A Defence of Mercator's Chart against the Censure of the Late Mr. West of Exeter: In a Letter to Charles Morton, M. D. Secret. R. S. from Mr. William Mountaine, F. R. S.

XIX. A Defence of Mercator's Chart against the Censure of the late Mr. West of Exeter: In a Letter to Charles Morton, M.D. Secret. R.S. from Mr. William Mountaine, F.R.S. To Doctor Morton, Secretary to the Royal Society. Dear Sir, Received your favour with Mr. Samuel Dunn's letter, touching Mr. West's method of constructing a nautical planisphere, referred to me by the Royal Society, which I now beg leave to return with the following account. As this island is situate by nature, not only for coasting trade, but foreign commerce, so every real improvement in the art of navigation has always met with the approbation and encouragement of the ingenious and sensible part of these kingdoms. The greatest single advantage that this important business ever received, was from the invention of the mariner's compass; and next to this, the projection of a true nautic practical chart claims place; — this last was performed by that great improver of navigation, Mr. Edward Wright, as appears by his book intitled "certain errors in navigation detected and corrected", published about the year 1599. In chapter 2d, of said book, he tells us, "that the errors in the plain chart had been complained "of by divers, as namely by Martin Cortese, Petrus Nonius, and even Gerardus Mercator seemeth to have corrected them, in his Universal Map of the World; yet none of them had taught any certain way how to amend such gross faults:" And, in his Preface, he declares, "that, by occasion of Mercator's map, he first thought of correcting so many and great absurdities in the common Sea Chart, but the way how this was by him done, he neither learnt of Mercator, nor of any man else." Wright's method (erroneously called Mercator's) was at this time then adopted, has continued ever since in use, and has been improved by some of the greatest mathematicians who have flourished since that time, and although sometimes attacked, yet it has been found impregnable. The first person (that I am aware of) who charged Mr. Wright with errors in his tables of rhumbs, is Simon Stevins, in his large volume of mathematical remembrances, which Wright himself plainly confutes in a subsequent edition of his book: now, Stevins does not condemn the principles, but only asserts that his tables have some faults in them, and endeavours to prove that the fourth rhumb at 78 deg. of longitude ought to have $61^\circ 26'$ of latitude, whereas Wright makes it only $61^\circ 14'$. Hence, the great difference is no more than 12 minutes; and what inconvenience can arise hereby to the mariner in such a run, was this the fact? But it turns out otherwise, for this difference is reduced to less than one minute (even according to Stevins own way) as evidently appears from Wright's answer in page 214. If every rhumb is then found to possess its true latitude in this chart at every degree and minute of longitude, without any sensible or explicable error (to make use of our author's own words) it follows, that the degrees of latitude are duly encreased, or that the table of meridional parts are true. The great Doctor Halley has given us a curious method of dividing the nautical meridian, and of performing the problems in sailing according to the true chart, in Philosophical Transactions, No. 219. by a method different from Mr. Wright's, but so nearly corresponding in practice, that this alone is a sufficient testimony in favour of my author. Our worthy brother Mr. John Robertson, in his excellent Elements of navigation, vol. II. page 358, expresses himself thus: "Now although a table thus made (Wright's table of meridional parts constructed to minutes) be abundantly sufficient for all nautical purposes, yet had the secants of smaller parts than minutes been taken, the table would have been more correct; and therefore Mr. Oughtred, Sir Jonas Moore, Doctor Wallis, Doctor Halley, and others, have been induced to find methods of constructing those tables with more accuracy than by the addition of secants to every minute. "But a table of meridional parts, constructed by the most accurate method, only shews that Mr. Wright's tables do no where exceed the true meridional parts by half a minute, and this only near the pole; for in latitudes as far as navigation is practicable the difference is scarce sensible."—About About the year 1720, a curvilinear sea chart made its appearance, said to be done by Henry Wilson, the publishers whereof represented Wright's chart as puzzling, difficult, and false. But these groundless assertions were rationally answered by Mr. Thomas Haselden, afterwards master of the Royal Academy at Portsmouth, in a letter and pamphlet addressed to Dr. Halley about the year 1722. In the year 1755 was published a book intitled, "The art of sailing upon the Sea", by W. E. which initial letters are sufficient to point out the ingenious author.—In page 74 he says, "It is demonstrable (by the method of fluxions) that the length of the part of the meridian line in Mercator's chart, which represents the difference of latitude of two places upon the globe, is equal to the difference of the log. tangents of half the complements of the two latitudes, multiplied into the number 2,30258509, and that product into the radius of the sphere". And in Scholium to his Fundamental principles, page 75. "In the few foregoing propositions, I have demonstrated the truth of the chief methods of sailing now in use; and deduced them from their genuine principles, and fixt them upon their proper foundations: By which the reader will be enabled to see that this theory is not founded upon false principles; but upon such as are solid and true; and consequently that all calculations built hereon may be depended on as exact". Notwithstanding these, Wright's method is charged with great imperfection by the late Mr. West of Exeter, in his posthumous work, referred unto by Mr. Mr. Dunn.—Mr. West therein declares that "the errors of the plain chart are corrected, in a great measure, by Mercator's or Wright's chart; tho' the latter is not a true projection of the sphere in any shape; nor indeed is it pretended to be such by Mr. Wright, one of its inventors". — The first part of this paragraph surely contains a contradiction; for how can the errors in the plain chart be in a great measure corrected by a projection that is not true in any shape? And in answer to the latter part,—Mr. Wright has no where made such concessions that I know of;—and further Mr. West blends Wright and Mercator together, when at the same time it does not appear that the latter ever published any principles of this kind of projection to the world. In the 20th article of the beforementioned book, Mr. West has laid down a method of constructing a nautical chart, which he asserts to be "the first representation of the terraqueous globe ever yet invented, in which the meridians, parallels and rhumbs, are justly and truly projected in right lines, for the latter cannot be so projected in Mercator."—If they cannot be so projected in Wright's, they cannot in his; for in both, the meridians are said to be right lines and parallel, and therefore the rhumbs must be right lines also, or how can they intersect the meridians so situate at equal angles? He also says in his scholium, that "It does not appear that Mercator or Wright ever thought of this projection; for the meridian line here is manifestly a line of tangents; whereas in their projection, it is a collection of secants." Vol. LIII. M What What Mercator's thoughts were upon this matter when he formed his universal map, I know not, as he has left us no account thereof; but what Wright's were, he has very plainly told us in his aforesaid book; and whether his primary conceptions, and preparative modulus, do not only take in the whole, but also the very manner, of Mr. West's construction, will better appear upon a due comparison of their respective methods, which I shall beg leave here to introduce in their own words. Mr. West's Proposition. "If a rectangular piece of paper be turned into the form of a right cylindrical tube, and a sphere be inscribed therein, so as that the axes of the sphere and cylinder do coincide, or that the equator be the line of contact between the said tube and sphere, and all the points of the spheric surface be projected or transferred to the concave surface of the tube, by right lines proceeding from the center of the sphere, and terminating in the said concave surface of the tube: And then, if the paper be opened, and stretched upon a plane, it will present a chart, in which the meridians, parallels of latitude, and rhumbs are all truly and geometrically projected in right lines". In Wright's Correction of errors, reprinted by Moxon in 1657, not having the original edition by me, in Chap. 2. we have the following account — "Whereas the spaces betwixt the parallels should in- crease more and more as you go from the equinoctial toward either of the poles, which Martin Cortese also noteth in his third book and second chapter of the art of navigation; But he omitted that wherein all the difficulty lyeth; that is, how much, or in what proportion those spaces should increase: Which that it may the better be perceived, I think it not unmeet first to shew by what kind of projection (or extension rather) the nautical planisphere may not unfitly be conceived to be geometrically made, after this manner. Mr. Wright's Method. Suppose a spherical superficies, with meridians, parallels, rhumbs, and the whole hydrographical description drawn thereupon, to be inscribed into a concave cylinder, their axes agreeing in one. Let this spherical superficies swell like a bladder (whiles it is in blowing) equally always in every part thereof (that is, as much in longitude as in latitude) till it apply, and join itself (round about, and all along also towards either pole) unto the concave superficies of the cylinder: each parallel upon this spherical superficies, increasing successively from the equinoctial towards either pole, until it come to be of equal diameter with the cylinder, and consequently the meridians still widening themselves, till they come to be so far distant every where each from the other, as they are at the equinoctial. Thus it may most easily be understood, how a spherical superficies may (by extension) be made a cylindric, and consequently a plane parallelogram superficies; "cies; because the superficies of a cylinder is no- thing else but a plain parallelogram wound about two equal equidistant circles that have one com- mon axtree perpendicular upon the centers of them both" &c — "So as the nautical planisphere may be defined to be nothing else but a parallelogram made of the spherical superficies of an hydrogra- phical globe inscribed into a concave cylinder, both their axes concurring in one; and the spherical su- perficies swelling in every part equally in longitude and latitude, till every one of the parallels there- upon be inscribed into the cylinder (each parallel growing as great as the equinoctial) or till the whole spherical superficies touch and apply itself every where to the concavity of the cylinder".— "In this nautical planisphere thus conceived to be made, all places must needs be situate in the same longitudes, latitudes, and directions or courses, and upon the same meridians, parallels, and rhumbs, that they were in the globe; because, that at every point between the equinoctial and the pole, we under- stand the spherical superficies whereof this plani- sphere is conceived to be made, to fswell equally as much in longitude as in latitude (till it join itself unto the concavity of the cylinder) so as hereby no part thereof is any way distorted or displaced out of his true and natural situation upon his meridian, parallel or rhumb, but only dilated and enlarged: the meridians also, parallels and rhumbs dilating and enlarging themselves likewise, at every point of latitude in the same proportion". By comparing these two modes of construction to- gether I think it is not very difficult to discover that Mr. West's derives its original from Wright's; for right lines lines drawn from the center thro' all the points in the spheric surface, and terminating in the concave surface of the tube, are secants, and the tube becomes a tangent line to all those respective secants: And, does not Wright's uniform dilatation, by the second law of motion, produce the same? West stops here, and gives us a chart at once; Wright calls these his geometrical lineaments only, by which he obtains a rectilinear planisphere, and from whence he demonstrates the principles upon which his table of meridional parts are founded. And that he does not esteem this as a chart completed, but only his apparatus, and preparative work, which requires yet to be applied and moulded into a true nautical chart, is evident from the next paragraph, "Now then (saies he) let us diligently consider of the geometrical lineaments, that is, the meridians, rhumbs, and parallels of this imaginary nautical planisphere, that we may in like manner express the same in the mariner's chart: For so undoubtedly we shall have therein a true hydrographical description of all places in their longitudes, latitudes, and directions, or respective situations each from other according to the points of the compass in all things correspondent to the globe, without either sensible, or explicable error". And hence he proceeds to the proof and application of these his lineaments, to the construction of his table of latitudes, as he calls it; which is, in this edition, computed to minutes of parallel-difference, but with a little contrivance in the calculus to reduce the same yet somewhat nearer the truth. Notwithstanding this care and nicety in computation, he is duly sensible that his increments of latitude calculated calculated to minutes, altho' without any sensible error, are yet not absolutely true, because they ought to flow with an uniform, and uninterrupted motion; He therefore cautiously guards against critical remarks hereupon, in the following paragraph: "In this table it was thought sufficient to use such exactness, as that thereby (in drawing the lineaments of the nautical planisphere) sensible error might be avoided. He that listeth to be more precise may make the like table to decades, or tens of seconds, out of Joachimus Rheticus his Canon Magnus Triangulorum: notwithstanding the geometrician that desireth exact truth, cannot be so satisfied neither; for whose sake and further satisfaction, I thought good to adjoin also this geometrical conceit of dividing a meridian of the nautical planisphere." "Let the equinoctial and a meridian be drawn upon a globe: Let the meridian, divided into degrees, minutes, seconds, &c. roll upon a straight line, beginning at the equinoctial, the globe swelling in the mean time in such sort, that the semidiameter thereof may be always equal to the secant of the angle, or arch contained between the equinoctial and semidiameter insisting at right angles upon the foresaid straight line: The degrees, minutes, seconds, &c. of the meridian, noted in the straight line, as they come to touch the same, are the divisions of the meridian in the nautical planisphere: And this conceit of dividing the meridian of the nautical planisphere may satisfy the curious exactness of the geometrician; but for mechanical use, the table before mentioned may suffice." More might be said in favour of Wright's chart, but I think it is altogether unnecessary; if his own principles principles and documents be duly considered; if almost general practice and experience for a century and an half past can have any weight; and, if the concurrent testimony and authority of so many eminent mathematicians who have handled this subject since his time, and some of them in a quite different method, can have any poise in the scale of reason. I have carefully endeavoured not to mistake the true sense and meaning of Mr. West's proposition in any part thereof; if I have not, I cannot pronounce what kind of chart may be formed from his tangent line being made the line of latitudes, or that meridian line whereupon the tangents are to determine the sections of their respective parallels: I shall only observe, that, if the meridians be right lines, and parallel to each other, the rhumbs must be right lines also; but by this tangential projection, these will be deflected from their true bearings, or make the angles of the courses too great, unless some expedient be devised to accommodate this error; and if the rhumbs be not right lines, such chart will then be embarrassed with more difficulties in practice than Mr. Wright's. Upon the whole, it seems as if the editor, confiding in Mr. West's abilities, hath, without examination, published this proposition (found amongst many other loose papers, none of which were, perhaps, ever intended for public inspection, as himself saies in his apology) just as he found it; and that the Reviewers in good opinion of both, and out of tenderness to the widow and family, the book being published for their benefit, have not so critically examined and compared it with what has been already done.—But, notwithstanding what is spoken in favour thereof, I suspect it will have little weight with the mariners, who very well know the value value of the Mercator's chart (as they call it) nor are they ever very easily induced to adopt new notions or inventions, and those contrary to what they are familiarized unto by constant practice. The Critical Reviewers do indeed hint as if this paper had been heretofore communicated by Mr. West to the Royal Society, and that in the following terms:—"Mr. West lays down the following very ingenious proposition, which, if we do not greatly mistake, we have seen, with little variation, in the Philosophical Transactions, communicated possibly by the same hand".—In this, I believe, they are mistaken, for I cannot find any thing like it in the transactions since the date 1746, the year in which it is said to be wrote. I am duly sensible of the frequent monitions, and sincere desire of the Royal Society, that its members may avoid all possible occasions of controversy; and whether this account has not a tendency thereunto, if it should, in other respects, be thought worthy of a place in the public Transactions, is submitted, with all due respect, to the determination of the Committee of papers. I have the honour to be, Sir, With the greatest esteem, The Royal Society's, Gainsford - Street, Southwark; Jan. 26th, 1763. and also your most obedient, and faithful humble servant, William Mountaine. XXI. An