Example.
V. Let Ll, the breadth of the zone, be 50°, lying between 10° and 60° north latitude; its longitude 110°, from 20° east of the Canaries to the center of the western hemisphere; comprehending the western parts of Europe and Africa, the more known parts of North America, and the ocean that separates it from the old continent.
And because Cx = Nn ⁄ Ll × MF × MT, add these three logarithms.
| Log. 0.8726650 (= 50° to radius 1) | -1.9408476 |
| Log. MF (sin. 35°) | -1.7585913 |
| Log. MT (tang. 55°) | 0.1547732 |
| Take the sum | -1.8542121 |
| from log. Nn (= .6923772) | -1.8403427 |
| the remainder | -1.9861306 |
| is the logarithm of Cx. And because 1: Cx ∷ MT : xt, to this adding the log. MT | 0.1547732 |
| The sum | 0.1409038 |
is the log. of xt = 1.383260; and xR (= xr = ½ Ll) being .4363325, Rt will be 0.9469275, rt = 1.8195925. Whence having fixed upon any convenient size for our map, the center t is easily found. As, allowing an inch to a degree of a great circle, or 50 inches to the line Rr, Rt the semidiameter of the least parallel will be 54.255 inches, and that of the greatest parallel 104.255 inches.
Again, making as radius to MF so the longitude 110° to the angle StV, that angle will be 63° 5´⅗. Divide the meridians and parallels, and finish the map as usual.
Note, The log. MT being repeated in this computation with a contrary sign, we may find xt immediately by subtracting the sum of the logarithms of Ll and MF from the log. of Nn.
VI. A map drawn by this rule will have the following properties:
1. The intersections of the meridians and parallels will be rectangular.
2. The distances north and south will be exact; and any meridian will serve as a scale.
3. The parallels thro’ z and y, where the line Rr cuts the arc Ll, or any small distances of places that lie in those parallels, will be of their just quantity. At the extreme latitudes they will exceed, and in mean latitudes, from x towards z or y, they will fall short of it. But unless the zone is very broad, neither the excess nor the defect will be any-where considerable.
4. The latitudes and the superficies of the map being exact, by the construction, it follows, that the excesses and defects of distance, now mentioned, compensate each other; and are, in general, of the least quantity they can have in the map designed.
5. If a thread is extended on a plane, and fixed to it at its two extremities, and afterwards the plane is formed into a pyramidal or conical surface, it may be easily shewn, that the thread will pass thro’ the same points of the surface as before; and that, conversely, the shortest distance between two points in a conical surface is the right line which joins them, when that surface is expanded into a plane. Now, in the present case, the shortest distances on the conical surface will be, if not equal, always nearly equal, to the correspondent distances on the sphere: and therefore, all rectilinear distances on the map, applied to the meridian as a scale, will, nearly at least, shew the true distances of the places represented.
6. In maps, whose breadth exceeds not 10° or 15°, the rectilinear distances may be taken for sufficiently exact. But we have chosen our example of a greater breadth than can often be required, on purpose to shew how high the errors can ever arise; and how they may, if it is thought needful, be nearly estimated and corrected.
Write down, in a vacant space at the bottom of the map, a table of the errors of equidistant parallels, as from five degrees to five degrees of the whole latitude; and having taken the mean errors, and diminished them in the ratio of radius to the sine of the mean inclination of the line of distance to the meridian, you shall find the correction required; remembering only to distinguish the distance into its parts that lie within and without the sphere, and taking the difference of the correspondent errors, in defect and in excess.
But it was thought needless to add any examples; as, from what has been said, the intelligent reader will readily see the use of such a table; and chiefly as, whenever exactness is required, it will be more proper, and indeed more expeditious, to compute the distances of places by the following canon.
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.
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:
| L. sin. 30° | = | -1.6989700 | ||||||
| L. sin. 80° | = | -1.9933515 | ||||||
| 2 L. sin. 55° | = | -1.8267290 | ||||||
| -1.5190505 | = | log. of | 0.330408 | |||||
| and | 2 L. sin. 25° | = | -1.2518966 | = | log. of | 0.178606 | ||
| Log. of the sum | 0.509014 | is | -1.7067297 | |||||
| Whose half | is | -1.8533648 | ||||||
the L. 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 6255.189 and 3255.189 respectively, and the angle STV is 63° 5´⅗, 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 fast 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 every-where 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 lines, as HI, 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 title, 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 rumb-lines on it being logarithmic spirals to their common pole t, as is partly represented in the figure: and the arithmetical 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
Rule.
To the logarithm of the number of minutes in the difference of longitude add the constant logarithm[29] -4.1015105, and to their sum the logarithm sine of the mean latitude, and let this last sum be S.
The cotangent of the mean latitude being T, and an arithmetical mean between half the difference of latitude and its tangent being called m, from the logarithm of T + m take the logarithm of T - m, and let the logarithm of their difference be D; then shall S - D be nearly the logarithm tangent of the angle, in which the ship’s course cuts the meridians.
Note, 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.