284. Miss Sarah Marks on the 
The special advantages of this instrument as a parallel 
ruler may best be seen by drawing what surveyors call the 
“ give and take line.” 
When two fields are divided by a curved boundary, it is 
often required to find a straight boundary which will divide 
the fields in the same proportions as before. This straight 
boundary is what is called the “ give and take” line. 
In order to understand the method of finding it, it will be 
necessary to recall the method of finding a rectangle which 
shall be equal to a given polygon. 
Fig. 4, 
A 
Let ABC DE (fig. 4) be the given polygon: produce C D 
both ways, and join A D and AC. 
Draw HF || to AD, and BG || to AC, and join AG, AF. 
Then AAFG is equal in area to the polygon ABCDH. 
For :.. A’s AF D and AED are on the same base A D and 
between the same ||’s A D, EF, 
.. they are equal. 
Similarly A AGC=A ABC. 
Having this triangle, it is easy to draw a rectangle equal 
to it. 
Now let A1234B (fig. 5) be the curved boundary of a 
field of which AC, BD are fixed boundaries; it is required 
to find a straight boundary to replace the curved one through 
A and B. 
Place the divided limb roughly parallel to AC, with its 
bevelled edge nearest to AC, and at a distance about equal 
to the breadth of either limb from it, and fix the points 
firmly in the paper. Now move the undivided limb till it 
cuts AC at A and 1 2 at 2; slide it along till it is on the point 
1, and mark the point a where it meets AC. Move it till it 
is against a and 3, and slide it along till it is against 2, mark- 
ing the point 6 where it meets AC produced, if necessary. 
Move it till it is on 6 and 4, slide it till it is on 3, and mark ¢, 
