Page:Encyclopædia Britannica, Ninth Edition, v. 17.djvu/267

Rh NAVIGATION 255 in modern tables. The principle is simply explained by fig. 5, where b is the pole and bf the meridian. At any point a a minute of longitude : a min. i of lat. : : ea (the semi-diameter of the parallel) : kf (the radius). Again ea : kf b
 * : kf : ki :: radius : sec. akf(sec. of lat.).

To keep this proportion on the chart, the points of latitude must increase in the same proportion as the secants of the arc contained between those points and the equinoctial, which was then to be done by the &quot; canon of triangles.&quot; l Fig. 5. Subsequent writers, including Gunter, Norwood, and Bond, give Wright the credit of having been the first to establish a correct proportion between the meridians and parallels for every part of the chart. This great improve ment in the principle of constructing charts was adopted slowly by seamen, who, putting it as they supposed to a practical test, found good reason to be disappointed. The positions of most places had been laid down erroneously, by very rough courses and estimated distances upon an entirely false scale, viz., the plane chart ; from this they were transferred to the new projection. 2 When Napier s Canon Mirificus appeared in 1614, Wright at once recognized the value of logarithms as an aid to navigation, and undertook a translation of the book, which he did not live to publish (see NAPIER). E. Gunter s tables (1620) made the application of the new discovery to navigation possible, and this was done by T. Addison in his Arithmetical Navigation (1625), as well as by Gunter in his tables of 1624 and 1636, which gave artificial sines and tangents, to a radius of 1,000,000, with directions for their use and application to astronomy and navigation, and also logarithms of numbers from 1 to 10,000. Several editions followed, and the work retained its reputation over a century. Gunter invented the sector, and introduced the meridional line upon it, in the just proportion of Mercator s projection. The third edition of Gunter s work was published in 1653, and the fifth edition in 1673, amended by Henry Bond, a practitioner in the mathematics, in the Bulwark near the Tower a thick octavo. A table of meridional parts is given, with instructions to construct it by the addition of secants as Wright did. The table has been found upon examination to be very correct. The degree is divided into 1000 parts. With the latitude left, course steered, and difference of longi tude made good, Bond found the latitude of ship, by projection on the chart, by the sector, or by the following rule : tan of course x proportional diff. of lat. = diff. long, radius And conversely, suppose latitude left 50, course 33 45, difference of longitude 5|= 330 ; then cot 30 45 x 330 j-. = 493 5, prop. diff. lat. , radius which, added to the meridional parts corresponding to 50, will give the number opposite 55, and 55 is the latitude. Various problems in sailing according to Mercator are solved arithmetic ally upon the tangents, without the table of meridional parts, which may also be done geometrically upon the tangent line of the cross-staff. The following important proposition is in Bond s own words: &quot;First we must know that the logarithmic tangents from 45 upwards do increase in the same manner as the secants added together do, if we account every half degree above 45 to be a whole degree of Mercator s meridional line ; and so the table of 1 The works of Wright passed through at least ten editions ; the last which the writer has seen was edited by Moxon in 1657. 2 W. Snellius, professor of mathematics at Leyden, published in 1624 a treatise on navigation, after Wright s plan. He mentions the name of Wright with others in the introduction, but, as he did not say what part he took from each, the division of the meridional line to minutes up to 70 was attributed to him, though Wright had done it up to 89 59 and published it in 1610. Justice was done to Wright s memory in the M&moires of the Academy of Sciences, Paris, 1753, p. 275. logarithmic tangents is a table of meridional parts to every two minutes of the meridian line, leaving out the radius.&quot; The way of using this proposition is as follows. The table begins at 45, and every 30 minutes is reckoned a whole degree ; therefore, when both latitudes are given, take the half of each increased by 45, subtract the tangent of the lesser sum from that of the greater, and divide the remainder by the tangent of 45 30 (radius omitted); the quotient will be the equal, or equinoctial, degrees contained between the two latitudes. Or multiply the aforesaid remainder by ten and divide by half the tangent of 45 30, and the quotient will be equal to the equinoctial leagues contained between the two latitudes. The logarithmic tangents are here treated as natural numbers, and the division done bv logarithms. Bond lays no stress on the above solution as being new ; it is merely used in lieu of a table of meridional parts. The subsequent history of the problem of meridional parts may most conveniently be added here rather than in its chronological place. An important letter from Dr Wallis, professor of geometry at Oxford, is given f with the Phil. Trans, for 1685, No. 176. The writer says that, the old inquiry about the sum or aggregate of secants having been of late renewed, he thought fit to trace it from its original, with such solutions as seemed proper to it. Archimedes and the ancients divided the curvilinear spaces as figs. 6 and 7. If they reckoned the first four it was too large ; if the last four, too small. As the segments increased in number the error diminished. The degrees of longitude decrease as the cosine of the latitude (which is the semidiameter of such parallel) to the radius of the globe or equator. By the straight lines &quot; each degree of longitude is increased above its due proportion, at such rate as the equator (or its radius) is greater than such parallel (or the radius thereof).&quot; The old sea charts represented the degrees of. - latitude and longitude all equal. &quot;Hereby, & among other inconveniences (as Mr Edward Wright observed in 1599), the representation of places remote from the equator were distorted.&quot; Wright advised that the degrees of latitude should be protracted in like proportion with those of longitude, that is, everywhere in such proportion as is the respective secant of such latitude to the radius &quot; (see Wright s p P ? P P explanation of this part, and fig. 5). Fig. 8 represents one quarter of the globe, the surface of which is opened out till the parallel LA becomes a straight line as la, and each of the four meridians reaches P, P, P, P. The equator is re presented by EE ; so that the position of each parallel on the chart should b at such distance from the equator &quot;as are all the secants (taken at equal distances in the arc) to so many times the radius,. . . which is equivalent to a projection of the spherical surface on the concave surface of a cylinder, erected at right angles to the plane of the equator,&quot; while each division of the meridian is equal to the secant of the latitude answering to such part, as fig. 9. This projection, if expanded into a plane, will be the same , as a plane figure whose base I is equal to a quadrantal arc extended (or a portion thereof), on which (as ordinates) are erected perpendiculars equal to the secants, answering to the respective points of the arc extended, as fig. 10. The first answers to the equator, the last to the pole infinite. &quot;For finding this distance answering to each degree and minute of latitude, Mr Wright added all the secants from the .... beginning to the position Fig. 9. required. The sum of all except the greatest (answering to the figure inscribed) is too little. The sum of all except the least (answering to the circumscribed) is too great which latter Mr Wright followed. It will be nearer the truth than either if we take the intermediate spaces ; instead of minutes, take |, 1, 2J, &c., or the double of these, 1, 3, 5, 7, &c., which yet, because on the convex side of the curve, would be rather too little. Either of these ways will be exact enough for a chart. If we would be more exact, Mr Oughtred directs, as did Mr Wright before him, to divide the arc into parts yet smaller than minutes, and calculate secants thereto. &quot; Wallis continued the subject and the doctrine of infinite