A Discourse Concernimg Gravity, and Its Properties, Wherein the Descent of Heavy Bodies, and the Motion of Projects is briefly, but Fully Handled: Together with the Solution of a Problem of Great Use in Gunnery. By E. Halley

A DISCOURSE Concerning GRAVITY, and its Properties, wherein the Descent of Heavy Bodies, and the Motion of Projects is briefly, but fully handled: Together with the Solution of a Problem of great Use in GUNNERY. By E. HALLEY. NATURE amidst the great variety of Problems wherewith She exercises the Wits of Philosophical men, scarce affords any one wherein the Effect is more visible, and the Cause more concealed than in those of the Phænomena of Gravity. Before we can go alone, we must learn to defend ourselves from the violence of its Impulse, by not trusting the Center of Gravity of our Bodies beyond our reach; and yet the Acuteest Philosophers, and the subtilest Enquirers into the Original of this Motion, have been so far from satisfying their Readers, that they themselves seem little to have understood the Consequences of their own Hypotheses. Des Cartes his Notion, I must needs confess to be to me Incomprehensible, while he will have the Particles of his Celestial matter, by being reflected on the Surface of the Earth, and so ascending therefrom, to drive down into their places those Terrestrial Bodies they find above them: This is as near as I can gather the scope of the 20, 21, 22, and 23 Sections of the last Book of his Principia Philosophiae; yet neither he, nor any of his Followers can shew how a Body suspended in libero æthere, shall be carried downwards by a conti- continual Impulse tending upwards, and acting upon all its parts equally: And besides the obscurity wherewith he expresses himself particularly, Sect. 23. does sufficiently argue according to his own Rules, the confused Idea he had of the thing he wrote. Others, and among them Dr. Vossius asserts the Cause of the Descent of heavy Bodies to be the Diurnal rotation of the Earth upon its Axis, without considering, that according to the Doctrine of Motion fortified with Demonstration, all Bodies moved in Circulo, would recede from the Center of their Motion; whereby the contrary to Gravity would follow, and all loose Bodies would be cast into the Air in a Tangent to the Parallel of Latitude, without the intervention of some other Principle to keep them fast, such as is that of Gravity. Besides the effect of this Principle is throughout the whole Surface of the Glob found nearly equal, and certain Experiment seems to argue it rather less near the Equinoctial, than towards the Poles, which could not be by any means, if the Diurnal rotation of the Earth upon its Axis were the cause of Gravity, for where the Motion was swiftest, the Effect would be most considerable. Others assign the Pressure of the Atmosphere, to be the Cause of this Tendency towards the Center of the Earth; but unhappily they have mistaken the Cause for the Effect, it being from undoubted Principles plain, that the Atmosphere has no other Pressure but what it derives from its Gravity, and that the Weight of the upper parts of the Air, pressing on the lower parts thereof, do so far bend the Springs of that Elastick Body, as to give it a force equal to the Weight that Compressed it, having of itself no force at all: And supposing it had, it will be very hard to explain the Modus, how that Pressure should occasion the Descent of a Body circumscribed by it, and pressed equally above and below, without some other force to draw, or thrust it downwards. But to demonstrate the contrary of this Opinion, an Experiment was long since shewn before the Royal Society, whereby it appeared peared that the Atmosphere was so far from being the Cause of Gravity, that the Effects thereof were much more Vigorous where the pressure of the Atmosphere was taken off; for a long Glass-Receiver having a light Down-feather included, being evacuated of Air, the Feather which in the Air would hardly sink, did in vacuo descend with nearly the same Velocity as if it had been a Stone. Some think to Illustrate this Descent of Heavy Bodies, by comparing it with the Vertue of the Loadstone; but setting aside the difference there is in the manner of their Attractions, the Loadstone drawing only in and about its Poles, and the Earth near equally in all parts of its Surface, this Comparison avails no more than to explain ignotum per aqua ignotum. Others assign a certain Sympathetical attraction between the Earth and its Parts, whereby they have, as it were, a desire to be united, to be the Cause we enquire after: But this is so far from explaining the Modus, that it is little more; than to tell us in other terms, that heavy bodies descend, because they descend. This I say, not that I can pretend to substitute any Solution of this Important Philosophical Problem, that shall more happily explicate the Appearances of Gravity, only it may be serviceable to those with whom the Credit of great Authors sways much, and who too-readily assent in verba magistri, to let them see that their Books are not always infallible: Besides the detection of Errors is the first and surest step towards the discovery of Truth. Tho' the Efficient Cause of Gravity be so obscure, yet the final Cause thereof is clear enough; for it is by this single Principle that the Earth and all the Celestial Bodies are kept from dissolution: the least of their Particles not being suffered to recede far from their Surfaces, without being immediately brought down again by vertue of this Natural tendency, which for their Preservation, the Infinite Wisdom of their Creator has Ordained to be towards each of their Centers; nor can the *Globes* of the *Sun* and *Planets* otherwise be destroyed, but by taking from them this power of keeping their parts united. The Affections or Properties of *Gravity*, and its manner of acting upon *Bodies falling*, have been in a great measure discovered, and most of them made out by *Mathematical demonstration* in this our *Century*, by the accurate diligence of *Galileus, Torricellius, Hugenius*, and others, and now lately by our worthy Country-man Mr. *Isaac Newton*, (who has an incomparable *Treatise of Motion* almost ready for the *Press*) which Properties it may be very material here to enumerate, that they may serve for a Foundation to all those that shall be willing to spend their Thoughts in search of the true Cause of this *descent* of *Bodies*. The first Property is, That by this principle of *Gravitation*, all Bodies do descend towards a Point, which either is, or else is very near to the Center of *Magnitude* of the *Earth* and *Sea*, about which the *Sea* forms itself exactly into a *Spherical surface*, and the *Prominences* of the *Land*, considering the Bulk of the whole, differ but insensibly therefrom. Secondly, That this Point or *Center of Gravitation*, is fixt within the *Earth*, or at least has been so, ever since we have any *Authentick History*: For a Consequence of its Change, tho' never so little, would be the over-flowing of the low Lands on that side of the *Globe* towards which it approached, and the leaving new Islands bare on the opposite side, from which it receded; but for this Two thousand years it appears, that the low *Islands* of the *Mediterranean Sea* (near to which the ancientest Writers lived) have continued much at the same height above the Water, as they now are found; and no *Inundations* or *Recesses* of the *Sea* arguing any such Change, are Recorded in History; excepting the *Universal Deluge*, which can no better way be accounted for, than by supposing this *Center of Gravitation* removed for a time, towards the middle of the then inhabited parts of the *World*; and a change of its place, but the two thousandth part of the Radius of this Globe, were sufficient to bury the Tops of the highest Hills under water. Thirdly, That in all parts of the Surface of the Earth, or rather in all Points equidistant from its Center, the force of Gravity is nearly equal; so that the length of the Pendulum vibrating seconds of time, is found in all parts of the World to be very near the same. 'Tis true at S. Helena in the Latitude of 16 Degrees South, I found that the Pendulum of my Clock which vibrated seconds, needed to be made shorter than it had been in England by a very sensible space, (but which at that time I neglected to observe accurately) before it would keep time; and since the like Observations has been made by the French Observers near the Equinoctial: Yet I dare not affirm that in mine it proceeded from any other Cause, than the great height of my place of Observation above the Surface of the Sea, whereby the Gravity being diminished, the length of the Pendulum vibrating seconds, is proportionably shortened. Fourthly, That Gravity does equally affect all Bodies, without regard either to their matter, bulk, or figure; so that the Impediment of the Medium being removed, the most compact and most loose, the greatest and smallest Bodies would descend the same spaces in equal times; the truth whereof will appear from the Experiment I before cited. In these two last particulars, is shewn the great difference between Gravity and Magnetism, the one affecting only Iron, and that towards its Poles, the other all Bodies alike in every part. As a Corollary; from hence it will follow, that there is no such thing as positive levity, those things that appear light, being only comparatively so; and whereas several things rise and swim in fluids, 'tis because bulk for bulk, they are not so heavy as those fluids; nor is there any reason why Cork, for instance, should be said to be light because it swims on Water, any more than Iron because it swims on Mercury. Fifthly, That this power encreases as you descend, and de- decreases as you ascend from the Center, and that in the proportion of the Squares of the distances therefrom reciprocally, so as at a double distance to have but a quarter of the force; this property is the principle on which Mr. Newton has made out all the Phenomena of the Celestial Motions, so easily and naturally that its truth is past dispute. Besides that, it is highly rational, that the attractive or gravitating power should exert itself more vigirously in a small Sphere, and weaker in a greater, in proportion as it is contracted or expanded, and if so, seeing that the surfaces of Spheres are as the Squares of their Radii, this power at several distances will be as the Squares of those distances Reciprocally, and then its whole action upon each Spherical Surface, be it great or small will be always equal. And this is evidently the rule of Gravitation towards the Centers of the Sun, Jupiter, Saturn and the Earth, and thence is reasonably inferred, to be the general principle observed by Nature, in all the rest of the Celestial Bodies. These are the principal affections of Gravity, from which the rules of the fall of Bodies, and the motion of Projects are Mathematically deducible. Mr. Isaac Newton has shewed how to define the spaces of the descent of a Body, let fall from any given height, down to the Center. Supposing the Gravitation to increase, as in the fifth Property; but considering the smallness of height, to which any Project can be made ascend, and over how little an Arch of the Globe it can be cast by any of our Engines, we may well enough suppose the Gravity equal throughout, and the descents of Projects in parallel lines, which in truth are towards the Center, the difference being so small as by no means to be discovered in Practice. The Opposition of the Air, 'tis true, is considerable against all light bodies moving through it, as likewise against small ones (of which more hereafter) but in great and ponderous Shot, this Impediment is found by Experience but very small, and may safely be neglected. Propositions concerning the Descent of heavy Bodies, and the Motion of Projects. Prop. I. The Velocities of falling Bodies, are proportionate to the times from the beginning of their falls. This follows, for that the action of Gravity being continual, in every space of time, the falling Body receives a new impulse, equal to what it had before, in the same space of time, received from the same power: For instance, in the first second of time, the falling Body has acquired a Velocity, which in that time would carry it to a certain distance, suppose 32 foot, and were there no new force, would descend at that rate with an equable Motion; but in the next second of time, the same power of Gravity continually acting thereon, superadds a new Velocity equal to the former; so that at the end of two seconds, the Velocity is double to what it was at the end of the first, and after the same manner may it be proved to be triple, at the end of the third second, and so on. Wherefore the Velocities of falling Bodies, are proportionate to the times of their falls, Q. E. D. Prop. II. The Spaces described by the fall of a Body, are as the Squares of the times, from the beginning of the Fall. Demonstration. Let A B (Fig.1. Tab.1.) represent the time of the fall of a Body, BC perpendicular to A B the Velocity acquired at the end of the fall, and draw the line AC, then divide the line A B representing the time into as many equal parts as you please, as b, b, b, b, &c. and through these points draw the lines bc, bc, bc, bc, &c. parallel to BC, 'tis manifest that the several lines, bc, represent the several Velocities of the falling Body, in such parts of the time as A b is of A B, by the former proposition. It is evident likewise that the Area A B C is the sum of all the lines bc being taken, according to the method of Indivisibles, infinitely many; so that the the Area ABC represents the sum of all the Velocities between none and BC supposed infinitely many; which sum is the space descended in the time represented by AB. And by the same reason the Areas A b c, will represent the spaces descended in the times A b; so then the spaces descended in the times AB, Ab, are as the Areas of the Triangles ABC, A b c, which by the 20th of the 6 of Euclid are as the Squares of their Homologous sides A B, A b, that is to say, of the Times: wherefore the descents of falling Bodies, are as the Squares of the times of their fall, Q. E. D. Prop. III. The Velocity which a falling Body acquires in any space of time, is double to that, wherewith it would have moved the space descended by an equable motion, in the same time. Demonstration, Draw the line EC parallel to AB and AE parallel to BC in the same fig. 1. and compleat the Parallelogram ABC E, it is evident that the Area thereof may represent the space, a Body moved equably with the Velocity BC, would describe in the time AB, and the Triangle ABC represents the space described by the fall of a Body, in the same time AB, by the second proposition. Now the Triangle ABC is half of the Parallelogram ABC E, and consequently the space described by the fall, is half what would have been described by an equable Motion with the Velocity BC, in the same time; wherefore the Velocity BC at the end of the fall, is double to that Velocity, which in the time AB, would have described the space fallen, represented by the Triangle ABC, with an equable Motion, Q. E. D. Prop. IV. All Bodies on or near the surface of the Earth, in their fall, descend so, as at the end of the first second of time, they have described 16 feet one inch London Measure, and acquired the Velocity of 32 feet two inches in a second. This is made out from the 25th proposition of the second part of that Excellent Treatise of Mr. Hugenius de Horologio Oscillatorio; wherein he demonstrates the time of the least Vibrations of a Pendulum, to be to the time of the fall of a Body, from from the height of half the length of the Pendulum, as the Circumference of a Circle to its Diameter; whence as a Corollary it follows, that as the Square of the Diameter to the Square of the Circumference, so half the length of the Pendulum vibrating seconds, to the space described by the fall of a body in a second of time: and the length of the Pendulum vibrating seconds, being found $39,125$, or $\frac{1}{4}$ Inches, the descent in a second will be found by the aforesaid Analogy 16 Foot and one Inch, and by the third Proposition, the Velocity will be double thereto; and near to this it hath been found by several Experiments, which by reason of the swiftness of the fall, cannot so exactly determine its quantity. The Demonstration of Hugenius being the Conclusion of a long train of Consequences, I shall for brevity sake omit; and refer you to his Book, where these things are more amply treated of. From these four Propositions, all Questions concerning the Perpendicular fall of bodies, are easily solved, and either Time, Height, or Velocity being assigned, one may readily find the other two. From them likewise is the Doctrine of Projects deducible, assuming the two following Axioms; viz. That a body set a moving, will move on continually in a right line with an equable motion, unless some other force or impediment intervene, whereby it is accelerated, or retarded, or deflected. Secondly, That a Body being agitated by two motions at a time, does by their compounded forces pass through the same points, as it would do, were the two motions divided and acted successively. As for instance, Suppose a body moved in the Line GF, (Fig.2.Tab.1.) from G to R, and there stopping, by another impulse suppose it moved in a space of time equal to the former, from R towards K, to V. I say, the body shall pass through the point V, tho' these two several forces, acted both in the same time. Prop. V. The Motion of all Projects is in the Curve of a Parabola: Let the line GRF (in fig.2.) be the line in which the Project is directed, and in which by the first Axiom it would move move equal spaces in equal times, were it not deflected downwards by the force of Gravity. Let GB be the Horizontal line, and GC a Perpendicular thereto. Then the line GRF being divided into equal parts, answering to equal spaces of time, let the descents of the Project be laid down in lines parallel to GC, proportioned as the squares of the lines GS, GR, GL, GF, or as the squares of the times, from S to T, from R to V, from L to X, and from F to B, and draw the lines TH, VD, XY, BC parallel to GF; I say the Points T, V, X, B, are Points in the Curve described by the Project, and that that Curve is a Parabola. By the second Axiom they are Points in the Curve; and the parts of the descent GH, GD, GY, GC, = to ST, RV, LX, FB, being as the squares of the times (by the second Prop.) that is, as the squares of the Ordinates, HT, DU, YX, BC, equal to GS, GR, GL, GF, the spaces measured in those times; and there being no other Curve but the Parabola, whose parts of the Diameter are as the squares of the Ordinates, it follows that the Curve described by a Project, can be no other than a Parabola: And saying, as RU the descent in any time, to GR or UD the direct motion in the same time, so is UD, to a third proportional; that third will be the line called by all Writers of Conicks, the Parameter of the Parabola to the Diameter GC, which is alwaies the same in Projects cast with the same Velocity: And the Velocity being defined by the number of feet moved in a second of time, the Parameter will be found by dividing the square of the Velocity, by 16 feet 1 inch, the fall of a body in the same time. Lemma. The Sine of the double of any Arch, is equal to twice the Sine of that Arch into its Co-sine, divided by Radius; and the Versed sine of the double of any Arch is equal to the square of the Sine thereof divided by Radius. Let the Arch BC (in fig.3.) be double the Arch BF, and A the Center; draw the Radii AB, AF, AC, and the Chord BDC, BDC, and let fall BE perpendicular to AC, and the Angle EBC, will be equal to the Angle ABD, and the Triangle BCE, will be like to the Triangle BDA; wherefore it will be as AB to AD, so BC or twice BD; to BE, that is as Radius to Co-sine, so twice Sine, to Sine of the double Arch. And as AB to BD, so twice BD or BC, to EC, that is as Radius to Sine, so twice that Sine to the Versed-sine of the double Arch; which two Analogies resolved into Equations, are the Propositions contained in the Lemma to be proved. Prop. VI. The Horizontal distances of Projections made with the same Velocity, at several Elevations of the Line of direction, are as the Sines of the doubled Angles of Elevation. Let GB (fig.2.) the Horizontal distance be = z, the sine of the Angle of Elevation, FGB, be = s, its Co-sine = c, Radius = r, and the Parameter = p. It will be as c to s; so z to \( \frac{sz}{c} \) = FB = GC, and by reason of the Parabola \( \frac{psz}{c} \) = to the square of CB, or GF,. Now as c to r, so is z to \( \frac{sr}{c} \) = GF, and its square \( \frac{zzrr}{cc} \) will be therefore \( \frac{psz}{c} \): which Equation reduced, will be \( \frac{psc}{rr} = z \). But by the former Lemma \( \frac{2sc}{r} \) is equal to the Sine of the double Angle, whereof s is the Sine: wherefore 'twil be as Radius to Sine of double the Angle FGB, so is half the Parameter, to the Horizontal rang or distance sought; and at the several Elevations, the ranges are as the sines of the double Angles of Elevation Q.E.D. Corollary. Hence it follows, that half the Parameter is the greatest Randon, and that that happens at the Elevation of 45 degrees, the sine of whose double is Radius,. Likewise that the Ranges equally distant above and below 45 are equal, are the sines of all doubled Arches, to the sines of their doubled Complements. Prop. VII. The Altitudes of Projections made with the same Velocity, at several Elevations, are as the versed sines of the doubled Angles of Elevation: As \( c \) is to \( s \) :: so is \( \frac{p s c}{r r} \) \( = GB \) to \( \frac{p s s}{r r} = BF \), and \( UK = RU = \frac{BF}{4} \), the Altitude of the Projection \( = \frac{p s s}{4 rr} \). Now by the foregoing Lemma \( \frac{2 s s}{r} \) to the versed sine of the double Angle, and therefore it will be as Radius, to versed sine of double the Angle FGB, so an 8th of Parameters to the height of the Projection VK; and so these heights at several Elevations are as the said versed sines, Q.E.D. Corollary. From hence it is plain, that the greatest Altitude of the perpendicular Projection is a 4th of Parameter, or half the greatest Horizontal Rang; the versed sine of 180 degrees being \( = 2 r \). Prop. VIII. The Lines GF, or times of the flight of a Project cast with the same degree of velocity at different Elevations, are as the sines of the Elevations. As \( c \) is to \( r \) :: so is \( \frac{p s c}{r r} = GB \) by the 6 Prop. to \( \frac{p s}{r} = GF \), that is as Radius to sine of Elevation, so the Parameter to the line GF; so the lines GF are as the sines of Elevation, and the Times are proportional to those Lines; wherefore the Times are as the Sines of Elevation: Ergo constat propositio. Prop. IX. Problem. A Projection being made as you please, having the Distance and Altitude, or Descent of an Object, through which the Project passes, together with the Angle of Elevation of the line of Direction; to find the Parameter and Velocity, that is (in Fig. 2.) having the Angle FGB, GM, and MX. Solution. As Radius to Secant of FGB, so GM the distance given given, to GL; and as Radius to Tangent of FGB, so GM to LM. Then LM—MX in heights, or + MX in descents; or else MX—ML, if the direction be below the Horizontal-line, is the fall in the time that the direct impulse given in G would have carried the Project from G to L=LX=GY; then by reason of the Parabola; as LX or GY, is to GL or YX, :: so is GL to the Parameter sought. To find the Velocity of the Impulse, by Prop.2. & 4, find the time in seconds that a body would fall the space LX, and by that dividing the line GL, the Quote will be the Velocity, or space moved in a second sought, which is always a mean proportional between the Parameter and 16 feet 1 inch. Prop. X. Problem 2. Having the Parameter, Horizontal distance, and height or descent of an Object, to find the Elevations of the line of direction necessary to hit the given Object; that is, having GM, MX, and the greatest Random equal to half the Parameter; to find the Angles FGB. Let the Tangent of the Angle sought be =t, the Horizontal distance GM=b, the Altitude of the Object MX=h, the Parameter=p, and Radius=r, and it will be, As r to t, so b to \(\frac{tb}{r}\)=ML and \(\frac{t b}{r}+\frac{b}{r}\) in ascents=LX, and \[ \frac{p tb}{r} + ph = GL \quad \text{quad.} = XY \quad \text{quad. ratione Parabola;} \quad \text{but} \] \[ \frac{b b}{r r} = GL \quad \text{quad. 47. 1. Euclid. Wherefore} \] \[ \frac{p tb}{r} + ph = bb + \frac{tt bb}{rr} \quad \text{which Equation transposed, is} \] \[ \frac{tt bb}{rr} - \frac{p tb}{r} + ph - bb, \quad \text{divided by } bb \text{ is} \] \[ \frac{tt}{rr} = \frac{pt + pb}{br} - 1. \quad \text{this Equation shews the Question to have two Answers, and the Roots thereof are} \quad \frac{t}{r} = \frac{p}{2b} + \frac{\sqrt{pp + 4pb}}{4bb} - 1 \quad \text{from which I derive the following Rule.} \] Divide half the Parameter by the Horizontal distance, and keep the Quote; viz. \( \frac{p}{2b} \) then say, as square of the distance given to the half Parameter, so half Parameter \( + \) double height descent to the square of a Secant \( = \frac{pp + 4ph}{4bb} \) the Tangent answering to that Secant, will be \( \sqrt{\frac{pp + 4ph}{4bb}} - i \) or \( rr \): so then the sum and difference of the afore-found Quote, and this Tangent will be the Roots of the Equation, and the Tangents of the Elevations sought. Note here, that in Descents, if the Tangent exceed the Quote, as it does when \( ph \) is more than \( bb \), the direction of the lower Elevation will be below the Horizon, and if \( ph = bb \), it must be directed Horizontal, and the Tangent of the upper Elevation will be \( \frac{pr}{b} \): Note likewise, that if \( 4bb + 4ph \) in ascents, or \( 4bb - 4ph \) in descents, be equal to \( pp \), there is but one Elevation that can hit the Object, and its Tangent is \( \frac{pr}{2b} \) and if \( 4bb + 4ph \) in ascents, or \( 4bb - 4ph \) in descents, do exceed \( pp \), the Object is without the reach of a Project cast with that Velocity, and so the thing impossible. From this Equation \( 4bb + 4ph = pp \) are determined the utmost limits of the reach of any Project, and the Figure assigned, wherein are all the heights upon each Horizontal distance beyond which it cannot pass; for by reduction of that Equation, \( b \) will be found \( = \frac{i}{p} - \frac{bb}{p} \) in heights, and \( \frac{bb}{p} - \frac{i}{p} \) in descents; from whence it follows, that all the Points \( h \) are in the Curve of the Parabola, whose Focus is the Point from whence the Project is cast, and whose Latus rectum, or Parameter ad Axem is \( = p \). Likewise from the same Equation may the least Parameter or Velocity be found capable to reach the Object. Object proposed; for \( b = \frac{1}{2} pp + ph \) being reduced \( \frac{1}{2} p \) will be \( \sqrt{bb + bb} \pm b \) in ascents in descents, which is the Horizontal rang at 45 degrees, that would just reach the Object, and the Elevation requisite will be easily had; for dividing the so found Semi-parameter by the Horizontal distance given \( b \), the Quote into Radius will be the Tangent of the Elevation sought. This Rule may be of good use to all Bombardiers and Gunners, not only that they may use no more Powder than is necessary, to cast their Bombs into the place assigned, but that they may shoot with much more certainty, for that a small Error committed in the Elevation of the Piece, will produce no sensible difference in the fall of the Shot: For which Reasons the French Engineers in their late Sieges have used Morter-pieces inclined constantly to the Elevation of 45, proportioning their Charge of Powder according to the distance of the Object they intend to strike on the Horizon. And this is all that need to be said concerning this Problem, of Shooting upon heights and descents. But if a Geometrical construction thereof be required; I think I have one, that is as easy as any can be expected, which I deduce from the forgoing Analytical Solution, viz. \( t = \frac{p + \sqrt{pp + ph - b^2}}{2b} \), and tis this. Having made the right Angle LDA, Tab. I. fig. 4. make DA, DF = \( p \), or greatest Rang, DG = \( b \) the Horizontal distance, and DB DC = \( h \), the Perpendicular height of the Object; and draw GB, and make DE thereto. Then with the Radius AC and center E sweep an Arch, which if the thing be possible, will Intersect the line AD in H; and the line DH being laid both waies from F will give the points K and L, to which draw the lines GL, GK; I say the Angles LGD, KGD are the Elevations required for hitting the Object B. But note that if B be below the Horizon, its descent DC = DB must be laid from A, so as to have AC = to AD + DC. Note likewise, that if in descents DH be greater than FD, and so K fall below D the Angle Angle KGD shall be the depression below the Horizon: Now this Construction so naturally follows from the Equation, that I shall need say no more about it. Prop. XI. To determine the force or Velocity of a Project, in every point of the Curve it describes. To do this we need no other præcognita, but only the third Proposition, Viz. that the Velocity of falling Bodies, is double to that which in the same time, would have described the space fallen by an equable motion: For the Velocity of a Project, is compounded of the constant equal Velocity of the impressed motion, and the Velocity of the fall, under a given Angle, viz. the complement of the Elevation: For instance, in Fig. 2, in the time wherein a project would move from G to L, it descends from L to X, and by the third Proposition has acquired a Velocity, which in that time would have carried it by an equable motion from L to Z or twice the descent L X; and drawing the line G Z, I say the Velocity in the point X, compounded of the Velocities G L and L Z under the Angle G L Z, is to the Velocity impressed in the point G, as G Z is to G L; this follows from our second Axiome; and by the 20 and 21 Prop. lib. i, conic. Midorigii, XO parallel and equal to G Z shall touch the Parabola in the point X. So that the Velocities in the several points, are as the lengths of the Tangents to the Parabola in those points, intercepted between any two Diameters: And these again are as the Secants of the Angles, which those Tangents continued make with the Horizontal line G B. From what is here laid down, may the comparative force of a Shot in any two points of the Curve, be either Geometrically or Arithmetically discovered. Corollary. From hence it follows, that the force of a Shot is always least at U, or the Vertex of the Parabola, and that at equal distances therefrom, as at T and X, G and B its force is always equal, and that the least force in U is to that in G and B, as B, as Radius to the Secant of the Angle of Elevation F G B. These Propositions considered, there is no question relating to Projects, which by the help of them may not easily be Solved; and tho' it be true that most of them are to be met withal, in Galileus, Torricellius and others, who have taken them from those Authors, yet their Books being Foreign, and not easy to come by, and their Demonstrations long and difficult, I thought it not amiss to give the whole Doctrine here in English, with such short Analytical Proof of my own, as might be sufficient to evince their Truth. The Tenth Proposition contains a Problem, untouch't by Torricellius, which is of the greatest use in Gunnery, and for the sake of which this Discourse was principally intended; It was first Solved by Mr. Anderson, in his Book of the Genuine use and effects of the Gunn, Printed in the Year 1674; but his Solution required so much Calculation, that it put me upon search, whether it might not be done more easily, and thereupon in the Year 1678 I found out the rule I now publish, and from it the Geometrical Construction: Since which time there has a large Treatise of this Subject Entituled, L'art de jeter les Bombes, been Published in France by Monsieur Blondel, wherein he gives the Solutions of this Problem by Messieurs Bout, Romer and de la Hire; But none of them being the same with mine, or in my Opinion more easy, and most of them more Operose, and besides mine finding the Tangent, which generally determines the Angle better than its Sine, I thought myself obliged to Print it for the use of all such, as desire to be informed in the Mathematical part, of the Art of Gunnery. Now these rules were rigidly true, were it not, as I said before, for the Opposition of the Medium, whereby not only the direct imprest Motion is continually retarded, by likewise the increase of the Velocity of the fall, so that the spaces described thereby, are not exactly as the squares of the times: But what this Opposition of the Air is, against several Velocities, Bulks, and Weights, is not so easy to determine. Tis certain that the weight of Air, to that of Water, is nearly as 1 to 800, whence the weight thereof, to that of any Project is given; tis very likely, that to the same Velocity, and Magnitude, but of different matter, the Opposition should be reciprocally as the weights of the Shott; as likewise that to Shott of the same Velocity and matter, but of different Sizes, it should be as the Diameters reciprocally: whence generally the Opposition to Shott with the same Velocity, but of differing Diameters, and Materials, should be as their Specifick Gravities into their Diameters reciprocally; but whether the Opposition, to differing Velocities of the same Shott, be as the Squares of those Velocities, or as the Velocities themselves, or otherwise, is yet a harder Question. However it be, tis certain, that in large Shott of Metal, whose weight many Thousand times Surpasses that of the Air, and whose force is very great, in proportion to the Surface wherewith they press thereon; this Opposition is scarce discernable: For by several Experiments made with all Care and Circumspection with a Morterpeice Extraordinary well fixt to the Earth on purpose, which carried a Solid Brass Shott of 4½ Inches Diameter, and of about 14 Pound weight, the Ranges above and below 45 Degrees were found nearly equal; if there were any difference, the under Ranges went rather the farthest, but those differences were usually less than the Errors committed in ordinary Practice, by the unequal Goodness and Dryness of the same sort of Powder, by the Unfitness of the Shott to the Bore, and by the Losses of the Carriage. In a Smaller Brass-Shott of about an Inch and half Diameter, cast by a Cross-Bow which ranged it, at most about 400 foot, the Force being much more Equal than in the Morterpeice, this difference was found more Curiously, and Constantly and most Evidently, the under Ranges out went the upper. From which Trials I conclude, that altho' in small and light Shott, the Opposition of the Air, ought and must be accounted for; yet in Shooting of Great and Weighty Bombs, there need be very little or no allowance made; and so these Rules may be be put in Practice to all Intents and Purposes, as if this Impediment were absolutely Removed. An Account of an Experiment shewn before the Royal Society, of Shooting by the Rarefaction of the Air: By Dr. D. Papin, R.S.S. Whereas ordinary Wind-Guns do their Effect by the Compression of the Air. Ottho Ghericke hath found a new Sort that shoots by Rarefaction; and he hath Publish'd that device at large in his Book about Pneumatick Experiments, but he doth not express how strong was the Effect. I have therefore had the Curiosity to try it myself by another Contrivance, which I take to be better than his: First, because I can make a Rarefaction much more perfect than he could do. Secondly, because his Device could not be used but for Guns of a small bore; but my way may be apply'd to the biggest bore that can be made by Workmen: So that one might by this means throw up vast Weights to a great distance. A A is a Pipe very equal from one end to the other. B B a small Pipe solder'd to a Hole near the end of the Pipe A A, and apply'd to the Plate of the Pneumatick Engine. C C C C some kind of Stool to bear up the hinder part of the Pipe A A. D. a piece of Lead fitted to the bore of the pipe A A. The pipe A A is to be shut at both ends by Valves outwardly apply'd, and so the said pipe A A, though never so big, may be exhausted of Air by means of the Pneumatick Engine: Which done, the Valve towards D must be suddenly open'd, so that the whole pressure of the Atmosphere acting upon the Lead D may drive it along the pipe A A with such