Chapter XII: Appendix (11)

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_, _h_ will be found = _¼p_ - _bb/p_ in _heights_, and _bb/p_ - _¼p_ 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_ proposed; for _bb_ = _¼pp_ ∓ _ph_ being reduced, _½p_ will be = √(_bb + hh_) ± _h_ {in ascents|in descents} which is the _Horizontal Range_ at 45 degrees, of a Project cast with the least Velocity 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 Mortar-pieces inclin'd constantly to the _Elevation_ of 45, proportioning their Charge of Pouder 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 can be expected, which I deduce from the foregoing _Analytical Solution_, _viz._

_t/r_ = _p/2b_ ± √((_¼pp ± ph - bb_)/_bb_),

and 'tis this, having made the right Angle GDF, (_Tab. 5. Fig. 3._) make DF = _½p_, or greatest Range, and GD = _b_ the Horizontal Distance, and DB = _h_ the perpendicular heighth of the Object; to be laid upwards from D, if the Object be above the Horizon, or downwards if below it. Parallel to GD draw FA, and make it equal to GB the Hypothenusal Distance of the Object; and with the Center A and Radius FB = _½p ± h_, sweep an Arch, which shall if the thing be possible, intersect the indeterminate Perpendicular DF in two Points K and L, to which draw the Lines, GL, GK; I say, the Angles DGK, DGL, are the Elevations requisite to strike the Object B.

_Demonstration._ The Square of FK or FL, is equal to FB_q_ - GB_q_: or (_½p ± h_)² - _bb_ - _hh_ or _¼pp ± ph - bb_, and therefore √(_¼pp ± ph - bb_) is = FK = FL, and by Consequence DK, DL = _½p_ ± √(_½pp ± ph - bb_). And as DG: DK and DL :: Radius: Tangents sought, which coincides with our Algebraical Expression thereof.

_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 LX; and drawing the Line GZ, I say, the _Velocity_ in the Point X, compounded of the _Velocities_ GL and LZ under the Angle GLZ, is to the _Velocity_ impress'd in the Point G, as GZ is to GL; this follows from our second _Axiom_, and by the 20 and 21 _Prop. lib. 1. conic. Midorgii_, XO parallel and equal to GZ 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 GB. 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_ discover'd.

_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 _Radius_ to the Secant of the _Angle_ of _Elevation_ FGB.

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'd 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 _Gun_, 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, Intituled, _L'Art dejetter les Bombes_, been Published by _Monsieur Blondel_, wherein he gives the Solutions of this _Problem_ by _Messieurs Buot_, _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 my self 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, but 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 easie 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 Shot; as likewise that to Shot of the same _Velocity_ and Matter, but of different _Sizes_, it should be as the _Diameters reciprocally_: Whence generally the _Opposition_ to Shot 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 Shot, 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 Shot 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 _Mortarpiece_, Extraordinary well fix'd to the Earth on purpose, which carried a solid Brass Shot of four Inches and a half _Diameter_, and of about fourteen Pound Weight, the _Ranges_ above and below forty five _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 Shot to the Bore, and by the Loosness of the Carriage.

In a smaller Brass-Shot of about an Inch and half Diameter, cast by a Cross-Bow which ranged it, at most about four Hundred Foot, the Force being much more equal than in the Mortarpiece, this difference was found more Curiously: and Constantly and most Evidently, the under Ranges out-went the upper. From which Trials I conclude, that although in small and light Shot, 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 put in practice to all Intents and Purposes, as if this Impediment were absolutely remov'd.

_A Proposition of general Use in the Art of Gunnery, shewing the Rule of laying a Mortar to pass, in order to strike an Object above or below the Horizon._

It was formerly the Opinion of those concerned in Artillery, that there was a certain requisite of Powder for each Gun, and that in Mortars, where the distance was to be varied, it must be done by giving a greater or lesser Elevation to the Piece. But now our later Experience has taught us that the same thing may be more certainly and readily performed by increasing and diminishing the quantity of Powder, whether regard be had to the Execution to be done, or to the Charge of doing it. For when Bombs are discharged with great Elevations of the Mortar, they fall too Perpendicular, and bury themselves too deep in the Ground, to do all that damage they might, if they came more Oblique, and broke upon or near the Surface of the Earth; which is a thing acknowledg'd by the Besieged in all Towns, who unpave their Streets, to let the Bombs bury themselves, and thereby stifle the force of their Splinters. A Second Convenience is, that at the extream Elevation, the Gunner is not obliged to be so curious in the direction of his Piece, but it will suffice to be within a Degree or two of the Truth; whereas in the other Method of Shooting he ought to be very curious. But a Third, and no less considerable Advantage is, in the saving the Prince's Powder, which in so great and so numerous Discharges, as we have lately seen, must needs amount to a considerable Value. And for Sea-Mortars, it is scarce practicable otherwise to use them, where the agitation of the Sea continually changes the Direction of the Mortar, and would render the Shot very uncertain, were it not that they are placed about 45 Degrees Elevation, where several Degrees above or under, make very little difference in the Effect.

In the precedent Discourse, I considered all the Propositions relating to the Motion of Projectiles, and gave a Solution to this Problem; _viz._ _To hit an Object above or below the Horizontal Line with the greatest Certainty and least Force._ That is, that the Horizontal distance of the Object being put = _b_, and the Perpendicular Heighth = _b_, the Charge requisite to strike the Object with the greatest Advantage, was that which with an Elevation of 45° would cast the Shot on the Horizontal Line, to the distance of √(_bb +hh_), when the Object was above the Horizon; or if it were below it, the Charge must be lesser, so as to reach on the Horizon, at 45° Elevation, no greater a Distance than √(_bb + hh_) - _h_; that is, in the one Case, the Sum of the Hypothenusal Distance of the Object from the Gun, and the Perpendicular Heighth thereof above the Gun; and in the other Case, when the Object is below the Horizon, the difference of the same _per_ 47. I _Eucl._ And I then shew'd how to find the Elevation proper for the Gun so charged, _viz._ As the Horizontal Distance of the Object, to the Sum or Difference of the Hypothenusal Distance and Perpendicular Height: So Radius to the Tangent of the Elevation sought. But I was not at that time aware that the aforesaid Elevation did constantly bisect the Angle between the Perpendicular and the Object, as is demonstrated from the Difference and Sum of the _Tangent_ and _Secant_ of any Arch being always equal to the _Tangent_ and _Cotangent_ of the half Complement thereof to a Quadrant. Having discovered this, I think nothing can be more compendious, or bids fairer to compleat the Art of Gunnery, it being as easie to shoot with a Mortar at any Object on demand, as if it were on the Level; neither is there need of any Computation, but only simply laying the Gun to pass, in the middle Line between the Zenith and the Object, and giving it its due Charge. Nor is there any great need of Instruments for this purpose: For if the Muzzle of the Mortar be turned truly Square to the Bore of the Piece, as it usually is or ought to be, a piece of Looking-glass Plate applied parallel to the Muzzle, will by its Reflection give the true Position of the Piece, the Bombardeer having no more to do, but to look perpendicularly down on the Looking-glass, along a small Thread with a Plumbet, and to raise or depress the Elevation of the Piece, till the Object appear reflected on the same Point of the _Speculum_, on which the Plumbet falls; for the Angle of Incidence and Reflection being equal, in this Case a Line at Right Angles to the _Speculum_, as is the _Axis_ of the Chase of the Piece, will bisect the Angle between the Perpendicular and the Object, according as our Proposition requires. So that it only remains by good and valid Experiments to be assured of the Force of Gunpowder, how to make and conserve it equal, and to know the Effect thereof in each Piece; that is, how far differing Charges will cast the same Shot out of it; which may most conveniently be engraven on the outside thereof, as a standing Direction to all Gunners, who shall from thence forward have occasion to use that Piece: And were this Matter well ascertained, it might be worth the while to make all Mortars of the like Diameter as near as may he, alike in length of Chase, Weight, Chamber, and all other Circumstances.

This Discovery that the utmost Range on an inclined Plane, is, when the _Axis_ of the Piece makes equal Angles with the Perpendicular and the Object; compared with what I have demonstrated of the same Problem in the aforesaid Discourse does lead to and discover two very ready Theorems; the one, to find the greatest Horizontal Range at 45° Elevation, by any Shot made upon any inclined Plane, with any Elevation of the Piece whatsoever: And the other to find the Elevations proper to strike a given Object, with any Force greater than what suffices to reach it with the aforesaid middle Elevation. Both which being performed by one single Proportion, may be very serviceable to such as are concerned in the Practice of Gunnery, but are unwilling to trouble themselves with tedious and difficult Rules. The two Propositions are these.

_PROP. I._

A Shot being made on an inclined Plane, having the Horizontal Distance of the Object it strikes, with the Elevation of the Piece, and the Angle at the Gun between the Object and the Perpendicular; to find the greatest Horizontal Range of that Piece, laden with the same Charge; that is, half the _Latus rectum_ of all the _Parabolæ_ made with the same _Impetus_.

_RULE._

Take half the Distance of the Object from the _Nadir_, and take the Difference of the given Elevation from that half; the Versed Sine of twice that Difference subtract from the Versed Sine of the Distance of the Object from the _Zenith_: Then shall the Difference of those Versed Sines be to the Sine of the Distance of the Object from the _Zenith_, as the Horizontal Distance of the Object strook, to the greatest Horizontal Range at 45°.

_PROP. II._

Having the greatest Horizontal Range of a Gun, the Horizontal Distance and Angle of Inclination of an Object to the Perpendicular, to find the two Elevations necessary to strike that Object.

_RULE._

Halve the Distance of the Object from the _Nadir_; this half is always equal to the half Sum of the two Elevations we seek. Then say, _As the greatest Horizontal Range is to the Horizontal Distance of the Object: So is the Sine of the Angle of Inclination or Distance of the Object from the Perpendicular, to a fourth Proportional; which fourth being subtracted from the Versed Sine of the Distance of the Object from the _Zenith_, leaves the Versed Sine of the Difference of the Elevations sought; which Elevations are therefore had by adding and subtracting the half Difference to and from the aforesaid half Sum._

I shall not need to speak of the Facility of these Solutions, I shall only observe that they are both General, without Exception or Caution, and derived from the Knowledge that these two Elevations are equidistant above and below the Line, bisecting the Angle between the Object and the _Zenith_.

_A Discourse concerning the Measure of the Airs Resistance to Bodies
moved in it. By the Learned _John Wallis_, S. T. D. and R. S. S._

1. That the Air (and the like of any other _Medium_) doth considerably give Resistance to Bodies moved in it; and doth thereby abate their Celerity and Force; is generally admitted. And Experience doth attest it: For otherwise, a Cannon Bullet projected Horizontally, should (supposing the Celerity and Force undiminished) strike as hard against a Perpendicular Wall, erected at a great distance, as near at hand; which we find it doth not.

2. But at what Rate, or in what Proportion, such Resistance is; and (consequently, at what Rate the Celerity and Force is continually diminished) seems not to have been so well examined. Whence it is, that the Motion of a Project (secluding this Consideration) is commonly reputed to describe a Parabolick Line; as arising from an Uniform or equal Celerity in the Line of Projection, and a Celerity uniformly accelerated in the Line of Descent; which two so compounded, do create a Parabola.

3. In order to the Computation hereof, I first premise this _Lemma_, (as the most rational that doth occur for my first footing,) That (supposing other things equal) the Resistance is proportional to the Celerity. For in a double Celerity, there is to be removed (in the same time) twice as much Air, (which is a double Impediment) in a treble, thrice as much; and so in other Proportions.

4. Suppose we then the Force impressed (and consequently the Celerity, if there were no Resistance) as 1; the Resistance as _r_. (which must be less than the Force, or else the Force would not prevail over the Impediment, to create a Motion.) And therefore the effective Force at a first Moment, is to be reputed as 1 - _r_: That is, so much as the Force impressed, is more than the Impediment or Resistance.

5. Be it as 1 - _r_ to 1; so one to _m_. (which _m_ is therefore greater than 1.)

6. And therefore the effective Force (and consequently the Celerity) as to a first Moment, is to be 1/_m_ of what it would be, had there been no Resistance.

7. This 1/_m_ is also the remaining Force after such first Moment; and this remaining Force is (for the same Reason) to be proportionally abated as to a second Moment; that is, we are to take 1/_m_ thereof, that is 1/_mm_ of the impressed Force. And for a third Moment (at equal distance of time) 1/_mmm_; for a fourth 1/_m_⁴; and so onward infinitely.

8. Because the length dispatched (in equal times) is proportional to the Celerities; the Lines of Motion (answering to those equal Times) are to be as 1/_m_, 1/_m_², 1/_m_³, 1/_m_⁴, _&c._ of what they would have been, in the same Times, had there been no Resistance.

9. This therefore is a Geometrical Progression; and (because of _m_ greater than 1) continually decreasing.

10. This decreasing Progression infinitely continued (determining in the same Point of Rest, where the Motion is supposed to expire) is yet of a finite Magnitude; and equal to 1/(_m_ - 1), of what it would have been in so much Time, if there had been no Resistance. As is demonstrated in my Algebra, _Chap._ 95. _Prop._ 8. For (as I have elsewhere demonstrated) the Sum or Aggregate of a Geometrical Progression is (_VR - A_)/(_R_ - 1) (supposing _V_ the greatest Term, _A_ the least, and _R_ the common Multiplier.) That is _VR_/(_R_ - 1) - _A_/(_R_ - 1). Now in the present Case, (supposing the Progression infinitely continued) the least Term _A_, becomes infinitely small, or = 0. And consequently _A_/(_R_ - 1) doth also vanish, and thereby the Aggregate becomes = _VR_/(_R_ - 1). That is (as will appear by dividing _VR_ by _R_ - 1;) _V + V/R + V/RR + V/R³ + &c._ = _VR_/(_R_ - 1);[14] (supposing the Progression to begin at _V_ = 1.) That is (dividing all by _R_, that so the Progression may begin at _V/R_ = 1/_m_:) _V_/(_R_ - 1) = _V/R + V/RR + V/R³ + &c._, That is, in our present Case (because of _V_ = 1, & _R_ = _m_:) 1/_m_ + 1/_mm_ + 1/_m_³ &c. = 1/(_m_ - 1). That is, (putting _n_ = _m_ - 1) 1/_n_ of what it would have been if there had been no Resistance.

11. This infinite Progression is fitly expressed by an Ordinate in the Exterior Hyperbola, parallel to one of the Asymptotes; and the several Members of that, by the several Members of this, cut in continual Proportion. As is there demonstrated at _Prop._ 15. For let _SH_, (_vid._ Fig. 4. Tab. 5.) be an Hyperbola between the Asymptotes _AB_, _AF_: And let the Ordinate _DH_ (in the Exterior Hyperbola, parallel to _AF_,) represent the impressed Force undiminished; or the Line to be described in such time, by a Celerity answerable to such undiminished Force. And let _BS_ (a like Ordinate) be 1/_m_ thereof; which therefore, being less than _DH_ (as being equal to a Part of it) will be farther than it from _AF_. In _AB_ (which I put = 1) let _Bd_ be such a Part thereof, as is _BS_ of _DH_. Now because (as is, well known) all the inscribed Parallelograms, in the Exterior Hyperbola, _AS_, _AH_, _&c._ are equal; and therefore their sides reciprocal: Therefore as _Ad_ = 1 - 1/_m_ (supposing _Bd_ to be taken, from _B_ towards _A_,) to _AB_ = 1, or as _m_ - 1 to _m_: so is _BS_ = (1/_m_)_DH_, to _dh_, which is therefore equal to 1/(_m_ - 1) of _DH_; that is (as will appear by dividing 1, by _m_ - 1,) to 1/_m_ + 1/_mm_ + 1/_m_³, _&c._ of _DH_.[15]

Or if _Bd_ be taken beyond _B_; then as _Ad_ = 1 + 1/_m_ to _AB_ = 1, or as _m_ + 1 to _m_, so is (1/_m_)_DH_ to _dh_, which is therefore equal to 1/(_m_ + 1)DH; that is (as will appear by like dividing of 1 by _m_ + 1;) = to 1/_m_ - 1/_mm_ + 1/_m_³ - _&c._ of _DH_.

12. Let such ordinate _dh_, or (equal to it in the Asymptote) _AF_, be so divided in _L_, _M_, _N_, _&c._ (by Perpendiculars cutting the Hyperbola in _l_, _m_, _n_, _&c._) as that _FL_, _LM_, _MN_, be as 1/_m_, 1/_mm_, 1/_m_³, _&c._ That is, so continually decreasing as that each Antecedent be to its Consequent, as 1 to 1/_m_, or as _m_ to 1. See _Fig. 5. Tab. 5._

13. This is done by taking _AF_, _AL_, _AN_, _&c._ in such proportion. For, of continual Proportionals, the Differences are also continually proportional, and in the same proportion. For let _A_, _B_, _C_, _D_, _&c._ be such Proportionals, and their Differences _a_, _b_, _c_, _&c._ That is, _A_ - _B_ = _a_, _B_ - _C_ = _b_, _C_ - _D_ = _c_, _&c._

Then, because A, B, C, D, _&c._ are in continual proportion, That is, A. B :: B. C :: C. D :: _&c._ And dividing (A - B). B :: (B - C). C :: (C - D). D :: _&c._ That is, _a_. B :: _b_. C :: _d_. D :: _&c._ And alternly _a. b. c._ _&c._ :: B. C. D. _&c._ :: A. B. C. _&c._ That is, in continual proportion as A to B, or as _m_ to 1.

14. This being done; the Hyperbolick Spaces _Fl_, _Lm_, _Mn_, &c. are equal. As is demonstrated by _Gregory San-Vincent_; and as such is commonly admitted.

15. So that _Fl_, _Lm_, _Mn_, _&c._ may fitly represent equal Times, in which are dispatched unequal Lengths, represented by _FL_, _LM_, _MN_, _&c._

16. And because they are in Number infinite (though equal to a finite Magnitude) the Duration is infinite: And consequently the impressed Force, and Motion thence arising, never to be wholly extinguished (without some further Impediment) but perpetually approaching to _A_, in the Nature of Asymptotes.

17. The Spaces _Fl_, _Fm_, _Fn_, &c. are therefore as Logarithms (in Arithmetical Progression increasing) answering to the Lines _AF_, _AL_, _AM_, &c. or to _FL_, _LM_, _MN_, &c. in Geometrical Progression decreasing.

18. Because _FL_, _LM_, _MN_, &c. are as 1/_m_, 1/_mm_, 1/_m_³, &c. (infinitely) terminated at _A_; therefore (by ¶ 10) their Aggregate _FA_ or _dh_, is to _DH_, (so much Length as would have been dispatched, in the same time, by such impressed Force undiminished) as 1 to _m_ - 1 = _n_.

19. If therefore we take, as 1 to _n_, so _AF_ to _DH_; this will represent the Length to be dispatched, in the same time, by such undiminished Force.

20. And if such _DH_ be supposed to be divided into equal Parts innumerable (and therefore infinitely small;) these answer to those (as many) Parts unequal in _FA_, or _hd_.

21. But, what is the Proportion of _r_ to 1, or (which depends on it) of 1 - _r_ to 1, or 1 to _m_; remains to be inquired by Experiment?

22. If the Progression be not infinitely continued; but end (suppose) at _N_, and its least Term be _A_ = _MN_; then, out of

_V_/(_R_ - 1) = 1/_m_ + 1/_mm_ + 1/_m_³, _&c._

is to be subducted _A_/(_R_ - 1) (as at ¶ 10.) that is (as by Division will appear)

_A_/_R_ + _A_/_R_² + _A_/_R_³ &c.

That is (in our present Case)

_a_/_m_ + _a_/_mm_ + _a_/_m_³ &c.

And so the Aggregate will be

(1-_a_)/_m_ + (1-_a_)/_mm_ + (1-_a_)/_m_³ &c. = (1-_a_)/_n_.

And thus as to the Line of Projection, in which (secluding the Resistance) the Motion is reputed uniform; dispatching equal Lengths in equal Times. Consider we next the Line of Descent.

23. In the Descent of Heavy Bodies, it is supposed that to each Moment of Time, there is superadded a new Impulse of Gravity to what was before: And each of these, secluding the Consideration of the Air's Resistance, to proceed equally (from their several beginnings) through the succeeding Moments. As (in the erect Lines) 1 1 1 1, _&c._ 1 1 1, _&c._ 1 1, _&c._ 1, _&c._ and so continually, as in the Line of Projection.[16]

24. Hence ariseth (in the transverse Lines) for the first Moment 1, for the second 1 + 1, for the third 1 + 1 + 1, and so forth, in Arithmetical Progression: As are the Ordinates in a Triangle, at equal distance.

25. And such are the continual Increments of the Diameter, or of the Ordinates in the exterior Parabola, answering to the interior Ordinates, or Segments of the Tangent, equally increasing; as is known, and commonly admitted.

26. If we take in the Consideration of the Air's Resistance; we are then, for each of these equal Progressions, to substitute a decreasing Progression Geometrical; in like manner (and for the same Reasons) as in the Line of Projection.

27. Hence ariseth, for the first Moment 1/_m_; for the second 1/_m_ + 1/_m_²; for the third 1/_m_ + 1/_m_² + 1/_m_³, _&c._[17] And such is therefore the Descent of a heavy Body falling by its own weight. The several Impulses of Gravity being supposed equal.

28. That is (in the Figure of ¶ 12) as _FL_, _FM_, _FN_, &c. in the Line of Descent, answering to _FL_, _LM_, _MN_, &c. in the Line of Projection.

29. But though the Progressions for the Line of Projection, are like to each of those many in the Line of Descent; it is not to be thence inferred, that therefore 1/_m_ in the one, is equal to 1/_m_ in the other: But in the Line of Projection (suppose) (1/_m_)_f_ (such a Part of the Force impressed, and a Celerity answerable:) in the Line of Descent, (1/_m_)_g_ such a Part of the Impulse of Gravity.

30. Those for the Line of Descent (of the some Body) are all equal, each to other: Because _g_ (the new Impulse of Gravity) in each Moment is supposed to be the same.

31. But what is the Proportion of _f_ to _g_ (that of the Force impressed, to the Impulse of Gravity in each Body) remains to be inquired by Experiment.

32. This Proportion being found as to one known Force; the same is thence known as to any other Force (whose Proportion to this is given) in the same uniform _Medium_.

33. And this being known, as to one _Medium_; the same is thence known as to any other _Medium_, the Proportion of whose Resistance to that of this is known.

34. If a heavy Body be projected downward in a perpendicular Line; it descends therefore at the Rate 1/_m_, 1/_mm_, 1/_m_³, _&c._ of _f_, (the impressed Force) increased by 1/_m_, 1/_m_ + 1/_m_², 1/_m_ + 1/_m_² + 1/_m_³, _&c._ of _g_ the impulse of Gravity, (by ¶ 7, and ¶ 27) Because both Forces are here united.

35. If in a perpendicular Projection upwards; it ascends in the rate of the former, abated by that of the latter. Because here the impulse of Gravity is contrary to the Force impressed.

36. When therefore this latter (continually increasing) becomes equal to that former (continually decreasing) it then ceaseth to ascend; and doth thenceforth descend at the rate wherein the latter continually exceeds the former.

37. In an Horizontal, or Oblique Projection: If to a Tangent, whose Increments are as _FL_, _LM_, _MN_, &c. that is as (1/_m_)_f_, &c. be fitted Ordinates (at a given Angle) whose Increments are as _FL_, _FM_, _FN_, &c. that is, as (1/_m_)_g_, &c. The Curve answering to the Compound of these Motions, is that wherein the Project is to move.

38. This Curve (being hitherto without a Name) may be call'd _Linea Projectorum_; the Line of Projects, or things projected; which resembles a Parabola deform'd.

39. The Celerity and Tendency, as to each Point of this Line, is determined by a Tangent at that Point.

40. And that against which it makes the greatest Stroke or Percussion, is that which (at that Point) is at right Angles to that Tangent.

41. If the Projection (at ¶ 27) be not infinitely continued, but terminate (suppose) at _N_, so that the last Term in the first Column or Series erect be _a_; and consequently in the second, _ma_; in the third, _mma_, &c. (each Series having one Term fewer than that before it:) Then (for the same Reasons, as at ¶ 22) the Aggregates of the several Columns (or erect Series) will be (1 - _a_)/_n_, (1 - _ma_)/_n_, (1 - _mma_)/_n_, and so forth, till (the Multiple of _a_ becoming = 1) the Progression expire.

42. Now all the Abatements here, _a_, _ma_, _mma_, &c. are the same with the Terms of the first Column taken backward. For _a_ is the last, _ma_ the next before it; and so of the rest.

43. And the Aggregate of all the Numerators is so many times 1, as is the Number of Terms (suppose _t_,) wanting the first Column; that is

_t_ - (1 - _a_)/_n_, or (_nt_ - 1 + _a_)/_n_;

and this again divided by the common Denominator _n_, becomes

(_nt_ - 1 + _a_)/_nn_.

And therefore ((_nt_ - 1 + _a_)/_nn_)_g_, is the Line of Descent by its own Gravity.

44. If therefore this be added to a projecting Force downward in a Perpendicular; or subducted from such projecting Force upward; that is, to or from ((1 - _a_)/_n_)_f_: The Descent in the first Case will be

((1 - _a_)/_n_)_f_ + ((_nt - 1 + a_)/_nn_)_g_;

and the Ascent in the other Case

((1 - _a_)/_n_)_f_ - ((_nt - 1 + a_)/_nn_)_g_.

And in this latter Case, when the ablative Part becomes equal to the positive Part, the Ascent is at the highest; and thenceforth (the ablative Part exceeding the positive) will descend.

45. In an Horizontal or Oblique Projection, having taken

((1 - _a_)/_n_)_f_,

in the Line of Projection, and thence (at the Angle given)

((_nt_ - 1 + _a_)/_nn_)_g_,

in the Line of Descent; the Point in the Curve answering to these, is the Place of the Project answering to that Moment.

46. I am aware of some Objections to be made, whether to some Points of the Process, or to some of the Suppositions. But I saw not well how to wave it, without making the Computation much more perplex'd. And in a Matter so nice, and which must depend upon Physical Observations, 'twill be hard to attain such Accuracy, as not to stand in need of some Allowances.

47. Somewhat might have been farther added to direct the Experiments suggested at ¶ 21, and 31. But that may be done at leisure, after deliberation had, which way to attempt the Experiment.

48. The like is to be said of the different resistance which different Bodies may meet with in the same _Medium_, according to their different Gravities (extensively or intensively consider'd) and their different Figures and Positions in Motion. Whereof we have hitherto taken no account; but supposed them, as to all these, to be alike and equal.

_POSTSCRIPT._

49. The Computation in ¶ 41, 42, 43, may (if that be also desired) be thus represented by Lines and Spaces. The Ablatives _a_, _ma_, _mma_, &c. (being the same with the first Column taken backward) are fitly represented by the Segments of _NF_ (beginning at _N_) in Figure 5 and 6, and therefore by Parallelograms on these Bases, assuming the common height of _Fh_, or _NQ_; the Aggregate of which is _Nh_, or _FQ_. And, so many times 1, by so many equal Spaces, on the same Bases, between the same Parallels, terminated at the Hyperbola: The Aggregate of which is _hFNQn_. From whence if we subduct the Aggregate of Ablatives _FY_; the remaining Trilinear _hQn_, represents the Descent.

50. If to this of Gravity, be joined a projecting Force; which is to the Impulse of Gravity as _hK_ to _hF_ (be it greater, less, or equal) taken in the same Line; the same Parallels determine proportional Parallelograms, whose Aggregate is _KQ_.

51. And therefore if this be a perpendicular Projection downwards; then _hKkn_ (the Sum of this with the former) represents the Descent.

52. If it be a Perpendicular upwards; then the difference of these two represents the Motion; which so long as _KQ_ is the greater, is Ascendent; but Descendent, when _hQn_ becomes greater; and it is then at the highest when they be equal.

53. If the Projection be not in the same Perpendicular, (but Horizontal, or Oblique) then _KQ_ represents the Tangent of the Curve; and _hQn_ the Ordinates to that Tangent, at the given Angle.

54. But the Computation before given, I take to be of better use than this Representation in Figure. Because in such Mathematical Enquiries, I choose to separate (as much as may be) what purely concerns Proportions; and consider it abstractly from Lines, or other Matter wherewith it is incumbred.

As to the Question proposed; whether the resistance of the _Medium_ do not always take off such a proportional part of the Force moving through it, as is the specifick Gravity of the _Medium_ to that of the Body moved in it: (For, if so, it will save us the trouble of Observation.)

I think this can by no means be admitted. For there be many other things of Consideration herein, beside the intensive Gravity (or, as some call it, the specifick Gravity) of the _Medium_.

A viscous _Medium_ shall more resist, than one more fluid, though of like intensive Gravity.

And a sharp Arrow shall bore his way more easily through the _Medium_, than a blunt-headed Bolt, though of equal Weight, and like intensive Gravity.

And the same Pyramid with the Point, than with the Base forward.

And many other like Varieties, intended in my ¶ 48.

But this I think may be admitted, namely, That different _Mediums_, equally liquid, (and other Circumstances alike,) do in such proportion resist, as is their intensive Gravity. Because there is, in such proportion, a heavier Object to be removed, by the same Force. Which is one of the things to which ¶ 33 refers.

And again: The heavier Project once in Motion, (being equally swift, and all other Circumstances alike) moves through the same _Medium_ in such proportion more strongly, as is its intensive Gravity. For now the Force is in such proportion greater, for the removal of the same resistance. And this Part of what my ¶ 32, insinuates.

But where there is a Complication of these Considerations one with another, and with many other Circumstances, whereof each is severally to be considered; there must be respect had to all of them.

[14]
R - 1_V_) _R_ (_V_ + _V/R_, _V/RR_, &c.
_VR_ - V
--------
+ V
+ V/VR
-------
+ V/R
+ VV/RRR
--------
+ V/RR
_&c_.

[15]
_m_ - 1) 1 (1/_m_ + 1/_mm_ + 1/_m_³ + &c.
1 - 1/_m_
-------
+ 1/_m_
+ 1/_m_ - 1/_mm_
--------------
+ 1/_mm_
+ 1/_mm_ - 1/_mmm_
----------------
+ 1/_mmm_
&c.

[16] 1 1 1 1 1 1 1 1 1 1 _&c._

[17]
1
---
_m_

1 1 ----- --- _m_² _m_

1 1 1 ----- ----- --- _m_³ _m_² _m_

1 1 1 1 ----- ----- ----- --- _m_⁴ _m_³ _m_² _m_

_An Instance of the Excellence of the _Modern Algebra_, in the
Resolution of the Problem of finding the _Foci_ of Optick Glasses
Universally. By _E. Halley_, S. R. S._

The Excellence of the _Modern Geometry_ is in nothing more evident, than in those full and adequate Solutions it gives to Problems; representing all the possible Cases at one view, and in one general Theorem, many times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the manner of the _Ancients_, might well become the Subjects of large Treatises: For whatsoever Theorem solves the most complicated Problem of the kind, does with a due Reduction reach all the subordinate Cases. Of this I now design to give a notable Instance in the Doctrine of _Dioptricks_.

This Dioptrick Problem is that of finding the _Focus_ of any sort of _Lens_, exposed either to converging, diverging, or parallel Rays of Light, proceeding from, or tending to a given Point in the _Axis_ of the _Lens_, be the _Ratio_ of _Refraction_ what it will, according to the Nature of the transparent Material whereof the _Lens_ is formed, and also with allowance for the thickness of the _Lens_ between the _Vertices_ of the two Spherical Segments. This Problem being solved in one Case, _mutatis mutandis_, will exhibit Theorems for all the possible Cases, whether the _Lens_ be _Double-Convex_ or _Double-Concave_, _Plano-Convex_, or _Plano-Concave_, or _Convexo-Concave_, which sort are usually call'd _Menisci_. But this only to be understood of those Beams which are nearest to the _Axis_ of the _Lens_, so as to occasion no sensible difference by their Inclination thereto; and the _Focus_ here formed, is by _Dioptrick Writers_ commonly call'd the principal _Focus_, being that of use in _Telescopes_ and _Microscopes_.

Let then (in _Fig. 7. Tab. 5._) BEβ be a double Convex _Lens_, C the Center of the Segment EB, and K the Center of the Segment Eβ, Bβ the thickness of the _Lens_, D a Point in the _Axis_ of the _Lens_; and it is required to find the Point F, at which the Beams proceeding from the Point D, are collected therein, the _Ratio_ of Refraction being as _m_ to _n_. Let the distance of the Object DB = DA = _d_ (the Point A being supposed the same with B, but taken at a distance therefrom, to prevent the coincidence of so many Lines) the _Radius_ of the Segment towards the Object CB or CA = _r_, and the _Radius_ of the Segment from the Object Kβ or K = ρ; and let Bβ the thickness of the _Lens_ be = _t_, and then let the Sine of the Angle of Incidence DAG be to the Sine of the refracted Angle HAG or CAφ as _m_ to _n_: And in very small Angles, the Angles themselves will be in the same proportion; whence it will follow that,

As _d_ to _r_, so the Angle at C to the Angle at D, and _d + r_ will be as the Angle of Incidence GAD; and again as _m_ to _n_, so _d + r_ to (_dn + rn_)/_m_, which will be as the Angle GAH = CAφ; This being taken from ACD which is as _d_, will leave (_m - nd - nr_)/_m_ analogous to the Angle AφD; and the Sides being in this Case proportional to the Angles they subtend, it will follow, that as the Angle AφD is to the Angle ADφ, so is the Side AD or BD to Aφ or Bφ: That is, Bφ will be = _mdr_/(_m - nd - nr_), which shews in what Point the Beams proceeding from D, would be collected by means of the first Refraction; but if _nr_ cannot be subtracted from _m - nd_, it follows that the Beams after Refraction do still pass on diverging, and the Point φ is on the same side of the _Lens_ beyond D. But if _nr_ be equal to _m - nd_, then they proceed parallel to the _Axis_, and the Point φ is infinitely distant.

The Point φ being found as before, and Bφ - Bβ being given, which we will call δ, it follows by a Process like the former, that βF, or the focal Distance sought, is equal to

_δρn_/(_m - δ + mρ_) = _f_.

And in the room of δ substituting

Bφ - Bβ = _mdr_/(_m - nd - nr_) - _t_,

putting _p_ for _n_/(_m - n_), after due Reduction this following Equation will arise,

(_mpdrρ - ndρt + nprρt_)/(_mdr + mdρ - mprρ - m - ndt + nrt_) = _f_.

Which Theorem, however it may seem operose, is not so, considering the great Number of _Data_ that enter the Question; and that one half of the Terms arise from our taking in the thickness of the _Lens_, which in most Cases can produce no great Effect; however it was necessary to consider it, to make our Rule perfect. If therefore the _Lens_ consist of _Glass_, whose Refraction is as 3 to 2 'twill be

(_6drρ - 2dρt + 4rρt_)/(_3dr + 3dρ - 6rρ - dt + 2rt_) = _f_.

If of _Water_, whose Refraction is as 4 to 3, the Theorem will stand thus

(_12drρ - 3dρt + 9rρt_)/(_4dr + 4dρ - 12rρ - dt + 3rt_) = _f_.

If it could be made of _Diamant_, whose Refraction is as 5 to 2, it would be

(_(10/3)drρ - 2dρt + (4/3)rρt_)/(_5dr + 5dρ - (10/3)rρ - 3dt + 2rt_) = _f_.

And this is the universal Rule for the _Foci_ of double Convex Glasses exposed to diverging Rays. But if the thickness of the _Lens_ be rejected, as not sensible, the Rule will be much shorter, _viz._

_pdrρ_/(_dr + dρ - prt_) = _f_,

or in Glass

_2drρ_/(_dr + dρ - 2rρ_) = _f_,

all the Terms wherein _t_ is found being omitted, as equal to nothing. In this Case, if _d_ be so small, as that _2rρ_ exceed _dr + dρ_, then will it be - _f_, or the _Focus_ will be Negative, which shews that the Beams after both Refractions still proceed diverging.

To bring this to the other Cases, as of converging Beams, or of Concave Glasses, the Rule is ever composed of the same Terms, only changing the Signs of + and -; for the distance of the Point of Concourse of converging Beams, from the Point B, or the first Surface of the _Lens_, I call a negative Distance or - _d_; and the Radius of a Concave _Lens_ I call a negative Radius, or - _r_ if it be the first Surface, and - ρ if it be the second Surface. Let then converging Beams fall on a double Convex of Glass, and the Theorem will stand thus

- _2drρ_/(- _dr - dρ - 2rt_) = + _f_,

which shews that in this Case the _Focus_ is always affirmative.

If the _Lens_ were a _Meniscus_ of Glass, exposed to diverging Beams, the Rule is

- _2drρ_/(- _dr + dρ + 2rρ_) = _f_,

which is affirmative when _2rρ_ is less than _dr - dρ_ otherwise negative: But in the Case of converging Beams falling on the same _Meniscus_, 'twill be

+ _2drρ_/(+ _dr - dρ + 2rp_) = _f_,

and it will be + _f_, whilst _dρ - dr_ is less than _2rρ_; but if it be greater than _2rρ_, it will always be found negative or - _f_. If the _Lens_ be double Concave, the _Focus_ of converging Beams is negative, where it was affirmative in the Case of diverging Beams on a double Convex, _viz._

- _2drρ_/(+ _dr + dρ - 2rρ_) = _f_,

which is affirmative only when _2rρ_ exceeds _dr + dρ_: But diverging Beams passing a double Concave, have always a negative _Focus_, _viz._

- _2drρ_/(+ _dr + dρ + 2rρ_) = - _f_.

The Theorems for converging Beams, are principally of use to determine the _Focus_ resulting from any sort of _Lens_ placed in a Telescope, between the _Focus_ of the Object-Glass and the Glass it self; the distance between the said _Focus_ of the Object-Glass, and the interposed _Lens_ being made = - _d_.

I here suppose my Reader acquainted with the Rules of Analytical Multiplication and Division, as that + multiplied by + makes the Product +, + by - makes -, and - by - makes +; so dividing + by + makes the Quote +, + by - makes -, and - by - makes +; which will be necessary to be understood in the preceding Examples.

In case the Beams are parallel, as coming from an infinite distance, (which is supposed in the Case of Telescopes) then will _d_ be supposed Infinite, and in the Theorem

_pdρr_/(_dr + dρ - prρ_)

the Term _prρ_ vanishes, as being finite, which is no part of the other infinite Terms; and dividing the Remainder by the infinite Part _d_, the Theorem will stand thus _pρr_/(_r + ρ_) = _f_, or in Glass, _2rρ_/(_r + ρ_) = _f_.

In case the _Lens_ were _Plano-Convex_ exposed to diverging Beams, instead of _pdρr_/(_dr + dρ - prρ_), _r_ being infinite, it will be _pdρ_/(_d - pρ_) = _f_, or _2dρ_/(_d - 2ρ_) if the _Lens_ be Glass.

If the _Lens_ be Double-Convex, and _r_ be equal to ρ, as being formed of Segments of equal Spheres, then will (_pdρr_)/(_dr + dρ - prρ_) be reduced to (_pdr_/(_2d - pr_))_f_; and in case _d_ be infinite, then it will yet be farther contracted to ½_pr_, and _p_ being = _n_/(_m - n_), the focal distance in Glass will be = _r_, in Water 1½_r_, but in Diamant ⅓_r_.

I am sensible that these Examples are too much for the compleat Analyst, though I fear too little for the less Skilful; it being very hard, if possible, in such Matters, so to write, as to give satisfaction to both; or to please the one, and instruct the other. But this may suffice to shew the extent of our Theorem, and how easie a Reduction adapts any one case to all the rest.

Nor is this only useful to discover the _Focus_ from the other proposed _data_, but from the _Focus_ given, we may thereby determine the distance of the Object; or from the _Focus_ and Distance given, we may find of what Sphere it is requisite to take another Segment, to make any given Segment of another Sphere cast the Beams from the distance _d_ to the _Focus_ _f_. As likewise from the _Lens_, _Focus_, and Distance given, to find the _Ratio_ of Refraction, or of _m_ to _n_, requisite to answer those _Data_. All which it is obvious, are fully determined from the Equation we have hitherto used, _viz._

_pdρr_ = _drf + dρf - prρf_,

for to find _d_ the Theorem is (_prρf_)/(_rf + ρf - pρr_) = _d_, the distance of the Object.

For ρ the Rule is

_drf_/(_pdr + df + prf_) = ρ.

But for _p_ will be

(_drf + dρf_)/(_dρr + fρr_) = _p_,

which latter determines the _Ratio_ of Refraction, _m_ being to _n_, as 1 + _p_ to _p_.

I shall not expatiate on these Particulars, but leave them for the Exercise of those that are desirous to be informed in Optical Matters, which I am bold to say are comprehended in these three Rules, as fully as the most Inquisitive can desire them, and in all possible Cases; regard being had to the Signs + and -, as in the former Cases of finding the _Focus_. I shall only shew two considerable Uses of them; the one to find the distance whereat an Object being plac'd, shall by a given _Lens_ be represented in a _Species_ as large as the Object it self, which may be of singular Use in drawing Faces and other things in their true Magnitude, by transmitting the _Species_ by a Glass into a dark Room, which will not only give the true Figure and Shades, but even the Colours themselves, almost as vivid as the Life. In this Case _d_ is equal to _f_, and substituting _d_ for _f_ in the Equation, we shall have

_pdrρ_ = _ddr + ddρ - dpρr_,

and dividing all by _dprρ_ = _dr + dρ - prρ_, that is, _2prρ_/(_r + ρ_) = _d_; but if the two Convexities be of the same Sphere so as _r_ = ρ, then will the distance be = _pr_; that is, if the _Lens_ be Glass = _2r_, so that if an Object be placed at the Diameter of the Sphere distant, in this Case the _Focus_ will be as far within as the Object is without, and the _Species_ represented thereby will be as big as the Life; but if it were a _Plano-Convex_, the same distance will be = _2pr_, or in Glass to four times the _Radius_ of the Convexity; but of this Method I may entertain the Curious at some other Time, and shew how to magnifie or diminish an Object in any proportion assign'd, (which yet will be obvious enough from what is here deliver'd) as likewise how to erect the Object which in this Method is represented inverted.

A Second Use is to find what Convexity or Concavity is required, to make a vastly distant Object be represented at a given _Focus_, after the one Surface of the _Lens_ is formed; which is but a Corollary of our Theorem for finding ρ, having _p_, _d_, _r_ and _f_ given; for _d_ being infinite, that Rule becomes

_rf_/(_pr - f_) = ρ,

that is in Glass _rf_/(_2r - f_) = ρ, whence if _f_ be greater than _2r_, ρ becomes Negative, and _rf_/(_f - 2r_) is the _Radius_ of the Concave sought.