ON COPYING PLANS.

It rarely happens that one copy only of a plan is required. When the lines and boundaries are regular, duplicates may be made by laying the plan upon the sheet of paper or vellum on which the copy is to be drawn, and pricking with a fine needle through all the angular points necessary to define the figures: the punctures being then connected by pencil lines, the plan is finished by drawing these in Indian ink.

This method is not suitable to the transfer of irregular or curved boundaries. An accurate and rapid way of copying these, or plans of small extent, is by means of the instrument called a copying glass. It consists of a large piece of plate glass set in a frame of wood, which can be inclined to any angle, in the same manner as a reading or music desk. On this glass the original plan and the fair sheet of paper are laid, and the frame being raised to a suitable angle, a strong light is thrown by means of tin reflectors or otherwise on the under side of the glass, whereby every line in the original plan is seen distinctly through the fair sheet. The copy is at once made in ink, and finished while being traced.

Plans of greater extent cannot be conveniently copied by means of the "copying glass." Moreover, being generally mounted on linen or other material which renders them opaque, they do not admit of the operation just described. In such cases, the plan is first traced in Indian ink on transparent tracing paper. This first This first copy is then carefully laid over the fair sheet; black-leaded or transfer paper being first placed under the tracing. All is steadied by numerous weights laid along the edges, or, by drawing pins fixed into the drawing board or table; a fine and smooth point is then passed over each boundary or mark

on the tracing with a pressure of the hand sufficient to cause a clear pencilled mark to be left on the fair sheet by the black-leaded or "transfer paper." The whole outline is thus obtained, and afterwards drawn in Indian ink in the usual way.

On Reducing Plans.

Plans may be reduced by means of the pentagraph, to any proportion wanted. The instrument consists of a jointed rhombus B D E F, made of brass, and having the two sides BD, BF extended

to double their length; the
side DE and the branch D A
are marked from D with suc-
cessive divisions, D O being
made to B O always in the
ratio of D P to B C. Small
sliding boxes for carrying A

a pencil, or a tracing point,

B

E

are placed at P and C, and secured in their positions by screws; the point O is made the centre of motion, and rests on a fulcrum or support of lead; and the tracer is fixed at C, while the pencil is lodged in P. From the property of similar triangles, the three points O, P, and C must range in the same straight line, which is divided at P in the ratio required. While the point C, therefore, is carried along the boundaries of any figure, the intermediate point P. will trace out a similar figure, reduced in the proportion of O C to OP or of O B to O D, the proportion required *.

By changing the relative positions of the tracer and pencil, the figure would be enlarged in the same proportion;

*LESLIE'S Geometry, page 431.

but errors are so much increased by enlarging plans with the pentagraph, that it ought not to be resorted to where accuracy is required, except in cases where the field-books are either lost, or do not afford the data required to plot on a larger scale, and when there are no means of repeating the survey.

CHAPTER II.

SURVEYING INSTRUMENTS.

The Diagonal Scale.

WHEN a line is to be divided into equal parts, so numerous and minute, that they would be indistinct, some method, not embodying direct subdivision, must be adopted in order to estimate those minute fractional parts. The diagonal scale, an ingenious application of the property of similar triangles, described in Euc. 4, VI., was among the earliest methods resorted to for this purpose.

d

In the annexed figure, the line a c is divided into any number of equal parts, or into n times a b, and a space equal to each of these parts is indirectly subdivided into n secondary parts, by means of diagonal lines, of any arbitrary length, raised from the points marking the primary divisions. These diagonals

decline, in their entire lengths

from the perpendicular by intervals equal to one of the primary parts, and they are cut transversely into n equal parts by equidistant lines parallel to a c. In the triangles

adb, fde, we have

bd: abde: ef,

but de is by construction equal to the nth part of bd, there

fore e f is equal the nth part of a b, or eƒ =

But by construction, a b =

1

=

x ab.

The Vernier Scale.

This subdivision by diagonals is still in general use, and constituted the first improvement in dividing astronomical and geodesic instruments*. In this part of its application it has been superseded by the vernier scale, the simplest and most ingenious of the methods hitherto invented for the minute subdivision of lines. It obtains this object by measuring the differences between the divisions of two approximating scales; one of which is fixed, and called the primary scale; the other moveable, and called the vernier.

If a space, on the primary scale, be divided into a given number of parts, equal to n-1 (in the figure equal to 9), and a space, equal in length to the first, be divided on the moveable seale into a number of parts equal to n (in the figure equal to 10), these latter parts will each be smaller than the first by the nth part (in the figure the 10th part) of a division on the primary scale.

For, let a = the length of a division on the primary scale,

b the length of a division on the moveable

=

that is, b, a division on the moveable scale, is smaller than a, a division on the fixed scale, by the nth part of a.

The edges of the two scales being applied to each other, so that the extreme end of the vernier, which is marked and is called the index, coincides with a division on the

*LESLIE'S Geometry.