MATHEMATICAL AND SURVEYING INSTRUMENTS. 5d 
from these last points, parallels to the basis: their intersections with the 
perpendiculars first erected, will determine the four corners of the perspec- 
tive representation, a‘d’, of the upper base of the prism. 
Pl. 4, fig. 63, exhibits the perspective representation of a compound body, 
viz. a pedestal surmounted by a cross. All the measurements necessary, are 
obtained from the plan abcd, and the geometrical elevation shown on the 
right hand side. By keeping in view the principles already explained, and 
observing the correspondence of lettering, the perspective construction will 
not be difficult. The line of height drawn on the right hand side serves to 
determine all the heights, as any geometrical height may be marked off upon 
it, and innumerable lines drawn from the relative top and bottom points: 
from the different intersections, the shortening in height for any plane in the 
picture may be determined. 
VI. OF THE MOST IMPORTANT MATHEMATICAL AND 
SURVEYING INSTRUMENTS. 
For the better elucidation of the preceding observations, the most 
important instruments used in geometrical drawing, and the various 
geodetical operations, have been represented on pil. 5, figs. 1-56. 
The simpler drawing instruments, as the simple compasses, the drawing 
compasses with movable lead tube and pen point, the drawing pen, the 
ruler, the scale, and the square, need not be mentioned here, as they occur 
in every box of mathematical instruments, and consequently are too well 
known to require description. The first of the rarer instruments to be men- 
tioned, is the hair compasses (figs. 1, 2), serving for very minute measure- 
ments. One foot, b, of a common compass, ab, is cut off, and a spring prolon- 
gation riveted at d, so that when it is attached to the head piece, it forms a 
leg, asin ab. The screw c can turn at é in the rivet, and has its nut in the 
upper leg. Turning this screw will cause a very slight motion of the spring 
joint to or from the other leg, independently of any motion in the joint. 
Very minute differences of measurements or lines may be thus appreciated. 
The repeated efforts necessary to divide lines into a certain number of 
equal parts by the ordinary method, have caused the invention of the propor- 
tional compasses. ‘These depend upon the geometrical principle, that in 
similar isosceles triangles the bases are as the sides. If, then, the legs of two 
such triangles are to each other as 1:2, the bases must be se likewise, and 
the smaller base will be one half the size of the larger. If we suppose two 
compasses, so united by their heads as to have a common joint, and the legs 
In the above mentioned ratio, all the conditions will be fulfilled, and the 
bisecting compasses will be the result. In this, the space between the points 
of the small legs will be just half of that between the points of the larger 
ones. 
As, however, other divisions besides bisections are required, the head 
of the compasses has been made movable, thus forming the proportional 
53 
