Protractor on the dial from 180°, as the dial is graduated from opposite zero marks to 90° each way. Since the spaces, both on the main scale and the Vernier scale, are numbered both to the right and to the left from zero, any angle can be measured. The readings can be taken either to the right or to the left, according to the direction in which the zero on the main scale is moved. Reading the Vernier scale: The bevel protractor Vernier scale may have graduations of 5′ (minutes) or 1/12°. Each space on the Vernier scale is 5′ less
Tropical geometry Tropical geometry Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry, using the tropical semiring instead of a field. Algebraic varieties can be mapped to a tropical counterpart and since this process still retain some geometric information about the original variety, it can be used to help prove classic results from algebraic geometry, such as Brill–Noether theorem using the tools of tropical geometry. The basic ideas of tropical analysis have been developed independently in the same notations by mathematicians working in various fields (see and