A manual of the principles and practice of road-making: comprising the location, construction, and improvement of roads, (common, macadam, paved, plank, etc.) and rail-roads. By W. M. Gillespie ...

108 THE LOCATION OF ROADS.
fulfil the conditions required; the length of the zigzag being
equal to the necessary length mn, and the lines AC and BI
being perpendicular to the top and bottom of the hill, and therefore the shortest possible distances to it.
Sub-case 2. When the line joining the perpendiculars is
equal to the developed line mn.
Fig. 50.
A C'_ B
Draw the perpendiculars AC and BD, as before, and join
their feet by the line CD. Then will the line ACDB be
shorter than any other line, (as AC'D'B, obtained by the construction of Case 2) for AC and BD, being perpendiculars, are
the shortest possible, and CD has a constant length, wherever
it may be placed.
A zigzag line from C'to E would not produce the shortest
line, for the same reasons as in Case 2.
Sub-case 3. When the line yoznzng the perpendiculars is
longer than the developed line mn.
From A, Fig. 51, draw AE parallel and equal to mn. Join EB.
From the point D (where this line intersects the edge of the
slope) draw DC parallel and equal to AE. Join CA. Then
will ACDB be the shortest line required.
For, AE, being equal to mn, cannot be shortened, and EB
is a straight line, and therefore the shortest possible line, as is