that

the feed is always reckoned by the males, and ro the females of a family, and (that) the name in a father's houfe could only be preferved by the male defcendants.'in which refpect Jefus Chrift was not a fon of David: therefore this connexion is deftroyed.

"The doctrines of chriftianity, especially the preceptive ones, taken from his own mouth, cannot to all appearance be affected by infifting on the character of David, whether good or bad; right forry would this author be if they refted, on any fuch dependance. The divine tendency of these doctrines is fo manifeft; they are fo refined in their own nature, that they will ever claim the most exalted reverence from mankind, to them and to their glorious publifher, independant on all foreign contingencies; on all former narrow tenets, and fuperftitious rites; which he has fo totally fuperfeded among his profeffors: and that they ever may is the fincere prayer of this abufed writer. He therefore hopes ari acquittal from this part of his indictment."

How far thefe profeffions are fincere, it would not become us to determine; but we cannot help diffenting from this writer's opinion, that chriftianity cannot be affected by fuch controverfies as this, concerning the character of king David; as we are firmly perfuaded, that the connection between the Old and New Teftament is fo infeparable, that whatever tends to weaken the authority of the one, muft infallibly impair the foundation of the other. However, on his own fuppofition, we shall take leave of this writer with a hint of advice, viz. that as we have no mean opinion of his abilities (whatever may be thought of his learning) we should be glad to see them more usefully employed.

We are not forry, therefore, to find him concluding with the following fenfible. declaration, viz. "Religious conteft is moft difagreeable of all other: fince it has ever been leaft decifive; and moft incentive of thofe propenfities which the genuine dictates of pure religion directly tend to fupprefs. The writer, therefore, at the fame time that he drops this controverfy, has no present intention ever more to engage in any of the like nature."

Before we entirely clofe this article, it may not be improper to mention the Author's poftfcript, in which he has treated Dr. Patten's miferable vindication of David with becoming fpirit and contempt; at the fame time that he does juftice to the candour of Mr. Harwood, author of the Converfion of a Deift: in the appendix to which, Mr. Harwood had inferted fome Remarks on the Hiftory of the Man after God's own Heart.

Mathematics.

Mathematics. With Eleven Copper-plates. By the late Ret. Mr. William Weft of Exeter. Revifed by John Rowe. 3s. in boards. Richardfon.

HIS fmall manual, of which every page fhews the hand

of a mafter, is a pofthumous work; and is published for the benefit of the Author's widow, by the ingenious Mr. Rowe, who fome years fince obliged the world with An Introduction to the Doctrine of Fluxions. The Editor informs us, that the present publication was felected from feveral mathematical papers, written at different times, few of which were finished, and none perhaps ever intended for public infpec

tion.

The first five pages contain a fhort introduction to the Auxionary calculus; the following thirty-eight, refpecting the application of that doctrine to the folution of problems de maximis et minimis; in which Mr. Weft has rendered that ufeful and difficult branch of the mathematics plainer, and more easy to be understood, than we remember to have feen it done by any other Writer; and, at the fame time, has demonftrated, in a very elegant manner, the common method of making the fluxion of a maximum or minimum

= 0.

The other part of this fmall performance confifts of tweuty-four miscellaneous queftions, with their folutions: feyeral of these are very curious, and the folutions ricat and elegant. But the twentieth propofition, which, from a note at the foot of the page, appears to have been written in the year 1746, is of fo interefting a nature, that it deferves particular notice. It is a new method of conftructing a fea-chart, in which the meridians, parallels of latitude, and rhumbs, are all truly and geometrically projected in strait lines.

It is well known, that every projection of the fphere will give true folutions to all nautical problems, but that only is properly adapted to the mariner's ufe, where the rhumb, or the path a fhip defcribes in failing, according to the direction of the compaís, is reprefented by a ftrait line, because there will then be no difficulty in determining exactly the bearing of any two places, or the true courfe that a hip must fail from one to the other.

• See Review, Vol. XIX. p. 528.
E

REV. July, 1759.

This

This rectilinearity of the rhumb-line is the chief property of Mercator's projection; and, to accomplish this very thing, employed the attention of the most celebrated navigators after the invention of the compafs; but was never brought to any degree of perfection, till performed by our ingenious countryman, Mr. Edward Wright: whofe invention will render his name famous to all pofterity.

This projection Mr. Weft has now perfected by the fole lowing propofition, which we shall give in his own words.

"If a rectangular piece of paper be turned into the form of a right cylindrical tube, and a fphere be infcribed therein, fo as that the axes of the fphere and cylinder coincide, or, that the equator be the line of contact between the faid tube and sphere, and all the points of the fpheric furface be projected or tranferred to the concave furface of the tube, by right lines proceeding from the center of the fphere, and terminating in the faid concave furface of the tube: and then, if the paper be opened and ftretched upon a plane, it will prefent a chart, in which the meridians, parallels of latitude, and rhumbs are all truly and geometrically projected in right lines. Quære the Demonftration?

DEMONSTRATION.

With regard to the meridians, it is evident, that they are all thrown into right lines in the tube, being all parallel to its axis and as the parallels of latitude are all projected in circles perpendicular to the faid meridians; fo, upon opening the tube, &c. as aforefaid, they muft neceffarily become right lines alfo. The only thing therefore that requires a demonftration is, that the rhumbs or loxodromics become right lines, when the paper tube is extended as above. In order to this, let the eye be supposed to be placed in the center of the sphere when infcribed in the tube, then every rhumb will appear to run round the concave tube in the manner of a bottle fcrew in infinitum; and the only thing to be proved is, that it keeps a parallel direction to itself every-where; or, that it makes the fame angle with all the meridians; or, that the projected rhumb makes the fame angle with the projected meridian, as the true rhumb makes with the true meridian on the furface of the fphere. These two angles do apparently coincide, with regard to the eye placed as aforefaid; that is, they are apparently equal to the eye in that fituation; and that they are also really equal is evident from this lemma, viz. That the real and apparent bignets of any angle are the fame, when the eye is placed perpendicularly over either of its fides,

or

or when a perpendicular, dropt from the eye to the plane of the angle, falls upon either of its fides. Now this is the very cafe with regard to both the angles in queftion; for the perpendicular from the eye falls on the angular point of the angle on the sphere; and a perpendicular from the eye falls on the meridian, which is one fide of the angle on the tube: confequently, the real and apparent bignefs of each of those angles is the fame; and therefore, as they appear equal, they are really fo. Q. E. D.

SCOLIU M.

"It does not appear, that Mercator, or Wright, ever thought of this projection; for the meridian line here is manifeftly a line of tangents; whereas, in their projection, it is a collection of fecants. It may be added, that Mercator's or Wright's chart is very faulty in the bearing of places; but in this it is as true and correct as upon the globe itfelf. I fhall therefore prefume to fay, that this naval planifphere, or fea-chart, is the most useful for the purposes of Navigation ever invented; it being better than Mercator's in one important refpect, and equal to it in all others.

"There are three projections of the fphere, the orthographic, the ftereographic, and the nautical; the two first of these are well known to mathematicians: the laft was invented for the purposes of navigation, though hitherto a very imperfect and defective invention. The errors of the plain chart are corrected, in a great measure, by Mercator's or Wright's chart; though this latter is not a true projection of the fphere in any fhape; nor indeed is it pretended to be fuch by Mr. Wright, one of its inventors, who reprefents it rather to be an extenfion of the fpherical furface upon the inner fide of the concave cylinder in, which it is inclofed. Suppofe (e. g.) the globe to be fo infcribed in a cylindric tube, as to touch it every-where in the equator, and confequently the axes of the globe and cylinder to coincide; then fuppofe the tube to be of hard and unyielding fubftance, as of marble or the like, and the globe to be of a foft fubftance, as a bladder, and to enlarge itfelf as that does when blown, until the globular furface becomes a cylindrical one, by applying itself to the internal or concave furface of the cylinder, both ways towards cach pole; Mr. Wright fuppofes all the parts of the fpherical furface to increase uniformly in this extenfion; or, fo as that the degrees of longitude and latitude every-where fhall ftill continue to bear the fame juft proportion to each other, i. e. as radius to fecant of the latitude. Whereas, the true projection (and which, I apprehend, will much better

E 2

anfwer

anfwer the purposes of navigation than either the plain chart or Mr. Wright's) is this, viz. Let the fphere be inscribed in a cylindric tube, as above; and let all the parts of the fpheric furface be transferred to the concave cylindric furface, by right lines drawn from the center of the fphere: the confequence of which is, that, when the cylinder is opened and fpread upon a plane, the meridians, parallels, and loxodromics will be all projected in right lines, as in Mercator's or Wright's chart, but in different proportions. And I take upon me to affert, that this is the first chart, or reprefentation of the terraqueous globe, ever yet invented, in which the meridians, parallels, and rhumbs, are juftly and truly projected in right lines; for the latter cannot be fo projected in Mercator,"

This fpecimen of Mr. Weft's knowlege of mathematical fubjects, and of his manner of treating them, will, we doubt not, excite the curious to a perufal of the whole performance.

It is to this gentleman the world is indebted for feveral sensible and pious Difcourfes, published fome time ago, on the Lord's Prayer. See Review, Vol. XIX. p. 227.

MONTHLY CATALOGUE,

For JULY, 1762.

POLITICA L.

Art. 1. A Letter to the Right Honourable the Earl of E******t, on the prefent critical Situation of the British Ministry. 8vo. 1S. Nicoll.

OF

I

FFERS fome reasonable conjectures relating to the fatal war in Germany, the unpromifing war in Portugal, the monftrous increafe of the national debt, and the importance of our behaving with a proper firmnefs, whenever we come to fettle the terms for a future peace, If you would be great and glorious in your administration, fays he, to Lord E-t, fly the steps of the late Mr, and act according to your confcience. If our arms are attended with the greatest fuccefs, without the nation's reaping the benefit of it at a peace, your reputation, like his, will blaze like a meteor for a time, and then vanish for ever. In short, my Lord, Mr. P―'s maxim, in his negociation, was, to reflore; let yours be, LO`RETAIN.

3

Art.

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