Wright's improvement on Mercator's chart soon be came known abroad, where it seems to have been received with greater readiness, but less candour than in England. In 1624, Snellius, professor of mathematics at Leyden, published a ireatise of navigation, on Wright's plan, which he entitled Typhus Batavus, but without making the slightest acknowledgment for the principles he had borrowed from it. Hence, several years after, we find some of the French writers, who generally show a very illiberal jealousy of English matheinaticians, confidently asserting, that Snellius was the real inventor of the cor rected charts. Montucla, however, has since admitted that this admirable improvement, though at first ascribed to -.Mercator, is entirely due to our countryman Edward Wright. Even ten years before the publication of the -works of Snellius, every case in navigation had been solved numerically by 'Mr. Ralph Handson, in his nautical questions subjoined to a translation of Pitiscus's Trigo nometry, by applying arithmetical calculations to Wright's table of meridional parts. Of the priority of Wright's claim, therefore, to the honour of this important inven tion, no person can any longer entertain the smallest doubt.
Another method of determining the change of longi tude on an oblique course, and founded partly on the principles of plane sailing, and partly on those of parallel sailing, was at this time proposed by Mr. Handson, and found to answer extremely well in latitudes at no great distance from the equator. Two kinds of approximation were suggested, both of which were altogether indepen dent of Wright's enlargement of the meridian: the first was obtained by taking an arithmetical mean between the cosines of the extreme latitudes, and the second by the same mean between their secants, and thus declining the difference of longitude from the departure, as given by plane sailing. A similar method, but neither so correct, nor yet more simple in its application, is still frequently employed by seamen, to whom it is known by the appel lation of middle-latitude sailing. According to this method of computing the difference of longitude, which seems to have been first described by Gunter, in an edition of his works printed in 1623, the cosine of the mean of the two latitudes is employed instead of the mean of their cosines, as recommended by Handson.
The inventor of the logarithms, as WC have alleady hinted, contributed greatly to facilitate the necessary cal culations for determining a ship's place by the methods hitherto employed for that purpose. A table of the common form of logarithms adapted to the natural num bers, as we have described in the history of that excellent invention, was computed by Briggs, and first given to the world in 1618. This table served as the basis of the lo garithms of (he natural sines and tangents, and was so applied by Gunter, who drew up from it a table of the logarithms of these trigonometrical quantities for every minute of the quadrant, which he published in 1620. To Gunter the seaman is also indebted for the very useful scale which still bears his name, on which he inscribed the logarithmic lines for numbers, as well as for the sines and tangents of arches, and by help of which, and a pair of compasses, every question in tligonometry may be solved with the utmost expedition, and very considerable ac curacy.
IIaving now reached a period in the history of navigation when the art, so far as the determination of a ship's place depends on her reckoning, was nearly as perfect as it is at the present day, before we proceed to describe the im provements which it afterwards received from astronomy by means of instruments of a proper construction to be used at sea, we shall briefly allude to a method of calcu lating the meridional parts, which is not only more refin ed in principle, but, in extreme cases, more correct in its application, than the mode employed by Wright. His method of enlarging the meridian, as we have already stated, consisted in the continued additions of the suc cessive secants or portions of that great circle, increasing in an arithmetical progression, from the equator to the pole, the secants of tho increasing arches being in each latitude proportional to the rate of convergency of the meridians; but NYC have now to remark, that the resulting sums of these secants, or meridional parts as they are called, will only be perfect when the portions of the me ridian for which they are calculated increase by infinitely small differences ; because, strictly speaking, the degree of convergency is not the same throughout the whole of any arch, however small, and consequently when the atches increase by finite differences, errors must gra dually accumulate in the results of a calculation founded on that supposition. This defect in Wright's method of enlarging the successive portions of the meridian was noticed very early, and was obviated to a certain extent, by diminishing the common difference between the arches to a single minute of the quadrant ; but it was not removed entirely, till Henry Bond announced, in an edition of Norwood's Navigation, which he published in 1650, that the sums of the natural secants, or the enlarge ments of the meridian on the principles of Wright's pro jection, follow the same law as the logarithmic tangents of half the complements of the latitudes. The demonstra tion of this curious property was not given by Bond, who seems to have stumbled on the discovery by accident ; but this defect was afterwards supplied by James Gregory, in his Exercitationes Geometric ce, printed at London in 1658. Another demonstration of it was also given by Barrow, in his Lectiones Geometricre, when he proves that if r denotes the radius, and h tlIC latitude of a place, the sum of the secants in question is analogous to the logarithm ofr-±---).', which is the same relation as that al ready noticed.—Lastly, Dr. Halley furnished a very beauti ful demonstration of this important proposition, deduced from the properties of the logarithmic spiral, and the principles of the stereographic projection of the sphere on the plane of the equator.