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[[page number; upper right corner]] 90 [[/page number; upper right corner]]

indeed more expeditions, to compute the distances of places by the following canon.
  [[underline]] Multiply the product of the cosines of the two given latitudes by the square of the sine of half the difference of longitude; and to this product add the square of the sine of half the difference of the latitudes; the square root of the sum shall be the sine of half the arc of a great circle between the two places given. [[/underline]] [V. my Trigomometrical M.S. page facing 2g. for this Theorem by the same gentleman]
  Thus, if we are to find the true distance from one angle of our map to the opposite, that is, from S to Q, the operation will be as follows:
       S. sin. 30°=-1,6989700
       S. sin. 80°=-1,9933515
       2S. sin. 55°=-1,8267290
                     _________
                    -1,5190505=log. of 0,330408
    and 2L. sin 25°=-1,2518966=log. of 0,178606
                     S. of the sum-----0,509014 is -1,7067297
                                 whose half is---  -1,8533648
the S.sin. of 45°31', the double of which is 91°2' or 5462 geographical miles.
    And seeing the lines TS, TQ, reduced to minutes of a degree are 6225,189 and 3255,189 respectively, and the angle STV is 63°.5 3/5, the right line SQ on the map will be 5594, exceeding its just value by 132' or 1/42 of the whole.
7. The errors on the parallels increasing faster towards the north, and the line SQ having, at last, nearly the same direction, it is not to be wondered that the errors in our example should amount to 1/42.  Greater still would happen, if we measured the distance from O to Q by a straight line joining those points; for that line, on the conic surface, lying everywhere at a greater distance from the sphere than the points O and Q, must plainly be a very improper measure of the distance of their correspondent points on the sphere.  And therefore, to prevent all errors of that kind, and confine the other errors in this part of our map to narrower bounds, it will be best to terminate it towards the pole by a straight line KI touching the parallel OQ in the middle point K, and on the east and west by the lines, as KI, parallel to the meridian thro' K, and meeting the tangent at the middle point of the parallel SV in H.  By this means too we shall gain more space than we lose, while the map takes the usual rectangular form, and the spaces GHV remain for the [[underline]] title [[/underline]], and other inscriptions.
VII.  Another, and not the least considerable, property of our map is, that it may, without sensible error, be used as a sea-chart; the rhumb-lines on it being logarithmic spirals to their common pole t, as is partly represented in the figure; and the arithematical solutions thence derived will be found as accurate as is necessary in the art of sailing.
    Thus if it were required to find the course a ship is to steer between two ports, whose longitudes and latitudes are known, we may use the following [[underline]] RULE.  To the logarithm of the number of minutes in the difference of longitude add the constant logarithm* -[[/underline]] 4,1015105, [[underline]] and to their sum the logarithm sine of the mean latitude, and let this last sum be [[/underline]] S.
    [[underline]] The Cotangent of the mean latitude being [[/underline]] T, [[underline]] and the arithmetical mean between half the difference of latitude and its tangent being called [[/underline]] M, [[underline]] and from the logarithm of [[/underline]] T+M [[underline]] take the logarithm of [[/underline]] T-M, [[underline]] and let the logarithm of their difference be [[/underline]] D; [[underline]] then shall [[/underline]] S-D [[underline]] be nearly the logarithm tangent of the angle, in which the ship's course cuts the meridian. [[/underline]]
   [[underline]] Note, [[/underline]] We ought, in strictness, to use the ratio of tx+xR to tx-xR instead of T+M to T-M; but we substitute this last as more easily computed, and very little different.
EXAMPLE 1. Let the latitudes, on the same side of the equator, be 10° and 60°; then the middle latitude and its complement are 35° and 55°, and half the difference of the latitudes is 25°: and the difference of longitude being 110, the operation will stand as below.
[[line across page]]
*This constant logarithm contains the reduction of the diff. of longitude to parts of radius unity, and to [[underline]] Brigg's [[/underline]] Modules.                         Log.

Transcription Notes:
Multiple instances of old style long "s" found throughout. Transcribed in modern double "s" format.