cent. Several sections of the Peninsula are divisable, and if one canal is made to pay, it will certainly provoke emulation before long. These are considerations which investors in these high-flown speculations will do well to take to heart; nevertheless, as political and commercial problems, they demand the earnest attention of our Government.

(To be Continued.)

F. N. NEWCOME.

MERCATOR'S CHART.

ITS HISTORY, USE AND ABUSE.

N the Sea or Plane Chart of the early navigators large areas of the ocean and coastlines were delineated as if the earth were a flat surface. On this basis the difference of longitude between any two meridians, taken at the equator, was used as the distance of those meridians in all latitudes, and hence the degrees of latitude and longitude being everywhere equal, the intersection of the parallels and meridians sub-divided the surface into a series of equal squares. On such a projection the distances of places in high latitudes would be too great, and the relative positions of places inaccurate, although for places near the equator the relative directions and distances would be tolerably well preserved. For the higher latitudes the distortion would be such that no distances except those on a meridian, and no directions or bearings except those trending east and west, or north and south, could be correct. Still, the Plane Chart, which answered tolerably well for a very limited area, was, to a certain extent, a necessity for the purposes of navigation; and its use became very general. But we must take into consideration that the navigators who became illustrious through their discoveries in the fifteenth and sixteenth centuries, were men well versed in all that related to planispheres and globes, as well as in the use of great circle sailing-for which purposes globes of large dimensions were in demand; and it is especially worthy of note that prior to 1800globular sailing" or sailing by the arch of a great

66

circle" occupied a prominent part of all works that treated on navigation.

The general defects of the Plane Chart were not, however, wholly unknown, even to the humblest navigator. The various treatises on the Art of Navigation," published during the time of the general use of that chart, and for long after it had been improved upon, invariably contained a chapter or two devoted to "sailing by the Flat or Plain Chart, and the uncertainties thereof," or "proving the Plain Chart false and not to be trusted to, but in very short voyages," &c.; and in these] days, looking back to the elaborate rules for the use of that chart, it is wonderful how navigation was carried on at all, except coastwise, or unless the seaman was well up in the various "sailings," and their application in connection with a globe of large size.

When, however, something is wanted, it generally happens that somebody is found to devise a method to supply the deficiency. What seamen chiefly wanted was a chart projection which represented, with accuracy, the relative positions of places as respects the loxodrome or rhumb line. By what means Gerard Kauffman, the Flemish map-maker, arrived at the knowledge that the plane chart could be converted into an efficient sea-chart only on the basis that the degrees of latitude should be gradually lengthened in advancing from the equator to the poles, is not on record. Certain it is, however, that in 1569 he published his Universal Map, under his Latinized name of Mercator; but as he never showed how the chart was constructed, and as the degrees were not increased in the true proportion, it is generally assumed that it was the result of the tentative process of comparing the artificial globe with the plane chart. It was reserved for Edward Wright, in 1590, after an examination of a Mercator's chart, to develop the correct principles on which such a chart should be constructed; and in 1599 he published his famous treatise, entitled The Correction of certain Errors in Navigation, wherein he shows the reason for the division of the meridional line, the method of constructing the table of meridional parts, and its uses in navigation. Having determined that "the distance of any parallel of latitude from the equator is expressed by the sum of the secants

of all the arcs between the equator and that parallel," his table of meridional parts was obtained by actually adding secant after secant through every minute of the quadrant, as follows :—

The mer. parts for 1' he made equal to the secant of 1'.

The mer. parts for 2' equal to the sum of the sec. of 1' and 2'. The mer. parts for 3' equal to the sum of the sec. of 1', 2^ and 3'.

The mer. parts for 4' equal to the sum of the mer. parts of 3′ and the sec. 4'; and so on by a constant addition of the secants.

It will not surprise seamen-who, as a class, are beyond all other classes most tenacious of old habits and customs-to be told that it took the best part of a century to bring into general use Mercator's chart, as projected by Wright, on "true" principles.

The Mercator Projection, so-called, may be said to result from the development, upon a plane, of a cylinder tangent to the earth along the equator, the various parts of the earth's surface having been projected upon the cylinder in such manner as to satisfy the following condition :-That the loxodrome, or ship's track on the surface of the sea under a constant bearing, shall appear on the development as a right line preserving the same angle of bearing with respect to the meridians intersected by it as that of the ship's track.

In order to realise this condition, the line of tangency, or the equator of the earth, being the circumference of a right section of the cylinder, will appear as a right line on the development, while different right-lined elements of the cylinder, corresponding to the series of terrestial meridians, will appear as a system of equidistant right lines, parallel to each other and perpendicular to. the right-lined equator on the development or Mercator chart, maintaining the same relative positions and the same distances. apart on that equator as the primitive meridians have on the terrestrial spheroid. Moreover, the series of terrestrial parallels will also appear as a system of right lines parallel to each other and to the equator, and will so intersect the right-lined meridiansas to form a system of rectangles, whose successive widths must be variable, increasing from the equator in such manner that the

required equality of angles shall be maintained, for corresponding elements of a ship's track, on the spheroid and on the chart representation.

The conception of Mercator's Chart as a cylinder unrolled is illustrated in the accompanying diagram, where, looking from the centre of the circle enveloped within the cylinder (a, b, 90, 90), the successive parallels (from 0° to 90°) would appear to be projected further and further from each other, with gradually increasing distance between them, while the distance from meridian to meridian, as indicated at the equator (E, E), would remain undisturbed from top to bottom.

In such a projection it is only near the equator that portions of the earth's surface will appear as they are on the globe, everywhere else the area is exaggerated, and the relative areas of countries in different latitudes are wholly inaccurate, -the shape is all that remains correct. Nevertheless, that which is of the most value to seamen is persistent on any part of Mercator's chart-the angle which a straight line joining any two places on the chart makes with the meridians is equal to that which the loxodrome or

rhumb line joining the same two places on the globe makes with the meridians.

After Wright's time formula for the computation of meridional parts or increased latitudes were investigated by Bond, Gregory, Halley, and many others, both in England and France; these were chiefly in relation to the earth as a sphere. Maclaurin, in 1742, first discussed the problem with reference to the spheroid.

The most approved approximate formulæ as resulting from investigation through the integral calculus, and embracing the correction for the spheroid, is that given by the U.S. Bureau of Navigation.

1

m =

a

M

L

log tan (45° + ) -a (esin Lesin3L......)

2

where m is the meridional part; L the latitude of the parallel; M = 2·3025851; a the equatorial radius; e the compression, here Bessel's; and e the meridional eccentricity.

The several co-efficients, with their logarithms, are

L

m = 7915'7055 log tan (45° + ;)

-(22'-9448 sin L + 0.05104731 sin3L)

in which the first term is the ordinary formulæ for computing meridional parts, the earth being considered as a sphere; while the second part, consisting of a series of negative terms, represents the correction for the meridional eccentricity of the earth.

The common Tables, for compression O, are those ordinarily found in works on navigation. Bowditch, Norie and Raper give no decimal place in the numbers, the nearest unit alone being retained. Robertson, Kerigan, Chambers, Lynn and Breusing give one decimal place. Caillet (old editions), Domke and Inman give two decimals. The Tables of Mendoza Rios, for compression T. This table, computed for each tenth minute of arc with two decimal places, was published in the Connaissance des Temps for 1793. It is given in Mendoza Rios' earlier editions of his work navigation,