⚫ bumble Converfation. And may a great Bleffing of God crown our • Labours! Let us go on and the Lord profper us. And again, The unholy Teacher, let him preach ever fo well, difcourfeth to little Purpose; ⚫ there will be no Life in his Doctrine, because his Life is fo deftitute of ⚫the Spirit of Holiness; he will fooner damn his own Soul than fave any Man's elfe. His Difcourfes, tho' arm'd with the most powerful Oratory, ⚫ will ferve to move no other Affection in his Hearers, than that of Indignation against his Hypocrify and Impudence, to hear him excellently declaim against a Vice, of which himself is notoriously guilty.

adly, The most excellent Prelate, the late Bishop of Salisbury.] The • Manners and the Labours of the Clergy, thefe are real Arguments which all People do both understand and feel; they have a much more con<« vincing Force, they are more vifibie and perjuhde more universally than Books can do, which are little read and lefs confidered. And indeed, ⚫ the Bulk of Mankind is fo made, that there is no Working on them, but by moving their Affections, and commanding their Esteem.

in Devotion, and Diligence in the Paftoral Care, are fallen under too vifible and too fcandalous a Decay. And whereas the Understanding the • Scriptures, and an Application to that facred Study was at first the dif ⚫tinguishing Character of Proteftants, for which they were generally nicknamed Gofpellers; thefe Holy Writings are now fo little ftudied, that fuch as are obliged to look narrowly into the Matter, find great Cause of • Regret and Lamentation, from the grofs Ignorance of fuch as are either in • Orders, or that pretend to be put in them. While Men imagine that ⚫ their whole Work confifts in publick Functions, they reckon that if they ⚫ either do these themselves, or procure and hire another person in Holy Orders to do there, that then they answer the Obligation that lies on them. And the Paftoral Care, the Inftructing, the Exhorting, the Admonishing, and Reproving, the Directing, and Conducting, the Vifiting and Comforting the People of the Parish, is generally neglected; while the Incumbent does not think fit to look after it, and the Curate thinks • himself bound to nothing, but barely to perform Offices, accord⚫ing to Agreement. And again, But what can we say, when we find soften the pooreft Clerks in the richest Livings? whofe Incumbents, not content to devour the Patrimony of the Church, while they feed themfelves, and not the Flock, out of it, are fo fcandaloufly hard in their Allowance to their Curates, as if they intended equally to farve both Cus rate and People." Your Conftant Reader, R. D.

An Answer to a Note in a Book of Mr. RUTHERFORTH'S, concerning the Cafe of ABRAHAM's offering up ISAAC; in Defence of Mr Warburton,

I

H. G. London, August 10, 1744.

'N a late Book of the Learned and Ingenious Mr Rutherforth's, entitled, An Effay on the Nature and Obligation of Virtue, I find, in a Note, in p. 314, 317, fome Exceptions against understanding the Command which God gave to Abraham to facrifice his Son to have been a Revelation

to him of the Sacrifice of Chrift for the Redemption of Mankind; and fhall attempt to confute them as briefly as I can.

Mr Warburton, whofe Interpretation "the mentioned is of this famous Command, begins his Proof of it from thefe Words of our Saviour's spoken to the Jews; your Father Abraham rejoiced to fee my Day, and he faw it and was glad. He takes it for granted, and Mr. R. proves, elaborately, that the Jews understood our Saviour as if he had affirm'd, that Abrabam and he were Cotemporaries, and Abraham had beheld him in perfon; which contains Mr W's Affertion, that they understood him to speak of feeing in the literal Senfe, and not figuratively. And there is very little Force in Mr R's Attempt to evade the obvious Confequence from hence. For there is no Reasons to affert the Jews wilfully perverted our Saviour's Meaning with design to have a pretence for ftoning him, but it is a mere groundless, not to fay moft improbable, Fancy; and though they were always captious in their Difcourfes with him, that has never been efteem'd an Objection of any Weight against the very common Argument, in plain Cafes, from their Apprehenfions of it, to his real Meaning. It was, in all probability, because his Words expreffed definitely, and without any Ambiguity, feeing properly and fenfibly, either exprefly, or by obvious Confequence, that they pafled the Conftruction they did on the other part of his Affertion, in which, indeed, not being duly qualified, and for want of Ears to hear, they were likely enough to misapprehend him.

It is very ftrange in Mr R. to affirm, univerfally, that a Man may not be faid to have feen that of which he has feen a Representation, whether the Yokes and Bonds by which a Prophet express'd a certain Captivity, might take the Name of that which they were defigned to make a ftrong Impreffion of, or no (which 'tis needlefs to difpute, or confider), fuch a Picture, on Refemblance, nevertheless, as Ifaac's Sacrifice was of our Saviour's might very well be filed it, or his Day, and with as much Propriety, for Inftance, as the Memorial of the Lord's Paffover might be called his Paffover; and Abraham's seeing the facrifice of his Son, be faid to be feeing that which it fignified to him, as they to eat of the Paffover, who partook of the Pafchal Lamb; or the Ifraelites in the Wilderness to drink of a certain Rock, which Rock was Chrift. And this certainly was our Saviour's meaning; that Abraham had feen a fenfible Representation of his Day; because the Word de ufed by the Evangelift, whether taken literally, or figuratively, always denotes a full Intuition; which was what alone, Abraham could have had of that which happened Ages after him. There is no Neceffity, nor Reafon, for expounding this Word by Believing. as Mr R. does, in this Text; these all died in Faith, not having received the Promifes, but having feen them afar off, and were perfuaded of them. The perfons here spoken of, law thofe Types, which reprefented promifed Bleffings (which, in our Times, have been accomplish'd, and fo we have received) and apprehended their whole Nature, and both primary and fecondary End; and believed, and did not doubt but that what they forefhewed would certainly come to pafs; But having Jeen them afar off, and were perfuaded of them; firft they faw and then believed; the one of thefe Acts (I mean in a Vulgar fenfe) was not the other, but laid the Foundation for, and propounded the Object of it.

Mr R. owns that our Saviour affirms his Redemption of Mankind was revealed to Abraham. And Mr W. has prov'd, not only that our Saviour

would

4

would not have appealed to it on the Occafion, and to the Perfons he did, if it had not been discoverable by them in the Jewish Scriptures; but that certainly fo important a Fact would not be omitted in them; after which he fhews that no other part of Abraham's History can admit of this Interpretation, but the Command in question. Which determines the Dif pute between Mr W. and Mr A. (to fay nothing of the Appendix to a Late Examination of Mr W's fecond Propofition in D.L.) without infifting on the fenfe of the Word de fo fully vindicated.

But whereas in the Second part of his Differtation Mr W. proves from our Saviour's Words, feveral Circumftances relating to this Tranfaction of Offering up Ifaac, Mr R. enquires, which of all these particulars could be found in the Hiftory of God's Command to Abraham to facrifice him, related in fuch a Manner that fuch Jews as ftudied their Scriptures might be well acquainted with the Facts and Circumftances to which our Saviour appeals in them? In which he,' certainly, betrays great Inattention to Mr W's Differtation, and the Courfe of his Argument. The Circumitance which Mr W. infifts on, and which alone his Argument requires to be contained in Mofes's Hiftory, is this; that Abraham faw a sensible Representation of Chrift's Sacrifice for the Redemption and Resurrection of Mankind; nor is it any where affirm'd by him, or neceffary to be, that the Jews might difcover this Circumftance, precifely and completely, in the Command to offer up Ifaac, before the Doctrine of Christ's fuffering for the Sins of Mankind did, or might well explain it to them; and which both had been, with fufficient plainnefs, preached to thofe Jews, whom our Saviour told that their great Ancestor had seen his Day, before he did, and especially was very foon after. And therefore they might well difcover it therein; for Mr W. by the foundest Rules of Logic and Criticism, has proved it to be: Has prov'd, that the Hiftory of this Command, fo circumftanc'd as it is, demands this Interpretation. The Application of this to the Argument, with which Mr R. concludes his Attack, is very obvious; and it appears, at firft fight that Mofes might, as industriously, as Mr W. any where contends, conceal from the Jews, the Knowledge of a future State; and yet thofe Jews our Saviour difcourfed with, perceive that Abraham's offering up his Son reprefented to him Chrift's Sacrifice for the Redemption of Mankind to eternal Happiness. And to both Mr W's Premiffes in proof of his Interpretation of this Cafe, and his Conclufion are very fafe, notwithftanding this threatning Dilemma, and all that Mr R. has advanced to the contrary. As, I hope, he will be very fenfible, and own with an Ingenuity worthy his Character, efpecially fince he has told us he might give this as an Inftance of the Patriarch's Knowledge of a future State; and feems to apprehend nothing to lie in his Way to doing it, but thofe Difficulties which I cannot but think, now, are very manifeitly and totally removed.

The Curve which the Moon, or any other SATELLITE, defcribes about the SUN, determined, on all poffible Suppofitions of DISTANCE and VELOCITY.

T

HAT the Moon's Path is a protracted Epicycloid, as was afferted in my last, needs no Demonftration: it following immediately from

See Gent. Mag. Vol. 13. p. 592.

the.

the Genefis of that Curve, and from the Periodical Motions of the Earth and Moon. Dr 'sGravefande, particularly, had confidered the Paths of the Earth and Moon in this Light, long before your Correfpondents had moved any Question on the Subject; for in his Inftitut. Philof. Neut. Tab. XVII. Fig. 2. the Curve which he delineates has the Form of an Epicycloid partly convex towards the Sun. It is true, indeed, this learned Author has here committed an Overfight, facing, beyond certain Limits, the pretrated Epicyeloid becomes every where concave towards the Centre of its Bafe, as fhall be demonftrated below. A Miftake which he was probably led into by a mechanical Conftruction that did not nearly enough exhibit the true Distances of the Earth and Moon from the Sun. I followed him, and unluckily drew after me Mr R. Yate; for which he more juftly owes me a Grudge than for Stealing his + Trencher and Bread. However, as foon as I had inveftigated a Theorem for finding the Point of Contrary Flexion in thefe Curves, the Error was corrected of Course: All is let to rights again, and the Moon reftored to her Path every where (tho' not equally) concave towards the Centre.

The Refult of my Enquiry was fhortly this. That if n reprefent the Number of Synodical Months in a Year, "x 1 (d) the Distance of the Earth from the Sun, and m the Distance of the Moon from the Earth & then will the Epicycloid have a Point of Contrary Flexion, or not, as d x 12 is greater or less than Unity.

Whence, if, in the Cafe of our Moon, we put 12, and the Distances of the Sun and Moon from the Earth to be as 327 to 1, m will be .0413, and d x m less than Unity, that is, there will be no Point of contrary Flexion. But if n, or m, or both, are any how increased till dy mi is greater than Unity, then will the Moon's Path be partly concave, partly convex. As if m= , that is, if other things remaining, the Moon was removed to the Distance of a little more than 121 Semediameters of the Earth, her Path would be convex for fome time before and after the ConJunction ; but if m was again diminished to the Convexity would va

nifh, and the Radius of Curvature in the Moment of Conjunction become infinite. And, by the Way, we fee how eafily 'sGravesande and others might be impofed on by a mechanical Defcription, if they happened not to fix the Pencil or defcribing Point very near the Centre of the Circle.

DEMONSTRATION.

Let the Circle AES whofe Centre is R, be the Bafe upon which a Cirele CAPQ whofe Centre is C, revolving defcribes with the point T the Curve TtH. Let the revolving Circle, from its firft Pofition, when TCR were in a right Line, come into the Pofition cEaq, and the defcribing Point T into the Place t. From t, through the Point of Contact E draw tEB meeting the Axis in B; join likewife CR which will pals thro' the fame point E; and draw GF, a Tangent to the Epicycloid in t, cutting CR in F.

Then because the Curve TtH is defcribed by the continual Rotation of Et about the Points of contact E, the Angle FtB is right; and if t is a Point of contrary Flexion, the Fluxion of the Angle tBF vanishes. Further, cE being to ct in a conftant ratio (as of 1 to m) the Sines of the Angles

† See Gent. Mag. Vel, 13. p. 471, 639,

Point

c tE, cEt will be in the same ratio; and the Sum of thefe Angles, or the Angle acE, will be in a given ratio to the Angle ARE, namely in the ratio of RA to CA (or of n to 1). Writing therefore for the Sines of the Angels ctE, cEt, to the Radius

and for

Unity, the Letters z, y,
the feveral Angles mentioned, the
Letters that ftand at their angu-
lar Points, we fhall have these
three Equations.

1. y=mz
2. tx EnR

3. B=EX R

Take the Fluxions, and because of the contrary Flexion, put B, that is ER; fubftitute likewife for t, E, their equals

H

9

G

by an easy Reduction it will be S

a

COROLLARIES:

B

1. The Sine z is found by Equ. 1. And the Sum of the Angles to which the Sines z, y belong, i. e. the Angle acE, is the Motion of the Satellite in its proper Orbit, from the Time that it was in Opposition at T; as ARE (= X acE) is the Motion of the Planet for the fame Time.

2. If m=1,y=1. That is, tEB touches both Circles in E, where two Points of contrary Flexion meeting are changed into a Point of Reflexion. If mo, the Curve is a Circle with the Semidiameter RC. If m is greater than Unity, it becomes an Epicycloid contracted with Nodes. And if dis infinite, the Base being now a strait Line, there is generated the protracted, contracted, or common Cycloid, according as m is teffer or greater than Unity, or equal to it: and in the protracted, the Sine of cEt.

It is obvious that the Path of the Earth is of the fame Species with that of the Moon, only less deviating from the Ecliptic Circle defcribed by their common Centre of Gravity. And that the fame Theory may be applied to the Satellites of Jupiter and Saturn. The Periodic Time of a Satellite compared with that of its Primary will give the quantity #, and its fartheft Heliocentric Elongation from the Primary will determine m; whence it may be readily feen to what fpecies the Epicycloid of that Satel lite belongs, after which the Questions concerning it will be purely Geometrical. I remain, Sir, &c. X. Y. Jan. 19, 1743-4.