« PreviousContinue »
from the observations made in Europe compared together; but the observations made at the Cape confirm it with the greatest evidence.
It is of importance to be assured of the longitude of the places where the observations were made. M. W. endeavoured to determine them the best he was able, by observations of the eclipses of Jupiter's satellites, made at the same places. And he sets down a list of all the observations of these satellites made at different places the last year. It is pity that Messieurs L'Abbe Chappe, and Rumoski, did not succeed in observing several eclipses of the satellites, at Tobieske and Selenginsk, the better to confirm the longitudes of those places. However it appears that the difference between the meridians of Greenwich and Tobieske, is scarcely more than 4h 32m 55s. That between the meridians of Greenwich and Selenginsk, to judge from the 3 immersions observed there, should be but 7h 6m 03, but from other considerations he thinks it must be 10 or 15m more. If the longitude of these places should be more exactly determined, he is persuaded that we should obtain the parallax of the sun to nearly the 10th of a second, so exact the observations made at Selenginsk and Tobieske and the Cape appear to be.
Mr. Planman at Cajaneberg, observed the transit as follows;
7 59 10 26 22
Mr. Planman made use of a telescope of 20 or 21 feet; the latitude of Cajaneburg is 64° 13' 30"; the difference of meridians between Greenwich and Cajaneburg is sufficiently determined by observations on the eclipse of the moon May 18, 1761, made at Stockholm and Cajaneburg.
XVIII. Remarks on the Censure of Mercator's Chart, in a posthumous Work of Mr. West of Exeter. By Mr. Samuel Dunn.* p. 66. Dated Sept. 4, 1762. Mr. D. wishes to know if any paper has been printed in the Phil. Trans., concerning a sphere being inscribed in a hollow cylinder, and swelling its surface to the sides of the cylinder, thus to construct a more true and accurate chart for the purposes of navigation, than that which was invented by Edward Wright, and has long gone under the name of Mercator. The reason why he asks this is, he says, because there is lately published a posthumous work of one Mr.
* This gentleman was a native of Crediton, in Devonshire, where he kept a mathematical school for several years; but afterwards removed to Chelsea, where he followed the same occupation. He was well skilled in nautical calculations, and was a good practical astronomer. Besides several papers inserted in the Phil. Trans., he was also the author of some separate treatises on mathematical subjects, and published an Atlas in folio, which has been held in much estimation. He died in good circumstances, and left an estate of about 30 pounds a year, to support a mathematical school in his native town, the first master of which was appointed in 1793.
West, of Exeter, revised by J.Rowe, in which it is strongly insisted on, that the graduation of Mercator's chart is erroneous, and that the same, if rightly correspondent with the loxodromiques or rhumbs, should be graduated as a line of natural tangents, from the equinoctial to the poles. Now this error might have passed the less observed, but the Critical Review of last month sets it forth as a masterly performance, and a thing of the greatest merit and importance in navigation.
That there is a respect due to Edward Wright for his invention, that his principles are true, that Mr. West, or his editor, and both (if both of the same opinion) are false, is most certain. That the characters and abilities of Dr. Halley, Sir Jonas Moore, Mr. William Jones, Mr. James Hodgson, Mr. Haselden, and many others, for they are almost numberless, both of higher and lower mathematicians, who have written on the certainty and utility of Wright's chart, that the characters and abilities of these able geometricians are attacked by Mr. West and his editor, and by the Critical Reviewers, is plain, and that this will have great weight with many not over well acquainted with geometry, is no less plain.
But there are other circumstances; Edward Wright himself gives the very same construction by his words, as Mr. West does, though his tables make out quite another thing, that is, both Wright and West say expressly, the sphere being in the hollow cylinder, and the equinoctial remaining fixed without swelling, while the other parts swell towards the poles, the chart will be formed. But in this, Wright has badly expressed his own thoughts, for his tables make it that the equinoctial must either swell or contract itself. And this is very excusable in Edward Wright, for at that time geometricians had no notion of fluxions, or the increase of magnitude by local motion. Mr. West and his editor have therefore fallen into this error; they have taken the words, but not the sense of Edward Wright, and the Critical Reviewers vindicate them, and make it as if this property had been communicated to the Royal Society by Mr. West. The proposed demonstration of this tangential property, at page 58 of Mr. West's book, is no demonstration at all; there is nothing more plain than that in order to have the meridians at equal distances, the degrees of latitude must be enlarged to the same proportion in every part, as the circular meridians are nearer towards the poles, which proportion is as the cosine of the latitude to the radius.
XIX. A Defence of Mercator's Chart against the Censure of the late Mr. West, of Exeter. By Mr. Wm. Mountaine, F.R.S. p. 69.
The greatest single advantage that the important business of navigation 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 avigation Mr. Edward Wright, as appears by his
book intitled Certain errors in navigation detected and corrected,' published about the year 1599. In chap. 2d, 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 though sometimes attacked, yet it has been found impregnable.
The first person 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 4th rhumb, at 78° of longitude, ought to have 61° 26′ of latitude, whereas Wright makes it only 61° 14'. Hence the great difference is no more than 12 minutes; and what inconvenience hence can arise to the mariner in such a run, were this the fact? But it turns out otherwise; for this difference is reduced to less than one minute (even according to Stevins's 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 is
Doctor Halley has given a curious method of dividing the nautical meridian, and of performing the problems in sailing according to the true chart, in the Philos. Trans., 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 this author. Our worthy brother Mr. John Robertson, in his excellent Elements of Navigation, Vol. 2, page 358, expresses himself thus: "Now though 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 shows 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 scarcely sensible."
About the year 1720, a curvilinear sea chart made its appearance, said to be done by Henry Wilson, the publishers of which 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 Sca," 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 on 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 the 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 fixed them on their proper foundations: by which the reader will be enabled to see that this theory is not founded on false principles; but on such as are solid and true; and consequently that all calculations built on it 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, Mr. West therein declares that "the errors of the plain chart are corrected, in a great measure, by Mercator's or Wright's chart; though 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. 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 book, Mr. West has laid down a method of constructing a nautical chart, which he asserts to be "the first representation of
* William Emerson.
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 situated 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."
What Mercator's thoughts were on this matter when he formed his universal map, I know not, as he has left us no account of it; 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 on a due comparison of their respective methods.
By comparing the two modes of construction together, it is not difficult to discover that Mr. West's derives its original from Wright's; for right lines drawn from the centre through 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 2d 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 which he demonstrates the principles on 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 (says 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 distance, 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 to minutes, though without any sensible error, are