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shat--the feed is always reckoned by the males, and rot the females of a family, and (that) the name in a father's house could only be preserved by the male descendants.' in which respect Jesus Christ was not a son of David: there. fore this connexion is destroyed.

“ The doctrines of chriftianity, especially the preceptive ones, taken from his own mouth, cannot to all appearance be aifected by insisting on the character of David, whether good or bad; right forry would this author be if they rested on any such dependance. The divine tendency of these doctrines is so manifest; they are so refined in their own nature, that they will ever clain the most exalted reverence from mankind, to them and to their glorious publisher, independant on all foreign contingencies, on all former narrow tenets, and superstitious rites; which he has so totally fuper-feded among his professors : and that they ever may is the fincere prayer of this abused writer. He therefore hopes ani acquittal from this part of his indictment.”

How far these profeßions are sincere, it would not become us to determine; but we cannot help dissenting from this writer's opinion, that christianity cannot be affected by such controverfies as this, concerning the character of king David; as we are firmly persuaded, that the connection between the Old and New Testament is so inseparable, that whatever tends to weaken the authority of the one, must 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 Mould be glad to see them more usefully employed.

We are not sorry, therefore, to find him concluding with the following sensible declaration, viz. “ Religious contest is moft difagreeable of all other : since it has ever been leaft decisive; and most incentive of those propersities which the genuine dictates of pure religion directly tend to suppress. The writer, therefore, at the fame time that he drops this controversy, has no present intention ever more to engage in any of the like nature.”

Before we entirely close this article, it may not be improper to mention the Author's portfcript, in which he has treated Dr. Patten's miserable vindication of David with becoming spirit and contempt ; at the same time that he does justice to the candour of Mr. Harwood, author of the Conversion of a Deift: in the appendix to which, Mr. Harwood had inserted some Remarks on the History of the Man after God's own Heart.

Marbematics.

TH

Mathematics. With Eleven Copper-plates. By tắe latë Řeta

Mr. William West of Exeter. Revised by John Rowe. 35. in boards. Richardfon.

HIS small manual, of which every page hews the band

of a master, is a posthumous work; and is published for the benefit of the Author's widow, by the ingenious Mr. Rowe, who some years since obliged the world with An Introduction to the Doctrine of Fluxions * The Editor informs us, that the present publication was selected from several mathematical

papers, written at different times, few of which were finished, and none perhaps ever intended for public inspection.

The first five pages contain a short introduction to the fuxionary calculus; the following thirty-eight, respecting the application of that doctrine to the solution of problems de maximis et minimis ; in which Mr. West has rendered that useful and difficult branch of the mathematics plainer, and more easy to be understood, than we remember to have seen it done by any other Writer ; and, at the fame time, has demonstrated, in a very elegant manner, the common method of makisig the Auxion of a maximum or minimum

= 0.

The other part of this frall performance confifts of tweuty-four miscellaneous questions, with their solutions : feveral of these are very curious, and the solutions neat 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 so interesting a nature, that it deserves particular notice. It is a new method of constructing a fca-charts 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 sphere will give true solutions to all nautical problems, but that only is properly adapted to the mariner's use, where the rhumb, or the path a ship defcribes in failing; according to the direction of the compaís, is represented by a strait lime, because there will then be no difficulty in determining exactly the bearing of any two places, or the true course that a Thip must fail from one to the other.

• See Review, Vol. XIX. p. 528. Rev. July, 1759.

E

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 compass; but was never brought to any degree of perfection, till performed by our ingenious countryman, Mr. Edward Wright : whose invention will render his name famous to all posterity.

This projection Mr. Weft has now perfected by the fole lowing proposition, 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 sphere be inscribed therein, so as that the axeś of the fphere and cylinder coincide, or, that the equator be the line of contact between the said tube and sphere, and all the points of the spheric furface be projected or tranferred to the concave surface of the tube, by right lines proceeding from the center of the sphere, and terminating in the faid concave surface of the tube : and then, if the paper be opened and stretched upon a plane, it will pre

sent a chart, in which the meridians, parallels of latitude, and rhumbs are all truly and geometrically projeted in right lines. Quære the Demonstration ?

DEMONSTRATION. “ With regard to the meridians, it is evident, that they are all thrown into right lines in the tube, being all paralel to its axis : and as the parallels of latitude are all projected in circles perpendicular to the faid meridians; so, upon opening the tube, &c. as aforesaid, they must necessarily become right lines also. 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 inscribed 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 same angle with the projected meridian, as the true rhumb makes with the true meridian on the surface of the sphere. These two angles do apparently coincide, with regard to the eye placed as aforesaid ; that is, they are apparently equal to the eye in that situation; and that they are also really equal is evident from this lemma, viz. That the real and apparent bignels of any angle are the fame, When the eye is placed perpendicularly over either of its fides,

or

or when a perpendicular, drope from the eye to the plane of the angle, falls upon either of its fides. Now this is the very case with regard to both the angles in question ; 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 lide of the angle on the tube: consequently, the real and apparent bigness of each of those angles is the same ; and therefore, as they appear equal, they are really so. Q. E. D.

SCOLIUM. “ It does not appear, that Mercator, or Wright, evet thought of this pro cetion; for the meridian line here is ma nifestly a line of tangents ; whereas, in their projection, it is a collection of secants. 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 itself. I shall therefore presume to say, that this naval planisphere, or sea-chart, is the most useful for the purposes of Navigation ever invented; it being better than Mercator's in one ima portant respect, and equal to it in all others.

“ There are three projections of the sphere, the orthographic, the stereographic, and the nautical; the two first of these are well known to mathematicians : the last was in-, vented 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 sphere in any shape; nor indeed is it pretended to be such by Mr. Wright, one of its inventors, who represents it rather to be an extension of the spherical furface upon the inner side of the concave Cylinder in which it is inclofed. Suppose (e.g.) the globe to be lo inscribed in a cylindrie tube, as to touch it every-where in the equator, and consequently the axes of the globe and cylinder to coincide; then fuppose the tube to be of hard and unvielding substance, as of marble or the like, and the globe to be of a fost substance, as a bladder, and to enlarge itself as that does when blown, until the globular surface becomes a cylindrical one, by applying itself to the internal or concave surface of the cylinder, both ways towards cach pole; Mr. Wright supposes all the parts of the spherical surface to increase uniformly in this extention; or, so as that the degrees of longitude and latitude every-where fhall Aill continue to bear the fame just proportion to each other, i.e. as radius to secant of the latitude. Whereas, the trtie projection (and which, I apprehend, will much better answer the purposes of navigation than either the plain chart or Mr. Wright's) is this, viz. Let the sphere be inscribed in a cylindric tube, as above ; and let all the parts of the spheric furface be transferred to the concave cylindric surface, by right lines drawn from the center of the sphere: 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 assert, that this is the first chart, or representation of the terraqueous globe, ever yet invented, in which che meridians, parallels, and rhumbs, are justly and truly projected in right lines ; for the latter cannot be fo projected in Mercator.”

answer

En 2

This specimen of Mr. West's * knowlege of mathematical subjects, and of his manner of treating them, will, we doubt not, excite the curious to a perusal of the whole performance.

Te is to this gentleman the world is indebted for several sensible and pious Discourses, published some time ago, on the Lord's Prayer See Review, Vol. XIX. p. 227.

MONTHLY CATALOGUE,

For JULY,

1762.

POLITICAL.

Art. 1. A Letter to the Right Honourable the Earl of

E******t, on the present critical Situation of the British Ministry. 8vo. I S.

Nicoll. O FFERS some reasonable conjectures relating to the fatal war

in Germany, the uripromising war in Portugal, the monftrous increase of the national debt, and the importance of our behaving with a proper firmness, whenever we come to settle the terms for a future peace. If you would be great and glorious in your adminiftration, says he, to Lord E-t, fly the steps of the late Mr, and act according to your conscience. If our arms are attended with the greatest success, 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 hort, my Lord, Mr. P-'s maxim, in his negocia:ion, was, to relore; let yours bc, Lo RETAIN.

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