MERCATOR (Gerard)

, an eminent geographer and mathematician, was born in 1512, at Ruremonde in the Low Countries. He applied himself with such industry to the sciences of geography and mathematics, that it has been said he often forgot to eat and sleep. The emperor Charles the 5th encouraged him much in his labours, and the duke of Juliers made him his cosmographer. He composed and published a Chronology; a larger and smaller Atlas; and some Geographical Tables; beside other books in Philosophy and Divinity. He was also so curious, as well as ingenious, that he engraved and coloured his maps himself. He made various maps, globes, and other mathematical instruments for the use of the emperor; and gave the most ample proofs of his uncommon skill in what he professed. His method of laying down charts is still used, which bear the name of Mercator's Charts; also a part of navigation is from him called Mercator's Sailing.—He died at Duisbourg in 1594, at 82 years of age.—See Mercator's Chart, below.

Mercator (Nicholas), an eminent mathematician and astronomer, whose name in High-Dutch was Haussman, was born, about the year 1640, at Holstein in Denmark. From his works we learn, that he had an early and liberal education, suitable to his distinguished genius, by which he was enabled to extend his| researches into the mathematical sciences, and to make very considerable improvements: for it appears from his writings, as well as from the character given of him by other mathematicians, that his talent rather lay in improving, and adapting any discoveries and improvements to use, than invention. However, his genius for the mathematical sciences was very conspicuous, and introduced him to public regard and esteem in his own country, and facilitated a correspondence with such as were eminent in those sciences, in Denmark, Italy, and England. In consequence, some of his correspondents gave him an invitation to this country, which he some time after accepted, and he afterwards continued in England till his death. He had not been long here before he was admitted F. R. S. and gave frequent proofs of his close application to study, as well as of his eminent abilities in improving some branch or other of the sciences. But he is charged sometimes with borrowing the inventions of others, and adopting them as his own. And it appeared upon some occasions that he was not of an over liberal mind in scientific communications. Thus, it had some time before him been observed, that there was an analogy between a scale of logarithmic tangents and Wright's protraction of the nautical meridian line, which consisted of the sums of the secants; though it does not appear by whom this analogy was first discovered. It appears however to have been first published, and introduced into the practice of navigation, by Henry Bond, who mentions this property in an edition of Norwood's Epitome of Navigation, printed about 1645; and he again treats of it more fully in an edition of Gunter's Works, printed in 1653, where he teaches, from this property, to resolve all the cases of Mercator's Sailing by the logarithmic tangents, independent of the table of meridional parts. This analogy had only been found to be nearly true by trials, but not demonstrated to be a mathematical property. Such demonstration seems to have been first discovered by Mercator, who, desirous of making the most advantage of this and another concealed invention of his in navigation, by a paper in the Philosophical Transactions for June 4, 1666, invites the public to enter into a wager with him on his ability to prove the truth or falsehood of the supposed analogy. This mercenary proposal it seems was not taken up by any one, and Mercator reserved his demonstration. Our author however distinguished himself by many valuable pieces on philosophical and mathematical subjects. His first attempt was, to reduce Astrology to rational principles, which proved a vain attempt. But his writings of more particular note, are as follow:

1. Cosmographia, sive Descriptio Cœli & Terræ in Circulos, qua fundamentum sterniter sequentibus ordine Trigonometriæ Sphericorum Logarithmicæ, &c, a Nicolao Hauffman Holsato; printed at Dantzick, 1651, 12mo.

2. Rationes Mathematicæ subductæ anno 1653; Copenhagen, in 4to.

3. De Emendatione annua Diatribæ duæ, quibus exponuntur & demonstrantur Cycli So<*>is & Lunæ, &c; in 4to.

4. Hypothesis Astronomica nova, et Consensus ejus cum Observationibus; Lond. 1664, in folio.

5. Logarithmotechnia, sive Methodus Construendi Logarithmos nova, accurata, et facilis; scripto antehae communicata anno sc. 1667 nonis Augusti; cui nunc accedit, Vera Quadratura Hyperbolæ, & Inventio summæ Logarithmorum. Auctore Nicolao Mercatore Holsato è Societate Regia. Huic etiam jungitur Michaelis Angeli Riccii Exercitatio Geometrica de Maximis et Minimis, hic ob argumenti præstantiam & exemplarium raritate<*> recusa: Lond. 1668, in 4to.

6. Institutionum Astronomicarum libri duo, de Motu Astrorum communi & proprio, secundum bypotheses veterum & re<*>ntiorum præcipuas; deque Hypotheseon ex observatis constructione, cum tabulis Tychonianis, Solaribus, Lunaribus, Lunæ-solaribus, & Rudolphinis Solis, Fixar<*>m & quinque Errantium, earumque usu præceptis et exemplis commonstrato. Quibus accedit Appendix de iis, quæ novissimis temporibus cœlitus innotuerunt: Lond. 1676, 8vo.

7. Euclidis Elementa Geometrica, novo ordine ac methodo fere, demonstrata. Una cum Nic. Mercatoris in Geometriam Introductione brevi, qua Magnitudinum Ortus ex genuinis Principiis, & Ortarum Affectiones ex ipsa Genesi derivantur. Lond. 1678, 12mo.

His papers in the Philosophical Transactions, are,

1. A Problem on some Points in Navigation: vol. 1, pa. 215.

2. Illustrations of the Logarithmo-technia: vol. 3, pa. 759.

3. Considerations concerning his Geometrical and Direct Method for finding the Apogees, Excentricities, and Anomalies of the Planets: vol. 5, pa. 1168.

Mercator died in 1994, about 54 years of age.

MERCATOR's Chart, or Projection, is a projection of the surface of the earth in plano, so called from Gerrard Mercator, a Flemish Geographer, who first published maps of this sort in the year 1556; though it was Edward Wright who first gave the true principles of such charts, with their application to Navigation, in 1599.

In this chart or projection, the meridians, parallels, and rhumbs, are all straight lines, the degrees of longitude being every where increased so as to be equal to one another, and having the degrees of latitude also increased in the same proportion; namely, at every latitude or point on the globe, the degrees of latitude, and of longitude, or the parallels, are increased in the proportion of radius to the sine of the polar distance, or cosine of the latitude; or, which is the same thing, in the proportion of the secant of the latitude to radius; a proportion which has the effect of making all the parallel circles be represented by parallel and equal right lines, and all the meridians by parallel lines also, but increasing infinitely towards the poles.

From this proportion of the increase of the degrees of the meridian, viz, that they increase as the secant of the latitude, it is very evident that the length of an arch of the meridian, beginning at the equator, is proportional to the sum of all the secants of the latitude, i. e. that the increased meridian, is to the true arch of it, as the sum of all those secants, to as many times the radius. But it is not so evident that the same increased meridian is also analogous to a scale of the logarithmic tangents, which however it is. “It does not appear by whom, nor by what accident, was discovered the| analogy between a scale of logarithmic tangents and Wright's protraction of the nautical meridian line, which consisted of the sums of the secants. It appears however to have been first published, and introduced into the practice of navigation, by Mr. Henry Bond, who mentions this property in an edition of Norwood's Epitome of Navigation, printed about 1645; and he again treats of it more fully in an edition of Gunter's Works, printed in 1653, where he teaches, from this property, to resolve all the cases of Mercator's Sailing by the logarithmic tangents, independent of the table of meridional parts. This analogy had only been found however to be nearly true by trials, but not demonstrated to be a mathematical property. Such demonstration, it seems, was first discovered by Mr. Nicholas Mercator, which he offered a wager to disclose, but this not being accepted; Mercator reserved his demonstration; as mentioned in the account of his life in the foregoing page. The proposal however excited the attention of mathematicians to the subject, and demonstrations were not long wanting. The first was published about two years after, by James Gregory, in his Exercitationes Geometricæ; from hence, and other similar properties there demonstrated, he shews how the tables of logarithmic tangents and secants may easily be computed from the natural tangents and secants.

“The same analogy between the logarithmic tangents and the meridian line, as also other similar properties, were afterwards more elegantly demonstrated by Dr. Halley, in the Philos. Trans. for Feb. 1696, and various methods given for computing the same, by examining the nature of the spirals into which the rhumbs are transformed in the stereographic projection of the sphere on the plane of the equator: the doctrine of which was rendered still more easy and elegant by the ingenious Mr. Cotes, in his Logometria, first printed in the Philos. Trans. for 1714, and afterwards in the collection of his works published 1732, by his cousin Dr. Robert Smith, who succeeded him as Plumian professor of philosophy in the University of Cambridge.”

The learned Dr. Isaac Barrow also, in his Lectiones Geometricæ, Lect. xi, Āppend. first published in 1672, delivers a similar property, namely, “that the sum of all the secants of any arc, is analogous to the logarithm of the ratio of r + s to r - s, viz, radius plus sine to radius minus sine; or, which is the same thing, that the meridional parts answering to any degree of latitude, are as the logarithms of the ratios of the versed sines of the distances from the two poles.” Preface to my Logarithms, pa. 100.

The meridian line in Mercator's Chart, is a scale of logarithmic tangents of the half colatitudes. The differences of longitude on any rhumb, are the logarithms of the same tangents, but of a different species; those species being to each other, as the tangents of the angles made with the meridian. Hence any scale of logarithmic tangents is a table of the differences of longitude, to several latitudes, upon some one determinate rhumb; and therefore, as the tangent of the angle of such a rhumb, is to the tangent of any other rhumb, so is the difference of the logarithms of any two tangents, to the difference of longitude on the proposed rhumb, intercepted between the two latitudes, of whose half complements the logarithmic tangents were taken.

It was the great study of our predecessors to contrive such a chart in plano, with straight lines, on which all, or any parts of the world, might be truly set down, according to their longitudes and latitudes, bearings and distances. A method for this purpose was hinted by Ptolomy, near 2000 years since; and a general map, on such an idea, was made by Mercator; but the principles were not demonstrated, and a ready way shewn of describing the chart, till Wright explained how to enlarge the meridian line by the continual addition of secants; so that all degrees of longitude might be proportional to those of latitude, as on the globe: which renders this chart, in several respects, far more convenient for the navigator's use, than the globe itself; and which will truly shew the course and distance from place to place, in all cases of sailing.

Mercator's Sailing, or more properly Wright's Sailing, is the method of computing the cases of sailing on the principles of Mercator's chart, which principles were laid down by Edward Wright in the beginning of the last century; or the art of finding on a plane the motion of a ship upon any assigned course, that shall be true as well in longitude and latitude, as distance; the meridians being all parallel, and the parallels of latitude straight lines.

In the right-angled triangle Abc, let Ab be the true difference of latitude between two places, the angle bAc the angle of the course sailed, and Ac the true distance sailed; then will bc be what is called the departure, as in plane sailing: produce Ab, till AB be equal to the meridional difference of latitude, and draw BC parallel to bc; so shall BC be the difference of longitude.

Now from the similarity of the two triangles Abc, ABC, when three of the parts are given, the rest may be found; as in the following analogies: As Radius : sin. course : : distance : departure; Radius : cos. course : : distance : dif. lat.; Radius : tan. course : : merid. dif. lat : dif. longitude.

And by means of these analogies may all the cases of Mercator's Sailing be resolved.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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MELODY
MENISCUS
MENSTRUUM
MENSURABILITY
MENSURATION
* MERCATOR (Gerard)
MERCURY
MERIDIAN
MERLON
MERSENNE (Martin)
MESOLABE