and the radius on paper of parallel, a1 is a1d/(a2 - a1), and the radius of any other parallel = radius of a1 ± the true meridian distance between the parallels. This class of projection was used for the 1/1,000,000 Ordnance map of the British Isles. The three maximum scale errors in this case work out to 0.23%, the range of the projection being from 50° N. to 61° N., and the errorless parallels are 59° 31´ and 51°44´. Where no great refinement is required it will be sufficient to take the errorless parallels as those distant from the extreme parallels about one-sixth of the total range in latitude. Thus suppose it is required to plot a projection for India between latitudes 8° and 40° N. By this rough rule the errorless parallels should be distant from the extreme parallels about 32°/6, i.e. 5° 20´; they should therefore, to the nearest degree, be 13° and 35° N. The maximum scale errors will be about 2%. The scale errors vary approximately as the square of the range of latitude; a rough rule is, largest scale error = L²/50,000, where L is the range in the latitude in degrees. Thus a country with a range of 7° in latitude (nearly 500 m.) can be plotted on this projection with a maximum linear scale error (along a parallel) of about 0.1%;[29] there is no error along any meridian. It is immaterial with this projection (or with any conical projection) what the extent in longitude is. It is clear that this class of projection is accurate, simple and useful. [Illustration: (From _Text Book of Topographical Surveying_, by permission of the Controller of H. M. Stationery Office.) FIG. 16.--South Africa on a conical projection with rectified meridians and two standard parallels. Scale 800 m. to 1 in.] In the projections designated by (c) and (d) above, absolute errors of length are considered in the place of errors of scale, i.e. between any two meridians (c) the absolute errors of length of the extreme parallels are equated to the absolute error of length of the middle parallel. Using the same notation h(z + c) - sin c = h(z + c´) - sin c´ = -h(z + ½c + ½c´) - sin ½(c + c´). L. Euler, in the _Acta Acad. Imp. Petrop._ (1778), first discussed this projection. If a map of Asia between parallels 10° N. and 70° N. is constructed on this system, we have c = 20°, c´ = 80°, whence from the above equations z = 66.7° and h = .6138. The absolute errors of length along parallels 10°, 40° and 70° between any two meridians are equal but the scale errors are respectively 5, 6.7, and 15%. The modification (d) of this projection was selected for the 1:1,000,000 map of _India and Adjacent Countries_ under publication by the Survey of India. An account of this is given in a pamphlet produced by that department in 1903. The limiting parallels are 8° and 40° N., and the parallel of greatest error is 23° 40´ 51´´. The errors of scale are 1.8, 2.3, and 1.9%.