste displayed and the judgment exercised, will be
the effect of the work when executed.

[Illustration: FIG. 51.]

Fig. 51 shows the manner of employing these lines. In representing water
by this means, the lines should be drawn from the shores towards the
middle of the stream or lake, and never from the middle outwards, for if
the latter mode of proceeding be adopted, the proper graduation of the
spaces between the lines becomes impossible. The shore line, or outline
of the water, should be a moderately thick line, and of uniform
thickness throughout. The first shading line may be of nearly the same
thickness as the shore line, and it must be drawn as near to it as
possible. Also this shade line, as well as all subsequent ones, must
follow exactly all the windings of the shore line; this is essential to
a correct expression. To effect this with accuracy, care should be taken
to make the space between the shore and the shade line a fine _white
line_. The second shade line must be drawn a little finer than the
first, and at a slightly increased distance from it. This gradual
diminution of the thickness of the lines, and increase of the spaces,
must be continued to the middle of the current. The last line in the
middle of a piece of water must always return to itself. When the
shading lines meet the margin of the drawing, they should terminate in
it, that is, they should be drawn out to the margin as though they had
been continued beyond and cut off.

These lines require to be drawn clean, and to do this the hand must be
kept steady. This steadiness may be obtained by taking a very short hold
of the pen, and resting the middle finger upon the paper. The lines, as
we have already said, should be drawn towards the body, the drawing
being turned about as required to facilitate this, and the last line
drawn must be always kept on the left of the one being drawn. By this
means the last line and the point of the pen are kept constantly in
sight. It is also important that the lines should be completed
successively, rather than that several should be carried on at once,
because if the latter mode of working be adopted, the eye is apt to
become confused by the different intervals, and an uneven distribution
of the lines is the result. A principle to be attended to is that every
line shall return to itself, spirals being altogether inadmissible. The
distance of the lines apart and their thickness are expressive of the
character of the object; thus, in a small pond, for example, they will
be fine and close together; in a large pond or a lake they will be
thicker and more widely spaced; and in the open sea they will be made to
present a bold appearance by increasing still more their thickness and
the distance between them.

[Illustration: FIG. 52.]


_Grass-land._--Various combinations of lines and dots are used,
conventionally, to represent certain natural features of common
occurrence. As far as convenient execution will allow, these signs are
made to resemble the objects denoted. Thus the sign for grass-land
consists of groups of short lines, arranged like tufts of herbage, as
shown in Fig. 52. Each tuft is composed of five or seven lines
converging towards a point situate below the base, the middle line being
the longest, and the outside ones mere dots. In drawing these groups,
the base must be kept quite straight, and parallel to the base of the
drawing whatever the shape of the enclosure may be. Beginners usually
experience considerable difficulty in keeping the base straight, the
tendency being to make it curved. Great care is needed to distribute the
groups evenly over the paper, and to avoid the appearance of being in
rows, for the latter arrangement is destructive of that natural aspect
which this sign otherwise possesses.

[Illustration: FIG. 53.]


_Swamps and Marshy Ground._--As the surface of marshy ground consists of
water and grass, a combination of the signs for these objects is
employed to represent it. An illustration of this is given in Fig. 53.
The lines representing the water should always be ruled parallel to the
base of the drawing, and they should be grouped in an irregular manner
so as to leave small islands interspersed throughout the locality. These
islands should be covered with grass, and to show them out more
distinctly, there should be nothing but water immediately around them.
The division between the land and the water should be sketched in
lightly before proceeding to rule in the lines. Sometimes dotted lines
are used for the water, but full lines are to be preferred. The addition
of a tree here and there improves the appearance of a drawing. A
distinction is frequently made between a swamp and a marsh by watering
the former more extensively than the latter. In drawing in marsh land,
care should be taken to make the fineness of the lines in accordance
with the scale of the map, as otherwise an offensive appearance will be
produced. This caution applies equally to all signs.

[Illustration: FIG. 54.]


_Sand and Gravel._--Sand and gravel are represented by dots, the dots
being made larger for the latter than for the former, as shown in Fig.
54. Simple as the operation of filling in these dots is, it is one that
requires some degree of care. Beginners are apt to mar the appearance
of their drawings by inattention in this respect. The dots should be
made in the manner already described when speaking of the dotted line,
that is, the point of the pen should be brought slowly down upon the
paper, and lifted without dragging it; and no dot should be made without
a deliberate intention respecting its position. All arrangement in rows
must be carefully avoided. In sand-hills, the slopes should be made
darker than the level parts by placing the dots closer together. Mud in
tidal rivers may be represented by very fine dots placed close together.

[Illustration: FIG. 55.]

[Illustration: FIG. 56.]


_Woodland._--Trees are generally shown in plan (as in Fig. 55). The
outline is circular in character, and, to have a good effect, it should
be made up of simple curves firmly drawn; small indentations should be
avoided as bad. A few touches of the pen are given on the interior and
towards the shadow. The latter is cast by parallel rays of light
inclined 45° to the horizon, and is detached from the outline of the
tree. When the scale is large, the shadow will be elliptical in form,
but in small scales it will become a simple circle. In representing
woodland, the trees and masses of trees should be disposed in every
possible variety of position, care being taken, however, to avoid all
regular figures and arrangements in lines. In parks and gardens, where
the arrangement of the trees is artificial, it is usual to represent a
grove in a rectangular form. Orchards are shown by placing single trees
with their shadows at the points of intersection of two sets of
equidistant parallel lines drawn at right angles to each other. These
lines are drawn in pencil and afterwards erased. Some draughtsmen prefer
to draw trees in elevation, as shown in Fig. 56. This method allows the
various kinds of trees to be distinguished on the plan, and gives scope
to artistic skill.

[Illustration: FIG. 57.]


_Uncultivated Land._--Uncultivated land, other than woodland, is
represented by drawing bushes in plan, similar to trees, but of smaller
dimensions, and mixing tufts of grass with them, as shown in Fig. 57.

[Illustration: FIG. 58.]


_Contour Lines._--Suppose a cone A B C (Fig. 58) cut at regular vertical
intervals apart by a series of horizontal planes 1, 2, 3. The
intersections of these planes with the surface of the cone will give
lines upon that surface; and it is obvious that the cone may be
represented in plan by the projection of these lines, as shown in the
figure. To obtain this projection, draw the horizontal line D E, and
from the apex of the cone and from the intersections of the cutting
planes let fall vertical lines. From the point where the line from the
apex meets the line D E as a centre, with radii equal to the distances
from this point to those where the lines from the sections meet D E,
describe circles. These circles will be the horizontal projections of
the lines on the surface of the cone produced by the cutting planes; and
these lines are called _contour lines_. Also it is obvious that, from
the plan of the cone so obtained, we may as readily project the
elevation, provided we know the vertical distance apart of the sections
denoted by the contour lines. To obtain the elevation, we have only to
draw horizontal lines at the given distance apart, and from the points
in D E erect perpendiculars to meet them. Lines drawn through the points
of intersection will give the elevation of the cone. To find the
inclination of the surface of the cone, upon _a b_, a portion of the
normal D E, as a base, erect a perpendicular _b c_, equal in height to
the distance of the sections apart, and join _a c_. The hypothenuse _a
c_ then represents that portion of the surface of the cone which is
included between the two contour lines, and of which the angle of
inclination is _b a c_. The space between two contour lines is called a
horizontal zone.

The cone being a regular figure, its contour lines are circles. For
irregular figures, the contour lines will be irregular curves. The
regular inclination of the surface of the cone causes the projections of
the contour lines to be at equal horizontal distances apart. But when
the inclination varies, the horizontal distance between the contour
lines also varies, the distance decreasing as the inclination increases.
Thus the method of representing objects in plan by contour lines, not
only gives the correct form of the object, but shows the relative
inclination of every portion of its surface. This may be clearly seen in
Figs. 59 and 60, the former of which is a representation in plan by
contour lines of an irregularly shaped object, and the latter an
elevation of the same object projected from the plan.

[Illustration: FIG. 59.]

[Illustration: FIG. 60.]

The system of representation by contour lines is generally adopted by
topographers to distinguish and define the variation of the surface of
the ground in regard to hill, valley, and plain. By intersecting a
mountain, for example, by a sufficient number of horizontal planes, its
correct form may be delineated, and the declivity of its surface
accurately shown. The relative declivity of any portion of its surface
is indicated by the difference in the horizontal distance of the curves
apart; and by constructing a triangle upon a normal to the upper curve
in the manner already described for the cone, the absolute slope at any
point between any two curves may be readily determined. The ground is
supposed to slope uniformly from one curve or contour line to the next.
Such, however, is rarely the case; but provided the curves are taken at
frequent intervals, the error is of no practical importance. Hollows are
represented in the same way; and whether the representation is that of a
hill or a hollow, is known from the other parts of the map. Thus, if
Fig. 59 represent a hill, the vertical projection will be as shown in
Fig. 60; but if it denote a hollow, the outer curve must be projected
highest, and the vertical section will be Fig. 60 inverted. In practice
the contour lines are numbered, the number of any contour indicating its
height above a plane of reference called a datum plane. The vertical
distance of the contour lines apart varies with the character of the
ground and the object of the survey; but it is seldom less than 25 feet.
The lines are obtained by the surveyor by fixing a number of points on
the same level by means of instruments.


SECTION IV.--COLOURS.

The preceding Section treats exclusively of representation by lines and
dots, or that mode of delineating objects and natural features known as
line or pen drawing. There is, however, another mode of representation
by means of colours that is fast coming into general use. This latter
mode is far more expressive than the former, and, besides affording a
wider scope for artistic effect, shows with greater distinctness and
precision the character of the object represented. For these reasons it
is almost always adopted for plans of estates and geological sections,
and also very frequently for other kinds of topographical as well as for
engineering and mechanical drawings. The colours used for this purpose
are not applied in the way the artist applies them; but they are laid on
in thin washes to produce a faint tint rather than a body of colour. The
process is called tinting or flat-washing, and though it cannot be
described as a work of art, considerable practice and skill are
requisite to execute it properly.


_Flat-tints._--A drawing to be coloured must be previously stretched and
gummed to the board, in the manner described in Section I. Unless the
paper be prepared in this way, it will remain blistered after being
wetted by the laying on of the tints. The lines of the drawing must be
very fine, and the ink, though black, should not be thick. Great care
should be exercised in drawing in the outlines, that there be always a
piece of clean paper between the hand and the drawing, for the least
degree of greasiness will prevent the colours from working freely.
Should the surface of the paper, however, from inattention to this
matter, or from accident, become slightly greasy, the defect may be
partially remedied by adding a little prepared ox-gall to the water with
which the colours are mixed. When all the outlines have been drawn in
and the pencil lines erased, the drawing is prepared for the colouring
by being _washed_. The washing is effected by passing a soft sponge well
saturated with clean water gently and rapidly over the surface. The
purpose of this washing is twofold; first, to remove those portions of
the ink which a wet brush would detach from the paper in laying on the
colours, and which, by becoming mixed with the tint, would injure its
purity; and second, to damp the surface of the paper in order to prevent
the colour from drying too rapidly. The latter is an important matter,
for if the tint which is being applied dries quickly, it is impossible
to unite the edges properly, and the tint, especially if the surface be
large, will have a cloudy and blotchy appearance. As the operation of
washing renders the paper too wet to immediately receive the colour, it
must be allowed to remain in a perfectly horizontal position for a short
time to dry, and during this time any tendency to dry unequally must be
corrected by means of blotting-paper. While the paper is drying, the
tints may be prepared.

To ensure satisfactory results, care must be taken in the preparation
and preservation of the tints. They should never be made by artificial
light, and a sufficient quantity should be made at first to cover all
the portions required, as it is very difficult to match a tint exactly.
When a drawing is several days in hand, it is best to prepare a fresh
tint for every coat, for the colours will change in the course of a day
or two, even if protected from the light. A few drops of water should be
added now and then, to make up for the loss by evaporation, especially
in warm weather. Tints left to dry upon the palette should never be wet
up again for use, but they should be washed clean out and a fresh tint
made; if this precaution be not attended to, the colour will not be
pure. When a tint is to be mixed, the end of the cake of colour should
be moistened and allowed to soften for a minute or two, as this will
cause it to rub smooth and free from fragments. The palette should then
be moistened and the end of the cake rubbed gently and evenly upon it
till a sufficient quantity of colour has been obtained, which may be
added to the requisite quantity of water by means of a brush. A
precaution necessary to be observed is never to rub one colour down upon
another, as it will probably be laid aside to dry with the other colour
on it. The brush used should be as large as the nature of the work will
allow, and it should be of the best sable hair; the quality is judged by
the length of the hair, the longest and stiffest being the best.
Draughtsmen frequently do all their work with a couple of sable brushes
attached to one holder, one being for colour and the other for water; in
this case the brushes should be of different colours to prevent
mistakes.

The art of laying on a flat-tint consists in allowing the coloured water
to flow equally over the paper, which thus becomes uniformly tinged. To
facilitate this, the surface of the drawing should be inclined towards
the draughtsman at an angle of about five degrees during the process of
laying on the colour. Having taken as much colour on the brush as it
will safely carry without dropping, the operation of applying it should
be begun in the upper left-hand corner, the brush being carried along
towards the right, so as to make the colour lie neatly along the upper
outline. The brush should then be struck unhesitatingly from right to
left and from left to right alternately, so as to bring the colour down
in horizontal bands or stripes, taking care not to pass the brush a
second time over the same surface during the same wash, and to control
it neatly within the proper limits. If the surface of the paper be in
this way kept well wetted with the colour, or if, in other words, a flow
of colour be kept in motion with the point of the brush, the tint can be
carried on with perfect continuity. It is important to keep as nearly as
possible the same quantity of colour in the brush until the lower
outline is nearly reached, when the quantity must be diminished so as to
finish at the lower outline without a great excess of tint, for the
excess must be taken up by a damp brush. No accumulations should be
allowed to take place anywhere, as on drying, these places would show a
darker tint. When the colour has once flowed over the surface, the tint
is finished, and must not, as we have said, be touched a second time,
for any attempt to remedy defects while the colour is drying will only
make them worse. Generally it will be found that the more quickly a tint
is laid on, the better is its appearance. A little practice will enable
the student to lay on a wash in the proper manner, but to keep within
the outlines is a matter of greater difficulty and one that requires
some dexterity in the handling of the brush. If the boundary should be
exceeded, a finger of the left hand should be instantly applied to brush
the colour back. Though the foregoing directions can be followed
strictly only on large surfaces, the principles involved in them must in
every case be observed.

The alternate or double tint consists of two colours applied
alternately, their edges being made to blend into each other. The
application of the double tint involves no particular difficulty. Having
prepared two tints of equal intensity and provided a brush for each, lay
on one of the colours at the upper outline of the figure, and before
this dries, take the brush charged with the other colour, and run round
its edge, allowing them to blend together. Repeat the first tint in the
same manner, and continue the tints alternately till the surface is
covered. The forms of the masses of each colour should be varied, and
not made in stripes or spots, but irregularly clouded.

All flat-tints should be made very light, and intensity of colour should
be produced by repeating the wash. As every surface looks better with
two washes than with only one, the strength of the tint should be such
as to allow two coats to be laid over the lightest parts. If the colours
have been laid on too dark, or the general effect be uneven and
disagreeable, the defect may be remedied by sponging. This operation
should be performed with a close-grained 6-inch sponge, and be commenced
at the upper end of the inclined board. A basin of clean water having
been provided, and an empty basin to receive the dirty water from the
sponge, first moisten all the white surface of the paper to prevent the
tint taken off by the sponge from adhering to it; then, having filled
the sponge with water, pass it gently to and fro across the sheet. Press
out the dirty water into the basin, refill the sponge, and repeat the
operation until hardly any tint comes off. Sponging after five or six
coats have been laid on generally improves the appearance of a drawing;
it softens down asperities, and makes the tints blend into each other;
the surface of the paper also takes the tints more readily after
sponging.

Small defects may frequently be remedied by a process called stippling.
This consists in making a number of dots with the point of a brush
containing an almost imperceptible quantity of colour. The process,
though a tedious one, produces a very beautiful effect, similar to that
of dotted engravings. Excesses beyond the boundary lines may be washed
out with the water-brush, and the stains removed by a piece of clean
blotting-paper. White spots left in a tint may be filled up, after the
tint is dry, with the point of the brush; but care must be taken not to
touch beyond the edges of the tint, as that would double the intensity
at the edges and produce a ring.

All flat surfaces in a drawing should be lighter or darker, in
accordance with their distance from the eye. In laying on flat-tints
when the surface is not in shade, it must be borne in mind (1) that all
surfaces which are parallel to the plane of the picture, and therefore
equally distant from the eye, should receive a tint of uniform
intensity; (2) that those surfaces which are farthest from the eye
should receive the darkest tint; and (3) that surfaces which are
inclined to the plane of the drawing should receive a tint of varying
intensity, the depth of the tint increasing as the surface recedes from
the eye. When the surfaces are in shade, the converse of these rules
holds good.


_Conventional Colours._--In representing objects by means of colours,
the natural colours of the objects are in some cases adhered to; and in
others, for the sake of greater distinctness, a conventional colour is
adopted. In engineering, architectural, and mechanical drawings, the
latter mode is nearly always resorted to, while in plans of estates the
former is very frequently employed. Unfortunately, practice is not
uniform among draughtsmen in the conventional use of colours; but the
following Table shows the colours mostly employed, and represents the
general practice.

  Carmine or crimson lake     For brickwork in plan or section to be
                              executed.
  Prussian blue               Flintwork, lead, or parts of brickwork to
                              be removed by alterations.
  Venetian red                Brickwork in elevation.
  Violet carmine              Granite.
  Raw sienna                  English timber, not oak.
  Burnt sienna                Oak, teak.
  Indian yellow               Fir timber.
  Indian red                  Mahogany.
  Sepia                       Concrete works, stone.
  Burnt umber                 Clay, earth.
  Payne’s grey                Cast iron, rough wrought iron.
  Dark cadmium or orange      Gun metal.
  Gamboge                     Brass.
  Indigo                      Wrought iron--bright.
  Indigo, with a little lake  Steel--bright.
  Hooker’s green              Meadow land.
  Cobalt blue                 Sky effects.
  And some few others occasionally for special purpose.

Sections are represented either by lines of the colour drawn with the
pen or the point of the brush, or by a darker shade of the colour. In
mechanical drawings, sections are frequently shown by ink lines drawn
over the colour.

In plans and maps, as we have said, some attempt is made to give the
true appearance of things. As this--which may be called the natural mode
of representation--allows more scope for artistic skill than the
conventional, a great deal must be left to the judgment and the taste of
the draughtsman. But there are general principles and features that may
be laid down and described, and such are the following:--


_Water._--For water, a flat-tint of pure indigo is used. To produce the
clear, transparent effect of water, there should be two coats of the
tint, which, to allow of this, must be very light coloured.


_Grass-land._--For grass or cleared land, a flat-tint of green is
employed. This tint is composed of indigo and gamboge, and should be of
a lively hue, which may be produced by giving predominance to the
gamboge. Care must always be taken in preparing greens for maps and
plans, that the blue be kept subordinate to the yellow; for a
predominance of the former colour produces a _cold_ quality, which is
utterly destructive of that _natural_ appearance it is intended to give.
The intensity of the tint for this and for other purposes should be such
as to distinguish it clearly from others, and to allow somewhat for
fading, without masking any of the details of the drawing; and it must
be clear and transparent. We may here remark that all tints which are
much extended should be _balanced_, that is, no one should obtrude
itself upon the eye by its relatively too great intensity.


_Marsh._--Marsh and swamp are represented, as in line drawing, by a
combination of the signs for water and grass-land. The tints are laid on
horizontally, that is, parallel to the base of the drawing. They are
not, however, laid on in bands or strips across the drawing, but are
made to project in irregular points from each side, with here and there
a long and narrow patch to represent an island. The land should cover a
larger portion of the space than the water, and it should be washed in
first, care being taken to make the white spaces left for the blue
colour resemble the green in form, which spaces should project their
horizontal points into the green as the latter projects its points into
the white. The outer limits of a marsh should consist of an outline of
projecting green points. The land portion of the marsh is finished by
drawing a light shading line of indigo and burnt sienna along the lower
edge of the green. This line must be drawn _upon_ the edge and not
_against_ it upon the white space. In washing in the water, care must be
taken not to overlay the edges of the green. A good effect is produced
by introducing a tree here and there upon the land.


_Sand and Gravel._--Sand is shown by a flat-tint of yellow ochre. Sand
and gravel are represented by dotting the flat-tint with burnt sienna by
means of the point of the brush held in a vertical position. Stones and
rocks in sand should be first outlined with the pen in burnt sienna and
sepia in equal proportions, and afterwards filled in with the brush with
the same colour.


_Mud._--In the survey of rivers, creeks, and coasts, it frequently
becomes necessary to show tracts of mud between the lines of high and
low water. For this purpose a flat-wash of sepia or Indian ink may be
used dotted with Indian ink of greater intensity. The dots in this case
must be very minute and thinly placed, and they should be evenly
distributed. A fine-pointed pen will be found more effective in putting
in these dots than the point of the brush.


_Woodland._--To represent woodland, a flat-tint of green is first laid
over the ground, as for grass-land. The groups and masses of trees are
next drawn in outline, in the manner described in the last Section, with
a hard and sharp lead pencil, or with a pen and pale ink. To fill in
these outlines, a colour made up of indigo and gamboge in the same
proportions as the ground tint, but of greater intensity, is laid on the
lower and right-hand portion of each tree and mass of foliage, so as to
occupy about two-thirds of the figure. The remaining portion, which will
be the side towards the light, is then touched with an orange tint
composed of gamboge and burnt sienna. It only remains to add the shadow.
As the light is supposed to enter the drawing in parallel rays from the
upper left-hand corner, the shadow of every object will surround its
lower and right-hand outlines. It is laid close up to the outline in
masses of foliage; but for single trees, as in orchards, it is detached.
The form of the shadow was described in the last Section. To produce the
shadow, the same tint is used as for the ground, two or three successive
applications being sufficient to increase the intensity to the requisite
degree; or a neutral tint may be used, composed of indigo, burnt sienna,
and a little lake. After the shadow has been put in, the outlines on
that side should be strengthened by going over them again with the pen.
By drawing the trees in elevation, an opportunity is afforded for the
display of artistic skill far greater than the foregoing method admits
of. When drawn in this way, the work partakes somewhat of the nature of
landscape painting.


_Cultivated Land._--Cultivated land is represented by a flat-tint of
burnt sienna.


_Uncultivated Land._--Uncultivated land or brushwood is represented by a
double tint of green, as for grass-land, and burnt sienna, as for
cultivated land, laid on in the manner already described for the double
tint. As this is the only double tint used, it may be made, if thought
desirable, with alternate green and crimson lake.


_Buildings._--Buildings, including all structures of masonry, as
bridges, locks, walls, and such like, are coloured with crimson lake,
and shadowed with a neutral tint composed of indigo, burnt sienna, and a
little lake, as given above for forest land.


_Roads and Streets._--Roads and streets, and generally all those
portions of a drawing not particularly described, are tinted with yellow
ochre.


_Fences._--Hedges are represented by green dots, varied in size for
bushes; stone or brick walls, by a line ruled in red; and wooden fences
by lines of neutral tint, either ruled or drawn in by hand, according as
the line is to be straight or otherwise. In every case the shadow must
be put in.

In determining the intensity of the various tints employed on a
topographical drawing, care must be taken that everything be “in
keeping.” A cardinal rule of art is that nothing shall unduly obtrude
itself; and in a coloured plan, _spottiness_, as it is called, should be
studiously avoided. Forest, brushwood, and cultivated land, should be
represented by tints of about equal intensity, and the same equality may
be observed for grass-land, marsh, water, and sand, but the intensity
should be less than in the former case. Tints that are of small extent
may be a little exaggerated in intensity for the purpose of giving them
greater distinctness, especially when the object represented is a
building. Gardens and orchards require a little exaggeration in depth of
tint, to distinguish them from the surrounding country; but care must be
taken not to make the distinction too marked. It will generally be found
conducive to a maintenance of “keeping,” to lay the lightest tints on
first.


SECTION V.--SHADING.

In mechanical and architectural drawings, shade lines must be considered
rather as embellishments than constituent parts of the drawing. They
are, however, frequently employed; and as their incorrect use may
deceive the eye with respect to the intention of the designer, it
becomes an important matter to know when to apply them with propriety.

[Illustration: FIG. 61.]

[Illustration: FIG. 62.]


_Application of Shade Lines._--As we have already explained, the light
is supposed to fall upon the objects in a drawing in parallel rays from
the upper left-hand corner for elevations, and from the lower left-hand
corner for plans. To determine whether or not a given line should be a
shade line, we have only to ascertain whether or not the light,
introduced in such a manner, falls upon that edge of the object which
the line represents. All those parts of a body upon which the rays of
light fall directly, are said to be _in light_; all those parts upon
which the rays of light do not fall directly, are said to be _in shade_;
and those parts of a surface which are deprived of light by another body
intercepting the rays, are said to be _in shadow_. These definitions
should be borne in mind. Lines representing the boundaries of surfaces
in light should be fine lines, and lines representing the boundaries of
surfaces in shade should be thick or shade lines. Let it be required,
for example, to determine the shade lines of the cube shown in elevation
in Fig. 61. The extreme rays of light falling upon the cube meet the
edges in _b_ and _c_; hence the surfaces _a b_, _a c_, are in light, and
the surfaces _d b_, _d c_, are in shade. The foregoing rule will thus
make _a b_ and _a c_ fine lines, and _d b_ and _d c_ shade lines. If the
cube were turned so that _a b_ should be at right angles to the rays of
light, the extreme rays would fall on the edges _a_ and _b_, and the
middle ray which now falls on _a_ would fall on the middle of the line
_a b_. The rays immediately beyond those which are arrested by the edges
_a_ and _b_, may be considered to pass along in contact with the
surfaces _a c_ and _b d_; and these surfaces must, therefore, be
regarded as in light. Thus we shall have in this case the lines _a b_,
_a c_, and _b d_, fine lines, and the line _c d_ a shade line. It is the
practice of some draughtsmen to make _a c_ and _b d_ in such cases a
medium line, and the practice has propriety to recommend it. The
foregoing explanations of the shade lines in the elevation of the cube,
render any further remarks concerning those in the plan, Fig. 62,
unnecessary. In practice, whether or not a surface is in light may be
determined by placing the set square of 45° against it.

[Illustration: FIG. 63.]

[Illustration: FIG. 64.]

The same principles are observed in the end elevation of the hollow
cylinder, shown in Fig. 63. The extreme rays meet the circumference in
the points _a_ and _b_; consequently the surface _a c b_ is in light,
and the surface _a d b_ is in shade. The middle ray meets the surface
perpendicularly at the point _c_, which will be the lightest part of
that surface; similarly, _d_ will be the darkest part. To show this, the
shade line must be gradually increased in thickness towards the point
_d_. The shading of the inner circle will be the converse of the outer.
Fig. 64 shows a plan of the same object.

[Illustration: FIG. 65.]


_Cylindrical Surfaces._--Let _a b c d_, Fig. 65, be a plan, and _k l n
m_ an elevation of a cylinder. The portion _a c b_ is in light, and the
portion _a d b_ is in shade, of which latter portion _a_ and _b_ are the
edges. From the points _a_ and _c_ draw vertical lines _e f_, _g h_.
Then will _e f_ be that part of the cylinder upon which the light falls
perpendicularly, or the lightest part, and _g h_ the edge of the surface
in shade, or that portion of the surface of the cylinder that would cast
a shadow upon the plane of projection. Hence this will be the darkest
part, and consequently it is obviously improper to make the line _k l_ a
shade line. This demonstration, which is given by Binns, shows that
shade lines must never be applied to cylindrical surfaces. If this
principle be observed, cylindrical may be readily distinguished from
flat surfaces.

[Illustration: FIG. 66.]


_Shading Lines._--Shade lines are applied only to the edges or
boundaries of surfaces; when lines are put upon a surface to show the
effects of light and shade, they are called shading lines. The use of
the latter is determined by the same principles as that of the former;
indeed, a shade line may be practically considered as an end view of a
number of shading lines. In Fig. 66, which is an elevation of a hexagon,
the surface _c_ is in shade, and to represent this surface correctly, it
must be made darker than the others. This darkening of the surface is
effected by drawing the shading lines heavier or closer together, or by
both of these means combined. The surface _b_ is in light, but the rays
fall upon it obliquely; the shading lines on this surface will therefore
be lighter and more widely spaced than on _c_. The surface _a_ is also
in light, and receives the rays normally, that is, the direction of the
rays is normal to the surface. Hence this surface will reflect most, or,
in other words, will be the lightest. This is shown by making the
shading lines still lighter, and spacing them still more widely than
those on _b_. The greatest care is needed in applying shading lines to
keep their thickness and the spacing regular, as an error in these
respects will frequently produce an effect quite opposed to what is
intended.

[Illustration: FIG. 67.]


_Shading Lines on Cylindrical Surfaces._--If the demonstration
previously given concerning shade lines on cylindrical surfaces be
understood, the application of shading lines to these surfaces will
present no difficulty. The darkest and the lightest part of the cylinder
having been determined, and in practice this can be accomplished with
sufficient exactness by the eye, the shading lines are applied according
to the principles explained above with respect to the hexagon. The first
shading line is drawn upon the darkest part; and each successive line on
each side of this first line is drawn lighter and spaced more widely
than the preceding. At the lightest part, a clear space is left to
represent the reflexion of the rays that occurs strongly there, and
beyond this part the shading is made equal to that of the corresponding
part on the other side. The thickening of the lines is effected by going
over them a sufficient number of times. Fig. 67 shows a vertical and a
horizontal cylinder shaded in this manner. In outline drawings of
machinery, this mode of shading with parallel lines is frequently
resorted to.

[Illustration: FIG. 68.]

[Illustration: FIG. 69.]

[Illustration: FIG. 70.]

[Illustration: FIG. 71.]

It will be evident, on reflection, that when the cylindrical body stands
parallel with the direction of the rays of light, as shown in Fig. 68,
the lightest part will be in the middle, and the shade will increase in
intensity as it approaches the edges. The shading of the interior of a
cylinder is, as we have already remarked when treating of shade lines,
the converse of that of the exterior. This is shown in the sectional
elevation, Fig. 69. When parallel with the direction of the rays of
light, as in Fig. 70, the internal shading is the same as the external.
On bright circular surfaces, such as that of a circular saw, or the
polished end of a shaft, the light is radiated from the centre, as shown
in Fig. 71. This mode of shading is strictly in accordance with the
appearance presented by such surfaces. It may be remarked here, that if,
through inadvertence, any part should be made too dark, the error may be
corrected by darkening all the other parts in a corresponding degree.

[Illustration: FIG. 72.]


_Shading Lines in Topographical Drawings._--The shading lines put upon
mechanical drawings are merely accessories used for purposes of
embellishment. But in topographical drawings, shading lines are applied
to give expression, and they constitute an essential element in the
representation. We have shown how undulations of the ground,
constituting hill and valley, are represented by contour lines. But it
is obvious that these lines furnish information respecting the
character of the surface only at those points through which they pass.
Thus we are necessarily left in ignorance of the irregularities existing
between any two successive contours. To supply this information which
the contours fail to give, shading is resorted to. Another important
object of hill shading is to represent the surface of the ground
conventionally in a manner that will immediately afford an idea of its
character without the aid of regular contours. The method adopted
consists in employing lines varying in their thickness and in their
intervals apart according to the slope of the ground to be represented.
This method is based upon the principle of the horizontal contours,
which is to give to the same vertical interval the same absolute amount
of shade, whatever the inclination of the ground may be. The shading
lines are used, as we have said, to fill in the features of the ground
between contours already fixed; and to ensure accuracy and uniformity in
the representation, a “scale of shade” is employed. The accompanying
Fig. 72 shows the standard scale of shade adopted by the Council of
Military Education, and made use of for all the Government surveys. The
second and the fifth columns of this scale show the spacing of the
hachures and their thickness for different angles of slope, while the
first and the last columns show the number of hachures to be
interpolated between contours at every 25 feet vertical intervals,
supposing the slope to be uniform. The slope is denoted both by the
number of degrees in the angle it makes with the horizontal, and by a
fraction showing the ratio of the vertical height to the base in a
right-angled triangle, the hypothenuse of which is the slope in
question.

The scale of shade is constructed for a horizontal scale of six inches
to the mile, and the amount of shade has been chosen with a view of
producing the best possible artistic effect. Of course, the most
satisfactory results, both artistically and practically, will be
obtained when the ground is delineated to this scale, but it can be
readily applied to any other scale. For example, the horizontal interval
for a slope of ¹⁄₂₀, corresponding to a vertical interval of 25 feet,
will be 20 × 25 = 500 feet, which, on a scale of six inches to a mile,
will be represented by a length equal to ⁵⁰⁰⁄₅₂₈₀ × 6 = 0·566 inches. In
this case, therefore, supposing the slope of the ground to be uniform
between two given contours 25 feet apart, we should represent it by
means of the hachures shown opposite a slope of ¹⁄₂₀, continued over a
space of 0·566 inch.

[Illustration: FIG. 73.]

In topographical drawings, the light is supposed to fall vertically upon
the surface; hence a level surface will reflect all the light that falls
upon it, while one of 45° will not reflect any.

The drawing of the hachures presents certain difficulties of execution
that can be overcome only by continued practice and careful attention to
the modes of proceeding which experience has proved to be the most
effectual. Thus an important rule is always to draw “from left to right
and downwards.” To allow this to be done, the drawing must be placed
with the summit of the hill to the left hand, and be turned round as the
work progresses. The hachures should always be commenced at the crest of
the hill, working outwards towards the foot of the slope. They should be
drawn firmly, and of a length varying from ¹⁄₄ inch to ³⁄₄ inch,
according to the width of the zone, that is, according to the greater or
less degree of the slope, as shown in Fig. 73, at _a_, _b_, _c_, _d_.
When the hill is steep, the lines are made short and thick, and when the
declivity is less, they are made longer and lighter, becoming fine and
clean as the level is approximated to. A difficulty with beginners is
to press upon the pen equally from the beginning to the end of the
stroke, the tendency being to press more heavily towards the end, thus
producing a whip-like appearance quite opposed to artistic effect, and
conveying a false impression of the character of the ground. A good
effect is produced by imparting a slightly tremulous motion to the pen
when drawing the hachures. The form of the hill being accurately defined
by the pencil contour lines, it is not necessary that the accessory
curves formed by the shading lines should be rigorously continuous, and
indeed a much better effect, artistically, is gained by avoiding such a
manner of drawing them. The various sets of lines must be placed
together, end to end, in such a way that the groups or sets shall not be
separated by a vacant space, nor overlap each other. Care must be taken
that the junctions of sets in two contiguous zones do not form a
continuous line from one zone to the other, but everywhere “break
joint.” Each zone must be filled in before the next lower one is
commenced, the drawing being turned as the work progresses to allow the
rule enunciated above of “from left to right and downwards” to be
complied with. The distance between the shading lines must be increased
or diminished according as the width of the zone varies, so as to divide
the space equally; and on reaching the part where the lines were begun,
the ends must be brought neatly together. As this can be most
satisfactorily accomplished where the lines come close together, it is
best to begin at the steepest part of the slope.

In taking a set of hachures round a sharp bend, as in the case of a spur
or a ravine, a practical difficulty occurs, which difficulty is
increased as the angle becomes more acute. The most effective way of
overcoming this difficulty is to draw a pencil line down the spur or
re-entering angle, as shown at A B and C D in Fig. 74, and to mark off
on this line, at the proper intervals, small arcs of the same radius, as
near as can be judged by the eye, as the curve of the contour line. The
sets of hachures on each side may then be drawn to these arcs. Guiding
lines, as _a b_, _c d_, _e f_, and _g h_, should be drawn at right
angles to the general direction of the contours to ensure the hachures
being correctly placed before and after rounding the angle. For this
method of carrying a set of hachures round a sharp curve, we are mainly
indebted to Lieut. R. Pulford’s ‘Theory and Practice of Drawing.’ When
this method is not employed, the hachures must be drawn on each side of
the angle first, and those for the angle filled in separately.

[Illustration: FIG. 74.]

Great care must be taken in filling in the zones formed by the contour
lines, that the drawing when finished do not present the appearance of
separate layers or bands; for such an appearance is not only quite
opposed to artistic effect, but it conveys a false notion of the
character of the ground. The successive zones are not separate portions
of the surface, but each is a continuation of the one adjoining it. The
great principle to be observed in this, as in all matters of hill
shading, is that changes of slope are gradual. When the contours are
only pencilled in as guide lines to be afterwards erased, the
above-mentioned defect may be avoided by drawing the hachures over them,
without reference to exact spacing. But when, as is usually the case in
regular surveys, the contours are inked in in dotted lines, the only
means of avoiding it is to space the hachures on each side of a contour
line at the same distance apart.

The student of map drawing should practise assiduously this system of
shading in detached portions before undertaking the delineation of a
complete hill. For such exercises, either a soft, medium-pointed steel
pen, or a quill may be used.


_The Vertical System of Shading._--The foregoing system of shading is
known as the _Horizontal_, and is now generally employed in this country
for all kinds of surveys. There is, however, another system much used
abroad, and frequently adopted here for engraved maps. In this system,
which is known as the _vertical_, the shading lines are made to radiate
from or converge into the curved parts of a hill, according as they
project or re-enter. Such lines are called _lines of greatest descent_;
they are supposed to describe the same course that water would describe
if allowed to trickle in streams down the slopes, and hence they exhibit
both the _direction_ and the _degree_ of the slope. Having the
horizontal sections given, we may obtain a complete knowledge of the
direction in which the ground slopes by drawing perpendicular to them
any number of lines of greatest descent; the degree of declivity is
expressed by purely conventional means. The means adopted for this
purpose are of two kinds. One depends upon the principle of vertical
illumination, in which the maximum quantity of light is reflected
upwards to the eye by a horizontal surface, and a minimum by a surface
inclined 45° to the horizon. This is the English and German convention,
and it lays more stress upon the proportions of black to white in
indicating the degree of slope, than upon the distance between the
shading lines. The other convention, which is the French, on the
contrary, makes its expression depend more upon the distance between the
lines of greatest descent than upon the shade of colour produced, though
in this also the tint is graduated from dark to light, according to the
degree of declivity.

A scale of shade is used for this system, founded upon the same
principles as that already given for the horizontal system. The scale
adopted is due originally to Major Lehmann, of the Saxon Infantry; but
it has received some modification to adapt it to the requirements of
practice. Fig. 75 shows Lehmann’s scale. It is constructed for every 5°,
from a level up to a slope of 45°, which is the steepest slope at which
earth will stand. Each division of the scale corresponding to a given
slope is subdivided into nine parts, to show the proportions of black to
white. For a level, the whole of these spaces are left white; for a
slope of 5°, the proportion is one black to eight white; for a slope of
10°, two black to seven white; and so on up to 45°, for which slope we
have all black. The longitudinal divisions of the scale below that
against the outer edge A B contain the same proportions of black to
white, but equally distributed to show the mode of applying it. Thus, in
the division _o p r s_, corresponding to a slope of 5°, the single black
space is, in E F G H, divided into two equal parts and distributed; in
G H I K, these two parts are again equally divided and distributed; and
so on throughout the other longitudinal divisions. If now the scale be
cut off along the line L M, the part L M C D will constitute a scale,
the graduated edge L M of which will furnish us with a means of marking
off the distance between the centres of the shading lines.

[Illustration: FIG. 75.

LEHMANN’S SCALE OF SHADE.]

To find the proportion of black to white in the foregoing scale for any
given slope:--Subtract the given inclination from 45° for a denominator,
and put the given inclination for a numerator. In the scale, as drawn in
the figure, the variations are by 5°; but it is obvious that a scale may
be drawn in the same manner to mark smaller variations, if thought
desirable.

In applying this method in the United States’ Coast Survey, it was
remarked that “this scale of shade does not represent slopes greater
than 45°, thereby limiting the graphic capabilities and effect of the
map. It also makes the slopes too dark as they approach the inclination
of 45°, and does not well represent slopes of less than 5°, which latter
it is often desirable and necessary to express distinctly.” The
following modification was therefore made:--

  -----------------+-------------
                   |Proportion of
        Slope.     +------+------
                   |Black.|White.
  -----------------+------+------
   2¹⁄₂° or  2³⁄₄° |   1  |  10
   5°     „  6°    |   2  |   9
  10°     „ 11°    |   3  |   8
  15°     „ 16°    |   4  |   7
  25°     „ 26°    |   5  |   6
         35°       |   6  |   5
         45°       |   7  |   4
         60°       |   8  |   3
         75°       |   9  |   2
  -----------------+------+------

By this scale, the slighter slopes are represented distinctly. For
slopes less than 16°, the shades are darker than in Lehmann’s scale;
this makes their difference more noticeable. Above 25° the shades are
lighter.

A further modification, which for ordinary purposes possesses the
advantages of simplicity and facility of application, has been made in
England, and very generally adopted. This modification consists in
fixing with accuracy only three proportions of black to white for three
medium slopes, as follows:--

  -------+-------------
         |Proportion of
   Slope.+------+------
         |Black.|White.
  -------+------+------
  Level  |  ..  | all
  15°    |   1  |  2
  22¹⁄₂° |   1  |  1
  30°    |   2  |  1
  45°    | all  | ..
  -------+------+------

A scale of shade may at once be constructed from this Table, by assuming
the thickness of the shading line for the medium slope of 22¹⁄₂°, which
thickness must be suited to the scale, and to the degree of fineness and
finish it is intended to give the drawing. Generally, if the lines have
such a relation to the scale of the drawing as to present a
well-connected appearance, it will be found that fewer shading lines and
a rather coarse texture will conduce more to clearness of expression
than a finer texture, which tends to produce a dryness of style. In
shading to this scale, it should be applied to the drawing wherever the
slope corresponds to one of the three on the scale. Intermediate slopes
are indicated by graduating the thickness of the shading lines. In all
cases a good deal must be left to correctness of eye and skill of hand.

In the French method, as we have said, the inclination is expressed by
the distances between the centres of the lines of greatest descent. The
limits of the slopes that can be represented by this method are, ¹⁄₁ or
45° for the greatest and ¹⁄₆₄ or 0° 53′ 43″ for the smallest. The
largest scale that will admit of conveniently drawing the lines of
greatest descent is ¹⁄₆₀₀ full size, or about 8³⁄₄ feet to a mile. The
vertical distance between the horizontal sections is generally taken as
1 yard. Hence to a scale of ¹⁄₆₀₀ the least width of zone will be ⁶⁄₁₀₀
inch, and the greatest ⁶⁄₁₀₀ × 64 = 3⁸⁴⁄₁₀₀ inches.

The distance between the shading lines is reckoned from centre to
centre, and is determined by the rule:--To the distance between the
upper and the lower curves of any zone add ³⁄₁₀ of an inch; a sixteenth
part of this sum will be the proper interval for the shading lines. The
distance is measured along the line of greatest descent. Thus, if the
inclination be ¹⁄₆₀ and the scale ¹⁄₆₀₀, the width of zone will be ·06 ×
60 = 3·60 inches, and by the rule we have

  3·60 + ·3   3·9
  --------- = --- = 0·244 inch.
      16       16

Another rule is:--To a fourth of the distance between the upper and the
lower curves of any zone, add ⁷⁵⁄₁₀₀₀ of an inch; a fourth part of the
sum will be equal to the interval.

The thickness or breadth of the lines is made to vary directly as the
inclination to assist in expressing the declivity. This thickness is
determined by the following rule. For a slope of ¹⁄₁ the thickness of
the shading lines is equal to ²⁄₃ of the distance between their
centres, and this thickness will diminish with the inclination down to
¹⁄₆₄, where the lines will be as fine as they can be drawn. In a slope
of ¹⁄₁ this rule will always make the breadth of the shading lines twice
that of the white space contained between them.

To represent declivities by the vertical system of shading a
considerable amount of practice is required. This practice should be
commenced by drawing repeatedly the scale of shade, and gradually
applied, as proficiency is attained, to the varying inclinations of a
hillside. Having the horizontal sections of the hill given, the degree
of slope should be written upon it in pencil in as many places as is
necessary. The distances between the centres of the shading lines may
then be marked off upon the upper curve of the zone from the scale of
shade, and the lines of greatest descent drawn through the points thus
determined. The exact proportion of black to white being then adopted,
the colour will express the degree of the slope, and the line of
greatest descent will show its direction.

The principle of making the shading lines longer on a gentle slope than
on a steep one should be adhered to generally; but in this matter much
must be left to the judgment and the skill of the draughtsman.
Frequently on slight inclinations it will be desirable to divide and
subdivide the zone by medial lines, as shown in Fig. 76, and on very
steep slopes the shading lines may be drawn over two or more zones. For
ordinary scales the extremes of length may be fixed at ¹⁄₆ of an inch on
the steepest slopes, and ³⁄₄ of an inch on the gentlest.

[Illustration: FIG. 76.]

It is not necessary to repeat the process of construction for every
line, such a mode of proceeding would be too laborious and slow. It will
be sufficient to determine the lines in this exact manner at those parts
where the greatest changes of slope occur. Thus a group should be
constructed in each zone where the slope is greatest and another where
it is least, after which a few intermediate ones may be put in. The
vacancies may then be filled in, taking care to graduate the changes in
passing from group to group. By this means we do not, of course, get a
mathematically exact representation of the surface, but it is
sufficiently accurate for practical purposes.

When the preparatory pencil lines have been drawn in and the spaces for
the shading lines laid off by dots, the shading should be commenced at
the steepest part of the upper zone. The lines should be drawn firmly
from curve to curve, taking care to make each row terminate evenly at
the lower edge; they must always be drawn downwards and from left to
right, proceeding in this direction round the zone till the point of
setting out is reached, where the joining must be carefully effected.
This can always be done most neatly where the lines are thickest, as we
have previously pointed out. The succeeding zones should be filled up in
the same manner. As changes must be gradual in every direction, care
must be taken to make the contiguous zones blend into each other. When
it is required to pass from a light zone to a darker one beneath it, the
lower ends of the lines in the light zone should be thickened a little,
so as to meet the upper ends of the lines in the dark zone with nearly
the same colour. The upper ends of these latter lines should also be
slightly lightened. The lines of one zone must not be continued into
those of the next. Even on a uniform slope such a prolongation of the
lines would produce a hard appearance, which should be avoided. But in
the case of a conical hill, like that shown in Fig. 77, it would give
rise to an error in principle; for soon after leaving the summit we
should have too few lines of descent. When the hill has been covered
with shading lines, the base and the summit must be softened off by
tapering the lower end of each line at the base, and the upper end of
each line at the summit. To give the taper to the latter, the drawing
should be turned upside down.

[Illustration: FIG. 77.]

When the curves are parallel or nearly so, the shading lines are
straight, and also nearly parallel. But when the curves depart widely
from each other, the shading lines will themselves have a slight
curvature, for being lines of greatest descent, they must be normal to
the curves. In such cases, a number of normals should be put in at short
distances with the pencil, as shown in Fig. 78, to serve as guides to
the shading lines. The foregoing directions for shading a hill apply
equally to the shading of a hollow, the shading lines in which are
converging.

[Illustration: FIG. 78.]

Occasionally short slopes steeper than the “natural slope” of 45° will
be met with. Such being exceptions to the law of slopes, are marked in
an exceptional manner. When the surfaces of these slopes are of earth,
they are shown by black lines drawn parallel to the horizontal curves,
and when of rock, by black lines drawn in all directions, not
intersecting, but abutting abruptly upon each other in short heavy
masses, as shown in Fig. 78.


_Shading in Colours._--Frequently in topographical drawings, and still
more frequently in mechanical drawings, colour is resorted to to produce
the effect of shading lines. As the principles according to which colour
is applied for this purpose are the same as those which determine the
use of shading lines, there remains little to be said on this matter
beyond describing the modes of applying the colour.


_Hill Slopes._--In representing slopes, the tint employed to give the
effect of that produced by the ink lines already described is composed
of indigo and burnt sienna, and is applied as a flat-wash. A little lake
is added to neutralize the greenish hue of this tint when it is to be
laid over sand or cultivated ground. The different degrees of intensity
required to express the inclination are produced by repeating the wash
over those parts which are darker than the rest. To accomplish this
neatly, the darker portions must be washed in first, so that the final
washings may cover the whole surface, and the edges of each successive
wash must be softened off or blended into the next with a brush and
clean water. In shading hills, the paper along the crest of the slope
should be first moistened with the water-brush, and before it dries, the
laying on of the colour should be begun on the moistened part, and
proceeded with down the slope. The effect of representing hills by this
method, which is a very expeditious one, is much improved by adding
light shading lines with the pen, either in pale ink, or a mixture of
indigo and burnt sienna. The ground is always covered with its
appropriate sign before the shading tint is laid on.


_Cylindrical Surfaces in Mechanical Drawings._--In shading cylindrical
surfaces and drawings generally, three methods are employed. One of
these is known as _softening off_, and is employed on finished drawings
of machinery. For shading by this method, a brush called a softener is
required; this has a brush at each end of the handle, one being larger
than the other. Having moistened the paper, and filled the smaller brush
with colour and the larger one with water, a narrow strip of colour is
laid along the darkest part of the cylinder, and immediately after,
while the colour is quite moist, the water-brush is drawn along one edge
of the strip and then in like manner along the other, so as to cause the
colour to flow over that portion of the surface which has been damped.
The brush is then wiped upon a cloth and drawn lightly down the edge to
take up the superfluous water. The colour should be light to begin with,
and the quantity to be taken in the brush must be determined by
experience. The same remark applies to the water-brush, for if too
little be used the colour will not spread sufficiently, and if too much,
the colour will be diluted and rendered uneven. These operations of
laying on the colour and softening off are continued until the
cylindrical appearance has been produced. Each succeeding coat should be
laid on before the preceding one is quite dry, as the colour will spread
more evenly over a damp surface. The previously applied coat must,
however, have been sufficiently absorbed not to wash up, or a clouded
appearance will be the result.

Another method, known as the French, consists in applying a narrow strip
of colour to the darkest part, and overlaying this with other strips,
each wider than the one previously laid on. To regulate the breadth of
the strips, a number of meridian lines are drawn upon the cylinder. When
shaded in this manner, the cylinder presents the appearance of a polygon
rather than that of a cylinder.

The third method, by reason of the facility it affords of producing
effect, is very suitable for large drawings and diagrams for
illustrating papers and lectures. In shading according to this method, a
thick line or a narrow strip of very thick and black Indian ink is laid
on the darkest part of the cylinder with the point of the brush. The
breadth of the strip will be regulated by the diameter of the object,
and it should be previously lined out in pencil. When dry, a damp brush
is passed over it so as to remove the sharp edges of the strip, and to
cause the ink to run slightly over the moistened surface of the paper.
The flat colour washes are then applied as required, the washes being
carried over the black strips, which will be further reduced in tone by
a portion of the ink mixing with the colour.

In shading, it will be found convenient to keep the light side of the
object next to the operator, as it is easier to wash towards the body
than from it with the water-brush. The brush should be held in as nearly
a vertical position as possible, as it is more easy, when that position
is observed, to keep within the boundary lines.




PART II.--APPLICATIONS.


SECTION I.--LETTERING, BORDERING, AND NORTH POINTS.


_Lettering._--The lettering of a plan, map, or drawing of any kind,
occupies a prominent and conspicuous position, and may be considered as
forming an essential part of the drawing. It is, therefore, obvious that
the character of the lettering, and the degree of finish introduced into
its execution, will have an important influence on the general
appearance of the drawing. Nothing detracts more from the value of a
map, considered as a work of art, than a bad style of lettering, while,
on the other hand, a well-chosen and well-executed style is both
pleasing to the eye, and produces on the mind an impression of accuracy
in the more important features of the work. Hence it is not merely
desirable, but necessary, that the draughtsman should acquire the
ability to form letters correctly and neatly, especially if he be
engaged on topographical drawings, into which lettering enters very
largely.

The formation of letters requires great attention and long practice. It
is not a matter in which much assistance is to be derived from
descriptions or written instructions of any kind; practice alone from
good models will give the requisite skill. The difficulty of forming the
letters correctly and of uniform dimensions may, however, be
considerably lessened by using guide lines drawn in pencil, to be
afterwards erased. Such lines are called _construction lines_, and the
mode of employing them is shown in Plate 4. A careful study of this
Plate will give the student a clear understanding of the use of these
lines, which could not be imparted by pages of description. A reference
to the letters B, E, and T, in connection with the construction lines
will show most readily the nature and the degree of assistance afforded
by the latter.

In making capitals, each letter must be sketched in pencil; the outline
must then be drawn in ink with a firm and steady line, and afterwards
filled up with the pen. In forming the small roman and italic letters,
three construction lines are drawn, the lower two to limit the height of
the ordinary letters, and the upper one to limit the height of such
letters as _d_ and _l_, and the capitals. The heavy parts of these
letters are made at once by a bold pressure of the pen. The curved
portions should be carefully distinguished from the straight. The
letters _a_, _c_, _g_, _o_, _s_, &c., for example, are composed wholly
of curved lines. They should be drawn symmetrically, and their width
should be only a little less than their height. The round portion of the
_g_ should not quite reach to the lower line. A perfect regularity
should be maintained throughout the letters, as the beauty of their
appearance depends greatly on this. Care must also be taken, in italic
writing, to keep the inclination the same everywhere. Manuscript
lettering should be more extended than the clear roman or italic type,
for crowding greatly mars its appearance.

The character of the letters employed should be in accordance with that
of the drawing upon which they are to appear. Thus for engineering and
mechanical drawings, there is nothing more suitable generally than the
plain block letter. But on drawings of a more artistic and ornamental
character, a more elaborate form of letter may and should be used. And
of these elaborate forms, there will always be one more suitable than
the rest to the particular character of the drawing. The choice of this
form is a matter to be left entirely to the judgment and the taste of
the draughtsman.

Another matter on which the draughtsman will have to exercise his
judgment is the _size_ of the letters employed. This must manifestly be
in accordance, first, with the character of the object denoted, and,
second, with the scale of the drawing. With regard to the former of
these conditions, it is obvious that propriety will demand a larger
letter for the city than the town, for the town than the village, for
the village than the farm, and for the mansion than the gate-lodge. This
propriety of relative importance must be everywhere observed. The
different types of lettering are arranged in the order of importance as
follows:--1, The upright capital; 2, the inclined capital; 3, the
upright roman, or ordinary small type; and 4, the small italic. The
draughtsman will have to exercise his judgment in suiting the size to
the scale of the map, but the following Table may be taken as a general
guide:--

  --------------+----------------------+-----------------------------
      Scale.    |   Height of Upright  |   Height of Small Roman.
                |       Capitals.      |
  --------------+----------------------+-----------------------------
  ¹⁄₆₀₀, or one |                      |
  inch to fifty |                      |
  feet.         |Six-tenths of an inch.|Twelve-hundredths of an inch.
  ¹⁄₂₆₄₀, or two|                      |
  feet to a     |                      |
  mile.         |Four     „      „     |Eight       „        „
  ¹⁄₅₂₈₀, or one|                      |
  foot to a     |                      |
  mile.         |Three    „      „     |Six         „        „
  ¹⁄₁₀₅₆₀, or   |                      |
  four inches   |                      |
  to a mile.    |Two      „      „     |Four        „        „
  --------------+----------------------+-----------------------------

The thickness of the capital should be one-seventh of the height.

As far as practicable, the lines of lettering should be parallel to the
base of the drawing. Frequently, however, cases will occur in which it
will be desirable to letter in other directions and in curved lines. In
writing along a curved or very irregular outline, the course of a river
or the boundary of an estate, for example, an agreeable effect is
produced by making the lines of lettering conform in some degree with
the outlines against which they are written.

The arrangement of the letters in titles and the effective disposition
of the words are also matters requiring great care and some taste. The
design and the execution of the title afford another opportunity of
enhancing the beauty of a drawing by a display of striking arrangement
and appropriate ornamentation. Plates 7 and 8 show some useful models
for plans, and Plate 25 contains some specimens of flourishes which may
frequently be introduced with pleasing effect. The form which the title
shall assume and the space which it shall occupy must be determined
before beginning to put it upon the drawing. To avoid erasures, it is
well to sketch roughly upon a piece of paper, a trial title, emendations
in which can be easily made. When found satisfactory, draw a vertical
centre line, which should pass through the middle letter of each line.
Apply this centre line to the centre line of the title on the drawing,
and lightly mark in with the pencil the position of each letter. When
this method is not adopted, the middle letters should be put in first
upon the centre line, and the others afterwards inserted from left to
right, and from right to left.

In maps, the title may be placed outside the border if it consist of one
line only, but if it occupy more than one line, it should be placed
within the border. Generally, it should be placed in one of the corners
of the map, and its size should bear some proportion to that of the map.
The letters composing the name of the locality, which is usually the
most important word, should not exceed in height three-hundredths of the
length of the short side of the border. The letters of the other words
will vary in size according to their relative importance.


_Borders._--Plain borders usually consist of two lines, the outer one
heavy, and the inner one light. The heavy line should be equal in
breadth to the blank space between it and the light line, and the total
breadth of the border, that is, of the two lines and the space between
them, should be one hundredth part of the length of the shorter side.
Ornamental corners may be made to embellish a drawing considerably, and
they afford some scope to the fancy and the taste of the draughtsman.
Several examples of borders and ornamental corners will be found in the
accompanying Plates.


_North Points._--The meridian or north and south line is an
indispensable adjunct to every topographical drawing. When the extent of
country represented is considerable, it is usual to make the top of the
map the north, and in such a case the side border lines are meridian
lines. Frequently, however, in plans, the shape of the ground does not
admit of this arrangement, and then it becomes necessary to mark a
meridian on some part of the map. This line is usually made a
conspicuous one, and its north extremity is often ornamented with some
fanciful device. The ornamentation of the meridian line should be in
keeping with the rest of the map. Plate 9 contains several examples
which may be adopted or modified as deemed desirable.


SECTION II.--SCALES.

To all drawings which do not show the full size of the objects
represented, it is necessary to affix the scale according to which the
objects are drawn. Such a scale is called a scale of lengths or
distances, because, by means of it, the distance from one point to
another is ascertained. The scale of distances does not contain very
minute subdivisions, and consequently is not suitable for use in
_constructing_ the drawing. For the latter purpose, another scale,
similarly but more minutely divided, is employed, and is known as the
scale of construction. A familiarity with the modes of constructing both
of these scales should be early acquired by the young draughtsman.


_Scales of Distances._--One means of denoting the scale of a drawing is
furnished by what is called its _representative fraction_, the
denominator of which shows how many times greater the actual length is
than that in the drawing. Thus a scale of ¹⁄₂₄ shows that 1 inch on the
drawing represents 24 inches on the object; in other words, that the
object is twenty-four times larger than the drawing. But in addition to
this representative fraction, it is usual to affix a graduated straight
line, termed a _scale_, for the purpose of conveniently measuring
distances upon it. It is manifest that the unit of length in this scale
must bear the same ratio to the real unit of length that a line in the
drawing bears to the line which it represents. Thus if the
representative fraction be ¹⁄₂₄, 1 inch on the scale will represent 2
feet.

Scales of distances are usually of such a length as to be a multiple of
10 linear units of some kind, as 100 miles, 50 chains, 20 feet; and this
length should also be such as to allow of long lines being taken off at
one measurement. To construct the scale, two light lines should be drawn
at a suitable distance apart, and below the lower of these lines and at
a distance from it equal to one-third of the space between them, a third
and heavy line should be drawn. The primary divisions may then be made
with the compasses in the following manner. Supposing the number of
divisions to be five, open the dividers to what appears to be the fifth
part of the line, and step this distance along the line; if the fifth
step exceed or fall short of the end of the line, close or open the
dividers ¹⁄₅ of the distance, and repeat the trial. This is the quickest
and, for large divisions, the most accurate method of dividing a line.
To render the divisions more distinct, draw a heavy line between the two
light lines in alternate divisions. The left-hand division must be
subdivided into the units or lesser measures of which it is made up. For
example, if the primary divisions are each of 10 feet, the subdivisions
will be feet; if they represent feet, the subdivisions will be inches,
and so on. The subdividing should be performed in the following manner.
Having erected a perpendicular of indefinite length from the left-hand
extremity of the scale, take with the compasses from any scale the
number of divisions into which it is required to divide the part. With
this distance in the compasses, strike, from the first primary or zero
division, an arc cutting the perpendicular, and join the point of
intersection to the centre from which the arc is struck. Thus we shall
have a right-angled triangle formed of the first primary division of the
scale, the perpendicular and the radius, the latter being the
hypothenuse (see Fig. 79). Mark on the hypothenuse the divisions to
which it was made equal, and from the points of division let fall
perpendicular lines upon the scale. These will divide the latter into
the required number of equal parts. The length of the hypothenuse should
be so chosen as to make an angle not greater than 50° with the base.

[Illustration: FIG. 79.]

The total length of the scale will be determined by the greatest length
which it is required to read off at once, and in the following manner.
Thus, let it be required to construct a scale of ¹⁄₂₄, = ¹⁄₂ inch to
the foot, to show 12 feet. Here ·5 inch : _x_ inches :: 1 inch : 12
inches; whence _x_ = 12 × ·5 = 6 inches. This distance of 6 inches must,
therefore, be set off upon the lines intended for the scale, and divided
in the manner described above. Again, to construct a scale of ¹⁄₁₀₅₆₀, =
6 inches to a mile, to show 100 chains. Since 6 inches represents 5280
feet or ⁵²⁸⁰⁄₆₀ = 80 chains, the proportion becomes 6 : _x_ :: 80 : 100;
whence _x_ = ⁶⁰⁰⁄₈₀ = 7¹⁄₂ inches. If the scale is ¹⁄₃₉₆₀ = 16 inches to
a mile, = 5 chains to an inch, and the distance to be shown is 30
chains, we have 1 : _x_ :: 5 : 30; or _x_ = ³⁰⁄₅ = 6 inches. In a scale
of 10 yards to the inch, for example, the representative fraction is 10
× 3 × 12 = ¹⁄₃₆₀. So, on the contrary, ¹⁄₃₆₀ = ³⁶⁰⁄₃₆ = 10 yards to the
inch. Sometimes it is required to construct a comparative scale, that
is, a scale having the same representative fraction, but containing
other units. Thus suppose, for example, we have a Russian plan on which
is marked a scale of _archines_ measuring a length of 50 archines. It is
required to draw upon this plan a comparative scale of yards, upon which
a distance of 50 yards may be measured. The Russian archine = ·777 yard.
Hence we have the proportion 50 : _x_ :: ·777 : 1, whence _x_ = ⁵⁰⁄·₇₇₇
= 64·35 archines. Measure off this length from the Russian scale, and
upon it construct the English scale in the manner already described.
This scale may then be used to measure distances on the plan.

Amongst Continental nations, decimal scales are usually employed, which
are far more convenient in practice than those involving the awkward
ratios of miles, furlongs, chains, yards, feet, and inches. The decimal
scale has also been adopted for the United States’ Coast Survey, the
smallest publication scale of which is ¹⁄₃₀₀₀₀; this is also the scale
of the new map of France.

In choosing a scale, regard must be had alike to the purposes for which
the drawing is intended, and to the nature and the amount of detail
required to be shown. Thus a larger scale is required in plans of towns
than in those of the open country; and the smaller and more intricate
the buildings, the larger should the scale be. Also a plan to be used
for the setting out of works should be to a larger scale than one made
for parliamentary purposes.

The following Tables, given by Rankine in his ‘Civil Engineering,’
enumerate some of the scales for plans most commonly used in Britain,
together with a statement of the purposes to which they are best
adapted.

HORIZONTAL SCALES.

  ----------------------------+--------+--------------------------------
     Ordinary Designation     |Fraction|             Use.
           of Scale.          |of real |
                              | Dimen- |
                              | sions. |
  ----------------------------+--------+--------------------------------
   1.--1 inch to a mile       | ¹⁄₆₃₃₆₀| Scale of the smaller Ordnance
                              |        | maps of Britain. This scale is
                              |        | well adapted for maps to be
                              |        | used in exploring the country.
   2.--4 inches to a mile     | ¹⁄₁₅₈₄₀| Smallest scale permitted by the
                              |        | Standing Orders of Parliament
                              |        | for the deposited plans of
                              |        | proposed works.
   3.--6 inches to a mile     | ¹⁄₁₀₅₆₀| Scale of the larger Ordnance
                              |        | maps of Great Britain and
                              |        | Ireland. This scale, being just
                              |        | large enough to show buildings,
                              |        | roads, and other important
                              |        | objects distinctly in their
                              |        | true forms and proportions, and
                              |        | at the same time small enough
                              |        | to enable the eye of the
                              |        | engineer to embrace the plan of
                              |        | a considerable extent of
                              |        | country at one view, is on the
                              |        | whole the best adapted for the
                              |        | selection of lines for
                              |        | engineering works, and for
                              |        | parliamentary plans and
                              |        | preliminary estimates.
   4.--6·366 inches to a mile | ¹⁄₁₀₀₀₀| Decimal scale possessing the
                              |        | same advantages.
   5.--400 feet to an inch    | ¹⁄₄₈₀₀ | Smallest scale permitted by the
                              |        | Standing Orders of Parliament
                              |        | for “enlarged plans” of
                              |        | buildings and of land within
                              |        | the curtilage.
   6.--6 chains to an inch    | ¹⁄₄₇₅₂ |}Scale answering the same
                              |        |}purpose.
   7.--15·84 inches to a mile | ¹⁄₄₀₀₀ |}Scales well suited for the
                              |        |}working surveys and land plans
                              |        |}of great engineering works, and
                              |        |}for enlarged parliamentary
                              |        |}plans.
   8.--5 chains to an inch,   | ¹⁄₃₀₆₀ | (Scale 8 is that prescribed in
       or 16 inches to a mile.|        | the Standing Orders of
                              |        | Parliament for “cross sections”
                              |        | of proposed railways, showing
                              |        | alterations of roads.)
   9.--25·344 inches to a mile| ¹⁄₂₅₀₀ | Scale of plans of part of the
                              |        | Ordnance survey of Britain,
                              |        | from which the maps before
                              |        | mentioned are reduced. Well
                              |        | adapted for land plans of
                              |        | engineering works and plans of
                              |        | estates.
  10.--200 feet to an inch    | ¹⁄₂₄₀₀ | Scale suited for similar
                              |        | purposes. Smallest scale
                              |        | prescribed by law for land or
                              |        | contract plans in Ireland.
  11.--3 chains to an inch    | ¹⁄₂₃₇₆ | Scale of the Tithe
                              |        | Commissioners’ plans. Suited
                              |        | for the same purposes as the
                              |        | above.
  12.--100 feet to an inch    | ¹⁄₁₂₀₀ | Scale suited for plans of
                              |        |towns, when not very intricate.
  13.--88 feet to an inch, or | ¹⁄₁₀₈₀ | Scale of the Ordnance plans of
       60 inches to a mile.   |        | the less intricately built
                              |        | towns.
  14.--63·36 inches to a mile | ¹⁄₁₀₀₀ | Decimal scale having the same
                              |        | properties.
  15.--44 feet to an inch, or | ¹⁄₅₂₈  | Scale of the Ordnance plans of
       120 inches to a mile.  |        | the more intricately built
                              |        | towns.
  16.--126·72 inches to a     | ¹⁄₅₀₀  | Decimal scale having the same
       mile                   |        | properties.
  17.--30 feet to an inch     | ¹⁄₃₆₀  |}
  18.--20 feet to an inch     | ¹⁄₂₄₀  |}Scales for special purposes.
  19.--10 feet to an inch     | ¹⁄₁₂₀  |}
  ----------------------------+--------+--------------------------------

VERTICAL SCALES.

  ------------+--------+------------------+-------------+---------------
    Ordinary  |Fraction| Horizontal Scales|Exaggeration.|
   Designation| of real|  with which the  |             |     Use.
   of Vertical| Height.| Vertical Scale is|             |
      Scale.  |        | usually combined.|             |
  ------------+--------+------------------+-------------+---------------
              |        |                  |      from   |
  1.--100 feet| ¹⁄₁₂₀₀ |¹⁄₁₅₈₄₀ to ¹⁄₁₀₅₆₀| 13·2 to 8·8 | Smallest scale
  to an inch  |        |                  |             | permitted by
              |        |                  |             | the Standing
              |        |                  |             | Orders of
              |        |                  |             | Parliament for
              |        |                  |             | sections of
              |        |                  |             | proposed
              |        |                  |             | works.
  2.--40 feet | ¹⁄₄₈₀₀ |¹⁄₄₈₀₀  to ¹⁄₃₉₆₀ | 10 to 8·25  | Smallest scale
  to an inch  |        |                  |             | permitted by
              |        |                  |             | the Standing
              |        |                  |             | Orders of
              |        |                  |             | Parliament for
              |        |                  |             | cross sections
              |        |                  |             | showing
              |        |                  |             | alterations of
              |        |                  |             | roads.
  3.--30 feet | ¹⁄₃₆₀  |¹⁄₃₉₆₀  to ¹⁄₂₃₇₆ | 11 to 6·6   |}Scales
  to an inch  |        |                  |             | suitable for
  4.--20 feet | ¹⁄₂₄₀  |¹⁄₃₉₆₀  to ¹⁄₂₃₇₆ | 16·5 to 9·9 |} working
  to an inch  |        |                  |             | sections.
  ------------+--------+------------------+-------------+---------------

The vertical scale, or scale of heights, is always much greater than the
horizontal scale or scale of distances, and the proportion in which the
vertical scale is greater than the horizontal, is called the
_exaggeration_ of the scale. This exaggeration is necessary, because the
differences of elevation between points on the ground are in general
much smaller than their distances apart, and would therefore, without
exaggeration, be unapparent, and also because, in the execution of
engineering works, accuracy in levels is of more importance than
accuracy in horizontal positions.


_Scales of Construction._--Scales of construction are intended to afford
means of measuring more minute quantities than scales of distances. Of
the former there are two kinds, known respectively as the _Diagonal_ and
the _Vernier_ scale. The diagonal is the more frequently employed. Its
construction involves no peculiar difficulty, as it consists simply of
an ordinary scale of distances, with the addition of a number of
parallel lines crossed by other parallel lines drawn diagonally from the
smaller points of division. An example will best show the construction
and mode of using this scale. Suppose it to be required to construct a
scale of 10 miles to the inch, showing furlongs diagonally; the scale to
measure 50 miles. Here 1 : 10 :: _x_ : 50, whence _x_ = 5 inches. Divide
this length of 5 inches into five equal parts, and the first part into
tenths to show miles, in the manner already described for scales of
distances. Then, since it is required to show furlongs or eighths of a
mile, eight equidistant parallel lines must be drawn above the scale, at
a convenient interval apart, as shown in Fig. 80. Produce the primary
points of division to meet the top parallel; and from the last secondary
point of division draw a line to the point in which the extreme primary
division meets the top parallel. Draw from the other points of division,
lines parallel to this one, and the scale will be complete. It will be
seen that the inclined lines are the diagonals of the rectangular
figures formed by the top and bottom parallels and vertical lines drawn
from the smaller points of division.

[Illustration: FIG. 80.]

To use this scale, suppose a length of 24 miles 5 furlongs is required.
Place one leg of the dividers upon the point in which the fourth
diagonal intersects the fifth parallel, and extend the other to the
point in which the primary division marked 20 intersects the same
parallel. In like manner, if the distance required be 33 miles 3
furlongs, it must be taken from the intersection of the third diagonal
with the third parallel, to the intersection of the primary division
marked 30 with the same parallel.

It is obvious that if a scale of feet showing inches diagonally be
required, twelve equidistant parallel lines must be drawn instead of
eight as in the foregoing example where furlongs are required. The
diagonal scale possesses the important advantages of accuracy and
distinctness of division which render it very suitable as a scale of
construction. Another practical advantage is that it is less rapidly
defaced by use than the other kinds, in consequence of the measurements
being taken on so many different lines.

The construction of the vernier scale is similar to that of the
graduated arcs of surveying and astronomical instruments. The principle
of the vernier is as follows. If a line containing _n_ units of
measurement be divided into _n_ equal parts, each part will, of course,
represent one unit; and if a line containing _n_ + 1 of these units be
also divided into _n_ parts, each part will be equal to (_n_ + 1)/_n_
units; and the difference between one division of the latter and one of
the former will be ((_x_ + 1)/_n_) - 1 = 1/_n_ of the original unit.
Similarly, the difference between two divisions of the one and two of
the other will be 2/_n_ of a unit, between three of the one and three of
the other, 3/_n_, and so on. Hence, to obtain a length of _x_/_n_ of a
unit, we have only to make a division on one scale coincide with one on
the other scale; the space between the two corresponding _x_th divisions
from this on both scales will be the required length of _x_/_n_ of a
unit. The same reasoning will evidently hold good if a length equal to
_n_ - 1 be taken.

[Illustration: FIG. 81.]

To show how the foregoing principle is applied in practice, we will take
an example. It is required to construct a scale of ¹⁄₁₀₀ to show feet
and tenths of a foot. Construct a scale in the ordinary way, and
subdivide it throughout its whole length, as shown in Fig. 81; then each
division will show one foot. Above the first primary division, draw a
line parallel to the scale and terminating at the zero point. From the
zero point, set off on this line towards the left a distance equal to
eleven subdivisions, and divide this distance into ten equal parts. Now,
as eleven divisions of the plain scale have been divided into ten equal
parts on the vernier, each division on the latter will represent ¹¹⁄₁₀ =
1·1 of that on the former; and as the divisions of the plain scale
represent feet, those of the vernier will represent 1·1 foot.
Consequently, the distances from the zero of the scale to the successive
divisions on the vernier are 1·1, 2·2, 3·3, 4·4, 5·5, 6·6, 7·7, 8·8,
9·9, and 11 feet. It will be seen that the divisions of the two scales
are made to coincide at the zero point.

The mode of using this scale will be seen from the following example.
Let it be required to take off a distance of 26·7 feet. From zero to the
7th division of the vernier is, as we have seen, 7·7 feet. Therefore, to
ascertain how far to the right of zero we must go to obtain the distance
of 26·7 feet, we must subtract 7·7 from that distance, which gives 19.
Thus to take off the distance, one leg of the dividers must be placed on
the 7th division of the vernier, and the other on the 19th division of
the plain scale. If the distance to be taken were 27·6 feet, one leg of
the dividers would have to be placed on the 6th division of the vernier,
and the other on the (27·6) - (6·6) = 21st division of the plain scale.

To construct a scale to show feet and inches, make the vernier equal to
thirteen divisions of the plain scale and divide it into twelve equal
parts. Each of these divisions will then represent ¹³⁄₁₂ = 1¹⁄₁₂ of a
foot.

Scales of construction may be purchased upon box-wood or ivory, but
where great accuracy is important, it is best to lay down the scale upon
some part of the drawing, as in such a case it expands and contracts
with the drawing under the influence of moisture.

Examples of scales of distances will be found on Plates 8 and 9.


SECTION III.--PLOTTING.

The transference of the measurements determined by the survey from the
field-book to the paper is termed _plotting_. The operations of plotting
are very simple, and the ability to perform them properly may be
acquired with a little practice. But their due performance demands the
same extreme care and attention as that of the operations in the field,
for it is obvious that the precautions taken to ensure accuracy in the
latter may be rendered nugatory by inaccurate plotting. The angular
instrument used in plotting is the protractor, and to ensure correct
results this instrument must be accurately divided. When, however, the
survey has been made without the aid of an angular instrument, the
protractor is not required in laying down the results. In such a case,
which frequently occurs in surveys of small extent, the lines, having
all been chained and registered in the field-book, are laid down
directly from the scale by means of an ordinary straight-edge and a pair
of compasses. The several methods of plotting and the various operations
involved have now to be considered.


_Reference Lines and Points._--The lines chained over in a survey and
recorded in the field-book are not usually the actual lines existing on
the ground, but imaginary straight lines chosen for the purpose of
referring other lines and points to them. They are, therefore,
appropriately termed _reference lines_, and all points situate in them
to which other lines are referred, in other words, all points in a
reference line in which other reference lines intersect it, are termed
_reference points_. Reference lines are generally made to form triangles
for facility of computation, and these triangles enclose the area to be
surveyed. But to determine the details included within them, it is
necessary to form other and smaller triangles within the larger ones
first laid down. The latter are, therefore, distinguished as Primary and
the former as Secondary triangles, and the lines of which they are
composed are called primary and secondary reference lines.


_Plotted Points._--In laying down a line of definite length upon paper,
the positions of its extremities are determined and marked by pencil
dots; such dots, or rather the points indicated by such dots, are termed
_plotted points_. The line is drawn by joining the plotted points.


_To Plot Reference Lines and Points._--To plot a reference line of a
given length when the position of neither of its extremities is given, a
light dot must be made upon the paper in a convenient part to indicate
the position of one extremity. The pencil-point mark should be as light
and well defined as possible, and hence it is essential that the pencil
used should be hard, and always kept pared to a fine conical point. The
scale must then be applied, in the direction of the line to be drawn,
with its zero point coincident with the plotted point. The scale should
be lightly but firmly held in this position with the left hand. The
distance of the other extremity of the line must then be found upon the
scale, and the eye placed directly over the line of the graduation; this
is necessary to the correct placing of the point, and it is well to
train the eye to trace accurately the prolongation of the line of
graduation upon the paper. A dot must be placed in the prolongation of
this line and close to the edge of the scale, to mark the position of
the other extremity. The reference line is then to be drawn between
these two plotted points with a sharp chisel-pointed pencil. Reference
lines, like reference points, should be well defined, but drawn as fine
and light as possible. The degree of fineness and lightness should be
such that when the detail is finely but firmly outlined, the reference
lines and points may not be visible, except on a close inspection of the
surface.

To plot secondary reference lines, as, for example, a number of offset
lines, apply the scale so that its edge may be parallel to and almost
over the pencil trace of the primary line and its zero point coincident
with the point at which the line begins. Care must always be taken to
place the zero of the scale at the beginning of the line, and not at the
end of it. At the distances recorded in the field-book as those at which
the offsets were taken, plot upon the line, in the manner described
above, the points indicating the extremities of the offset lines. All
other points, such as stations and intersection of fences, roads, and
streams, should be plotted at the same time. Around all stations, a
light, hand-drawn circle should be placed, and intersections marked by
small cross lines. This being done, place the offset scale so that the
zero may coincide with the plotted point in the reference line and the
edge be perpendicular to the line. To bring the edge into this position,
the end of the scale should be placed parallel to the reference line;
this is, of course, assuming the scale to be perfectly rectangular, as
it ought to be. The other extremities of the lines may then be plotted
in the same manner as those of the primary lines.

To illustrate the foregoing remarks, let it be required to plot the
following portion of a field-book.

               +------+
  {|     00    | 1346 | or 000 Line 1.
  {|     75    | 1170 |
  {|     96    | 1000 |
  {|    105    |  650 |
  {|     66    |  400 |
  {|     00    |  000 |
               +------+
  Line 3. From | 1946 |  last.
               +------+
               |   ↰  |
               +------+
               | 1946 |  00  |}
               | 1420 |  88  |}
               | 1200 | 144  |}
               | 1000 | 110  |}
               |  600 |  75  |}
               |  520 |  50  |}
               |  000 |  00  |}
               +------+
  Line 2. From | 2504 | last.
               +------+
               |  ↙↑  |
               +------+
               | 2504 |  00  |}
               | 2000 |  65  |}
               | 1790 |  95  |}
               | 1610 | 115  |}
               | 1440 |  87  |}
               | 1220 | 110  |}
               | 1000 |      |}
               |  850 |  28  |}
               |  420 | 100  |}
  Line 1.      |  000 | 00 or 1346
               |      | Line 3.
               +------+
  Begin at south corner and range N.

Having laid down in a convenient part of the paper the beginning of Line
1, place the edge of the scale so that the zero may coincide with the
plotted point and the edge be parallel to one of the meridians. Holding
the scale firmly in this position, plot the distance 2504 links, and
join the plotted points. Then, without removing the scale, plot upon
this reference line the distances at which the secondary or offset lines
were taken, that is at 420, 850, 1220, 1440, 1610, 1790, and 2000 links.
In this case we have nothing but offsets; had there been stations or
intersections of fences, roads, or streams, these would have had to be
plotted at the same time, and distinguished by an appropriate mark. Line
2 begins at the end of Line 1, and returns on the left at an acute
angle, as indicated by the arrow in the field-book. But as the magnitude
of this angle is not known, the survey having been taken with the chain
alone, the exact direction of Line 2 must be ascertained in the
following manner. Take the length of the line in the compasses, and with
this distance as a radius, strike an arc from the end of Line 1. Also
with the length of Line 3 as a radius, strike an arc from the beginning
of Line 1, at which point Line 3 closes, intersecting the former. The
point of intersection will be the end of Line 2. Join this point to the
end of Line 1, and plot the offset reference points upon this line in
the manner already described for Line 1. Line 3 begins at the end of
Line 2, and terminates at the beginning of Line 1; hence both
extremities being determined, we have only to join these points, and to
plot the offset reference points as before. Had a split line been taken
for the sake of accuracy from the angle A C B to the base Line 1, the
intersection of this line should have been plotted upon Line 1, and the
position of the end of Line 2 found by striking an arc from this point
also with a radius equal to the length of the split line. If the
distances have been correctly measured in the field and correctly taken
from the scale, the three arcs will intersect at the same point. If they
do not so intersect, the error must be noted in an error-sheet, and
corrected in the field before proceeding further. It may be remarked
here that on account of the irregularities of the surface chained over,
all measured lines are liable to be recorded a little too long. One link
in ten chains may be allowed for this source of error. Assuming,
however, that the primary reference lines “close” properly, the offset
lines may be plotted in the manner already described, and the detail or
boundary lines drawn in, which in the present instance are hedges. Fig.
82 shows the survey as laid down in the foregoing notes.

When the survey has been made with the aid of an angular instrument, the
method of plotting the primary reference lines differs somewhat from the
foregoing. In this case, the paper should be first covered with a number
of parallel straight lines ruled about an inch and a half apart to
represent magnetic meridians. The first station may then be marked upon
one of these meridians in a convenient part of the paper. To lay down
the first reference line, apply the protractor to this meridian with its
centre point coincident with the plotted point, and from the bearing
recorded in the field-book, lay off the given angle. Join the two
plotted points and produce the line indefinitely; and upon this line lay
off a distance equal to the length of the measured line. The second
reference line must be drawn in the same manner, from the end of the
first, by laying off from that point the recorded angle. But as the end
of the first line will probably not fall upon a meridian, the protractor
will have to be moved up to the point by means of a parallel ruler
adjusted to the nearest meridian. A more convenient and a more accurate
method, however, is to make the left of the drawing represent the north
while plotting, and to use the [T]-square instead of the parallel ruler.
All subsequent lines are plotted in the same way. Instead of covering
the paper with meridians before commencing to plot, it may, in some
cases, be found more convenient to draw with the set and [T] squares a
short meridian through the point as required.

[Illustration: FIG. 82.]

To lay down Fig. 82 in this manner, having fixed the first station A,
the length of the first primary reference line A B may be laid off upon
the meridian, because in this case the bearing being due north, the
reference line will be coincident with the meridian. This done, the
protractor is to be placed over the plotted point B and the second
bearing laid down. Having plotted the length of this line in the point
C, the third reference line C A will be determined both in length and
direction by the plotted points C A, which should be joined and the line
measured to ascertain whether its length corresponds with the measured
distance. If these do not correspond, the angle must be replotted and
the lengths laid off anew to discover the source of the error. Assuming,
however, that the lines close properly, the offsets and other secondary
points may be next plotted in the manner previously described.

Angles may be more accurately laid down by means of a table of _natural
sines_ and _cosines_ and a linear scale than by means of a protractor.
This is especially true when the angles are subtended by long lines, as,
for example, lines of 3, 4, and even 6 feet. In such cases, a protractor
is of little use. This mode of laying down angles is also convenient in
some cases where angles have been taken, but some of the sides not
measured. In using the table, it must be remembered that the radius of
the sines and cosines is taken as unity; therefore, to find the sine and
cosine for any other radius, the sine and cosine in the tables must be
multiplied by that radius. To lay down the angle A B C in Fig. 82, the
reduced cosine of the angle should be plotted from B in the point _a_,
according to some scale. The scale length of the reduced sine should
then be scribed from _a_, and the scale length of the radius scribed
from B. A line drawn from B through the point of intersection of the
scribes will lay down the angle corresponding to the sine and cosine in
the table. Suppose the radius chosen to be 5 chains, the angle being 32°
30′. The cosine of 32° 30′ is ·8434, which multiplied by 5, the assumed
radius, = 4·2170. Lay off this distance from B on the base A B. The sine
of 32° 30′ is ·5373, which multiplied by 5, = 2·6865. From the point
_a_, which is distant 4·2170 chains from B, with a radius equal to
2·6865 chains, describe an arc; and from the point B, with a radius
equal to 5 chains, describe another arc. From B draw a line through the
intersection of these arcs, and lay off upon it the measured length of
1946 links as recorded in the field-book.

If only the length of the base A B and the magnitudes of the angles
A B C and B A C were given, the lengths of the sides B C and A C would
have to be calculated by trigonometrical formulæ. This method of
calculating the lengths of the sides of triangles and plotting them with
the beam compasses, like chained triangles, is the most accurate for
laying down the great or primary triangles of a survey.

When it is required to plot according to this principle a solitary
angle, as, for example, that between a station line and the meridian, a
circle should be drawn with as large a radius as practicable round the
station at which the angle is to be laid down. The distance between the
points at which the two lines enclosing the angle cut that circle is
then found by multiplying the radius by the _chord_ of the angle, that
is, by twice the sine of half the angle.

It sometimes happens, particularly in extensive surveys, that all the
angular points of some triangles cannot be plotted upon the same sheet
of paper. In such cases, the plot of the outlying points and the sides
of the triangles may be laid down in the following manner. Plot the
intersected triangles independently and trace them on tracing paper.
Then, having drawn a fine line upon both sheets to represent the sheet
edge, lay the points on the trace corresponding to those already plotted
on the first sheet down upon, and make them to coincide with, the
latter. Secure the trace in this position and trace the sheet edge line
upon it. The intersected lines may now be plotted on the fair sheet with
a pricker at points outside the sheet edge line. Next apply the trace to
the second sheet and make the sheet edge lines coincide. Having secured
the trace in this position, the points and the intersected lines on this
second sheet may be plotted upon the fair paper by means of the pricker.


_To Plot Traverse Reference Lines._--In plotting a traverse survey in
which the angles have been measured from a fixed line of direction, the
magnetic meridian, the direction of the lines may be all laid down at
the first angular point. An example will best show the method employed
in this case. It is required to lay down the traverse shown in Fig. 83.

[Illustration: FIG. 83.]

In a convenient part of the paper draw the straight line N S to
represent the magnetic meridian, and plot upon it the first station A.
Set the protractor with its centre accurately placed over this point and
its 360th and 180th divisions coinciding with the meridian. Holding the
instrument securely in this position, lay off around it all the bearings
as entered in the field-book, numbering them in the order in which they
were taken. Against each of these numbers it is well to place the page
of the field-book on which the measurement of the angle and the survey
of the line are entered. The plotting must now be commenced by laying
down the first line through the first bearing and determining its
length from the recorded measurements. The direction of the second line
has next to be transferred from the first station A to the extremity of
the first line, or the second station B. This is accomplished by means
of the parallel ruler, by placing the edge of the ruler through the
plotted point and the point marked as bearing 2, and extending it till
the same edge intersects the point B. A line is then to be drawn from
this point and its length laid off from the field-book as before. The
direction of the third line will then be transferred from the first
angular point to the end of the second line, or station C, in the same
manner. This will be continued for all the lines in the traverse, and if
all the measurements have been correctly laid down, not only will the
last line pass through the point A, but it will be of the same length as
the chained line. Also the bearings taken from A to E and H will pass
through these latter stations. These proof line bearings should be laid
down at the same time as the reference line bearings, from which they
should be distinguished by some sign. The directions of the reference
lines should be consecutively transferred, and the length of each line
should be plotted in its proper place before the direction of the next
is transferred. To ensure the work closing properly, great care must be
taken to plot the points accurately and to draw the pencil lines fine.

[Illustration: FIG. 84.]

The degree of accuracy to be attained will depend in a great measure
upon the extent of the traverse. With long lines the difficulties
increase, and with a great number of angles the chances of error are
multiplied. If the angles are carefully taken, it is probable that
seconds have been read off in several instances, and these if neglected,
especially upon long chain lines, may lead to an error of some
importance. Also when the lines are long, the parallel ruler becomes
practically useless, and some other system has to be adopted. One way of
overcoming these difficulties is to draw a parallel to the first
meridian through every third or fourth angle; in such a case, great care
must be observed in drawing the parallels. A more easily practicable
method, however, is to use the [T]-square in the manner already
described. If the left-hand edge of the drawing board be made the north,
the blade will determine meridional lines, and by laying the straight
side of a semicircular protractor against the edge of the blade, its
zero will be adjusted to the fixed line of direction. The first bearing
having been laid down, the line is drawn and made the scale length of
the chain line; the blade of the square is then pushed to the station
thus found, and the next bearing set off. This operation is repeated
until all the lines have been laid down. If the work closes properly,
the plotting of the secondary lines may be proceeded with.

The most accurate method of plotting a traverse is by rectangular
co-ordinates, or, as it is usually termed, Northings, Southings,
Eastings, and Westings, because the position of each station is plotted
independently, and is not affected by the errors committed in plotting
previous stations. This method consists in assuming two fixed lines or
axes crossing each other at right angles at a fixed point, computing the
perpendicular distances or co-ordinates of each station from those two
axes, and plotting the position of each station by means of the [T] and
set squares and a linear scale. The meridian is usually made to
represent one of the axes, and in this case the co-ordinates parallel to
one axis will be the distances of the stations to the north or south of
the fixed point, and those parallel to the other axis will be their
distances to the east or west of the same point. Let N S, Fig. 84,
represent the meridian, and A B the first bearing taken, and the first
line measured. The angle in this case is N A B = θ. If θ is an acute
angle, the second station B is to the north of the first station A; if
it is an obtuse angle, B is to the south of A. If the angle θ lies to
the right of the meridian, B is to the east of A; if to the left, to the
west. Thus it will be seen that if the northernly and easternly
directions are considered positive, the southernly and westernly
directions will be negative. From the foregoing it is manifest that the
co-ordinates of B are as follows:--

  Northing A _a_ = _b_ B (or if negative, southing) = A B × cos. θ.

  Easting  A _b_ = _a_ B (or if negative, westing)  = A B × sin. θ.

To plot the point B, draw through the point A, with the aid of the
[T]-square, a horizontal line. Multiply the chained length of the line
A B by the sine of the angle N A B as entered in the field-book, and set
off this distance along the horizontal line. From the point thus
determined, erect, with the aid of the set square, a perpendicular,
which will be parallel to the meridian. Multiply the chained length of
A B by the cosine of the angle N A B, and set off this distance along
the perpendicular line. The point thus determined will be the position
of the second station B, which may then be joined to A by a straight
line.

[Illustration: FIG. 85.]

The mode of laying down the survey in Fig. 85 will now be obvious.
Having determined the position of the second station B in the manner
just described, draw a horizontal line through B and determine the third
station C in the same way. The fourth station D being to the left of the
meridian passing through C, _c d_ is a westing and is to be considered
as negative. Therefore the horizontal line through C must be drawn to
the left of that station, and the station D determined in the same
manner as the preceding stations.

The results of all these calculations should be entered in a book, in
four columns, for northings, southings, eastings and westings
respectively. Also in four other columns should be entered the total
northing or southing, and easting or westing, of each station from the
first station, computed by adding all the successive northings and
subtracting the southings made in traversing to the station, the result
being a northing if positive, and a southing if negative. The same
treatment is applied to the eastings and westings. This affords a means
of testing the accuracy of the work. It is also obvious that the
position of the last or any station may be determined by this means
without plotting the intermediate stations.

Let it be observed that both θ and sin. θ are positive or negative
according as that angle lies to the east or to the west of the meridian;
and that the cosines of obtuse angles are negative.


_To Plot Detail._--By “detail” is meant outlines or objects whether
natural or artificial, such as fences, walls, rivers, canals, roads,
lakes, water margins, beach marks, seas, or imaginary boundary lines. In
plotting from the entries of measurements for detail, these measurements
should be laid down upon the paper in the order and manner indicated in
the field-book. The mode of plotting the perpendicular reference lines
by means of which the position of the detail is fixed has already been
fully described and illustrated. The proper connections for detail, as
shown by the field-book, should be made by drawing a firm pencil line
through the detail points with the aid of an offset scale adjusted
successively to the adjacent points. All such connections should be
clearly and elegantly made. When all boundaries, roads, and streams have
been drawn and inked in, tracings should be taken in small portions of
all that has been laid down for the use of the “examiner.” The duties of
the examiner are to make on the ground the necessary corrections for
omissions and detail in error; to give, in position and character,
woods, water, marsh, commons, vegetable and geological features, and
permanent artificial structures; and to furnish the descriptive names of
places and things, or any other desirable information. The topographical
character of mountains, marshes, bogs, rough pasture, woods and water,
should be drawn in character on trace and tinted. If required, hill
sketching should also be supplied on the examiner’s trace. On being
returned to the office, the plotter should replot from the field notes
the detail corrected, and transfer the details from the trace to the
map.


_To Plot Contours._--The student who has made himself familiar with the
methods of laying down angles, and plotting reference lines and points,
will find no difficulty in laying down contour traces. When the contour
points have been surveyed with the chain, the contour is obtained by
drawing a free line of feature through the plotted points. But when the
contour points have been surveyed by measuring magnetic angles to known
points, such angles must be laid down at these points, and produced to
meet in the contour point. The drawing of contours differs from the
drawing of ordinary detail insomuch as each contour point is shown by a
small dot, and each carrying point by a similar dot surrounded by a
small hand-drawn circle to distinguish it. The former should be so
plotted as to be distinguishable in the trace or contour line, which
should be readily traceable, but not conspicuous. The line joining
adjacent points should be true lines of feature. Colour is usually
employed for these lines, and it is well to give them a broken or
somewhat undefined character. When the French system is adopted, contour
lines are drawn continuous, a broad but faint line of colour.


_To Plot Sounded Points in Submerged Districts._--When the angles have
been measured on dry land with the theodolite, these angles should be
laid down at the dry-land points, and the lines produced to meet in the
sounded point. But when the angles have been measured on the water with
a sextant, a station pointer is required. The arcs of the pointer should
be adjusted to read the measured angles, and the instrument applied to
the plotted points of the observed objects so as to bring the hair lines
accurately to their respective object points. The sounded point may then
be correctly plotted through the centre of the pointer. If the angles
have been measured by the magnetic compass, that is, if the angles are
those made with the magnetic meridian, the angles should be laid down at
the plotted points of the observed land objects, and the lines produced
to meet in the sounded point. Instead of the station pointer, a piece of
tracing paper may be used in the following manner. Draw three straight
lines radiating from one point so as to make with each other angles
equal to the measured angles. Lay the paper on the plan and move it
about till the three lines traverse the observed objects. The point from
which they diverge will then mark the position of the sounded point,
which may be plotted by being pricked off.

The sounded point determined by angles measured with the sextant may
also be plotted by describing circles on the land-object lines as
chords, to contain segmental angles equal to the measured angles. Such
circles will intersect in the common land-object point and the sounded
point. To plot the sounded point in this manner, requires the solution
by construction of the problem, “to describe on a given line a segment
of a circle that shall contain a given angle.” But this method is
generally found too tedious in practice.


_Errors and Error-sheets._--There is a tendency, as we have previously
remarked, for the measured lengths of lines to be a little too long, by
reason of the irregularities of the surface. It is usual to allow for
this source of error 1 in 1000 in fair open country, and 1¹⁄₂ in 1000 in
close country. When the measurements differ by an amount exceeding these
limits, the pencil trace should not be drawn between the reference
points, but the line should be entered on an “office error-sheet.” The
error-sheet should show the number of the plot-sheet, the triangle, the
book and page in which the measurements are entered, and the scale and
measured lengths of the line. To ascertain the source of the error,
other lines referenced to the reference point or points of the line in
error should be plotted, and the apparent source should be entered on
the error-sheet. If the lines referred to the same point be found to
plot to another point in the reference line, the scale measurement of
this point should also be entered. And if the reference point in error
be not directly surveyed in the survey of their respective lines, the
measurements for reference and the arithmetical reductions will have to
be examined. Besides the office error-sheet, there should be a field
error-sheet for each book and triangle, upon which should be entered the
book, the page, and the line in error, and some indication of the source
of the error. This sheet will be forwarded with the field-book to the
surveyor for correction. The following are examples of a common and very
good form of error-sheet, but it may be varied in many ways if thought
desirable:--

OFFICE ERROR-SHEET.

  _Plotter’s Name_____________                       _Date_ ____________
  -----+------+--------+---------+--------+----------+------------------
  Book.|Lines.| Scale  |Reference|Apparent|Triangles.|   Observations.
  -----+      |Measure-|  Points | Correc-|          |
  Page.|      |  ment. |of Lines.| tions. |          |
  -----+------+--------+----+----+--------+----------+------------------
   200 |      |        |1118|1551|  1651  |          |Examine reference
  -----+ 1953 |  1907  +----+----+--------+          |point and line.
    23 |      |        |2094|4020|  4020  |          |
       |      |        |    |    |        |          |
   200 |      |        |2814|1308|        |  [Illus- |
  -----+ 3000 |  3056  +----+----|  3056  |   tra-   |Examine line.
    28 |      |        |4020|3082|        |   tion]  |
       |      |        |    |    |        |          |
   200 |      |        |1551| 85 |  1651  |          |Examine reference
  -----+ 1314 |  1323  +----+----+--------+          |point and line.
    42 |      |        |4020|3028|  4020  |          |
  -----+------+--------+----+----+--------+----------+------------------

FIELD ERROR-SHEET.

  _To A. B., Surveyor._                              _Date_ ____________
  -----+------+----------+----------------------------------------------
  Book.|Lines.|Triangles.|                   Observations.
  -----+      |          |
  Page.|      |          |
  -----+------+----------+----------------------------------------------
       |      |          |_Examine reference points and line._
   200 |      |          |1651
  -----+ 1953 |          |---- Reference point (August 10th, 1874).
    23 |      |          |4020
       |      |          |
   200 |      |  [Illus- |
  -----+ 3000 |   tra-   |_Examine line._ Line 3056 (August 10th, 1874).
    28 |      |   tion]  |
       |      |          |_Examine reference points and line._
   200 |      |          |1651
  -----+ 1314 |          |---- (August 12th, 1874).
    42 |      |          |4020
       |      |          |                 Corrected in the field, A. B.
  -----+------+----------+----------------------------------------------


_To Plot Vertical Sections._--In plotting a vertical section, a fine and
firm horizontal line is first drawn to represent the _datum line_. The
reference points are then plotted upon this line from the level-book by
means of a linear scale, in the manner already described for plotting
such points. The reference points to be plotted upon the datum line are
the chain lengths entered in the field-book in the column headed
Distances. These distances are the points at which the levels were
taken, and between them, unless otherwise stated in the field-book, the
ground is supposed to slope uniformly. Moreover, these distances are
assumed to be measured horizontally, and therefore care must be taken to
ascertain whether or not they were so measured in the field; if not,
they must be reduced before plotting, or the section will be too long.
Having plotted the reference points on the datum line, a perpendicular
must be erected from each of them, and a length laid off upon this
perpendicular equal to the vertical height above the datum line
indicated by the entry in the column of the level-book headed Reduced
Levels against the distance to which the perpendicular corresponds. To
render the differences of altitude more apparent, these vertical
distances are plotted to a much larger scale than the horizontal, as
explained in a preceding Section. To erect the perpendiculars, the [T]
and set square furnish the most convenient means. The detail points thus
determined upon the perpendiculars represent the points in the surface
of the ground at which the levels were taken, and by joining these
points we obtain the surface line. An example will clearly show the
method pursued. Let it be required to lay down the section from the
following level-book:--

  -------+-------+-----+-----+-------+-------+-------------------------
   Back  | Fore  |Rise.|Fall.|Reduced| Dis-  |         Remarks.
  Sights.|Sights.|     |     |Levels.|tances.|
  -------+-------+-----+-----+-------+-------+-------------------------
   feet. | feet. |feet.|feet.| feet. |chains.|
         |       |     |     | 100·00|   ..  |{B.M. north-west corner
         |       |     |     |       |       |{of church tower.
    16·41|   9·37| 7·04|  .. | 107·04|  230  |
    19·36|  10·43| 8·93|  .. | 115·97|  465  |
    16·42|  19·36|  .. | 2·94| 113·03|  640  |
     8·36|  14·36|  .. | 6·00| 107·03|  794  |
     9·37|  12·49|  .. | 3·12| 103·91| 1030  |
    11·64|  19·76|  .. | 8·12|  95·79| 1200  |
    19·46|   9·32|10·14|  .. | 105·93| 1564  |
    17·64|  13·62| 4·02|  .. | 109·95|  ..   |{Centre of road at 123
         |       |     |     |       |       |{links.
    18·76|  12·64| 6·12|  .. | 116·07| 1823  |
    19·84|  16·92| 2·92|  .. | 118·99| 1964  |
    19·76|  11·64| 8·12|  .. | 127·11| 2100  |{Forward station ☉ at
         |       |     |     |       |       |{corner of wood.
    17·64|  19·72|  .. | 2·08| 125·03| 2250  |
     9·73|  18·64|  .. | 8·91| 116·12| 2376  |
     8·64|  17·64|  .. | 9·00| 107·12| 2590  |
    18·76|  16·24| 2·52|  .. | 109·64| 2700  |
  -------+-------+-----+-----+       |       |
   231·79| 222·15|49·81|40·17|       |       |
   222·15|       |40·17|     | 100·00|       |
  -------+       +-----+     +-------+       |
         |       |     |     |       |{Difference between datum and
     9·64|       | 9·64|     |   9·64|{last reduced level, or height of
         |       |     |     |       |{B above A.
  -------+-------+-----+-----+-------+---------------------------------

[Illustration: FIG. 86.]

Draw the datum line D L, Fig. 86, and set off along it the distances
230, 465, 640, 794, &c., links; these points will be the reference
points for the perpendiculars. Erect a perpendicular from each of these
points, and lay off, to a suitable scale, upon these lines successively
the vertical heights 100, 107·04, 115·97, 113·03, 107·03, &c. The points
thus determined will be the surface detail points, and by joining these
we shall obtain the surface line. Then will A D L B represent a section
of the ground between A and B. A description of objects on the surface
worthy of notice should be written over such objects.

In working sections, where great accuracy is required, larger scales are
employed, and the levels are taken at more frequent intervals. Thus, in
a railway working section, for example, the levels are taken at every
chain’s length, and also over every little undulation in the surface of
the ground. In preparing such sections, vertical lines are drawn in blue
at every chain’s length up to the surface of the ground from the datum
line, and on each vertical is written the reduced height above datum
from the column of reduced levels in the level-book.

Sections, especially working sections, are usually drawn upon ruled, or,
as it is called, “section” paper, the nature of which we have already
described. This method, which was introduced by Mr. Brunel, possesses
many practical advantages, inasmuch as it obviates the necessity of
plotting the “distances” and erecting perpendiculars, the latter already
existing. It also greatly facilitates the computation of the contents of
a given section. Its chief defect lies in the difficulty of making the
horizontal lines coincide when joining the sheets end to end. Of course
scales are not required upon section paper.


_To lay down Gradients._--The method of laying down the gradients of
railways and roads usually adopted in practice consists in applying one
end of an extended silken thread to the section at the point in which
the road commences, and the other end in such a position that the thread
may cut the profile of the earth’s surface so as to leave equal portions
of space above and below the thread, as nearly as can be judged by the
eye. The cuttings from the parts above the thread will then furnish
sufficient materials to form the embankments in the spaces below. This
is called “balancing” the cuttings and embankments. When the first
gradient has been determined in this way, it may be found unfavourable
to the second in respect to the extent of cuttings and embankments; in
such a case it must be modified to suit the requirements of the latter.
In this way the gradients must be modified successively until the
compound result evidently gives a _minimum_ of cuttings and embankments,
due regard, of course, being had to the limits imposed by the nature of
the case, both with respect to the ruling gradient and the proper
heights for bridges.


_To Plot a Section from a Contour Map._--The mode of plotting a section
from a contoured plan was explained when treating of contour lines. The
contour map used for this purpose should give the features of the
surface configuration in sufficient detail without serious error. Having
drawn a line of section on the map and a datum line upon the fair paper
for the vertical section, the points in which the section line
intersects the contours should be measured on the scale of the map from
a zero point in that line, and the measurements plotted upon the datum
line. Perpendiculars should then be drawn through these plotted points,
and on these perpendiculars the reduced altitudes of their respective
contour points should be plotted. A line drawn through these surface
plotted points will be the surface line. When the horizontal scales of
the map and the section are the same, the contour plane lines may be
drawn on the paper for the section parallel to the section line on the
map, and perpendiculars raised to intersect them from the points on the
map in which the section line intersects the contours, in the manner
previously described. The points of intersection with the parallel lines
will be the surface contour points in the vertical section. For
practical purposes, the parallel lines and the perpendiculars are only
temporarily drawn in pencil until the surface trace shall have been
obtained and drawn in ink, with the datum line.


SECTION IV.--CIVIL ENGINEERS’ AND SURVEYORS’ PLANS.

In the preceding Sections the manner of laying down plans has been fully
described and the principles involved in the operations minutely
explained. It now only remains, therefore, to direct attention to
certain matters relating to the preparation of plans, which are
necessitated by the circumstances of particular cases.

Civil engineers’ plans usually consist, if we except harbour surveys, of
a representation, to a rather large scale, of long and narrow tracts of
country through which it is proposed to construct a means of
communication, such as a railway, a road, or a canal. They do not differ
essentially from other plans, the survey being taken in the ordinary
way, and the plan laid down according to one or other of the methods
described in the preceding Section. The width of railway surveys varies
from five to twenty chains, at the option of the engineer. An important
matter demanding careful attention is to survey, plot, and number all
fields and other enclosures, houses and other buildings situate within
the _limits of deviation_, that is, the boundaries of the space beyond
which it is not proposed to deviate the line of railway. The object of
numbering every separate enclosure, road, building, or other object on
the plan, is that they may be the more readily described in a book
prepared for that purpose and called the Reference Book. Parish and
county boundaries are shown by dotted lines, as explained in a former
Section. Frequently it is necessary, in consequence of the smallness of
the scale adopted for the plan, to give enlarged drawings of certain
portions. In these cases, whenever practicable, the enlarged plan should
be placed directly under or over the small plan to which it refers, as
such an arrangement is not only more pleasing to the eye, but is far
more convenient for reference than one in which there is no relation of
position between the two plans. The proposed railway should be
represented by a full and heavy line, and the limits of deviation shown
by strong dotted lines. The names of the different parishes through
which the line passes should be conspicuously written, and the name of
the county placed at the top of each sheet; the sheets also should be
distinctly numbered. It is not usual to distinguish different kinds of
fences on plans of engineering projects, as on estate plans to a large
scale; on the former it is sufficient to distinguish between fenced and
unfenced lines of division of land, marking the former by plain, and the
latter by dotted lines. It is almost needless to remark that a scale of
distances should accompany every plan.

The section should be drawn to the same horizontal scale as the plan,
and the exaggeration of the vertical scale should be such as to show
distinctly the irregularities of the surface. The horizontal datum line
of the section should have marked on it a scale of distances
corresponding with those marked along the centre line of the plan, in
order that corresponding points on the plan and the section may be
readily found, and great care should be taken that horizontal distances
on the plan and on the section exactly agree. Cross sections are
longitudinal sections of existing lines of communication which the
proposed work will cross; they may cross the centre line of the proposed
work either at right angles or obliquely. Cross sections may also be
required where the ground slopes sideways; in general they should be
ranged accurately at right angles to the centre line, and they should be
plotted without exaggeration, that is, their vertical and horizontal
scales should be the same as the vertical scale of the longitudinal
section. All cross sections should be plotted as seen by looking
_forward_ towards them along the centre line.

To distinguish the nature of the soils passed through, sections are
frequently coloured, as shown in Plate 21. The information given by this
means concerning the character of the strata is of very great value to
the engineer or to the contractor, inasmuch as it enables them to
predicate with some degree of certainty the amount of labour that will
be required in executing the proposed work. It is, therefore, highly
important that the draughtsman correctly represent the character of the
strata. The conventional modes of representing these features are shown
on Plate 20, which should be carefully studied and copied.

It is necessary that the engineering draughtsman should be acquainted
with the “Standing Orders” of Parliament relating to the preparation of
plans and sections, in order that he may fulfil the conditions therein
laid down. And as most of the important details involved by the
exigencies of practice in the preparation of such plans and sections are
prescribed by these Standing Orders, we will give so much of them as
relates directly to the matters under consideration; by so doing, the
details will be clearly and fully described, and the requirements of the
law concerning them authoritatively made known.


_Nature of the Documents required._--“In cases of bills relating to
engineering works, a plan and also a duplicate thereof, together with a
book of reference thereto, and a section and also a duplicate thereof,
as hereinafter described, shall be deposited for public inspection at
the office of the clerk of the peace for every county, riding or
division in _England_ or _Ireland_, or in the office of the principal
sheriff clerk of every county in _Scotland_, and where any county in
_Scotland_ is divided into districts or divisions, then also in the
office of the principal sheriff clerk in or for each district or
division in or through which the work is proposed to be made,
maintained, varied, extended or enlarged, or in which such lands or
houses are situate, on or before the 30th day of _November_ immediately
preceding the application for the bill.”

“In the case of railway bills, the ordnance map, on the scale of one
inch to a mile, or where there is no ordnance map, a published map, to a
scale of not less than half an inch to a mile, or in Ireland, to a scale
of not less than a quarter of an inch to a mile, with the line of
railway delineated thereon, so as to show its general course and
direction, shall, on or before the 30th day of _November_, be deposited
at the office of the clerk of the peace, or sheriff clerk, together with
the plans, sections, and book of reference.”

“In cases where the work shall be situate on tidal lands within the
ordinary spring tides, a copy of the plans and sections shall, on or
before the 30th day of _November_, be deposited at the office of the
Harbour Department, Board of Trade, marked ‘Tidal Waters,’ and on such
copy all tidal waters shall be coloured blue, and if the plans include
any bridge across tidal waters the dimensions as regards span and
headway of the nearest bridge, if any, above and below the proposed new
bridge, shall be marked thereon, and in all such cases such plans and
sections shall be accompanied by a published map or ordnance sheet of
the country, over which the works are proposed to extend, or are to be
carried, with their position and extent, or route accurately laid down
thereon.”

“In the case of railway bills, a copy of all plans, sections, and books
of reference, and the aforementioned published map with the line of
railway delineated thereon so as to show its general course and
direction, is required to be deposited at the office of the Board of
Trade, and at the Private Bill Office of the Houses of Parliament; and
in cases where any portion of the work is situate within the limits of
the Metropolis, a copy of so much of the plans and sections as relates
to such portion of the work is required to be deposited at the office of
the Metropolitan Board of Works. Also a copy of so much of the plans and
sections as relates to each parish in or through which the work is
intended to be made, or in which any lands or houses intended to be
taken are situate, together with a copy of so much of the book of
reference as relates to such parish, is required to be deposited with
the parish clerk of each such parish in _England_, with the
school-master of each such parish in _Scotland_, and with the clerk of
the union within which such parish is included in _Ireland_.”


_Plans._--“Every plan required to be deposited shall be drawn to a scale
of not less than four inches to a mile, and shall describe the line or
situation of the whole of the work (no alternative line or work being in
any case permitted), and the lands in or through which it is to be made,
maintained, varied, extended or enlarged, or through which every
communication to or from the work shall be made; and where it is the
intention of the parties to apply for powers to make any lateral
deviation from the line of the proposed work, the limits of such
deviation shall be defined upon the plan, and all lands included within
such limits shall be marked thereon; and unless the whole of such plan
shall be upon a scale of not less than a quarter of an inch to every one
hundred feet, an enlarged plan shall be added of any buildings, yard,
courtyard or land within the curtilage of any building, or of any ground
cultivated as a garden, either in the line of the proposed work or
included within the limits of the said deviation, upon a scale of not
less than a quarter of an inch to every one hundred feet.”

“In all cases where it is proposed to make, vary, extend or enlarge any
cut, canal, reservoir, aqueduct or navigation, the plan shall describe
the brooks and streams to be directly diverted into such intended cut,
canal, reservoir, aqueduct or navigation, or into any variation,
extension or enlargement thereof respectively, for supplying the same
with water.”

“In all cases where it is proposed to make, vary, extend or enlarge any
railway, the plan shall exhibit thereon the distances in miles and
furlongs from one of the termini; and a memorandum of the radius of
every curve not exceeding one mile in length shall be noted on the plan
in furlongs and chains; and where tunnelling as a substitute for open
cutting is intended, such tunnelling shall be marked by a dotted line on
the plan.”

“If it is intended to divert, widen or narrow any turnpike road, public
carriage road, navigable river, canal or railway, the course of such
diversion, and the extent of such widening or narrowing shall be marked
upon the plan.”

“When a railway is intended to form a junction with an existing or
authorized line of railway, the course of such existing or authorized
line of railway shall be shown on the deposited plan for a distance of
eight hundred yards on either side of the proposed junction, on a scale
of not less than four inches to a mile.”


_Book of Reference._--“The book of reference to every such plan shall
contain the names of the owners or reputed owners, lessees or reputed
lessees, and occupiers of all lands and houses in the line of the
proposed work, or within the limits of deviation as defined upon the
plan, and shall describe such lands and houses respectively.”


_Sections._--“The section shall be drawn to the same horizontal scale as
the plan, and to a vertical scale of not less than one inch to every one
hundred feet, and shall show the surface of the ground marked on the
plan, the intended level of the proposed work, the height of every
embankment and the depth of every cutting, and a datum horizontal line,
which shall be the same throughout the whole length of the work, or any
branch thereof respectively, and shall be referred to some fixed point
(stated in writing on the section), near some portion of such work, and
in the case of a canal, cut, navigation, turnpike or other carriage road
or railway, near either of the termini.”

“In cases of bills for improving the navigation of any river, there
shall be a section which shall specify the levels of both banks of such
river; and where any alteration is intended to be made therein, it shall
describe the same by feet and inches, or decimal parts of a foot.”

“In every section of a railway, the line of the railway marked thereon
shall correspond with the upper surface of the rails.”

“Distances on the datum line shall be marked in miles and furlongs, to
correspond with those on the plan; a vertical measure from the datum
line to the line of the railway shall be marked in feet and inches, or
decimal parts of a foot, at each change of the gradient or inclination;
and the proportion or rate of inclination between each such change shall
also be marked.”

“Wherever the line of the railway is intended to cross any turnpike
road, public carriage road, navigable river, canal or railway, the
height of the railway over or depth under the surface thereof, and the
height and span of every arch of all bridges and viaducts by which the
railway will be carried over the same, shall be marked in figures at
every crossing thereof; and where the roadway will be carried across any
such turnpike road, public carriage road or railway, on the level
thereof, such crossing shall be so described on the section; and it
shall also be stated if such level will be unaltered.”

“If any alterations be intended in the water level of any canal, or in
the level or rate of inclination of any turnpike road, public carriage
road or railway, which will be crossed by the line of railway, then the
same shall be stated on the said section, and each alteration shall be
numbered; and cross sections in reference to the said numbers, on a
horizontal scale of not less than one inch to every three hundred and
thirty feet, and on a vertical scale of not less than one inch to every
forty feet, shall be added, which shall show the present surface of such
canal, road or railway, and the intended surface thereof when altered;
and the greatest of the present and intended rates of inclination of
such road or railway shall also be marked in figures thereon; and where
any public carriage road is crossed on the level, a cross section of
such road shall also be added; and all such cross sections shall extend
for two hundred yards on each side of the centre line of the railway.”

“Wherever the extreme height of any embankment, or the extreme depth of
any cutting, shall exceed five feet, the extreme height over or depth
under the surface of the ground shall be marked in figures upon the
section; and if any bridge or viaduct of more than three arches shall
intervene in any embankment, or if any tunnel shall intervene in any
cutting, the extreme height or depth shall be marked in figures on each
of the parts into which such embankment or cutting shall be divided by
such bridge, viaduct or tunnel.”

“Where tunnelling, as a substitute for open cutting, or a viaduct as a
substitute for solid embankment, is intended, the same shall be marked
on the section.”

“When a railway is intended to form a junction with an existing or
authorized line of railway, the gradient of such existing or authorized
line of railway shall be shown on the deposited section, and in
connection therewith, and on the same scale as the general section, for
a distance of eight hundred yards on either side of the point of
junction.”

Besides the information thus written on the plan, it is useful to the
engineer, though not prescribed, to have the levels of important points
either written or shown by means of contour lines, especially when the
plan is to be used in selecting a line of railway. The results of trial
pits and borings may also be written on the plan, and the estimated cost
of each part of the work placed opposite to its position on the paper.


_Working Sections._--For working sections the horizontal scale adopted
is usually three or four chains to the inch, and the vertical scale 30
or 40 feet to the inch. A working section should show the level of the
ground, the level of the proposed work, and the height of embankment or
depth of cutting at every point of the ground where the level has been
taken, these quantities being found by calculation, not by measurement
on the paper. The position and levels of all “bench marks” should also
be clearly indicated. At every crossing of road, river or stream of any
kind, should be inserted some remark respecting the work to be
constructed, with a reference to the number of the working drawing
prepared. The latter may be a special drawing, as for a bridge, or a
general drawing, as for a level crossing and gates. The results of
boring should also be shown on the working section. As soon as the works
of construction have been determined upon, notes should be inserted from
the working drawings, or other sources, of the angles of skew at which
the line crosses roads or streams, the spans of arches on the square and
on the skew, the rise of the arch, the depth of the arch stones, and of
the puddle, if any be used, and, if the works be on an inclined plane,
the rise or fall from centre to centre of the piers. Similar memoranda
should also be made of girder bridges, culverts, and other works
occurring along the line. To all working drawings the acting engineer
always affixes his signature.

Besides an acquaintance with the “Standing Orders,” the engineering and
surveyor’s draughtsman should possess a knowledge of the Regulations of
the Local Government Board, for these have to be complied with in the
preparation of plans relative to main sewerage, drainage, and
water-supply. These Regulations are as follows.


_Boundary Maps._--In cases in which a special district is proposed to be
formed for the adoption of the Local Government Act, a map must be
submitted, accompanied by a written description of the proposed
boundary, designated by letters from point to point, commencing from a
fully and clearly defined point on the north side of the map marked by
the letter A and a written description, then proceeding eastward by
natural or other well-defined features, until the description closes
upon the point started from. The name of the proposed district must be
printed on the map, with the area in acres. The population and the
number of houses, with the rate of increase as ascertained at the two
last decennial periods upon which the census was taken, must be given,
and a duplicate or tracing of the map furnished.


_Maps for Division into Wards._--A map of the entire district must, in
this case, be submitted, with the main boundary distinctly defined, and
the name of the district clearly printed thereon. The proposed division
into wards must be by lines, clearly defined on the map of the district;
brooks, roads, footwalks, streets, fences, or other easily recognizable
lines of division may be adopted. Such lines must be defined on the map
by a margin of colour. The proposed boundary-lines must be described in
the same manner as in the boundary map, and the name or number of the
ward, with the relative areas, population, and rateable value must be
given. A duplicate map or a tracing must be deposited at the Local
Government Act Office for future reference.


_Plans of Proposed Works._--It is in all cases necessary, upon
application being made by Local Boards for the Secretary of State’s
sanction to a loan for the execution of works, that plans, sections,
detailed estimates, and specifications be submitted with the
application, accompanied with the information relative to area,
population, number of houses, and rateable value of the district
required for boundary maps. Tracings of such plans and sections, and
copies of the estimates and specifications must be sent in for filing at
the Local Government Act Office.


_General Plan._--A general plan exhibiting the area which will be
affected by the proposed works must be laid down to a scale of not less
than _two feet to a mile_. It should have figured upon it the levels of
the centres of all the streets and roads at their intersections and
angles, and at every change of inclination; also, where a district is
near the sea, it should show the high and the low tide level of the sea,
and where there is a river, the summer and the flood-water levels should
be recorded. Permanent bench marks having reference to the surface
levels should be cut on public buildings, or other permanent and
suitable objects, throughout the district, and clearly marked on the
plan. Sections should accompany this plan, upon which the levels of the
cellars should be shown. Such a plan may be used for showing the lines
of main-sewers and drains, lines of water-pipes, and gas-mains. The
lines of main-sewers and drains should have the cross-sectional
dimensions and the gradients distinctly marked upon them. The
dimensions of water and gas pipes should also be shown in figures or in
writing.


_Detailed Plan._--A detailed plan for the purposes of house-drainage,
paving, the sale and purchase of property, or other purposes of a like
character, must be constructed to a scale of not less than _ten feet to
a mile_. Upon this plan must be exhibited all houses and other
buildings, bench marks, the levels of streets and roads, of cellars, of
the sea at high and low tide level, and the summer and the flood level
of rivers. Three feet by two feet will be a convenient size for the
sheets of this plan, and by representing the marginal lines of the
sheets upon the general plan, and numbering the sheets to correspond,
the general plan will become a very useful index.

As it may occasionally be desired to carry out works piecemeal, with a
view to save the time which would be occupied in the preparation of a
complete plan from actual survey, it is sufficient in the first instance
to furnish a general plan of streets and roads only, with the surface
levels and those of the deepest cellars, and the proposed scheme of
works shown thereon, after which the works can proceed in sections. But
with each separate application for sanction to a loan, a correct plan
and section or sections should be submitted, accompanied by detailed
estimates and specifications. It must, however, be understood that the
complete plan of the entire district must be proceeded with, so that,
when the works are finished, the Local Board and the office of the Local
Government Act may possess a proper record.


_Mining Plans._--The plotting of mining surveys is performed in the same
manner as the surface traverse surveys already described. Before
proceeding to lay down the plan, it is well to divide the paper into
squares of 10 chains side, or 10 acres area, by two sets of lines
crossing each other at right angles, one of which sets should represent
meridians. This operation should be performed with scrupulous care; and
to ensure accuracy, beam compasses should be used to lay off the
divisions. The lines should be finely drawn in colour to distinguish
them from other lines to be put upon the drawing. Care must be taken to
get the plan fairly upon the paper, so that the conformation of the
outline or boundary to the edges of the paper may have a pleasing effect
to the eye; and the direction of the meridians must be determined
according to this condition. By having the plan thus divided into
squares of 10 acres each, the quantity of ground worked out may be
approximatively estimated at any time by inspection, and the total
quantity may be readily computed in the same manner. Besides dividing
the plan itself in this way, it will be found extremely useful to have a
sheet of tracing paper, or better, tracing cloth, divided into squares
of 316·228 links side, or 1 acre area, in black lines; each of such
squares being subdivided into 4 by lines in colour, to show quarters of
an acre. To find the area of any portion of exhausted or of unworked
ground, it is only necessary to lay this divided sheet over such portion
and to count the squares and quarters included in it.

By reason of the variations of the magnetic meridian, the date of a
survey should always be written on the plan; and as the plan of
underground workings is laid down piecemeal as the workings progress,
often extending over a period of many years, care must be taken to
reduce all bearings to the original meridian. Unless these matters are
strictly observed, serious errors may result.

When two or more veins of mineral are being worked one above the other,
and are placed upon the same plan, they are distinguished by means of
colour. It matters not what colours are employed for the several
separate workings so long as they are distinct from each other. Also the
mode of applying the colour, whether with the brush or with the pen, is
entirely a question of taste. It may, however, be observed that as
mining plans are constantly being added to, it is very difficult to
avoid a patchy appearance when the colour is laid on with the brush.
Plate 33 shows the manner in which mining plans are got up.


_Estate and Town Plans._--Plans of estates and towns, including as they
do only a limited area and requiring great distinctness of detail, are
laid down to a large scale; for the form and character of the detail
are, on such plans, of equal importance with its position. With such a
scale as is required in these cases, it is possible, not only to clearly
distinguish natural and artificial features, but to introduce means of
producing pictorial effect into their representation. The nature of
these means may be seen in the examples of plans appended to this work.

The manner of showing the various kinds of fences has been already
described. Trees are usually shown in elevation for the sake of artistic
effect; but care must be taken to give them such dimensions as will
accord with the scale of the drawing. Houses and other buildings are
shown in plan of the correct form, and washed for distinction in light
red for dwelling-houses, dark grey for outhouses, and light grey for
public buildings. Dark grey is also used for all wooden and iron
buildings to distinguish them from those constructed of the ordinary
materials, brick and stone. But besides such distinctions, others are
needed to indicate the character of natural features and artificial
constructions. These are obtained either by showing the object roughly
in elevation, or by some purely conventional means. The signs of this
character that are likely to be frequently required on estate plans are
shown on Plate 15. The manner of representing water, which has been
described in a preceding Section, will be found illustrated in detail on
Plate 11. Plates 10 and 14 show various kinds of trees; in this form
they may be introduced very effectively into plans of estates.

The several stages which a plan passes through in the office are shown
on Plate 2. If the plan is to be coloured, the colouring must be done
before the lettering is put on. Plate 3 shows a plan lightly coloured,
as used by surveyors, solicitors, and others; and Plate 17 shows a
finished plan in colour. The methods of laying on the colours and the
principles involved in the operations have been fully described and
explained in a former Section. In Plate 13 is given a town plan showing
a proposed street improvement. Such a plan must be laid down to a large
scale, and the details in and near the part affected must be drawn in
clearly and accurately. The uncoloured portion represents the plan as
prepared for lithographing. When pink colour is used to show the
proposed street, the buildings should be coloured in black by a light
wash of Indian ink. Yellow or any other bright tint may be used for the
proposed street, the object being merely to distinguish it clearly.
Existing streets should be coloured in yellow ochre, except when that
colour is used for the proposed street, in which case burnt sienna may
be used.

The Plates relating to this Section are Nos. 2, 3, 10, 11, 13, 14, 15,
17, 19, 20, 21, and 33.


SECTION V.--MAP DRAWING.

The principles and practice of map drawing, being in the main identical
with those of ordinary plan drawing, have been generally explained and
described in the preceding Sections. In the present Section, therefore,
we have only to direct attention to such details as belong especially to
the former class of topographical representations. These details relate
chiefly to the selecting of objects and features on the surface of the
ground whose character entitles them to special notice, and therefore to
distinct delineation; to the practical methods of sketching such objects
and features in the field, and to the means and the manner of
reproducing them on the finished map. The first and the last of these
questions have been treated by Mr. James in his Handbook of Topography,
and the second by Lieut. R. S. Smith of the United States’ Army, in so
concise and yet so complete a manner that we have not hesitated to avail
ourselves of their labours rather than attempt to offer any instructions
of our own. The following is, therefore, worthy of respectful attention.


_Single Stroke Streams._--In inking in streams, begin at the _source_
and draw downwards towards yourself, increasing the pressure on the pen
as you descend. The use of the steel pen in drawing single stroke
streams is very objectionable. Even soft steel pens are apt to cut the
surface of the paper, and in sharp bends it is quite impossible to
ensure an even width of line with the best yet made; by re-inking, much
time is lost, and frequently a rough jagged line is the result. The
common quill pen finely pointed will work well on any sort of paper.


_Double Line Streams and Rivers._--In maps of a small scale from 8 to 32
miles to the inch, it is usual to darken the north-western bank,
supposing the light to fall from the N.W. corner of the map; but on maps
of a large scale it is usual to attend strictly to the height of the
banks, and the draughtsman should carefully represent the exact nature
of each bank on his field-sketch or plane-table sheet.


_Colouring Streams or Rivers._--Single stroke streams may be inked in
with either a dark line of Prussian blue, or a light line in Indian ink
may first be drawn and a streak of Prussian blue or cobalt run neatly
along it. Cobalt is much used both in single and double stroke
streams--it is certainly the prettiest and most lasting blue we have,
and the preference should be given it, as it imparts a high finish to
MS. maps.

In maps which contain much hilly ground, the streams should be drawn in
with light ink and a very fine pen at first, and be re-drawn with dark
ink or dark blue after the shading of the hills. Large rivers on all
maps published in England are now coloured with a flat-wash of cobalt or
Prussian blue. Some draughtsmen prefer shading rivers according to
bends, and keeping the shade as falling from the N.W., but this system
cannot be carried out on maps of a large scale, where the height of the
bank is correctly represented.


_Islands and Sand-banks, Sandy and Pebbly Beds of Rivers._--Islands
which are only visible at low water, on well-coloured maps, are usually
first washed over with a light shade of burnt or raw sienna, or a
mixture of raw sienna and light red; the last-mentioned colour does not
easily mix with water, and should not be used if any other can be
substituted. After the tint is dry, dot finely with light Indian ink or
dark burnt sienna. Sand-banks are coloured in the same way. Sandy beds
may be similarly treated, omitting the dotting if pressed for time.
Pebbly beds should first be tinted with a mixture of burnt or raw
sienna, then dip into a dark shade of burnt sienna any coarse camel-hair
brush, and splitting the brush by drawing it between the forefinger and
thumb, dot in the tinted portions. Care must be taken to avoid having
too much colour in the brush, or the dots will run into each other and
make ugly daubs.

Another very easy and successful method of dotting in sand or pebbly
beds is with a tooth-brush. First, on tracing or any other thin paper,
trace out exactly the limit of the tinted portion requiring dotting, cut
out these portions from the trace and place it correctly over the
original, dip a tooth-brush lightly into a saucer of colour of the
required depth of tint, and holding it in the left hand over the
uncovered portions, with the forefinger of the right hand or the blade
of a pen-knife, gently splutter the colour from the brush; when it is
necessary to cover a large space with dots, this will be found the
simplest and most speedy way of doing it.


_Roads and Pathways._--The main or trunk roads in any country should be
very distinctly represented by double and perfectly parallel ink lines,
coloured between the lines with lake or carmine. District roads
metalled, or those made between chief towns, should be shown by a single
line coloured with lake or carmine. Unmetalled roads and paths by only a
single line in burnt sienna.

The same system should be carried out in roads in a mountainous country,
and the draughtsman should give, either on the map or in the column of
remarks, such information regarding roads as is likely to be useful to
travellers or military authorities.


_Mountain Passes._--On large scale maps these should be distinctly
marked, and the windings of the road correctly shown. Along the _Pass_,
write the name in small lettering, and state whether it is practicable
for horses, or fit only for men on foot. On maps of a small scale, it
will be sufficient to show the pass by a zigzag line across the hollow,
with a note as above.


_Fords and Ferries, Toll-gates._--Fords should be carefully noted and
the name and depth of water during the rainy and dry seasons given, if
possible. The number of boats at every ferry should be correctly
ascertained, and noted on the map. Toll-gates may be shown on roads with
a light line drawn across the road, and the words “Toll-gate” be clearly
written on the side of the road.


_Encamping Grounds_, _Mile Stones_, _Wells_, _Springs and Tanks_, should
be correctly shown _and named_ on all maps.


_Telegraph Lines and Stations_ must be shown on all maps drawn to a
scale of four miles to an inch and upwards, by the usual symbol. On maps
of a small scale, show by dots or a thin line of yellow, giving a
reference under the title. The _Stations_ are of the first importance,
and should be represented by the symbol.


_Railways, Stations, and Termini._--Railways are represented by a strong
black line, with or without thin lines drawn at right angles to the main
line. They should of course be very carefully and accurately laid down,
as they form the chief feature in any country.

Stations are shown by well-defined circles, and the name given in plain
lettering. Termini are best shown by blocks representing the size of the
buildings according to the scale of the map.

Except on maps of a very large scale, jungle should not be shown over
_hilly_ ground. Representing such objects as trees, jungle or brushwood,
over plains or flat lands, on all ordinary scale maps, is very
necessary, and the exact limits of the jungle or waste should be
surveyed and correctly given by a dotted line, but over hilly ground it
would be impossible to do so without impairing the beauty and hiding the
features of the hill drawing. If it is actually necessary to make it
known that the hills have jungle on them, let a foot-note to the effect
be inserted amongst the remarks or under the title of the map. Under the
head of remarks or notes, it is always very necessary to state the kind
of jungle which exists in the surveyed tracts, for the information of
speculators and timber merchants, and for the guidance of the
lithographer or engraver. Notes on maps, whether statistical or
geographical, can never be too full; they are useful in supplying at
once information which could be obtained only from reading reports, and
frequently they render topographical details intelligible where there
might otherwise be doubt or misconception. They can be recorded in any
spare corner or blank space on the map.


_Size of Cities, Towns and Villages, and the different ways of
representing them._--It is of the utmost importance that all maps of a
large scale should show the size accurately of cities, towns and
villages. If the scale admits of it, the several blocks or groups of
houses with the roads between them should be correctly drawn.


_Sketching, Shading, and Copying Hills._--In sketching hills, always
begin by fixing--1st. _The drainage_; 2nd. Those features which are most
prominent, such as peaks, rocks, ledges of rock running with the strata
of the hills, trees remarkable for some peculiarity in shape or size so
as to be recognized from various positions, and any other objects likely
to help the eye in filling in the details; and lastly, sketch the
details, beginning always with the ground nearest yourself.

Endeavour to portray your ground faithfully--1st. By preserving the
direction and bend of streams as in nature; 2nd. By giving the run of
the ridges correctly; 3rd. By fixing the peaks, ledges of rock,
precipitous falls and flats carefully; 4th. By showing the saddles or
depressions between peaks, which can only be done by giving the peaks on
either side sufficient relief in shading; 5th. By attending strictly to
the true breadth of valleys; 6th. By suppressing all hollows with a
suitable depth of tint; 7th. By careful representation of the banks of
streams in the valleys; and lastly, by finishing shades and touches, in
which is comprehended the retouching with brush or pen work the entire
piece, strengthening the shade of the higher ridges and peaks to show
their relative heights, and suppressing the white tint along the ridges.

Many excellent draughtsmen are in the habit of leaving the ridge of
mountain ranges quite white; this is evidently a mistake, for, unless
the ridge of any range of hills is of one uniform height from end to
end, it cannot correctly be left white. Thus a wrong impression is
conveyed of the surface of the ridge, the white streaks look harsh and
are displeasing to the eye, and a stiff and unartistic look is given to
the finish of the drawing.

The means of communication, whether by roads or minor tracks, are
important, both for civil and military purposes, and should be carefully
inserted in the map. This can generally be done with facility in a hilly
country, as the fixed marks will be visible in sufficient number along
the road, so that the latter may be drawn in at once by plane-table
operations along the line of communication to be surveyed. In flat
countries, or where the view is circumscribed, it may be necessary to
resort to measurements and plotting; but should any case occur where the
fixed points of reference are far apart, the traverse system must be
resorted to, and the road should be plotted from computed co-ordinates.


_Field Sketching._--Field sketches are made with the lead pencil, and
may be drawn upon every page of the compass-book, or upon the alternate
pages, at the option of the topographer. In the former case, the
bearings and distances are recorded upon the drawing; in the latter, the
record occupies the left-hand page, and the sketch the opposite one. The
page for sketching should be ruled in squares, with blue or red ink,
forming thus an indeterminate scale, the length of the sides of the
squares being assumed at pleasure, according to the nature of the
ground. Both the record and the sketch are read from the bottom of the
page upward. Suppose the stations of the survey to be 100 feet apart;
then, assuming the side of the square to be 100 feet, commence the
sketch at the bottom of the page--in the centre, if the survey promises
to be tolerably straight; if otherwise, at some point to the right or
left of the centre, the reason for which will be explained directly. Let
the bearing from the first station, the starting point or zero, be N.
10° E. Draw a line from the bottom of the page upward; the side of the
square being assumed 100 feet, number the stations upon the squares as
far as the line is run, say 325 feet, and write the compass angle down
along this line. Let the bearing from the second station, or No. 1, be
N. 1° W.; draw a line, making, as nearly as can be judged by the eye,
the proper angle with the last bearing, and proceed as before. When the
page is exhausted, commence with a vertical line at the bottom of the
next one, marking upon it the remainder of the old bearing, and making,
by the eye, a new series of approximate protractions as before. If it
can be foreseen, as in most cases it can, that the line of survey will
be very crooked, bending, for example, from left to right, then commence
the bearing at the bottom of the page accordingly, beginning at a point
on the extreme right, and running it diagonally to the left, so as to
make due allowance for the great deflection anticipated in the next
bearing. Such cases may be foreseen in running around an inclosure, or
in following a curving stream or ridge. The advantages of the system of
squares in sketch books completely overbalance the one disadvantage,
which is, that the diagonal bearings will not make exact distances upon
the squares, while the vertical and horizontal ones will. It will be
remembered that the surveying book is designed to be exact only in its
_record_ and the general features of the ground, and that a slight
change of scale is not material, as it can be made exact when the survey
is protracted upon the map. By these approximate protractions, any page
of the book of survey conveys a very just notion of the bearings and
distances, and of the relative positions of the general features of the
ground. The first station being at the bottom of the page, note down, in
the space between it and the second one, all the features of the ground
passed over by the line of survey; as to whether it is cultivated,
forest, marsh, &c.; whether it is crossed by streams, ditches, &c., and
their width; if it rises or falls; about what degree of slope, &c. On
both sides of the line introduce, according to the scale, and their
distances, as judged by the eye, all topographical objects within sight,
such as buildings, roads, streams, hills, &c., &c., drawing them to the
scale if possible, and if they cannot be got upon the page, describing
briefly their nature and position. In sketching hills endeavour to
project as many horizontal curves as possible, which should be lightly
put in, and then the shading lines may be drawn over them. The degree of
slope should be frequently written down in numbers upon the sketch. The
names of localities, streams, hills, farms, &c., should also be entered.

Thus far we have supposed a measured line upon the ground, to which the
situation and dimensions of objects might be referred. It is much more
difficult to embody the relative positions and dimensions, where all is
left to the eye. Here a cultivated judgment is of the greatest value.
Practice alone can make a good sketcher under such circumstances. Rules
must, from the nature of the case, be few and general. In the first
place, all objects within the field of vision are presented to the eye
in _perspective_, whereas the sketch is to be a _plan_. The apparent
diminution of dimensions in distant objects must therefore be corrected
on the plan. For example, the windings of a crooked stream, or a road,
in perspective, are much exaggerated in retiring into the distance; they
must therefore be _straightened out_ in the sketch more and more, as
they are more removed. 2nd. In looking at variously placed hills from a
somewhat elevated station, the eye will in some cases look directly, or
perpendicularly, at the face of some slopes, while in others, the
surface of the slope, if prolonged, will pass through the eye, and will
not be seen in its true dimensions, though its inclination may be
judged. In sketching the shapes of hills, bodies of water, masses of
forest, &c., these facts must be taken into consideration, and to ensure
skill, eye sketches of a small portion of ground having well-marked
features must be frequently made, and compared with measurements of the
same features. In sketching a single hill, the best station is at the
summit. First endeavour to represent the lowest horizontal curve of its
surface; then a medial one; then the form of the level space at the
summit, or the highest horizontal curve. Others may then be introduced
between these, until the ground is sufficiently expressed. The angles of
inclination should be frequently noted down in numbers; all accidents of
ground, such as ravines, rocks, &c., should be carefully placed, and all
other objects, such as houses, fences, trees, &c., should be put down in
their proper relative positions and dimensions. Having thus prepared a
skeleton of horizontal curves, numbered as to inclination and heights,
the sketch will always serve a useful purpose without any lines of
greatest descent. After sufficient practice in this method, the eye will
become so cultivated as to enable the draughtsman to express the form of
ground by lines of descent at once, the mind conceiving the position of
the horizontal curves, and thus supplying the necessary data for the
shading lines, the relative thickness and length of which for the
different slopes is a matter very easy of acquirement. But this should
not be attempted until the method by horizontal sections is thoroughly
mastered.

It is easy thus to make a sketch of a single hill, but when there are
many, and the general face of the country is sloping also, the
difficulties of representing the connection of the different hills at
their bases are considerable. In such cases the direction and lengths of
the valleys, or water-courses if there are any, must first be noted,
bearing in mind the illusions of perspective in both its effects,
previously mentioned. Then establish the positions of the different
summits, marking down their relative heights, after which put in the
other objects to be represented, such as roads, trees, buildings, &c.,
referring their positions to each other, and correcting them where they
are found to disagree. Horizontal curves present the readiest means to
the beginner in sketching declivities. When, after some practice, the
form of a body suggests, as it always will, its horizontal sections,
then it will be time to resort at once to the lines of greatest descent.
The greatest difficulties to be overcome in the practice of
eye-sketching are, 1st, that of converting a perspective view into a
_plan_, in all its true proportions; and 2nd, in forming a just
conception of the intersections of different slopes _at their bases_.
Hence the rule, to project first upon the sketch, all the lowest lines,
or water-courses, and then the highest parts or summits. Then the middle
lines and objects may be placed, and the sketch filled up by referring
all others to those three groups which may be regarded as determined.

The lead pencil for field drawing should be moderately hard, and the
general tone of the drawing should be rather light. The shading of
slopes ought not to overpower by its depth the distinctness of other
objects, and the pencil should be so used and of such a quality as not
to be easily defaced by rubbing.

We have already described some of the duties of the “examiner” in
verifying and supplying detail in the field. The following fuller
exposition of those duties and the methods of performing them is taken
from an excellent little treatise on Land Surveying, by John A. Smith,
C.E.


_Examination of Maps in the Field._--For the purpose of the examination,
the “examiner” should be furnished with an elegant and accurate trace,
ink copy, of the plotted detail of the district, and he should be
provided with a suitable sketch case, lined with prepared ass skin,
pencil, linear scale, chain, &c., and labourers. The trace copy, in one
sheet, should be in extent not more than can be conveniently secured in
the sketch case. It is desirable that the marginal detail on the trace
copy shall be common to the adjoining sheets for examination. If the
district be extensive, and if there be no more than one examiner engaged
on the examination, adjoining sheets should not be given to the same
examiner, that the character of the examiner’s work may be ascertained
by independent examinations of the same marginal detail. In the
examination of the detail representation on a map the “examiner” should
be mainly guided by a few leading considerations; these are:--

1. The position of a straight line, or detail, on the map will be
correct when its actual and plotted position on the ground and map makes
equal angles with another known line and intersects it in a known point,
the position of which line and point on the ground has been previously
ascertained to be correctly represented on the map.

2. The line, or detail, will be correctly laid down--given in magnitude
and position--when its position and length on the ground and map are
ascertained to correspond accurately.

From 1 and 2 it will be seen--

_a._ That the point of intersection of two given straight lines on the
ground, and the corresponding point on the map, will be a given point on
the map, if the corresponding lines on the map be ascertained to be
correctly laid down in position. And,

_b._ That any two points being given or correctly determined, the
straight line terminating in them will be a given line. Further,

_c._ That a straight line traced or drawn through given points, is given
in position. It should be kept in view that lines may be more accurately
traced, and to a greater distance, with the naked eye, when the party
tracing is rather above than below the level of the field on which the
trace shall be made.

It may be also seen that a point on the map which is the common point of
intersection of three straight lines drawn through well-defined points
in the detail will be a given point, if lines traced through the
corresponding detail points on the ground be found to have a common
point of intersection. And further, that the correct determination of
two such points on the map determines, as already stated, the position
of a straight line through these points. The determination, in the above
manner, of three such common points of intersection correctly determines
the representation of a given triangle. In the examination the sides of
the triangle determined by intersections, as above, should be measured
on the ground, to ascertain and verify the accuracy of the
determinations of the angular points on the trace or map. The production
of detail lines, and lines traced through plotted points, should be
taken up in the chain measurements of the sides of this triangle.
Through these verified points straight lines should be traced, and drawn
in pencil, to well-defined points in the detail, such as the buttals of
fences, the corners of houses and walls, gate piers, &c. On these lines
the intersected and neighbouring detail should be examined by chain and
scale measurements. In the measurement of the lines the internal and
adjacent external detail should be very carefully examined, and
corrected on the map where found in error. The examination of the detail
should be carried forward by the production and intersection of given
lines, and also by chain measurements from given points, to verify the
position of the detail or other points on the map. This examination
should be continued to the limits of the trace sheet. In remote parts of
the trace and district, lines of verification should be drawn, traced on
the ground and measured with the chain to verify the scale measurements
by the examination. These lines should be long, and in situations
affording few facilities for the accurate determination on the map of
the position of the plotted detail by other modes of examination.

The straight line passing through the extremities, or other well-defined
points in curved detail, should be regarded as a detail line, and the
position of the intermediate curved detail verified by ordinates or
tangents. Buildings and adjacent detail should be carefully examined by
productions, &c., because of the greater difficulties these details
usually present to the surveyor and plotter, and the consequent
liability to small errors in the position of some of the plotted points,
which affect the direction of lines determined on them.

Among the Plates appended to this work will be found several examples of
map drawing suitable for reference. Plate 16 shows the signs used on
ordinary maps and charts. Plates 29 and 30 contain signs used chiefly
upon Indian and colonial maps; and Plates 31 and 32 give the signs
employed upon military maps, with a section and a plan of
fortifications. These signs should be neatly drawn and their dimensions
suited to the scale of the map, the same remark applying to these as to
trees in elevation. Plate 1 is a plan showing the principal characters
of work used in mapping. This plan has been very carefully compiled and
drawn to render it suitable as a plan of reference. Plate 12 illustrates
the construction and colouring of hills according to the several methods
described in the preceding Sections. Other examples, with rocky cliffs,
will be found on Plate 14. Plate 18 contains a piece of the Ordnance map
drawn to a scale of one inch to the mile, and furnishes an example of
finished work. Upon the same Plate will be found a piece of chart
showing soundings, intended as a reference for hydrographers and others
engaged in marine surveys. And Plate 28 shows the manner in which
geological maps are prepared. The whole of these examples will be found
worthy of careful study as specimens of the draughtsman’s art.

The Plates relating to this Section are Nos. 1, 10, 12, 14, 18, 28, 29,
30, 31, and 32.


SECTION VI.--MECHANICAL AND ARCHITECTURAL DRAWINGS.

It is not within the scope of the present work to explain and to
illustrate the principles according to which mechanical drawings are
executed. These must be studied in special treatises on Projection. The
several methods of giving expression and embellishment to this class of
drawings have, however, been fully described, and the principles upon
which these methods are founded carefully explained. It now remains for
us to add a few general remarks and some detailed instructions on the
practical application of these principles and methods.

Before commencing the delineation of any machine, the draughtsman should
make himself thoroughly acquainted with its character; that is, he
should ascertain the nature of the work it is designed to perform, the
means by which it performs that work, and the manner of its
construction. This preliminary study is necessary to enable him to
obtain a good general idea of the more important parts, which he will
have to give prominence to in the drawing, and to understand the nature
of the various connections between the numerous pieces of which the
machine is composed. The dimensions of the several parts must be
carefully taken, and when drawing from actual machinery, rough sketches
should be made to serve as a guide in getting out the complete drawing.
The dimensions should be clearly marked upon such sketches. As a general
rule, it is best to begin with the ground line and position of main
driving shafts, from which dimensions may be taken in every direction.
The manner of writing the dimensions, whether upon the rough sketch or
upon the complete drawing, should always be thus ← 2′ 6″ → for lateral,
and thus

    ↑
  5″³⁄₈
    ↓

for vertical dimensions. To enable the draughtsman to take these with
accuracy, he should be provided with a pair of callipers for measuring
the diameters of shafts, a plumb-line for obtaining lateral distances
when the objects are not in the same horizontal plane, and a two-foot
rule.

The chief point to be attended to in commencing the drawing of a machine
is to obtain the correct positions of the centre lines of its principal
component parts, especial regard being had to the centres of motion.
These centre lines have been explained in a former Section. Having laid
down these lines accurately in their relative positions, separate
sketches may be made on a large scale of each part of the machine, and
the details of each part constructed upon each corresponding centre line
in succession, until the whole machine is built up. The centre lines
should be drawn in red, and the dimensions should be laid off on each
side of them. It will frequently be necessary to take a careful section,
to obtain sufficient information from which to draw the plan and the
elevation.

With respect to the written dimensions on a drawing, it may be remarked
that they cannot be too full or too numerous. Indeed, without complete
written dimensions a drawing is almost useless; for though a scale may
and should in all cases be attached, great labour would be required to
make use of the drawing by means of the scale only. Every dimension
which an engineer is likely to require to know should, therefore, be
plainly written. Nor is it sufficient to give a dimension once only, as
on the plan, for example, and to omit it on the elevation or on the
section. It should never be necessary to refer to another drawing to
find a dimension. The lettering should be clearly executed, and the
direction of the lettering should be the same as that of the figuring,
an example of which has been given; that is, it should read from the
front or from the right-hand side of the drawing.

If a drawing is to be coloured, the lettering, and all dark lines, such
as shade lines, must be left till after the colour has been applied. On
all coloured drawings the draughtsman should endeavour to obtain a
bright, clear tint by repeating the washes a sufficient number of times.
In preparing a flat-wash the tint should be mixed up slightly darker
than is required, and the solid colouring matter allowed to settle
before using. The solution, being poured off without disturbing the
sediment, will give a perfectly clear and pure tint. Tints for colouring
perspective drawings should always be prepared in this manner. The
methods of laying on flat-washes and of shading by colours have been
described in former Sections. The following additional remarks on colour
shading are taken from Worthen’s ‘Cyclopædia of Drawing.’

A means of adding considerably to the definiteness of a coloured
mechanical drawing, and of promoting, in a remarkable degree, its
effective appearance, is obtained by leaving a very narrow margin of
light on the edges of all surfaces, no matter what may be the angles
they form with the surfaces that join them. This should be done
invariably; we do not even except those edges which happen to have
shadows falling upon them. In such cases, however, this margin, instead
of being left quite white, should be slightly subdued. The difficulty of
achieving this effect of imparting a clear, regular, unbroken appearance
to these lines of light seems very formidable, and, indeed, almost
insuperable. The hand of the colourist may be as steady and confident as
a hand can be, and yet fail to guide the brush, at an almost
inappreciable distance from a straight or a circular line, with that
precision and sharpness so requisite for the production of this
beautiful effect. We shall, however, explain a novel and an effective
method of arriving at this most desirable result.

Suppose the object about to receive the colour to be the elevation of a
long flat rod or lever, on the edge of which a line of light is to be
left. Fill the drawing pen, as full as it will conveniently hold, with
tint, and draw a broad line just within, but not touching, the edge of
the lever exposed to the light. As it is essential to the successful
accomplishment of the operation that this line of colour should not dry,
even partially, before the tint on the whole side of the lever has been
laid on, it will be well to draw the pen a second time very lightly
along the line, so as to deposit as much tint as possible. Immediately
this has been done, the brush, filled with the same tint, should be
passed along so as to join the inner edge of this line of colour and the
whole surface of the lever to be filled in. By this means a distinct and
regular line of light is obtained without sacrifice of time. A still
more expeditious way of colouring such surfaces is to draw a second
line of colour along and in contact with the opposite edge of the lever
or other object, and to fill in the intermediate space between the two
wet lines with the brush. In this way a clear, uniform outline to the
tint is obtained. The blades of the drawing pen must not be sharp, and
care must be taken not to press heavily upon it, as otherwise the blades
will leave their course visible--an unsightly betrayal of mechanical
means to obtain such regularity in the colouring. Flat circular surfaces
may be treated in the same way, by using the pen compass instead of the
drawing pen. When such surfaces are large it will be judicious to colour
them in halves or in quadrantal spaces, but great care must be taken to
join the parts neatly. The lines of junction may be obliterated by
slightly washing them, or by laying a very light tint over the whole
surface, taking care in crossing the lines of junction to rub them
lightly with the brush.

The line of light upon cylindrical objects may be beautifully produced
by the same means. To indicate this line with perfect regularity is
highly important, for if strict uniformity be not maintained throughout
its length, the object will appear crooked or distorted. Having marked
in pencil the position of the light, and filled the drawing pen with a
just perceptible tint, draw a line of colour on one side of the line of
light. Then, with the brush filled with the same tint, fill up the space
unoccupied by the shade tint, within which the very light colour in the
brush will disappear. The portion of the surface on the other side of
the line of light being treated in the same way, the desired effect, of
a stream of light, clear and mathematically regular, will be obtained.
The effectiveness and expeditiousness of this method will be most
noticeable on long circular rods of small diameter, where a want of
accuracy is more immediately perceptible. The extreme depth of shade, as
well as the line of light, may, on such rods, be marked by filling the
pen with dark shade tint, and drawing it exactly over the line
representing the deepest part of the shade. On either side, and joining
this strip of dark colour, another, composed of lighter tint, is to be
drawn. Others successively lighter should follow, until, on one side,
the line of the rod is joined, and, on the other, the lightest part of
the rod is nearly reached. The line of light is then to be shown, and
the faint tint used at this part of the operation spread with the brush
lightly over the whole of that portion of the rod situate on either side
of this line, thus blending into smooth rotundity the graduated strips
of tint drawn with the pen.

For the correct representation of a building, plans, sections, and
elevations are required. The plan is usually a horizontal section of the
building close above the ground floor. The position and the dimensions
of the walls and the rooms of a house are shown by this means. As the
walls are shown in section in the plan, sections of the various walls
must, of course, be supplied before the plan can be drawn. It is usual
to colour the section of the walls in a ground plan; but not
unfrequently a dark wash of Indian ink is preferred to colour. The
number of sections required will depend upon the regularity of the
building; but generally it will be found that two half-sections are
sufficient. These two half-sections are usually placed side by side,
separated by a single line. The lines on which they are constructed must
be drawn distinctly on the plan, and lettered. The section is then
described as “Section” or “Half-section” on A B, &c. Usually the line of
section is broken in plan, and the section is then said to be on A B,
C D, one half being on A B and the other half on C D. Separate sections
to larger scales are required for the details of construction, such as
joints of rafters, mouldings to windows, and other parts needing
distinct representation. Elevations generally represent the whole of one
side of the building, and every side that differs from the rest must
have its own elevation. Such elevations are termed Front, Back, and End
Elevations, or North, South, East, and West Elevations. In order to show
the foundations, a section of the ground is sometimes given with an
elevation; in such a case the level of the ground should be shown by a
distinct line. Sometimes the portions of the structure below the ground
are shown by dotted lines. Such portions should not be coloured. In
getting out the drawings the plan should first be drawn, then the
sections, and finally the elevations. The colouring of elevations will
afford the student an opportunity of applying the knowledge he may have
acquired from a former Section of this work, and of displaying his
artistic taste.

In the accompanying Plates will be found examples of colouring
mechanical and architectural drawings. These should be studied in
conjunction with the Section on colouring in the first part of this
work. Plate 22 shows a piece of marine engine carefully coloured to
indicate the material of which the several parts are made, and Plate 23
contains a piece of permanent way, consisting of wrought-iron rail and
bolt, cast-iron chair and wooden sleeper and block, and an elevation of
a skew bridge, accurately coloured and shaded in accordance with the
principles already explained. It is not within the scope of this work to
treat the subject of projection, whether orthographic, isometrical, or
perspective; but we have given examples of each of these for the purpose
of illustrating the remarks and instructions on colouring given in the
Section referred to above. Thus Plate 24 is a perspective drawing, such
as are frequently made by architects, requiring a high degree of skill
and taste on the part of the colourist. And Plate 27 contains two
isometrical views of a building. These examples are intended to serve as
models of finished colouring.

The Plates relating to this Section are Nos. 22, 23, 24, and 27.


SECTION VII.--COPYING AND REDUCING.

Duplicates of drawings are very frequently required; so frequently,
indeed, and in such numbers, that their production constitutes a large
portion of the work executed in every drawing office. Generally, these
duplicates are required to the same scale as the original drawing; but
often it becomes necessary to reduce or to enlarge the scale to render
the drawing suitable to the purpose for which it is intended. The
various means and methods by which such duplicates are produced are,
therefore, important matters to the draughtsman, and especially to the
young draughtsman, whose labours in the drawing office will for a long
time be confined almost exclusively to their employment. These means and
methods will now be described.


_Drawing from Copy._--Drawing from copy is rarely resorted to for the
purpose of obtaining duplicates, the process being too slow for
practical requirements. But it constitutes the principal means, after
the study of projection, by which pupils in the office are initiated
into the art of producing drawings. A few hints concerning the best
modes of proceeding in these operations will, therefore, be serviceable,
both to the instructor and the instructed.

First draw a horizontal and a vertical line through the middle, each
way, of the sheet upon which the copy is to be made; draw also similar
lines upon the copy. As these lines divide the paper equally, they may,
for the sake of distinction, be called “divisional lines.” If the centre
lines are not shown on the copy, these must next be drawn in lightly
with the pencil, great care being taken to place them correctly. The
position of these centre lines relatively to the divisional lines may
then be transferred by means of the dividers from the copy to the fair
sheet, upon which they must be drawn finely but distinctly. Sometimes it
will be necessary to draw other lines upon the copy, and to transfer
them in like manner to the fair sheet. The details may then be drawn in
upon these centre lines, by transferring to them the measurements taken
from the centre lines of the copy. In taking measurements from a centre
line through an object that has both sides alike, the dividers should be
turned over to ascertain whether the distance on the other side of the
centre line is the same, so as to prove the accuracy of the drawing with
respect to the centre line. All measurements must be taken in the exact
direction of the distance to be measured, and be transferred in the same
direction, or an obviously incorrect distance will be the result. In
making the mark, the point of the dividers should not be pushed into the
paper, a just visible mark being all that is required; care must also be
taken, when using the compasses, not to press the leg into the paper, as
the holes thus made render circles and arcs inaccurate, are unsightly at
all times, and completely destroy the unbroken appearance of a tint on
a coloured drawing by retaining the colour. When drawing in circular
details with the pencil, it will be well to place a small hand-drawn
circle around the centre for reference when inking in; also, when a
curve is struck from several centres, a temporary pencil line to
represent the radii should be drawn from the centres to their respective
arcs.

When two or more views of the same objects are given, they should be
worked upon simultaneously; because, having once drawn in the centre
lines, one measurement may be applied to the corresponding part in each
view, and so time and trouble saved.

In copying maps and plans by this method of drawing from copy, both the
copy and the fair sheets are divided up into small squares, by drawing a
number of other lines parallel to the divisional lines described above.
The intersection of detail with these lines may then be readily and
correctly transferred from the copy to the fair sheet.


_Copying by Tracing._--Tracing furnishes the most expeditious means of
multiplying drawings. When a tracing is required in outline only, the
usual way is to fasten the sheet of tracing paper with ordinary drawing
pins over the drawing to be traced; the sheet of tracing paper should be
sufficiently large to allow the pins to be clear of the drawing. If the
sheet is not large enough for this, strips of thin paper, with one edge
gummed to the tracing paper and the other to the board, may be used.
When this method is not practicable, the pin holes may be effaced to
some extent by turning the drawing upside down, and pressing back the
edges of the holes with the flat end of a pencil, after the tracing has
been removed. If the tracing is to be coloured, it must be stretched on
the board, or it will never lie flat after being moistened; and if the
colouring is to be applied before the tracing is removed from the
drawing, it is essential that the tracing paper be larger than the
drawing, so that it may be cut off without injury to the latter. When
there is not sufficient time to stretch the tracing paper, the tendency
to buckle up when drying may be greatly lessened by placing weights
around any part immediately after the colouring has been laid on. If the
tracing is to be mounted, the colouring should be applied after
mounting. When tracing cloth is used, a much better appearance will be
produced by applying the colour to the back of the tracing.

In performing the stretching process, the sponge must not be applied
directly to the tracing paper, but to a piece of clean white paper laid
over it; sufficient moisture will pass through to the tracing paper in a
few seconds. Sometimes, when the sheet is small, merely breathing upon
it will be found sufficiently effective. As tracing paper is thus
greatly affected by the breath, it has been recommended to entirely
finish both circles and lines within a small area at a time, when
copying a drawing, as if all the circles were put in first, as on a
drawing, many of them might be out of position before the lines could be
drawn. This recommendation is, however, of doubtful value. When tracing
from another tracing, a piece of white paper should be placed beneath
the copy to render the lines distinct.

A tracing may be made upon ordinary drawing paper by means of the _glass
drawing board_. This consists of a sheet of plate glass let into a
wooden frame about 3 inches wide flush with the face, the inner edges of
the frame being rebated for this purpose. This copying board is placed
on a table in front of a window, and supported at an angle of about 25°,
so as to get a strong light beneath, which light may be increased by
placing a sheet of white paper upon the table to reflect upwards. The
original drawing being pinned down to this board with a sheet of drawing
paper or parchment over it, the finest lines will be plainly visible,
and the drawing may be traced in the same manner as upon tracing paper.
To alter the light, the angle of the board may be changed. This method,
which is coming extensively into use, is a very convenient one for
copying plans and maps.


_Copying by Transfer._--Copying by transfer has superseded the method
already described as “drawing from copy.” Transfer paper, as employed
for this purpose, may be made in the following manner. Take half an
imperial sheet of very thin paper, such as tissue paper, and having
stretched it upon a board, rub some common blacklead powder well into
it. Then, having removed the dust and superfluous blacklead, well rub
the sheet with a cotton rag to prevent its soiling the paper when used
for transferring a drawing. A sheet of transferring paper prepared in
this way will last for years. Red transfer paper, which is principally
used by lithographers, is prepared in the same manner with red ochre.

To transfer a drawing, the sheet of transfer paper is laid with its
prepared face upon the paper which is to receive the drawing, and over
this is placed a tracing of the drawing to be copied, carefully pinned
down. The straight lines of the tracing may then be transferred to the
drawing paper below by going over them with a style or other pointed
instrument that will not cut the tracing. For the regular curves and
circles, it will be sufficient to mark the centres by a small cross,
thus, ×, and the radii by short lines. Other curves may be transferred
by means of the French curve. By this means a copy of the original
drawing is obtained in black or red lines, which may be afterwards inked
in. Though three distinct operations are required in this process,
_making the tracing_, _transferring_, and _inking in_, a drawing can be
much more rapidly copied by means of it, than by measuring off with the
dividers, as in drawing from copy.


_Reducing and Enlarging._--It is evident that in drawing from copy, the
drawing may be reduced or enlarged at pleasure, since it is only
necessary to take half or twice the dimensions as required. Usually
proportional compasses are employed for this purpose. When reducing by
scales, it is obviously not essential to use the same scale as that to
which the original is made; the dimensions on one scale may be readily
transferred to any other, and the student will do well to make himself
familiar with the operation.

For reducing or enlarging plans, several means are employed: one of
these is known as the method of squares, and is illustrated on Plate 26.
In the preceding remarks on drawing from copy, it was shown how in
copying to the same scale, both the copy and the fair sheet were divided
into squares of equal size, and how the intersections of the detail with
the lines forming these squares on the copy were transferred by
measurement to corresponding points on the fair sheet. It is obvious,
therefore, that if the squares on the latter be larger or smaller than
those on the former, as the intersections will be transferred to the
same relative positions on the fair sheet as they occupy on the copy,
the plan, or other drawing, will be enlarged or reduced accordingly.
This is the principle upon which drawings are reduced by this method.
Proportional compasses are required in the operations.

Drawings may also be rapidly reduced or enlarged by means of instruments
called the _Pantograph_ and the _Eidograph_. Both of these instruments
are shown on Plate 26. The following very complete description of the
pantograph and the eidograph is given in an excellent work on
‘Mathematical Drawing Instruments,’ by W. F. Stouley, of Holborn,
London.

“The pantograph, as represented on the plate, consists of four rules of
stout brass, which are jointed together in pairs, one pair of rules
being about double the length of the other. The free ends of the shorter
pair are again jointed to the longer in about the centre. It is
important that the distance of the joints on each of the short rules
should exactly correspond with the distance of the joints on the
opposite longer rules, so that the inscribed space should be a true
parallelogram. To enable the instrument to work freely and correctly,
all the joints should be perfectly vertical, and with double axes. Under
the joints casters are placed to support the instrument, and to allow it
to move lightly over the paper. One of the long rules has a socket fixed
near the end, which carries a tracing point when the instrument is used
for reducing. The other long rule, and one of the shorter rules, have
each a sliding head fitted upon it, which is similar to one of the heads
of a pair of beam compasses. Each head has a screw to clamp it in any
part of the rule, and carries a perpendicular socket, which is placed
over the edge of the rule in a true line with the joints. Each socket is
adapted to hold either a pencil holder, tracing point, or fulcrum pin,
as may be required. The rules upon which the heads slide are divided
with a scale of proportions: 1--2, 11--12, 9--10, &c., which indicate
as one is to two, as eleven are to twelve, as nine are to ten, &c.

“A loaded brass weight, which firmly supports a pin that fits exactly
into either of the sockets, forms the fulcrum upon which the whole
instrument moves when in use.

“The pencil holder is constructed with a small cup at the top, which may
be loaded with coin or shot to cause the pencil to mark with the
required distinctness.

“Arrangement is made to raise the pencil holder off the drawing. This is
effected by a groove down one side of the pencil holder, in which a cord
is fixed, passing from the pencil along the rules, turning the angles
over small pulleys, and reaching the tracing point, where it may be
readily pulled by the hand to raise the pencil. This will be found
especially convenient when the pencil is required to pass over any part
of the copy not intended to be reproduced.

“The pantograph is set to reduce drawings in two ways, termed
technically the _erect manner_ and the _reverse manner_. It will be
necessary to give full details of each manner, particularly in relation
to the scales engraved upon the instrument, which are not very
intelligible; indeed comparatively few professional men are sufficiently
acquainted with them to avail themselves of their full value.

“By the _erect_ manner of setting the pantograph, the reduced copy will
appear erect; that is, the same way as in the original. The general
position of the parts of the instrument set in this manner is shown in
the plate, where it will be seen that the fulcrum pin is placed in the
socket of the sliding head upon the outside long rule, and the pencil
holder in the socket upon the short central rule. By this method of
setting the instrument, it will reduce in any of the given proportions
not exceeding half-size, technically from 1--2. The scales engraved upon
the rules that accord with the erect manner of setting are those which
have 1 for the first proportion; as 1--2, 1--3, 1--4, &c. The other
scales may be used, but will not accord with the reading, except through
arithmetical deductions, the results of which may be given more clearly
by the following complete Table than by rules with exceptions.

TABLE OF REDUCTIONS BY THE PANTOGRAPH IN THE ERECT MANNER, THE FULCRUM
BEING PLACED IN THE SOCKET UPON THE OUTSIDE RULE, AND THE PENCIL UPON
THE CENTRAL RULE.

  ------------------+----------------
  Reading given upon|Reduces in the
     the Scales.    |Proportion of
  ------------------+----------------
          1--2      |    1 to 2
          1--3      |    1 „  3
          1--4      |    1 „  4
          1--5      |    1 „  5
          1--6      |    1 „  6
          1--7      |    1 „  7
          1--8      |    1 „  8
          1--9      |    1 „  9
          1--10     |    1 „ 10
          1--11     |    1 „ 11
          2--3      |    2 „  5
          3--4      |    3 „  7
          4--5      |    4 „  9
          5--6      |    5 „ 11
          6--7      |    6 „ 13
          7--8      |    7 „ 15
          8--9      |    8 „ 17
          9--10     |    9 „ 19
         10--11     |   10 „ 21
         11--12     |   11 „ 23
  ------------------+----------------

“In the above Table the readings which are given with the proportions
are given to show clearly which proportions agree with the erect scales;
many of those that do not agree with the reading are very useful, as
2--3, which is often required to reduce a drawing from a scale of 20 to
one of 50.

“In the _reverse_ manner of setting the pantograph, the reduced copy
appears reversed, or upside down, to the original. The fulcrum pin is
placed in the socket upon the short central rule, and the pencil holder
is placed in the socket upon the outside rule. This is generally the
most convenient way of using the pantograph for large drawings, as the
original and copy come edge to edge, and need not overlap each other,
which is often compulsory in the erect manner; the range of scale is
also much greater, as the proportions include the unit proportions of
the erect scale, and continue in ratios up to full size.

“The following Table will give the readings of the instrument which
accord with the reverse setting, and those which may be used to this
setting, obtained by calculation.

TABLE OF REDUCTIONS BY THE PANTOGRAPH IN THE REVERSE MANNER, THE FULCRUM
BEING PLACED IN THE SOCKET ON THE CENTRAL RULE, AND THE PENCIL IN THE
SOCKET UPON THE OUTSIDE RULE.

  ------------------+------------------
  Reading given upon| Reduces in the
     the Scales.    | Proportion of
  ------------------+------------------
          1--2      |  1 to 1 full size
          1--3      |  1 „  2
          1--4      |  1 „  3
          1--5      |  1 „  4
          1--6      |  1 „  5
          1--7      |  1 „  6
          1--8      |  1 „  7
          1--9      |  1 „  8
          1--10     |  1 „  9
          1--11     |  1 „ 10
          2--3      |  2 „  3
          3--4      |  3 „  4
          4--5      |  4 „  5
          5--6      |  5 „  6
          6--7      |  6 „  7
          7--8      |  7 „  8
          8--9      |  8 „  9
          9--10     |  9 „ 10
         10--11     | 10 „ 11
         11--12     | 11 „ 12
  ------------------+------------------

“The above Table and the previous one give the proportions for
reductions, the tracing point being in every instance considered upon
the outside rule. If it were required to produce an enlarged copy, which
the pantograph will do but very imperfectly, the pencil and tracer would
have to change places; the proportions of course would read the same.

“In using the pantograph some care is required in setting the fulcrum
weight in the best position to allow easy action of the instrument over
the space required. It should always be roughly tried over the boundary
before commencing the copy.

“The ordinary pantograph will in no instance work over a large drawing
at one operation, but it may be shifted about as required, using care,
and testing the copy after the fulcrum is moved, to see that the tracer
and pencil correspond in those parts already produced, that the
pantograph will reach in its shifted position. The fulcrum weight being
generally made with needle points to attach it to the drawing will be
found very difficult to shift so short a distance as is frequently
required. This may be easily remedied by attaching with gum a piece of
indiarubber over each of the sharp points, when it is required to be
used for large drawings. The rubber will hold the paper sufficiently if
the pantograph work freely in the joints and casters, as it should do.

“In copying the buildings which frequently occur in plans of estates,
&c., a straight slip of transparent horn will be found very convenient
to guide the tracing point. Some draughtsmen have the horn cut with an
internal angle, by which one side and one end of a building may be
traced without shifting the horn.

“Architects and mechanical engineers seldom use the pantograph; however,
it may perhaps be sometimes used with advantage for tracing in the most
difficult and tedious parts of a drawing with a precision impossible by
hand. This applies particularly to such parts as are frequently
repeated, as capitals, trusses, bosses, tracery, &c., upon drawings to
very small scales. In these instances it is only necessary to make a
detail sketch, say six times the size required, and to place the fulcrum
weight in such position that the pencil will pass over the parts
required to be filled in, the tracer at the same time resting on a
corresponding part of the detail sketch, which may be placed in position
under the tracing point, and be held sufficiently by two lead weights.
For a second ornament on the same drawing, the detail may be shifted
without moving the fulcrum.

“To follow the outline of any object of the ornamental class, or for the
reduction of mechanical drawings to a size suitable for wood or other
engravings, the strip of horn will be found particularly useful; indeed,
to obtain any degree of precision, it will be better, generally, to let
the tracer follow a guiding edge placed over the original for that
purpose. French curves are particularly useful, although perhaps only a
small piece may be available at once. The tracer may rest on the surface
until another part of the curve is found to correspond with the
continuation of the line.

“In some old pantographs a guide is fixed to the tracing point. The
guide is a kind of handle similar to a drawing pencil, the point of
which is hinged to the point of the tracer. This gives a convenient and
firm hold of the point, and appears to the author a useful appendage.

“Pantographs have been made in many shapes unnecessary to describe, as
they are all of one principle--that of a parallelogram jointed at the
four corners; the principal difference being in the position of the
points and fulcrum in relation to the parallelogram. One thing is
essential in every construction,--that is, that the fulcrum, tracer, and
pencil should always be in a true line when the instrument is set for
use. The parallelogram may be in any position on the instrument, to the
fancy of the maker.

“The _Eidograph_ was invented by Professor Willis in 1821. It is a most
ingenious and exact instrument, for many purposes superior to the
pantograph, within the range of its working powers, which, however, may
be considered to be limited to reducing or copying off, between the full
size of the original and one-third of the size; for greater reductions,
the balance of the various parts is thrown so far out that it appears
clumsy to use, and is really inferior to the pantograph. The great merit
of the eidograph is, that within its range it reduces conveniently and
exactly in all proportions; for instance, we may reduce in the
proportion of 9 to 25 as readily as 1 to 2. It is also in every way
superior to the pantograph in freedom of action, there being no sensible
friction on the single fulcrum of support, and in its movement it covers
a greater surface of reduction.

“It is somewhat curious that an instrument of such great merit should be
little known in the profession, where its uses would be so constantly
convenient. This may partly be attributed to the very few published
descriptions which are to be found in works treating on mathematical
instruments. It is not intended, however, to infer that there are not
many eidographs in use, but that the writer presumes they are
comparatively little known, from his personal acquaintance with
professional men, and from the number of large pantographs that are made
and sold to perform work that could be done so much more exactly and
conveniently by the eidograph. This remark will not apply to the small
pantograph, which is less expensive than a small eidograph, and answers
perfectly for the reduction of small plans--as, for instance, those
frequently attached to leases and conveyances.

“The details of the construction of the eidograph are as follows:--The
point of support is a heavy, solid, leaden weight, which is entirely
covered with brass; from the under side of the weight three or four
needle points project, to keep it in firm contact with the drawing. Upon
the upper side of the weight a pin, termed a fulcrum, is erected, upon
which the whole instrument moves. A socket is ground accurately to fit
the fulcrum, and attached to a sliding box, which fits and slides upon
the centre beam of the instrument. The sliding box may be clamped to any
part of the beam by a clamping screw attached. Under the ends of the
beam are placed a pair of pulley wheels, which should be of exactly
equal diameter; the centre pins of these revolve in deep socket fittings
upon the ends of the beam. The action of the two wheels is so connected
as to give them exact and simultaneous motion. This is effected by means
of two steel bands, which are attached to the wheels. The bands have
screw adjustment to shorten or lengthen them, or to bring them to any
degree of tension. Upon the under side of each of the pulley wheels is
fixed a box, through which one of the arms of the instrument slides, and
is clamped where required. At the end of one of the arms a socket is
fixed to carry a tracing point, at the end of the other arm a similar
socket is fixed for a pencil. The pencil socket may be raised by a lever
attached to a cord, which passes over the centres of the instrument to
the tracing point. The two arms and beam are generally made of square
brass tubes, and are divided exactly alike into 200 equal parts, which
are figured so as to read 100 each way from the centre, or by the
vernier cut in the boxes through which the arms and beam slide they may
be read to 1000.

“There is a loose leaden weight which fits upon any part of the centre
beam, packed in the box with the instrument. The weight is used to keep
the instrument in pleasant balance when it is set to proportions which
would otherwise tend to overbalance the fulcrum weight.

“In the above details it will be particularly observed that the pulley
wheels must be of exactly equal diameters. It is upon this that chiefly
depends the accuracy of the instrument, the periphery of these wheels
being the equivalent to the parallelogram, which has been already
described as the essential feature of the pantograph. The adjustment of
the wheels to size, by turning in the lathe, is, perhaps, the reason the
results of the eidograph are more exact than those of the pantograph,
which has no equivalent compensation for the always possible inaccuracy
of workmanship.

“From the details just given, the general principle of the eidograph may
be easily comprehended. Thus, the wheels at each end of the beam being
of equal size, the steel bands connecting them being adjustable, so as
to bring the wheels into any required relative position, it follows,
that if the arms fixed to the wheels be brought into exact parallelism,
they will remain parallel through all the evolutions or movements of the
wheels upon their centres; consequently, if the ends of the arms be set
at similar distances from the centres of the wheels, any motion or
figure traced by the end of one arm will be communicated to the end of
the other, provided the fulcrum of support be placed also at a similar
distance from the centre of one of the wheels.

“To adjust, or ascertain if the eidograph is in adjustment, is very
simple, from the reason that when the arms are parallel the adjustment
is perfect for all proportions. The manner of ascertaining this is as
follows: place all the verniers at zero, which will bring them to the
exact centres of the arms and the beam, place the arms at about right
angles with the beam, then mark simultaneously with the tracer and
pencil point, turn the instrument round upon its fulcrum, so that the
pencil point be brought into the mark made by the tracer; then, if the
tracer fall into the mark made by the pencil the instrument is in
adjustment. If it should not fall into the same mark, the difference
should be bisected, and the adjusting screws on the bands should be
moved until the tracer fall exactly into the bisection, which will be
perfect adjustment.

“When the eidograph is in adjustment, if the three verniers be set to
the same reading on any part of their scale, the pencil point, fulcrum,
and tracer will be in a true line. If it should not be so, it will show
the dividing of the instrument to be inaccurate. Thus we have a simple
way of testing the eidograph in every important particular.

“The divisions upon the eidograph do not positively indicate the
reductions required to be performed by the instrument, but merely give a
scale, which, with the assistance of the vernier, divides the beam and
arms into 1000 parts. To obtain the quantity to which the verniers are
to be set, it is necessary either to apply to a table of proportions
relative to divisions, or to simple arithmetic, as will be shown. A
printed table is very generally placed inside the lid of the box in
which the instrument is packed, which contains part of the following
proportions:

TABLE FOR REDUCING OR ENLARGING PROPORTIONS.

  ------------+---------
  Proportions.|Divisions
              | on Bars.
  ------------+---------
  As 1 is to 2| 33·333
   „ 1   „   3| 50
   „ 1   „   4| 60
   „ 1   „   5| 66·666
   „ 1   „   6| 71·428
   „ 1   „   7| 75
   „ 1   „   8| 77·777
   „ 1   „   9| 80
   „ 1   „  10| 81·818
   „ 2   „   3| 20
   „ 2   „   5| 42·857
   „ 3   „   4| 14·285
   „ 3   „   5| 25
   „ 4   „   5| 11·111
   „ 5   „   6|  9·09
  ------------+---------

“The table here given answers for the general purposes of reducing, such
as the bringing of a plan from one chain scale to another, the
quantities of which are found by the following rule:


“_To find the quantity equal to any given proportion for the setting of
the eidograph._--Subtract one sum of the proportion from the other, and
multiply this difference by 100 for a dividend; add the two sums of the
proportion together for a divisor: the quotient from the working of this
will give the number to which the arms and beam are to be set.

“For instance, let it be required to reduce a drawing in the proportion
of 3 to 5.

     5 - 3 = 2
           × 100
             ---
  5 + 3 = 8) 200 (25

“The centre beam is to be set to 25 on the side nearest the pencil
point, the pencil arm is also set to the 25 nearest the pencil point,
and the tracer arm is set to the 25 farthest from the trace. If it were
required to enlarge in the same proportion, each side would have to be
set at the opposite 25.

“To clearly illustrate the subject, it may be well to give another
example. Let it be required to reduce an ordnance plan of five feet to
the mile to a scale of three chains to the inch. First, we must have
like terms, therefore to reduce both proportions to feet to the inch
will, in this instance, be the most simple way; thus:

  5 feet to the mile   =  88 feet to the inch.
  3 chains to the inch = 198         „
              198 - 88 = 110
                           100
                         -----
         198 + 88 = 286) 11000 (38·461

“If the slides of the instrument be set to 38·46, it will be,
practically, sufficiently near.”

Photography is also frequently resorted to for the purpose of reducing
and enlarging drawings. The results are satisfactory within certain
limits of size; for it is obvious that when the drawing is large, the
parallel lines will converge in the photograph, for reasons which will
be understood from the laws of perspective. For enlarging small and
intricate drawings, photography is very useful. In preparing drawings
for reduction by this process, all lines and shadows should be put in in
Indian ink only. For optical reasons, colour cannot be reproduced by
photography, and as certain colours produce an effect which might not be
anticipated by the inexperienced, it will be well to warn such against
these effects, to prevent disappointment at the results obtained. Thus
blue, for instance, shows very indistinctly, and yellow surfaces in
coloured drawings come out very dark.


_Drawings for Lithographers and Engravers._--The drawings required by
the lithographic draughtsman are simply outline drawings or tracings,
with the shaded drawing for reference when such is required. The shaded
drawing should be traced when in outline only with a fine-pointed
pencil, not too hard. The engraver prefers such a tracing to the drawing
itself, unless he can have the latter before it is shaded. He will,
however, require the shaded drawing as a guide in copying in the
shadows. As the drawing always gets soiled under such circumstances,
unless protected, it is prudent to place it upon a board of the exact
size, with a glass over it to fit, the glass being kept in its place by
a strip of paper pasted round the edge. The drawing will not be required
at all if only an outline engraving is to be made. In that case, the
lines that are to be shade lines must be indicated on the pencil
tracing; a dot in red ink on each of such lines will be sufficient.

A scale should always be put upon lithographs and engravings, instead of
merely stating that it is drawn to some particular scale, because the
paper just before receiving the impression is damped, and consequently
expands. For this reason, no engraving is of the same size as the
original drawing; and as the degree of moisture varies, no two
engravings from the same plate ever are exactly equal in size. Hence the
necessity for drawing the scale is obvious.

The Plate relating to this Section is No. 26.


TRIGONOMETRICAL FORMULÆ.


_To compute the Sides of Triangles._--Let A B C be the angles of a plane
triangle, and _a_ _b_ _c_ the sides opposite. Then, for right-angled
triangles, we have

  _b_ = _a_ sin. B
  _b_ = _c_ tan. B  or _b_ = _c_ cot. C
  _c_ = _a_ cos. B     _c_ = _b_ cot. B
  _c_ = _b_ tan. C

and for oblique-angled triangles we have

        _a_ sin. B
  _b_ = ----------,
          sin. A

        _a_ sin. C
  _c_ = ----------.
          sin. A


_To compute the Areas of Triangles._--When two sides and the included
angle are known, _a_ and _b_ representing the two sides and θ the
included angle,

      _a_ _b_ sin. θ
  A = --------------.
            2

To find by logarithms the area in acres and decimals of an acre,

  Log. A = log. _a_ + log. _b_ + log. sin. θ - 15·30103.

When two angles and the included side are known, β and θ being the
angles and _a_ the included side,

      _a_² sin. β sin. θ
  A = ------------------.
        2 sin. (β + θ)

To find by logarithms the area in acres and decimals of an acre,

  Log. A = 2 log. _a_ + log. sin. β + log. sin. θ - log. sin. (β + θ)
  - 15·30103.

When the three sides are known, _a_ _b_ _c_ being the three sides and
_s_ their half sum,

  A = √(_s_(_s_ - _a_)(_s_ - _b_)(_s_ - _c_)).

To find by logarithms the area in acres and decimals of an acre,

  Log. A =

  log. _s_ + log. (_s_ - _a_) + log. (_s_ - _b_) + log. (_s_ - _c_)
  ----------------------------------------------------------------- - 5.
                                   2




INCLINED MEASURE.


TABLE SHOWING THE REDUCTION IN LINKS AND DECIMALS OF A LINK TO BE MADE
PER CHAIN FOR EVERY HALF DEGREE OF INCLINATION FROM 3° TO 30°.

(100 × versed sine of the inclination.)

  ------+----------
  Angle.|Reduction.
  ------+----------
   °  ′ |
   3   0|   0·15
   3  30|   0·19
   4   0|   0·24
   4  30|   0·31
   5   0|   0·38
   5  30|   0·46
   6   0|   0·55
   6  30|   0·64
   7   0|   0·75
   7  30|   0·86
   8   0|   0·97
   8  30|   1·10
   9   0|   1·23
   9  30|   1·37
  10   0|   1·53
  10  30|   1·67
  11   0|   1·84
  11  30|   2·01
  12   0|   2·19
  12  30|   2·37
  13   0|   2·56
  13  30|   2·76
  14   0|   2·97
  14  30|   3·19
  15   0|   3·41
  15  30|   3·64
  16   0|   3·87
  16  30|   4·12
  17   0|   4·37
  17  30|   4·63
  18   0|   4·89
  18  30|   5·17
  19   0|   5·45
  19  30|   5·74
  20   0|   6·03
  20  30|   6·33
  21   0|   6·64
  21  30|   6·96
  22   0|   7·28
  22  30|   7·61
  23   0|   7·95
  23  30|   8·29
  24   0|   8·65
  24  30|   9·01
  25   0|   9·37
  25  30|   9·74
  26   0|  10·13
  26  30|  10·51
  27   0|  10·90
  27  30|  11·30
  28   0|  11·71
  28  30|  12·11
  29   0|  12·53
  29  30|  12·96
  30   0|  13·40
  ------+----------




CURVATURE AND REFRACTION.


TABLE OF CORRECTIONS IN FEET AND DECIMALS OF A FOOT.

  ---------+----------+-----------+---------------
  Distance |Curvature.|Refraction.|Correction for
  In Miles.|          |           | Curvature and
           |          |           |  Refraction.
  ---------+----------+-----------+---------------
     ¹⁄₄   |     ·04  |     ·01   |      ·03
     ¹⁄₂   |     ·17  |     ·02   |      ·15
     ³⁄₄   |     ·37  |     ·05   |      ·32
    1      |     ·67  |     ·09   |      ·58
    1¹⁄₂   |    1·50  |     ·21   |     1·29
    2      |    2·67  |     ·38   |     2·29
    2¹⁄₂   |    4·17  |     ·60   |     3·57
    3      |    6·00  |     ·86   |     5·14
    3¹⁄₂   |    8·17  |    1·17   |     7·00
    4      |   10·67  |    1·52   |     9·15
    4¹⁄₂   |   13·55  |    1·93   |    11·62
    5      |   16·67  |    2·38   |    14·29
    5¹⁄₂   |   20·18  |    2·88   |    17·30
    6      |   24·01  |    3·43   |    20·58
    6¹⁄₂   |   28·18  |    4·03   |    24·15
    7      |   32·68  |    4·67   |    28·01
    7¹⁄₂   |   37·52  |    5·36   |    32·16
    8      |   42·69  |    6·10   |    36·59
    8¹⁄₂   |   48·19  |    6·88   |    41·31
    9      |   54·02  |    7·72   |    46·30
    9¹⁄₂   |   60·20  |    8·60   |    51·60
   10      |   66·70  |    9·53   |    57·17
  ---------+----------+-----------+---------------




INDEX.


  A.

  Angle, to bisect, 16
  ----, to construct, equal to a given angle, 17
  ----, to draw a line making a given, 15
  ---- of light in mechanical drawings, 48
  ---- ---- in topographical drawings, 54
  Arch, Gothic, equilateral, to draw, 23
  ----, ---- lancet, to draw, 24
  ----, ---- obtuse, to draw, 24
  ----, ---- ogee, to draw, 25
  ----, ---- Tudor, to draw, 24
  ----, Moorish horse-shoe, to draw, 24
  ----, semi-elliptical, to construct, 23
  Architectural drawings, 121
  Arcs, centres of, to be marked, 10
  Areas to triangles, to compute, 142


  B.

  Bisecting an angle, 16
  Blacklead paper, 8
  Book of reference, 97, 101
  Books for field sketching, 115
  Borders and corners, 69
  Borings, 104
  Bottle indiarubber, 9
  Boundary maps, 104
  Bows, 2
  ----, spring, 2
  Broken lines, 30
  Brushes for tinting, 41
  Buildings--plans, sections, and elevations, position of, on
  drawing, 125


  C.

  Carbonic paper, 8
  Cartridge paper, 6
  Centre lines, 10
  ---- ----, care to be taken in placing correctly, 10
  Centre of circle, to find the, 18
  Centres of arcs to be marked, 10
  Cinquefoil, Gothic, to draw, 26
  Circle, to describe, through three given points, 17
  ----, to draw a tangent to, 17
  ----, to draw radii of, the centre being inaccessible, 18
  ----, to find the centre of, 18
  Circles, concentric drawing, 10
  Cities, to represent size of, 112
  Civil engineers’ plans, 96
  Cleaning drawing pen, 3
  Cleaning off drawings, 9
  Cleanliness, importance of, 9
  ----, precautions to be taken to ensure, 9
  Cloth, tracing, 7
  ----, ----, Sager’s vellum, 7
  ----, ----, sizes of, 7
  Colour for buildings, 47
  ---- for cultivated land, 47
  ---- for fences, 47
  ---- for grass-land, 45
  ---- for gravel, 46
  ---- for marsh, 45
  ---- for mud, 46
  ---- for roads, 47
  ---- for sand, 46
  ---- for streets, 47
  ---- for water, 45
  ---- for woodland, 46
  ---- for uncultivated land, 47
  Colouring cylindrical objects, 124
  ---- drawings, 11, 122
  ---- rivers and streams, 110
  Colours, 39
  ----, conventional, Table of, 44
  ---- for sections, 44
  Compasses, 2
  ----, manner of using, 2
  ----, pencil leg of, 2
  ----, points of, 2
  ----, removing movable leg, 2
  Competition drawings, paper for, 6
  Concentric circles, drawing, 10
  Construction of scales, 11
  Continuous cartridge paper, 6
  ---- tracing paper, 7
  Contour lines, 37
  Contours, to plot, 90
  Conventional colours, Table of, 44
  Copying by tracing, 128
  ---- by transfer, 129
  ---- drawings, 11, 126
  ---- from tracing, 11
  Corners and borders, 69
  Cross sections, 98, 102
  Cultivated land, to represent, 32
  Curvature and refraction, Table for correction of, 143
  Curved lines, 29
  ---- ----, to draw, 29
  Cutting off drawings, 7, 11, 12
  Cylinders, various methods of shading, 64
  Cylindrical objects, to colour, 124
  ---- surfaces, shade lines, 50
  ---- ----, shading lines, 51
  Cyma recta, to draw, 25
  ---- reversa, to draw, 25


  D.

  Dented drawing-board, to remedy, 14
  Detail plotting, 89
  Dimensions of drawing table, 2
  ---- to be written on drawings, 122
  Distances, scales of, 70
  Dividing a line into equal parts, 10
  Dotted lines, 31
  Dotting pen, 31
  ----, regular, to produce, 111
  Drawing, copying and reducing, 126
  ----, inking in to commence from top of, 10
  ----, stretching paper for, 6
  Drawing concentric circles, 10
  ---- from copy, 127
  ---- lines, 10
  Drawing board, dented, to remedy, 14
  ---- office, essentials of, 1
  ---- ----, gaslights, 2
  ---- ----, position of windows, 1
  ---- ----, skylights unsuitable, 1
  ---- paper, sectional, 8
  ---- ----, to join sheets of, 12
  ---- papers, 5
  ---- ----, sizes of, 5
  ---- pen, 3
  ---- ----, cleaning, 3
  ---- ----, more than one required, 3
  ---- ----, setting, 3
  ---- ----, supplying with ink, 3
  ---- table, dimensions of, 2
  ---- ----, position of, 2
  Drawings, cleaning off, 9
  ----, competition, paper for, 6
  ----, colouring, 11
  ----, cutting off, 7, 11, 12
  ----, ink for, 8
  ----, margin to be left, 11
  ----, mechanical and architectural, 121
  ----, parchment for, 8
  ----, to colour, 122
  ----, to preserve rolled, 12
  ----, to reduce or enlarge, 130
  ----, to remove grease spots from, 10
  ----, to varnish, 14
  ---- for lithographers and engravers, 141
  ---- for specifications for letters patent, 8
  Dusters, 2


  E.

  Eidograph, 136
  ----, method of setting, 139
  ----, table for setting, 139
  ----, to adjust, 138
  Elevation of trees, 36
  Ellipse, to draw, 21
  Elliptical arch, to construct a semi-, 23
  Encamping grounds, 112
  Engravers, drawings for, 141
  Enlarging by instruments, 131
  ---- by scales, 130
  ---- by squares, 130
  Equal parts, to divide a line into, 15
  Equidistant and parallel lines, to draw, 28
  Equilateral arch, to draw, 23
  ---- triangle, to construct, on a given base, 16
  Erasure of ink lines, 11
  ---- of pencil marks, 9
  Error-sheets, 91
  ----, examples of, 92
  Errors, 91
  Essentials of drawing office, 1
  Estate plans, 107
  Examination of maps and plans, 117


  F.

  Ferries, 111
  Field-book, example of, 80
  Field sketching, 114
  Fir-graining, 32
  Flat-tints, 40
  Formulæ, trigonometrical, 142


  G.

  Gaslights in drawing office, 2
  Glass-paper to erase ink lines, 11
  Glue for mounting paper, 6
  Gothic cinquefoil, to draw, 26
  ---- equilateral arch, to draw, 23
  ---- lancet arch, to draw, 24
  ---- obtuse arch, to draw, 24
  ---- ogee arch, to draw, 25
  ---- quatrefoil, to draw, 26
  ---- trefoil, to draw, 25
  ---- Tudor arch, to draw, 24
  Gradients, to lay down, 95
  Graining, fir, 32
  ----, oak, 32
  ----, wood, 32
  Grass-land, to represent, 34
  Gravel, to represent, 35
  Grease spot, to remove from drawing, 10


  H.

  Hexagon, to describe a regular, 21
  Hills, representation of, 38
  ----, sand, to represent, 36
  ----, shading, horizontal system, 53
  ----, ----, vertical system, 57
  ----, sketching, shading, and copying, 113
  Horizontal zones, 38
  Horse-shoe arch, to draw, 24


  I.

  Importance of cleanliness, 9
  Inclined measure, Table for correction of, 143
  Indian ink, 8
  ---- ----, preparation of, for drawing, 9
  ---- ----, quality of, 9
  Indiarubber, native or bottle, 9
  ----, vulcanized, 9
  Ink for drawings, 8
  ----, Indian, 8
  ----, preparation of, 9
  ----, quality of, 9
  ---- lines, erasure of, 11
  ---- ----, to avoid smearing, 10
  ---- slab or saucer, position of, when in use, 10
  Inking in to commence at top of the drawing, 10
  Instruments, 2
  ----, quality of, 2
  Islands, 110


  J.

  Joining sheets of paper, 12, 13
  Jungle, 112


  L.

  Lakes, outline of, 30
  Lancet arch, to draw, 24
  Land, cultivated, to represent, 32
  ----, uncultivated, to represent, 37
  Lead-pencil marks, erasure of, 9
  Lettering, 66, 122
  ----, position of, on plans and maps, 69
  Letters, arrangement of, in titles, &c., 68
  ----, kinds to employ, 67
  ----, mechanical construction of, 66
  ----, size of, 67
  Level-book, example of, 94
  Line, dividing into equal parts, 10
  ----, regular pentagon on a given, 20
  ----, to bisect a given straight, 15
  ----, to construct a square on a given, 19
  ----, to divide a, into equal parts, 15
  ----, to draw a, making a given angle, 15
  ----, to erect a perpendicular to, 15
  Lines, drawing, 10
  ----, broken, 30
  ----, centre, 10, 122
  ----, combinations of, 31
  ----, curved, to draw, 29
  ----, curved and straight, 27
  ----, contour, 37
  ----, dotted, 31
  ----, ink, to avoid smearing, 10
  ----, parallel and equidistant, to draw, 28
  ----, reference, 78
  ----, section, 29
  ----, ----, to draw, 28
  ----, shade, application of, 48
  ----, shading, 50
  ----, ----, in topographical drawings, 52
  ----, straight, difficulties in ruling, 27
  ----, ----, to draw, 27
  ----, wavy, 33
  ---- of greatest descent, 57
  ---- of uneven thickness, 30
  Lithographers, drawings for, 141
  Local Government Board, regulations of, 104


  M.

  Machine-made paper, 6
  Machinery, rough sketches of, 121
  Manner of using compasses, 2
  Map drawing, 109
  Maps, boundary, 104
  ----, field, examination of, 118
  ----, signs used in, 120
  ---- for division into wards, 104
  Margin to drawings, width of, 11
  Marshy ground, to represent, 35
  Mechanical drawings, 121
  Mile stones, 112
  Mining plans, 106
  Moorish horse-shoe arch, to draw, 24
  More than one drawing pen required, 3
  Mountain passes, 111
  Mounting paper, glue for, 6
  ---- ---- on stretchers, 13
  ---- tracings, 129
  Mud in rivers, to represent, 36


  N.

  Native indiarubber, 9
  Needle to erase ink lines, 11
  North points, 69
  Northings and southings, 87


  O.

  Oak-graining, 32
  Obtuse arch, to draw, 24
  Ogee arch, to draw, 25
  Orchards, to represent, 36
  Outline of lakes, 30
  ---- of ponds, 30
  ---- of rivers, 30
  Oval, to construct, the width being given, 18


  P.

  Pantograph, 131
  ----, methods of setting, 132
  ----, tables for setting, 133, 134
  ----, to use, 134
  Paper, blacklead, 8
  ----, carbonic, 8
  ----, cartridge, 6
  ----, continuous cartridge, 6
  ----, drawing, 5
  ----, ----, to join, 12
  ----, ----, sizes of, 5
  ----, glue for mounting, 6
  ----, machine-made, 6
  ----, sectional, 8
  ----, stretching for drawing, 6
  ----, tracing, 7
  ----, ----, continuous, 7
  ----, ----, preparation of, 7
  ----, ----, sizes of, 7
  ----, ----, to join, 13
  ----, transfer, 8
  ----, ----, preparation of, 129
  ----, to join sheets of, 12
  ----, to mount on stretchers, 13
  ---- for competition drawings, 6
  ---- for large plans, 6
  Parabola, to draw, base and height being given, 21
  Parallel and equidistant lines, to draw, 28
  Parchment for drawings, 8
  Parliamentary plans and sections, 100
  ---- standing orders, 98
  Paste, 14
  Patent, drawings for specifications, 8
  Pathways, 111
  Pen, dotting or wheel, 31
  Pencil marks, erasure of, 9
  ---- leg of compasses, 2
  Pencils, 4
  ----, pointing, 4
  ---- for field sketching, 115
  Pentagon, to describe, on a given line, 20
  Perpendicular, to erect a, 15
  Plan of trees, 36
  Plans, civil engineers’ and surveyors’, 96
  ----, estate and town, 107
  ----, large, paper for, 6
  ----, mining, 106
  ----, railway, 97
  ----, parliamentary, 100
  Plotting, 77
  ---- angular surveys, 81
  ---- contours, 90
  ---- detail, 89
  ---- sounded points in submerged districts, 90
  ---- traverse reference lines, 84
  ---- vertical sections, 92
  Pointing pencils, 4
  Points of compasses, 2
  Ponds, outline of, 30
  Position of drawing table, 2
  ---- of ink slab or saucer when in use, 10
  ---- of windows in drawing office, 1
  Precautions to be taken to ensure cleanliness, 9
  Preparation of colours for tinting, 40
  ---- of ink for drawing, 9
  ---- of stretchers, 13
  ---- of tracing paper, 7
  ---- of transfer paper, 129
  Preserving drawings, rolled, 12


  Q.

  Quality of Indian ink, 9
  ---- of instruments, 2
  Quatrefoil, Gothic, to draw, 26


  R.

  Radii of circle, to draw, the centre being inaccessible, 18
  Railway plans, 97
  ---- sections, 102
  ---- stations and termini, 112
  Railways, 112
  Rectangle, to construct, similar to a given rectangle, 20
  Rectangular co-ordinates, to plot by, 87
  Reducing and enlarging drawings, 126, 130
  ---- by instruments, 131
  ---- by scales, 130
  ---- by squares, 130
  Reference, book of, 97, 101
  ---- lines and points, 78
  ---- lines, secondary, 79
  Refraction and curvature, Table for correction of, 143
  Regulations of Local Government Board, 104
  Removing movable leg of compasses, 2
  Rivers, beds of, 110
  ----, mud in, to represent, 36
  ----, outline of, 30
  ---- and streams, colouring, 110
  ---- ----, inking in, 109
  Roads, 111
  Rolled drawings, to preserve, 12
  Roman cyma recta and cyma reversa, to draw, 25
  Roofs, to draw, 30
  Rough sketches of machinery, 121
  Ruling straight lines, difficulties in, 27
  Running water, to represent, 33


  S.

  Sager’s vellum tracing cloth, 7
  Sand, to represent, 35
  ---- banks, 110
  ---- hills, to represent, 36
  Scales, 70
  ----, choice of, 72
  ----, construction of, 11, 70, 75, 76
  ----, diagonal, 75
  ----, vernier, 75
  ----, Tables of, 73, 74
  ---- of construction, 74
  ---- of distances, 70
  ---- of shade, English, 59
  ---- ----, Lehmann’s, 57
  ---- ----, standard, 53
  ---- ----, United States’, 59
  Sectional drawing paper, 8
  Section lines, 29
  ---- of water, to represent, 30
  Sections, colours for, 44
  ----, cross, 98, 102
  ----, parliamentary, 101
  ----, railway, 102
  ----, to plot, from contour map, 96
  ----, working, 94, 103
  ---- of wood, 32
  ----, vertical, to plot, 92
  Semi-elliptical arch, to construct, 23
  Setting drawing pen, 3
  Shade lines, application of, 48
  ---- ----, cylindrical surfaces, 50
  Shading, 48
  ---- cylinders, various methods, 64
  ---- hills, horizontal system, 53
  ---- ----, vertical system, 57
  ---- ----, rounding curves, 55
  ---- lines, 50
  ---- ----, cylindrical surfaces, 51
  ---- ---- in topographical drawings, 52
  ---- in colours, 63
  ---- ----, cylindrical surfaces, 64
  ---- ----, hill slopes, 63
  Sides of triangles, to compute, 142
  Sizes of drawing papers, 5
  ---- of tracing cloth, 7
  ---- of tracing paper, 7
  Sketches, rough, of machinery, 121
  Sketching, field, 114
  Skylights unsuitable for drawing office, 1
  Sounded points, to plot, 90
  Specifications for letters patent, drawings for, 8
  Spring bows, 2
  Springs, 112
  Square, to construct, equal to ¹⁄₂, ¹⁄₄, &c., of a given square, 19
  ----, to construct, in any proportion to a given square, 20
  ----, to construct, on a given line, 19
  ----, to construct, which shall be a multiple of a given square, 19
  Standing orders of Parliament, 98
  ---- water, to represent, 29
  Stations, railway, 112
  ----, telegraph, 112
  Straight-edge, thickness of, 5
  Straight line, to bisect a, 15
  ---- lines, difficulties in ruling, 27
  ---- ----, to draw, 27
  ---- and curved lines, 27
  Stretchers, mounting paper on, 13
  ----, preparation of, 13
  Stretching paper for drawing, 6
  ---- ----, glue for, 6
  Supplying drawing pen with ink, 3
  Surveyors’ plans, 96
  Swamps, to represent, 35


  T.

  Table, drawing, position of, 1
  ----, ----, size of, 1
  Table for correction of curvature and refraction, 143
  ---- for correction of inclined measure, 143
  ---- for setting eidograph, 139
  ---- of conventional colours, 44
  Tables for setting pantograph, 133, 134
  Tangent, to draw, to a circle, 17
  Telegraph lines and stations, 112
  Thickness of straight-edge, 5
  Tinting, 39
  Tints, art of applying, 40, 41, 43
  ----, brushes for applying, 41
  ----, double or alternate, 42
  ----, flat, 40
  ----, preparation of, 40
  Toll-gates, 111
  Topographical drawings, shading lines in, 52
  To remove grease spots from drawings, 10
  Town plans, 107
  Towns, to represent size of, 112
  Tracing, copying from, 11
  ----, to copy by, 128
  ---- cloth, 7
  ---- ----, Sager’s vellum, 7
  ---- ----, sizes of, 7
  ----- paper, 7
  ---- ----, continuous, 7
  ---- ----, preparation of, 7
  ---- ----, sizes of, 7
  ---- ----, to join sheets of, 13
  Tracings, to mount, 129
  Transfer paper, 8
  ---- ----, preparation of, 129
  Transferring, to copy by, 129
  Traverse plotting by rectangular co-ordinates, 87
  ---- reference lines, to plot, 84
  Trees in plan, 36
  ---- in elevation, 36
  ----, to represent, 36
  Trefoil, Gothic, to draw, 25
  Triangle, equilateral, to construct, 16
  ----, to construct, the length of base and angles at base being
  given, 17
  ----, to construct, the lengths of the sides being given, 16
  Triangles, primary and secondary, 78
  ----, to compute the areas of, 142
  ----, to compute the sides of, 142
  Trigonometrical formulæ, 142
  Tudor arch, to draw, 24


  U.

  Uncultivated land, to represent, 37


  V.

  Varnishing drawings, 14
  Vellum tracing cloth, Sager’s, 7
  Vertical sections, to plot, 92
  Villages, to represent size of, 112
  Vulcanized indiarubber, 9


  W.

  Washes, art of applying, 40, 41, 43
  ----, brushes for, 41
  ----, double or alternate, 42
  Water, running, to represent, 33
  ----, standing, to represent, 29, 33
  ---- in section, to represent, 30
  Wavy lines, 33
  Wells, 112
  Whatman’s drawing papers, 6
  Wheel pen, 31
  Width of margin to drawings, 11
  Windows, position of, in drawing office, 1
  Wood-graining, 32
  Wood sections, 32
  Woodland, to represent, 36
  Working sections, 103


  Z.

  Zones, horizontal, 38


THE END.


  LONDON: PRINTED BY WILLIAM CLOWES AND SONS, STAMFORD STREET AND
  CHARING CROSS.




PLATES.


[Illustration: PLATE 2.

  PLAN SENT TO BE COPIED

  PLAN TRACED BY JUNIOR HAND

  PLAN SHEWING WRITING GAUGED

  PLAN FINISHED

  _B. Alexander, Lith._
  E & F. N. Spon. London & New York.]


[Illustration: PLATE 3.

  PLAN OF ESTATE
  AT
  HASLINGTON,
  BUCKS.
  1874.

  _B. Alexander, Lith._
  E & F. N. Spon. London & New York.]


[Illustration: PLATE 4.

  ROMAN CAPITALS

  ITALIC CAPITALS _Angle 60 degrees_

  ROUND HAND _Angle 53 degrees_

  MECHANICAL CONSTRUCTION OF LETTERS &c.

  _8 Lines 6 Lines 4 Lines 3 Lines 2 Lines_

  _B. Alexander, Lith._
  E & F. N. Spon. London & New York.]


[Illustration: PLATE 5.

  OPEN STONE LETTERS

  Do. Do. _WITH ORNAMENT_

  ORNAMENTAL LETTERS.

  EGYPTIAN _WITH ORNAMENT_

  ROMAN _WITH ORNAMENT_

  Do. SMALL _WITH ORNAMENT_

  OLD ENGLISH _WITH ORNAMENT_

  Do. SMALL _WITH ORNAMENT_

  FIGURES _WITH ORNAMENT_

  _B. Alexander, Lith._
  E & F. N. Spon. London & New York.]


[Illustration: PLATE 6.

  ALPHABETS.

  OLD ENGLISH _Capitals._

  Do. Do. _Small._

  GERMAN TEXT _Capitals._

  Do. Do. _Small._

  GOTHIC _Capitals._

  Do. _Small._

  CHURCH TEXT _Capitals._

  Do. Do. _Small._

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 7.

  PLAN OF AN ESTATE
  CALLED
  THE LOWER CEDARS
  NEAR THE TOWN OF
  BIRMINGHAM
  1874.

  PLAN OF PROPERTY
  Situate in the Parish of
  HAMMERSMITH,
  MIDDLESEX.

  Plan of a Valuable Freehold Estate
  known as
  FROGNAL
  NEAR
  HAMPSTEAD HEATH.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 8.

  PLAN OF LAND TO BE LAID OUT
  IN BUILDING PLOTS
  BELONGING TO THE LATE
  SIR JOSEPH PAXTON,
  IN LEASES OF 99 YEARS,
  SUBJECT TO THE ANNEXED CONDITION.

  Scale of Chains.

  Scale 6 Chains to an Inch

  Scale of Feet.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 9.

  Scale of Chains.

  _Scale of Feet._

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 10.

  ORDINARY BEECH.

  WOOD.

  FIRS.

  LIGHT BRUSHWOOD.

  FIRS & OTHER TREES.

  COCOA & PALM TREES
  _For Colonial Plans._

  SWAMP.

  MARSH.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 11.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 12.

  CONSTRUCTION OF HILLS.

  FINISHED HILLS.

  CONTOUR HILLS.

  FINISHED HILLS _IN COLOR_.

  CONTOUR HILLS _IN COLOR_.

  HILLS _IN CHALK_.

  BROKEN HILLY COUNTRY.

  HILLS _IN COLOR_.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 13.

  PLAN SHEWING PROPOSED NEW STREET.

  _Half Black for Lithography_

  _Half Color for Paper._

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 14.

  ASH

  BEECH

  BIRCH

  CEDAR

  CYPRESS

  ELM

  FIR

  MOUNTAIN ASH

  OAK

  PLANTATION FOREST

  PINE

  POPLAR

  SYCAMORE

  THORN

  WEEPING WILLOW

  YEW

  MOUNTAINS

  ROCKY CLIFF

  _Do. IN COLOR_

  _Do. IN COLOR_

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 15.

SIGNS USED IN PLANS.

  _Fence_                         _Farm Buildings_

  _Fence with Bank_               _Churches_

  _Fence with Hedge_              _Windmills_

  _Footpath_                      _Water Mills_

  _Bridle Paths_                  _Lime Kiln_

  _Occupation Road_               _Sunk Road_

  _Public Road_                   _Raised Road_

  _Wall_                          _Quarry_

  _Parish Boundary_               _Inn_

  _Hamlet Boundary_               _Sand Pits_

  _County Boundary_               _Rocks_

  _Parish & County Boundary_      _Mud_

  _Railway_                       _Gas Works_

  _Tramway_                       _Glass Works_

  _Stream_                        _Iron Works_

  _Brook_                         _Column_

  _Ditch_                         _Old Castle_

  _Mineral Waters_                _Covered Passage_

  _Canal_                         _Saw Mill_

  _River_                         _Stone Windmill_

  _Ponds_                         _Wooden Windmill_

  _Lake_                          _Cotton factory_

  _Wooden Fence_                  _Woollen Factory_

  _Post and Rail or thus_         _Well_

  _Chains & Post_                 _Dry Well_

  _Hurdle Fence_                  _Salt Works_

  _Gates_                         _Field Wall_

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 16.

SIGNS USED IN MAPS AND CHARTS.

  _Fortress_                      _Rocks_ _always covered_

  _Citadel_                       _Reef of Rocks_

  _Fortified Castle_              _Sand_ _never covered_

  _Walled Town_                   _Sand_ _sometimes covered_

  _Open Town_                     _Shoals_ _always covered_

  _Country Town_                  _Mud Bank & Beach dry at Low Water_

  _Little Town_                   _Rocky Ledges which cover and uncover_

  _City_                          _Sandy Beach dry at Low Water_

  _Episcopal City_                _Lighthouses (position of)_

  _Borough or Corporation_        _Can Buoys_

  _Light Ship_                    _Nun Buoys_

  _Light House_                   _Mooring Buoy_

  _Anchorage for Ships_           _Buoys with Beacons_

  _Do for Coasters_               _Coral Reefs_

  _Wreck_                         _Kelp_

  _Stopping Places_               _Fish Weir_

  _Head of Navigation_            _Swampy Land_

  _Floating Light Vessel_         _Rocks_

  _Harbour_                       _Rocks dry at Low Water_

  _Telegraph_                     _Viaduct_

  _Signal House_                  _Tunnel_

  _Buoys_                         _Railway Bridge_

  _Channel Marks_                 _Bridge over Stream_

  _No Current_                    _Pontoon Bridge_

  _Direction of Current_          _Bridge (Brick or Stone)_

  _Rocks_ _sometimes covered_     _Foot Bridge_

  _Rocks_ _never covered_         _Towns_

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 17.

  Plan
  OF
  MELTON HALL ESTATE
  Situate in the
  COUNTY OF LINCOLN,
  1874.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 18.

  PIECE OF ORDNANCE MAP.

  PIECE OF CHART SHEWING SOUNDINGS, &c.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 19.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 20.

  SAND & GRAVEL.         RED SANDSTONE.         CLAY & GRAVEL.

  LIMESTONE.             BOG.                   CLAY SLATE.

  MILLSTONE.             GRANITE.               SANDSTONE & COAL.

  SPOIL BANK.            PURE GRAVEL.           CLAY CALCAREOUS.

  PEAT & CLAY.           SANDSTONE.             SHELL MARL.

  LIMESTONE WITH         BOG WITH TIMBER.       SANDSTONE WITH GYPSUM.
  MINERAL VEINS.

  MUD.                   SANDSTONE COARSE &     CLAY SLATE WITH MINERAL
                         FINE.                  VEINS.

                       COMBINATION OF THE ABOVE.

  Spoil Bank             Clay                   Clay Calcareous
  Limestone              Sand                   Yellow Clay
  Bog                    Gravel                 Rock
                         Red Sandstone

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 21.

  PARLIAMENTARY RAILWAY SECTION.

  DITTO, _SHEWING GEOLOGICAL STRATA_.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 22.

  PIECE OF MARINE ENGINE OF H. M. S. S. “RESEARCH”.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 23.

  SKEW BRIDGE.

  PERMANENT WAY.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 24.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 25.

  Ornamental Writing.

  London

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 26.

  METHODS OF REDUCING OR ENLARGING.

  _BY SQUARES_

  _BY EIDOGRAPH_

  _BY ENGLISH PENTAGRAPH_

  _BY FRENCH PENTAGRAPH_

  _and by Photography_.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 27.

  ISOMETRICAL VIEW OF BUILDING.

  EXTERIOR.

  INTERIOR.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 28.

  Depositary Rocks. 1-18

  Igneous Rocks. a-b

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 29.

  SIGNS USED IN INDIAN & COLONIAL PLANS.

  _Telegraph_                     _Hills with Peaks_

  _Trunk Road (Metalled)_         _D^{o}. in Contour_

  _District D^{o}._               _Scarped Face of Hill_

  _D^{o}. Unmetalled_             _Ravine_

  _Ford_                          _Garden_

  _Nullah or Khall_               _Paun Garden_

  _Bund or Embankment_            _Village_

  _Old Bank of River_             _Ferry_

  _Hedges with Trees_             _Flying Bridge_

  _River with Islands_            _Haut or Bazar_

  _River with Sand Bank_          _Mud Fort_

  _Stone Sluice Gate_             _Pucka Fort_

  _Lake or Tank_                  _Deserted Village_

  _Salt Pans_                     _Principal Survey Stations_

  _Cultivated Ground_             _Secondary D^{o}._

  _Burial Ground_                 _Flag for Surveying_

  _Sand Hills_

  _Ridge of Hills_

  _High Ground_

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 30.

  SIGNS USED IN INDIAN & COLONIAL PLANS.

  _Salt Golah_                    _Jhow Jungle_

  _Salt Chowkey_                  _Jheel_

  _Silk Factory_                  _Tamarind Trees_

  _Indigo Factory_                _Bamboo Jungle_

  _Sugar Factory_                 _Salt Waste_

  _Post Office_                   _Salt Waste with Jungle_

  _Dak Bungalow_                  _Mangoe Tope_

  _Police or Thana Sta^{n}._      _Cocoa Nut Trees_

  _Magistrate’s Kutcherry_        _Betel Nut Trees_

  _Stone or Pucka Houses_         _Trees generally_

  _Mahomedan Mosque_              _Date Trees_

  _Hindoo Temple_                 _Palm or Tar Trees_

  _Telegraph Tower_               _Mangrove_

  _Signal Staff_                  _Brides_

  _Boundary Pillar_               _Rail Road_

  _Wooden Boundary Post_

  _Pucka Well_

  _Kucha Well_

  _Forest Jungles_

  _Low Jungles_

  _Grass Jungles_

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 31.

  MILITARY SIGNS AND FORTIFICATIONS.

  _Tête du Pont_                  _Passable for Troops_

  _Vedettes_                      _Infantry Engaged_

  _Military Pits_                 _Cavalry Encamped_

  _Mines_                         _Village Inundated_

  _Infantry Encamped_             _Sand Bags_

  _Village Burnt_                 _Intrenchment_

  _Caltrop or Crows Feet_         _Cannon_

  _Trenches_                      _Cavalry Engaged_

                                  _Park_

                                  _Palisades_

  _Barrier_

  _Mortar & Shells_

  _Block House_

  _Field Piece and Limber_

  _Ammunition Waggon and Limber_

  SECTION OF GLACIS, DITCH, RAMPART, &C.

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]


[Illustration: PLATE 32.

  MILITARY SIGNS AND FORTIFICATIONS.

  _Infantry Column_               _Impassible for Artillery_

  _Cavalry Line_                  _Abatis_

  _Chevaux de frise_              _Redoubt_

  _Mortar Battery_                _Cavalry Column_

  _Impassible for Cavalry_        _Guns on March_

    _Gained_                      _Gun Battery_
  _Field of Battle_
    _Lost_                        _Impassible for Infantry_

  _Sentinel_                      _Baggage Waggons_

  _Infantry Line_                   _Fortified_
                                  _Castles_
  _Guns in Position_                _Closed_

  _Palisades_                     _Redan or Fleche_

  _Rifle Pits_

  GABIONS                         FASCINES

  PLAN OF BASTIONED FORT WITH LUNETTES.

  _B. Alexander, Lith._
  E & F. N. Spon. London & New York.]


[Illustration: PLATE 33.

  PORTION OF MINING PLAN SHEWING PILLAR WORKINGS.

  _AS SHEWN BY HAND ON PAPER._

  _AS SHEWN BY ENGRAVING OR LITHOGRAPHY._

  _B. Alexander, Lith._
  E. & F. N. Spon. London & New York.]




  Transcriber’s Notes


  Inconsistencies in spelling, hyphenation, lay-out, the use of quote
  marks, diacriticals and accents, etc. have been retained.

  Depending on the hard- and software used to read this document, some
  elements may not display as intended. The browser version of this
  document (available from www.gutenberg.org) gives best results, and
  contains larger versions of several illustrations.

  The minor differences between the Table of Contents and the headings
  in the text have not been standardised.

  Several plates have been printed with poor colour overlays, this has
  not been remedied.

  Some double-page plates have been re-combined, but the area between
  the pages may not be exactly as printed in the original work.

  Page 136, Professor Willis: should be Professor (William) Wallace;
  Professor (Robert) Willis invented other drawing tools.

  Plate 26, pentagraph (2×): as printed in the source document.


  Changes made:

  Some missing punctuation has been added silently, some minor
  typographical errors have been corrected silently.

  Some tables, calculations, formulas, etc. have been re-arranged for
  better readability.

  Page 15: apply a straight-edge F changed to apply a straight-edge D

  Page 75, Fig. 75: Lehman’s Scale changed to Lehmann’s Scale

  Page 83: the third reference line C D changed to the third reference
  line C A

  Page 132: the _erect manner_ and the reverse manner changed to the
  _erect manner_ and the _reverse manner_.

  Heading PLATES added after Index.