The Mathematics of the Longitude - Department of Mathematics

Similarly, from the triangle IBH,
Hence from (1) and (2),
θ = φ + x . (2)
a = 2 x , (3)
or the body's altitude is twice the angle between the mirrors I and H (or
between their normals). The altitude a is zero when x is zero, that is when
the mirrors I and H are parallel. In Figure 5.5, IO is parallel to the fixed
direction HD; O is the zero-point of the scale. The angle OIP can thus be
found from the reading on the graduated arc and, by (3), the body's altitude
is twice this angle.
The arc of the sextant is generally about one-sixth of the circumference of a
circle (hence the name "sextant"), but instead of having 60 divisions each
representing one degree, the arc is divided into 120 divisions. In this way,
the altitude is read directly from the scale without the necessity of applying
the factor 2 of equation (3). With the aid of sub-divisions and a vernier,
altitude can be read with a first-class instrument to one-tenth of a minute of
arc.
5.3 The Vernier Scale
In figure 5.7, 10 divisions of the so-called vernier scale V equal 9 divisions
of the main scale S. It follows that the interval between divisions on V
equals 0.9 times the interval between divisions on S. When the zero points
of V and S are aligned, as shown in Figure 5.7 (A), the 1 mark on V will lie
to the right of the 1 mark on S by 0.1 times the main-scale interval;
similarly, the 2 mark on V will lie to the right of the 2 mark on S by 0.2
times the main-scale interval, and so on. Suppose now the main scale S is
moved to the right of the fixed vernier V by an arbitrary distance, say, 0.6
times the main-scale interval as shown in Figure 5.7 (B). The zero of the