SECTOR, is also a mathematical instrument of great use in finding the proportion between quantities of the same kind, as between lines and lines, surfaces and surfaces, &c. for which reason the French call it the compass of proportion.
The great advantage of the sector above common scales, &c. is, that it is adapted to all radii, and all scales. For, by the line of chords, sines, tangents, &c. adapted to any radius betwixt the length and breadth of the sector, when opened.
The sector is founded on the fourth proposition of the sixth book of Euclid, where it is demonstrated, that similar triangles have their homologous sides proportional; an idea of its foundation or theory may be conceived from what follows. Let the lines AB, AC, represent the two legs of the sector, and AD, AE, two equal sections from the centre. If now the points BC and DE be connected, the lines BC and DE will be
Sector. be parallel: therefore the triangles ADE, and ABC, will be similar, and consequently the sides AD, DE, AB, BC, proportional; that is, as AD : DE :: AB : BC; whence, if AD be the half, third, or fourth part of AB, DE will be a half, third, or fourth part BC. Whence it follows, that if AD be the chord, sine, tangent, &c. of any number of degrees to the radius AB, DE will be the same to the radius BC.