they wind around each pole an infinite number of times but reach the pole in a finite distance. The pole-to-pole length of a loxodrome (assuming a perfect sphere) is the length of the meridian divided by the cosine of the bearing away from true north. Loxodromes are not defined at the poles. Etymology and historical description The word loxodrome comes from Ancient Greek λοξός loxós: "oblique" + δρόμος drómos: "running" (from δραμεῖν drameîn: "to run"). The word rhumb may come from Spanish or Portuguese rumbo/rumo ("course" or "direction") and Greek ῥόμβος rhómbos, from rhémbein. The 1878 edition of The Globe Encyclopaedia of Universal Information describes a loxodrome line as: Loxodrom′ic Line is a curve which cuts every member of a system of lines of curvature of a given surface at the same angle. A ship sailing towards the same point of the compass describes such a line which cuts all the meridians at the same angle. In Mercator's Projection (q.v.) the Loxodromic lines are evidently straight. A misunderstanding could arise because the term "rhumb" had no precise meaning when it came into use. It applied equally well to the windrose lines as it did to loxodromes because the term only applied "locally" and only meant whatever a sailor did in order to sail with constant bearing, with all the imprecision that that implies. Therefore, "rhumb" was applicable to the straight lines on portolans when portolans were in use, as well as always applicable to straight lines on Mercator charts. For short distances portolan "rhumbs" do not meaningfully differ from Mercator rhumbs, but these days "rhumb" is synonymous with the mathematically precise "loxodrome" because it has been made synonymous retrospectively. As Leo Bagrow states: "the word ('Rhumbline') is wrongly applied to the sea-charts of this period, since a loxodrome gives an accurate course only when the chart is drawn on a suitable projection. Cartometric investigation has revealed that no projection was used in the early charts, for which we therefore retain the name 'portolan'." Mathematical description For a sphere of radius 1, the azimuthal angle , the polar angle (defined here to correspond to latitude), and Cartesian unit vectors , , and can be used to write the radius vector as Orthogonal unit vectors in the azimuthal and polar directions of the sphere can be written which have the scalar products for constant traces out a parallel of latitude, while for