ADVERTISEMENT

To all LOVERS of INGENUITY.

TH HE Author intending to publish the next Quarterly Book in JUNE enfuing, defires that fuch Gentlemen and Ladies, &c. as think fit to encourage this Undertaking, either by promoting and recommending the Sale thereof, or by fending fuch Subjects either in Verfe or Proje, as are Suitable to this Defign, would please to direct their Letters to Mr. THO, GENT, Printer, near the Star in StoneGate, YORK, Poft paid, before the End of May next, where they will be carefully forwarded to the Author.

Whoever fend either Enigmas, or Arithmetical Queftions, are defired to give the Solution along with them, to prevent all Miftakes, or Ambiguity in the Terms: And I do affure the Buyers of this Treatife, that, in Process of Time, they shall have Questions in all the Branches of Mathematicks, viz. Arithmetical, Nautical, Geographical, Mechanical, Trigonometrical, Aftronomical, Algebraical, and in Fluxions, together with exponential Equations, and the Conftruction of Algebraic Equations, and finding their Loci; to fuit the Taftes of all Perfons, and render the Work generally usefull.

Omne tulit punctum, qui mifcuit utile dulci.

Mifcellanea

[6]

Mifcellanea Curiofa, &C.

DEFINITIONS Preparatory to

MATHEMATICAL PHILOSOPHY.

C

ENTRIPETAL Force is that, whereby a Body is drawn, impell'd, or in fome Way or other tends to a Centre.

Or this Sort is Gravity, whereby a Body tends to the Centre of the Earth; the Force Magnetic, or that whereby Iron

tends to the Centre of the Loadstone; the attraction or stretching of the Cord to retain the Stone that is whirl'd round in a Sling. Hither alfo is to be refer'd that Force, whatfoever it is, whereby the Planets are continually held back from rectilineal Motions, and compell'd to revolve in Curvilinear ones. The Quantity of this Centripetal Force, is of three Sorts, viz. abfolute, accelerating, and moving.

THE abfolute Quantity of the Centripetal Force, is the Measure of the fame, greater or leffer, acco:ding to the Efficacy of the Central Caufe, which propagates it from the Centre all round about. Thus the Strength of Magnets is different, and greater ceteris paribus, in the greater Magnet, and leffer in the lefs. The attraction or tenfion of the Cord greater in the Circumvolution of a greater S one, than in that of a lefs; and in the fwifter Circumro ation of the fame Stone, than in a flower; and

thus we may conceive, that the Gravitation of Bodies to the Sun, which is a Body fo much greater than the Earth, is greater at an equal Diftance than the fame is towards the Earth.

THE accelerating Quantity of the Centripetal Force is the Measure of it in divers Distances from the famé Centre ; which is proportional to the Velocity which it produceth in a given Time.

As the Virtue of one and the fame Magnet (in which confequently the abfolute Quantity remains the fame) is greater in a lefs Distance than in a greater; the gravi tating Force on the Surface of the Earth, is fomething greater about the Poles than about the Equator. It is greater alfo near the Surface of the Earth, than at a greater Diftance from the Centre. But this accelerating Force (which is diftinctly to be noted) is at equal Diftances from the Centre every where the fame, and this in all Bodies, whether they be heavy or light, great or fmall, folid or fluid: That is to fay, if you do here abstract from the Refiftance of the Air. Which Thing is proved by the equally fwift Defcent of all falling Bodies in Tubes emptyed of Air; and from the Motion of all Pendulums, of what Matter or Magnitude foever, vibrating together in like Circles or Cycloids.

THE moving Quantity of the Centripetal Force, is the Measure of the fame, proportional to the Motion, which it generates, in a given Time.

THIS Force is the Propenfion of the whole Body towards the Centre, which is eftimated by the Quantity of the Force contrary thereto, which is required to the hindering its Defcent, which is called the Weight of the Body, and is greater in a Body which is greater; and greater in the fame body, by how much it is nearer to the Earth. The abfolute Quantity therefore, of the Force we are treating of, is defined from the Magnitude, or at least from the Strength and Efficacy of the Central Body. The Accelerating is that Force as perpetually decreafing in the Increase of the Diftance, and on the contrary. The moving Force is the Weight itfelf; which arifeth from the Body or Mafs drawn into the accelerating Force. From whence the abfolute Force being given, the moving Force in a given Body will be as the accelerating;

lerating; and the accelerating being given, it will be as the Body. These three Forces therefore are referr'd to three Things; to Bodies, to the Places of Bodies, and to the Centre of Force. The motive Force refpects the Body, and the Endeavour and Propenfion thereof to the Cen tre, as compounded of the Endeavours and Propenfions of all the Parts. The accelerating refers to the Place of the Body in the Medium, as the Efficacy of the fame abfolute Force according to divers Diftances, from the Centre; and the abfolute Force refpects the Centre, or Central Body itfelf, as endow'd with fome Power, without which the moving Forces are not propagated round about; whether that Power or Caufe be the Central Body (as the Magnet in the Centre of the Magnetic Force, or the Earth in the Centre of the gravitating Force) or be fome other Thing which doth not appear. At leaff, this is a Mathematical Conception, and fufficient for our prefent Purpofe; for we do not yet confider the Phyfical Caufes of thofe Forces. The accelerating Force therefore is to the Motive, as Swiftnefs is to Motion; for from the fame, as multiply'd into the Quantity of the fame Matter, the Moving Force arifes; like as the Quantity of Motion arifes from the Celerity multiply'd into the Body. For the Sum of the Actions of the accelerating Force upon each Particle of the Body, is the moving Force of the whole; from whence where the accelerating Gravity is the fame, the moving Gravity, or Weight, is as the Body. And in the fame Body where the Acceleration is diminished, as in the Upper Regions, the Weight is likewife diminished. Thus where the accelerative Force is twofold lefs, the Weight of a Body twofold or threefold lefs, will become fourfold or fixfold lefs. Furthermore, we call Impulfes and Attractions accelerative and motive in the fame Senfe. And we ufe the Words [attraction, impulfe, and propenfion to a Centre,] indifferently and promifcuoufly; we at prefent confidering thefe Forces not Phyfically but Mathematically, as was faid before, and now fay again, to caution our Readers from underftanding us, as meaning and defining Phyfical Caufes or Reafons of Motions, or attributing to Centres, which are Mathematical Points, true and proper Phyfical Force; when at any Time we fay that the Centres draw, or have Force in them.

Axioms

Axioms or Laws of Motion.

1. EVERY Body perfeveres in its own prefent State,

whether it be that of Reft, or uniform direct Motion; unless it be compelled by fome Force imprefs'd, to change that State.

Projectiles hold on their Motion, fo far as they are not hinder'd by the Refiftance of the Air, or their own Gravity. A Top, whofe Parts by cohering continually draw themselves from the rectilinear Motion, ceafeth not to be whirl'd about, fo far as it is not retarded by the Air, or the unevennefs of the Surface, on which it turns. But the greater Bodies of Planets and Comets maintain their Motions, whether Progreffive or Circular, much longer in Spaces less refifting. This Law of Motion is indeed the fundamental Law of all, and is moft evident from the merely Paffive Nature of Matter; which makes it naturally as impoffible for a Body of it felf to stop its own Motion once begun, as it is for it to move itself originally.

(2.) All Motion is of itself rectilinear.

For Motion cannot be conceived, but it must be direct ed and determin'd towards fome Place or other, and it will by the Law foregoing keep the fame Direction which it firft had, until it be hinder'd or put out of its Way by lome extrinfic Cause.

And confequently, whenever any Body is mov'd in a Curve, that Curvature muft needs proceed from External Force; and therefore muft ceafe when that Force ceafeth: Which, when it doth, then by this and the foregoing Law, the Motion will be continued in a right Line, which is the Tangent of the Curve, at the very Point of the faid ceafing Force, or in the laft Rectilineal Direction. Thus it is in a Stone wheel'd about in a Sling, which flipping out of the Sling is not now carried forward in its former Circle or any Circle at all, but in a' Tangent of the former Circle; where indeed, by Reafon of the Force of Gravity, compounded with the projectile Force, it defcribés a Patabolic Line: But of this afterwards.