To HIT a MARK, As well upon Ascents and Descents, As upon the Plain of the Horizon: Experimentally and Mathematically DEMONSTRATED.

BY ROBERT ANDERSON.

LONDON: Printed for Robert Morden, at the Atlas near the Royal Exchange in Cornhil. MDCXC.

To the Right Honorable Sir Henry Goodrick, Kn t and Baron t, Lieutenant General of their MAJESTIES Ordnance, And One of His MAJESTIES Most Honorable Privy-Council, AND To the Honorable IOHN CHARLETON, Esq Master-Surveyor of Their MAJESTIES Ordnance, AND ALSO, To the rest of the Principal Officers of the same: This New Work Is Humbly Dedicated By Your most Observant, ROB. ANDERSON.

To all Ingenious Persons that delight in the TRUE USE OF Great Artillery.

IN the Year 1673. I publish'd my Genuine Use of the Gun, viz. To strike a Mark at demand, within the reach of the Piece Analytically, upon the Principles of Galileus. In the Year 75. Monsieur Blondell propounded that Problem to the Royal Aca­demy of Sciences of France to be resolved Geometrically, Mons. Buot, Romer the Sweed, and Monsieur de la Hire set upon the Mathema­tical part, whilst Monsieur Mariote and Perrault, with others, im­ployed themselves about Experiments of several sorts, yet not of the Gun, which is the chief end of that Problem, Monsigneur the Dauphin being present.

In the Year 86. Mr. Halley, in Transaction Number 179. gives a Resolution of the same Problem.

All these purely Mathematical, without the consideration of any Accidental Impediments.

Galileus tells us the beginning of the Parabola will be deformed, by reason of the Impulse of Fire: And the latter end, by reason of the Resistance of the Air, which amounts to very little, saith he.

Cavalierus bids us begin the Parabola where the Force leaves the thing projected. And this Line I call the Line of Impulse of Fire, and take it for a right Line for ease of Calculation; although I believe the thing projected moves as it can so far as the Impulse of Fire, or violent [Page] shake of the Engin is upon it, and the more irregular the thing projected is, the longer the Line will be before it passeth into its Parabola.

In these last 15 Years I have made more than a Thousand Experi­ments (and am ready to make as many more to illustrate these Truths) altho I set down so few; by which I find the Impulse of Fire, and so carry the Work on upon the Principles of Galileus, viz. To hit a Mark not only upon the Plain of the Horizon, but upon Ascents and Descents in the Parabola, according to Mathematical Principles. For ease of the Mortar-Piece, I have made a Table of Horizontal Ranges, and the like may be done for Ascents and Descents, if it were necessary; Also, by the Experiments of Mr. Eldred and my own compar'd together, I have made Tables of Ranges for Long Guns to 8 degrees of Elevation, which I am apt to believe comes near the matter; that so when proved and approved, the said Ranges may be graved upon the Guns; so their Use will become very easie and exact. Who ever go about to prove or disprove these things, I would have them qualified with these three Qualifications, viz. Art, Care, and Honesty, and not to stand a tip-toe to see farther than Nature and Art have made them capable. To make the Work more perfect, I have given a short Discourse of Granadoes, Fusees, Carcasses, and Fire-Balls. The service of War being ended, we turn our Guns into Pipes of Musick, playing the Charms of a Soldiers delight, in Consorts of Musical Instruments, where Sound and Time are actually employed.

Last of all, Guns in Geometrical Proportion, beautiful in shape, and delightful in sound, which might delight the Minds of noble Souls. These things I have done for the Truth, which has been my Recreation, and at my own Charge, and now may be useful, if faithfully put in practice, which I am apt to believe will not be corrected in this Age, nor perhaps in the next.

In the Mortar-Piece, I verily believe, there cannot be three Minutes got; nor in the Long Guns ten, perhaps not five Minutes, taking every thing in its right sense, which I shall ever endeavour to do, whilst

I am R. A.

ON THE Most Ingenious AUTHOR, Mr. ROB. ANDERSON, and his New Work.

AMongst the several Parts of Mathematical Elements, the Sections of a Cone, and Algebra, have ever been ac­counted the most abstruse and curious: But the adapt­ing these to explicate the Theories of heavy Bodies forc'd from Engins, and the like, is that wherein the Age preceding ours, exceeded all that went before it, by the almost immense and universal Wit of Galileus: His Scholar Torricellio succeeded him, and with most admirable Felicity and Learning render'd those matters easie, which his Master had made Mathematical. But, To hit a Mark (within the Pieces reach) in any Angle above or under the Horizontal level of the Piece, was what they could not over­come. And Torricellius does professedly avow the Difficulties thereof to be almost insuperable.

But this, our Author, by his extraordinary Abilities in Ma­thematicks, happy Address, and indefatigable Pains and Dili­gence, has not only overcome, but further added many useful things, to render this New Science profitable.

Those who have written on this Subject since the Author's Printing of his Genuine Use of the Gun, &c. 1673. have all con­firm'd what he first publish'd to the World. Nor can this Do­ctrin meet with any Opposition, except from those who nei­ther understand the Laws of Motion, nor the Nature of that Line wherein a Bullet moves.

[Page]His long and great Experince (having in Twenty Years made many Thousand Shots, with greater Accuracy each, perhaps, than Men of common Practice ever have made one); his great Curiosity, Expence, and Labor; of all which this present Piece of his is an undoubted Instance; but most of all his Zeal to be­nefit the Publick, without regard of profiting himself, renders him meritorious, not only of the Thanks and Praises of his Friends, but as he truly is an Honor to our Age and Nation, the same is due to him from all Mankind. The matter of this present Tract being wholly new, it could not reach that Length at which Collectors aim, who write the Histories (as I may say, the Life and Death) of such or such a Science: But it truly seems to be all that was wanting to complete the Art.

Now having its Foundation laid in Scientifique Certitude, confirmd by much and long Experience, and thus explain'd by Numbers; it neither needs nor can have greater Commenda­tion, other than reading of the Book it self, from which I shall no longer keep you.

B. M.

TO Hit a MARK.

PROPOSITION I.

GALILEUS Dialogo Quarto, Prop. I. Projectum dum fertur motu composito ex horizontali aequabili, & ex naturaliter accelerato deorsum, lineam Semiperabolicam deseribit in sua latione.

If a thing be projected in a Motion compounded of an equal Horizontal, and naturally accelerated motion downward, it describeth in its passage a Semiparabolical Line.

And from Page 180 to 190, Discourses of the Accidental Impediments, viz. Of the Roundness of the Earth; The Impediment of the Air; and The Impulse of Fire. Here I shall endeavour to enquire what the Quality and Quan­tity of these accidental Impediments are.

PROP. II.

EXperiments made with a Mortar-piece well fixed, the Diameter 4 Inches, the length of the Chase 2 Diameters, the Chase and Ball turned very fit.

On Wimbleton Heath, August 24. 1676.
½ at 15 deg. of Elevation Ranged the Shot 1336 1357
1378
¾ at 75 deg. of Elevation Ranged the Shot 1232 1337
1442
½ at 10 deg. of Elevation Ranged the Shot 539 538
537
¾ at 80 deg. of Elevation Ranged the Shot 514 507
500
Septemb. 17. 1677.
1 at 15 deg. of Elevat. Ranged the Shot
901
2 at 75 deg. of Elevat. Ranged the Shot
848
3 at 00 deg. of Elevat. Ranged the Shot
125
The Shot fell lower than the Centre of the Mouth of the Piece
2.35
[Page 2]
Lo Specchio ustorio, Bonaventura Cavasieri, Pag. 99.

Dico adunque, che I gravi spinti dal proiciente à qualsivoglia banda, fuorche per la perpendicolare all' Orizonte, separati che siano da quello, & esclusa l'impedimento dell' ambiente, descrivono una linea curva, insensibilinente differente dalla Parabola.

I say then, that heavy Bodies being projected by any force, except it be per­pendicular to the Horizon, and excluding the Impediment of the Medium, do after their separation from that force describe a curve Line insensibly differing from a Parabola.

PROP. III. PROB. I.

TWO Ranges made upon the plain of the Horizon, equi-distant above and below 45 deg. of Elevation, with the same Piece, Ball, and quantity of Powder being given. To find the Line of impulse of Fire, and greatest Range in the Parabola.

[figure]

Let R be the Axis on which the Piece moves, in Fig. 1.

The Range RI at 15 deg. of Elevation
901 the half 450.5=B
The Range RX at 75 deg. of Elevat.
848 the half 424=C
The sine of the Angle RFZ
=S
The sine of the Angle R nr
= s
The Line of Impulse of Fire RF
the thing sought =Z

Then as Radius: Z::S:RZ in the Diag. that is [...] in Diag. and [...] in the Diag. then [...], Galileus Dialog. quart. Prop. oct. that is, RB−ZS=RC−Z s, that is [...], that is, [Page 3] Half the difference of the Ranges multiplied by Radius, divided by the difference of the sine Complement of Elevations, the Quote is the Radius RF=Z the thing sought, the Line of Impulse of Fire.

Here we take RO to be equal to OI, which is not exactly so; yet neverthe­less it is sufficient for the purpose.

Example.

½ The difference of the Ranges in Radius 2650000 6.423246
The difference of the sines of 15 and 75 deg. 70711 sub 4.849487
The Radius of the Circle RF=Z the Line of Impulse of Fire 37.48 1.573759

To find the greatest Range in the Parabola.
As the Radius 90 10.000000
is to the sine of 15 9.412996
so is the Radius RF 37.48 1.573759
to R r 9.7 0.986755
As the Radius 90 10.000000
is to the sine of 75. 9.984944
so is the Radius RF 37.48 1.573759
to RZ 36.2 1.558703

½ the Range on the Horiz. at 75° 424 ½ the Range on the Horizon at 15° 450.5
R r sub 9.7 RZ subst. 36.2
ND 414.3 FV 414.3
The Range at 75° NE, in Parab. 828.6 equ. to the Ran. FG at 15° in Parab. 828.6

Then the double of NE or FG 1657.2 the greatest Rang in the Parabola.

PROP. IV. PROB. II.

THE Piece put parallel to the Horizon at R, Ranged the shot RMC, that is, the horizontal distance AC, the descent RA equal to MB, with the Range at 15 deg. of Elevation upon the plain of the Horizon RI being given. To find RM the Line of Impulse of Fire.

The Range RI at 15 deg. of Elevat. 901 the half 450.5=B
The Piece put parallel to the Horizon Ranged the shot 125 AC=A
RA the height of the Piece above where the shot fell 2.35=C
The sine of the Angle RFZ =S
Radius =R
RM=AB the Line of impulse of Fire, the thing sought =Z

[Page 4]Then [...] in the Diag. [...] in the Diag. Then A−Z its square A 2−2AZ+Z 2, that square divided by 8C is [...] in Diagr. so then [...], that is RA 2−2RAZ+RZ 2=8CBR−8CSZ, that is, [...]. In Numbers thus, Z 2=231.84 Z=7155.6 its square Root is 36.67=Z=RF the Line of Impulse.

Here note, if A 2−2AZ+Z 2 be divided by C, the Quote will be the double of the greatest Range in the Parabola; if by 2C the greatest Range, if by 4C the half of the greatest Range, if by 8C a quarter of the greatest Range in the Pa­rabola FV.

PROP. V. PROB. III.

THE horizontal distance, and the descent being given. To find the dou­ble of the greatest Range in the Parabola.

BM, 2.35. AC, 125 less AB, 3667. Their remains BC, 88.33 being given. To find the greatest Range in the parabola.

BC 88.33   1.946108
    squared 3.892216
BM 2.35 sub 0.371068
The double of the greatest Range which we call G 3320   3.521148
The half thereof is 1060 the greatest Range.

PROP. VI. PROB. IV.

THE horizontal distance, the descent, and the double of the greatest Range in the Parabola being given. To find the Line of impulse of Fire.

AC 125=A
RA equal to MB 2.35=C
G the double of the greatest Range 3320 being given.
To find RM the Line of impulse=Z.  

Then A−Z, squared is A 2−2AZ+Z 2=GC, that is, [...]

In Numbers thus Z 2=250Z−7823. Whose square Root is 36.68=Z=RF the Line of Impulse.

A method to extract the Roots of square Equations.

Take half the number of the coefficient. To the square of that half, add or sub­stract the absolute number, according to the Sign + or −, then extract the [Page 5] square Root of that sum or difference, which Root added to or substracted from the half coefficient, the sum or difference will be the Root of the Equation.

Example.

Take half of 250 (which is called the Coefficient) 125 its square is 15625 from that square substract the absolute Number 7823 by the sign−, the re­mainder is 7802, the Root thereof 88.32, substracted from 125 the remainder is 36.68 the Root required.

In these Experiments at 0 degree of Elevation, and at 15 deg. of Elevation there could be very little resistance of the Medium; therefore it is the impulse of Fire carrying the Ball in a right line, so far as the blast of the Piece is upon it, which after its separation from that force, passeth into the Curve of a Parabola.

By this means the point N is removed from the point F, which is one, if not the chief reason, that RX and RI are not equal upon the plain of the Ho­rizon.

PROP. VII. PROB. V.

[figure]

THE three last Propositions, viz. the 4th. 5th. and 6th. may be done Geo­metrically. Thus, if (in the 4th.) BC in the Diagr. be made equal to 8C in the equation, and CD in the Diag. be equal to B in the equation, then CE is a mean proportion. If EA in the Diag. be equal to A in the equation, and AF in the Diag. be equal to half the Coefficient in the equation, then HA and GA will be the Roots of the equation, and GA equal to 36.67.

If we take mean proportions between 2A and R, and 8C and S in the Coefficient part of the equation, the construction will be more compleated.

In the 5th.

If BC in the Diagr. be made equal to BM in the 5th. Prop. and CE in the Diagr. be equal to BC in the 5th. Prop. Then CD in the Diagr. a 3d. proportion will be equal to G the double of the greatest Range in the Parabola.

In the 6th.

If BC in the Diagr. be made equal to C in the equation, and CD in the Diag. be equal to G in the equation, then CE is a mean proportion, make EA in the Diag. equal to A in the equation, then IE, and KE are the Roots of the equation, and KE=36.67.

PROP. VIII. PROB. VI.

IF two shot be made at the same degree of elevation, with different quanti­ties of Powder, the corresponding parts of the Ranges will be in propor­tion, sufficiently near the truth. And to find the parts simile.

Iune the 5th. 1677. On Wimbleton-Heath, a Mortar-piece whose diameter of bore 4 inches, and the length of the Chase 2 diameters, charged with 4 ounces of Powder, and laid to 15 deg. of elevation, Ranged its shot to the horizontal distance of 659 paces.

By this, and what is done by the Experiments before cited, the qualifications of all Mortar-pieces are known, being of the same kind.

[figure]

Example.

As RI is to RF fig. 1. the Range at 15 deg. of elevation 901 2.954725
the Line of impulse 37.48 1.573759
so is AB to AD fig. 3. the Range at 15 deg. of elevation 659 2.818885
the Line of impulse 27.41 1.437919
As RI is to FG fig. 1. the Rang. at 15 deg. of elevat. 901 on the Hor. 2.954725
the Rang. at 15 deg. of elevat. 8286 in the Parab. 2.918345
so is AB to DE fig. 3. the Rang. at 15 deg. of elevat. 659 on the Horiz. 2.818885
the Rang. at 15 deg. of elevat. 606 in the Parab. 2.782505

DE 606 the Range at 15 deg. of elevat. in the Parabola, the double of it is 1212 Paces the greatest Range at 45 deg. of elevat. in the Parabola.

PROP. IX. PROB. VII.

THE Line of impulse, the greatest Range in the Parabola, and the An­gle of elevation being given. To find the Range upon the plain of the Horizon.

[figure]

In the right angled Triangle ABC, there is given the Angle BAC, AD the line of impulse, and G the double of the greatest Range in the Para­bola, with this qualification, as G:DC::DC:CB. To find AB the hori­zontal Range.

BAC=S sine 70 93969
R= 100000
G= 2424
AD= r 27.41
DC=Z 2304.89

Then, as R: r+Z::S:BC in diag. that is [...] in diag. and [...] in diag. by Lemma to the 23 Prop. De motu projectorum Liber Secundus, Torricel. Therefore [...] That is Z 1R=ZGS+G rS. In Numbers thus, Z 2=2277.80856Z+62434.7326296, the Root is 2304.89=DC. Then

DC 2304.89+DA27.41=AC2332.3 3.367784
sine 70 9.534052
AB 797.69 2.901836

A Second way by Lines thus, Geometrically.

In Fig. X. If AC be made Radius, and CB the sine of the Angle of

[figure]

elevation, and AD the Line of impulse, draw DE paral­lel to CB, then DE will be a fourth proportion by 12.6 Euclid. Then make VL in Fig. Y. equal to DE in Fig. X, and LE in Fig. Y, equal to the double of the greatest Range; the Line LN will be a mean proportion, 13.6. Euclid. Further, if AC in Fig. X. be made Radius, and CB the sine of the Angle of elevation, and AD the greatest Range from the point D draw DE parallel to BC, DE will be a fourth proportion; Lastly, make OL in Fig. Y. equal to DE in Fig. X. then draw the Line NOM, and it will be the Line required, viz.=Z in the equation, and DC in the first Fig. of this Proposition.

A Third way by Logarithms.

G=2424 3.384533 G=2424 3.384533
S=93969 4.972985 S=93969 4.972985
r=27.41 1.437909   8.357518 from
  9.795427 from 2R=200000 5.301030 subst.
R=100000 5.000000 subst. OL 1138.9 3.056488 subst.
  4.795427 rest LN+Radius 12.397713 from
LN= 2.397713 half LON tang. 12.22.30 9.341225 rest
    LON sine 12.22.30 9.331041 subst.
    LN+Radius 12.397713 from
    ON 1165.9 3.066672 rest
    LO+ON 2304.8 viz.=Z in the Equation, and DC in the first Fig. of this Proposition.

A Fourth Way shall be shewed, where we come to calculate the horizontal Ranges.

PROP. X. PROB. VIII.

THE Horizontal distance, the Line of Impulse, the double of the greatest Range in the Parabola being given. To find the Angle of Elevation, to hit an Object at the Horizontal distance given.

[figure]

In the Right angled triangle ABC, AB the Horizontal distance, AD the Line of Impulse, the Line G the double of the greatest Range, with this qua­lification, as G:DC::DC:CB being given. To find the Angle BAC, the angle of Elevation, to hit the Object at B, at the Horizontal distance AB.

AB=H= 264  
G= 568.867 To find the Roots Z=487.3=DC Paces.
AD= r 6.534 Z=302.8=DC

Then [...] in the Diagr. r+Z=AC in Diagr. squared is [...] I. Euclid. that is [...]

That Equation in Numbers.

−Z 4+323609.663689Z 2+4228931.085087852Z=22540483202.613561987516

The Roots are had, by the usual method of extracting mixt Powers, which will be very easie to any ingenious Person, with a little practice; the lesser Root is greater than EB, and lesser than DC at 45 deg. of Elevation. The greater Root DC, is greater than DC at 45 deg. of Elevation, and lesser than DC, at the Complement of the lower Elevation from the Zenith.

DC 302.8 DC 487.3
AD 6.534 AD 6.534
AC 309.334 AC 493.834

As AC 309.33 2.490422
is to the Radius 90 10.000000
so is AB 264 2.421604
to the sine of the angle BAC 31° 24′ 30″ 9.931182
As AC 493.83 2.490422
is to the Radius 90 10.000000
so is AB 264 2.421604
to the sine of the angle BAC 57° 41′ 9.728027

On Wimbleton-Heath,
1689. Iune 17. 1. at 10 deg. of Elevation the shot Ranged 610
  2. at 78.56 of Elevation the shot Ranged 616
  3. at 15 deg. of Elevation the shot Ranged 680
  4. at 73.57 of Elevation the shot Ranged 724
August 13. 1. at 15 deg. of Elevation the shot Ranged 694
  2. at 15 deg. of Elevation the shot Ranged 680
  3. at 73.57 of Elevation the shot Ranged 711
  4. at 73.57 of Elevation the shot Ranged 682
  1. at 31.24 of Elevation the shot Ranged 1953
  2. at 31.24 of Elevation the shot Ranged 1932
  3. at 57.41 of Elevation the shot Ranged 1975
  4. at 57.41 of Elevation the shot Ranged 2020
  1. at 31.24 of Elevation the shot Ranged 1077
  2. at 31.24 of Elevation the shot Ranged 1105
  3. at 57.41 of Elevation the shot Ranged 1077
  4. at 57.41 of Elevation the shot Ranged 1135

Here note, If the Horizontal distance be greater than the greatest Range of the Piece; the Object is beyond the reach of the Piece.

[Page 11]

On Wimbleton-Heath, June 26. 1689.
1. at 15 deg. of Elevation the shot Ranged 1149
2. at 15 deg. of Elevation the shot Ranged 1197 −1173
3. at 75 deg. of Elevation the shot Ranged 1104
4. at 75 deg. of Elevation the shot Ranged 1102 −1103

In Fig. I.

½ the difference of the Ranges in Radi 3500000 6.544068
the difference of the Sines of 15 and 75 deg 70711 subst. 4.849487
the Radius of the Circle RF the line of Impulse 49.5 1.694581
As the Radi 90 10.000000
is to the sine of 15° 9.412996
so is the Radi RF 49.5 1.694581
to R r 12.8 1.107577
As the Radi 90 10.000000
is to the sine of 75° 9.984944
so is the Radi RF 49.5 1.694581
to RZ 47.8 1.679525

½ the Rang. on the Hor. at 75° 551.5 ½ the Rang. on the Horiz. at 15° 586.5
R r subst. 12.8 RZ subst. 478
ND 538.7 FV 538.7
the Ran. at 75° NE in the Par. 1077.4 equ. to the Ran. FG at 15° in Par. 1077.4

As RI is to RF Fig. I. the Range at 15 deg. of Elevation 901 2.954725
the Line of Impulse 37.48 1.573759
In this Experiment the Range at 15 deg. of Elevation 1173 3.069298
the Line of Impulse 48.8 1.688332

Here may be seen, the fair agreement of the Parabola, with Experiments:

For 49.5 less 48.8 is but seven Tenths.

Upon Ascents and Descents. PROP. XI. PROB. IX.

THE angle of Elevation, the double of the greatest Range, the Ascent or Descent being given. To find the Horizontal distance.

[figure]

In the right angled Triangle ABC, the Angle BAC, the double of the great­est Range G, and the Ascent BE, with this qualification; as G:AC::AC:CE being given: To find AC=Z.

Then, as R:Z::S:BC in Diagr. that is [...] in Diagr. Then [...] in Diag. Then [...] in Diag. that is Z 2R=ZGS−GPR.

G= 2424
BE=P= 120
the sine of 80=S= 98581
Radius=R= 100000
AC=Z= 2258.37

In Numbers thus,

Z 2=2387.17944Z−290880 whose Roots are 2258.37=AC=Z, and 128.8=AC=Z.

[Page 13]By Logarithms thus,

G=2424 3.384533 G=2424 3.384533
P=120 2.079181 S=98481 4.993352
½ 5.463714   8.377885 from
LO+Radius is 12.731857 2R=200000 5.301030 subst.
  HO= 1193.6 3.076855 subst.
  LO+Radius 12.731857 from
  sine LHO 9.655002 rest
  co-sine LOH 9.950405
  HO=1193.6 3.076855
  HL= 1064.8 3.027260
  HO+HL=2258.4=AC=Z  
  HO−HL=128.8=AC=Z  

Radius 10.000000
AC 2258.4 3.353801
BCA sine 10° 9.239670
AB 392.17 2.593471
AB the Horizontal distance.  

By Lines thus,

[figure]

In Fig. Z, if VL be made equal to P, and LE equal to G, then LO will be a mean Proportion. Further, in Fig. X, if AC be made equal to Radius, and BC equal to the sine of the Angle of Elevation; as also, AD be made equal to ½ G, that is equal to the greatest Range in the Parabola, then DE will be a fourth Proportion. Further, from the point O in Fig. Z, make OH equal to DE in Fig. X, continue the Line OH to I. Lastly, set one point of the Compasses in H, and at the distance HL, sweep the Arch ILR, then are IO and RO the Lines required.

PROP. XII. PROB. X.

THE Angle of Elevation, the double of the greatest Range, and the Descent being given. To find the Horizontal distance.

[figure]

In the right angled Triangle ABC, the Angle BAC, the double of the greatest Range G, and the descent BE, with this qualification as G:AC::AC:CE being given. To find AC=Z.

Then, as R:Z::S:BC in Diagram that is [...] in Diagr. Then [...] in Diag. Then [...] in Diagr. that is Z 2R=ZGS+GPR.

G= 2424
BE=P= 120
The sine of 80=S= 98481
Radius=R= 100000
AC=Z= 2503.37

In Numbers thus,

Z 2=2387.17944Z+290880, whose Root is 2503.37.

By Logarithms thus, [...]

[Page 15]

Radi 90 10.000000
AC 25034 3.398530
BCA sine 10 9.239670
AB 434.7 2.638200 AB the Horizontal distance.

By Lines thus.

In Fig. X and Z, if VL be made equal to P, and LE equal to G, then LO will be a mean Proportion. Further, in Fig. X, if AC be made equal to Radius, and CB equal to the sine of the Angle of Elevation; as also, AD be be made equal to ½G then DE will be a fourth Proportion. Again make LH in Fig. Z, equal to DE in Fig. X, draw the Line OHI, set one foot of the Compasses in H, and at the distance HL strike the Arch LI, then IO will be the Line required.

PROP. XIII. PROB. XI.

THE Line of Impulse, the double of the greatest Range in the Parabola, the Ascent or Descent, and the Angle of Elevation being given. To find the Horizontal distance.

In the right angled Triangle ABC, the Angle

[figure]

BAC, the double of the greatest Range G, the Line of Impulse AB, and the perpendicular Ascent BE, with this qualification, as G:DC::DC:DE, being given. To find DC=Z.

Then R: r+Z::S:BC, that is [...] in Diag. [...] in Diagr. then [...] in Diag. that is [...] i. e. [...]

G= 2424
BE=P= 120
AD= r= 2741
sine of 80=S= 98481
Radius=R= 100000
DC=Z= 2288.67

In Numbers, thus,

Z 2=2387.17944Z−225447.4115496 whose Roots are 2288.67 and 98.51.

[Page 16]By Lines thus,

[figure]

Upon an Ascent. In Fig. Y, if AC be made Radius, and CB the sine of the Angle of Elevation, and AD the Line of impulse, and DE drawn pa­rallel to CB, then DE will be a fourth Proportion. In Fig. Z, if OD be made equal to DE in Fig. Y, and DE in Fig. Z be made equal to the dou­ble of the greatest Range in the Parabola, then DC will be a mean Propor­tion. Further, if RC be made equal to P the perpendicular Ascent, and CB equal to the double of the greatest Range, then CA will be a mean Proportion. Again, in Fig. Y, if AC be made equal to the Radius, and CB equal to the sine of the angle of Elevation, and AD the greatest Range, then DE will be a fourth Proportion, make AE in Fig. Z, equal to DE in Fig. Y, then LA and HA are the Lines desired.

By Logarithms thus,

G Log. 3.384533 G Log. 3.384533 G Log. 3.384533  
r 1.437909 S 4.993352 P 2.079181  
S 4.993352   8.377885   ½ 5.463714  
  9.815794 2R 5301030 AC is 2.731857 subst.
R sub. 5.000000 AF 1193.6 3.076855 DC+Ra. 12.407897 from
  4.815794       9.676040 si. of DAC
DC 2.407897 ½       9.944661 si. of DCA
Lastly.     AC 2.731857  
AD 27.41     AD from 12.676518  
DC 2288.7     AF subst. 3.076855 1193.6
AC 2316.1 3.364757   rest 9.599663 si. of AFD
ABC 10 sine. 9.239670   add AF 9.962590 si. of DAF
AB 402.2 2.604427     3039445 1095.1
the Horizontal distance desired.   AL the Number 2288.7 de.

PROP. XIV. PROB. XII.

THE Line of Impulse, the Descent, the greatest Range in the Parabola; the Angle of Elevation being given. To find the Horizontal distance.

[figure]

In the right angled Triangle ABC, the Angle BAC, AD the Line of Impulse, and G the double of the greatest Range in the Parabola being given, with this qualification G:DC::DC:CE. To find AB the Horizontal distance.

Then R: r+Z::S:BC that is [...] in Diagr. [...] in Diagram.

Then [...] in Diagr. that is [...] that is [...].

G= 2424
BE=P= 120
the Sine of 80=S= 98481
Radius=R= 100000
AD= r= 27.41
DC=Z= 2528.11

In Numbers thus,

Z 2=2387.17944Z+356312.5884504 the Root is 2528.11.

[...]
[...]
[...]
[...]

[Page 18]By Lines thus,

[figure]

In Fig. Y, if AC be made Radius, and CB, the sine of the Angle of Ele­vation; and AD the Line of impulse; then DE will be a fourth Proportion. In Fig. X. If OD be made equal to DE in Fig. Y, and DE in Fig. X, be made equal to the double of the greatest Range, then DC will be a mean Proportion. In Fig. X, if DR be equal to the Descent, and DB equal to the double of the greatest Range, DA will be a mean proportion. Then draw the Line AC. Further, in Fig. Y. If AC be Radius, and DC the sine of the Angle of Elevation, and AD the greatest Range, then DE will be a fourth proportion. In Fig. X, draw CF (at right Angle to AC) equal to DE in Fig. Y, then set one point of the Compasses in the point F, and take the distance FC, sweep the Arch CL, then LA will be the Line desired.

By Logarithms thus,

G Log. 3.384533 G Log. 3.384533 G Log. 3.384533
S 4993352 S 4.903352 P 2.079181
r 1.437909   8.3 [...]7885   ½ 5.463714
9.815794 2R sub. 5.301030 DA is 2.731857
R subst. 5.000000 GF 1193.6 3.076855    
DC ½ 4.815794        

DC+Radi. is 12.407897 from      
DA 2.731857 subst.      
Tang. DAC 9676040.      
sine DAC 9631992 subst. AD 27.41  
DC+Radi. 12.407897 from DC 2528.1  
AC 2.775905 subst. AC 2555.5 3.407476
FC+Radi. 13.676855 from ACB sine 10 9.239670
Tang. CAF 10.300950 AB 443.8 2.647146
sine CAF 9.951528 subst.      
AF 1334.5 3.125327      
CF 1193.6        
AL 2528.1 the Numb. desired.      

Ascents. Descents.
AB 402.2 with the Line of Impulse. AB 443.8 with the Line of Impulse.
AB 392.2 without the Line of Impul. AB 434.7 without the Line of Imp.
  10.0 the Horizontal difference.   9.1 the Horizontal difference.

PROP. XV. PROB. XIII.

THE greatest Range, the Horizontal distance, Perpendicular Ascent, and Line of Impulse being given. To find the Angle of Elevation; to hit the said Ascent at the Horizontal distance given.

In the right angled Triangle ABC: the double

[figure]

of the greatest Range in the Parabola G, the Hori­zontal distance AB, the Ascent BE, the Line of Im­pulse AD, with this qualification G:DC::DC:CE, being given. To find the Angle BAC.

Then [...] in Diag. Further, [...] in Diagr. that squared is [...] that +H 2 is equal to the square of r+Z, that is r2+2Z r+Z 2 that is [...]. 47. I. Euclid. that is [...]

G= 2424
AB=H= 392.2
BE=P= 120
AD=R= 27.41
Radius=R= 100000
DC=Z= 2290.6

In Numbers thus, Z4−5294016Z 2−322110040.32Z=−984013457269.2544.

Whose Root is 2290.6=DC for the upper Elevation.

Then,

DC 2290.6  
AD 27.41  
AC 2318.01 subst. 3.365113
Radius 90 10.000000
AB 392.17 2.593474
Co-sine of the Angle ACB 80 o 15′ 40″ 9.228361
The Angle of Elevation to hit the Object at E.  

PROP. XVI. PROB. XIV.

THE double of the greatest Range in the Parabola, the Horizontal distance, the Perpendicular Ascent or Descent being given. To find whether the Object be within the Reach of the Piece or not.

The double of the greatest Range

[figure]

EY, the Horizontal Distance AB, the Perpendicular Ascent BC being given. To find whether the Object C be within the Reach of the Piece or not. Let AF be the greatest Range, AE half thereof, Let ELF be a Simi-Parabola, then all the Projections from the Point A made with the same force will touch the Parabola ELF. Toricel. de Mot. Proj. lib. 2. Prop. 30.

Then EY equal to 2AF, or 4AE in EO equal to the square of OL. Appol. Coni. Lib. 1. Prop. 11. Therefore if the Horizontal Distance AG be equal to OL, it is in the farthest extent of the Projection; if it be less as AB, the Point C is within; but if greater, as AI, the Point N is without, therefore beyond the Reach of the Piece, that is, if half the greatest Range, less the Altitude of the Object in Ascents, more in Descents, be multiplyed by the Parameter (equal to the double of the greatest Range) whose square Root, if less than the Hori­zontal distance, the Object is out of the Reach of the Piece.

AB= 392.17 OE= 486 2.686636
EY= 2424 EY= 2424 3.384533
AF= 1212     6.071169
AE= 606 OL= 1085.4 3.035584
AO= 120      
AB= 392.17 less than OL=1085.4 therefore the Object at C is within the Reach of the Piece.

If any find the XV. Proposition tedious, they may make use of the Tables in my genuine Use of the Gun; for the difference is not much, as appears in the Close of the XIV. Proposition; the difference is 10 in the Ascent and 9 in the Descent, in that place; and in the XV. Proposition it is 16 minutes.

By the Tables in my genuine Use of the Gun.
Greatest Rang. G 1212 sub. 3.083503 greatest Range G 1212 sub. 3083503
Greatest Rang. G 2000 3.301030 greatest Range G 2000 3.301030
Horizont. dist. H 392.17 2.593474 Perpend. Ascent P 120 2.0 [...]9181
Horizont. dist. H 647.15 2.811001 Perpend. Ascent P 198 2.296708

Then entering the Tables with the Horizontal distance 647.15 and Perpendi­cular Ascent 198, and you will find the Elevations 80 deg. and 27. deg.

[Page 21]Thus have I given a Method to strike any place at demand within the Reach of the Peice, as well upon Ascents and Descents, as upon the plain of the Horizon, by Experimental and Mathematical demonstration. It remains now to say something of Mortar-Pieces, Granadaes, Fusees, Bombs, Carcasses, and Fire-Balls, &c.

About 70 or 80 degrees of Elevation is the best place to work a Mortar-Piece for service: Therefore I have given Rules from the Elevation given, to find the Horizontal distance, which is as necessary as the Converse. These Exam­ples are of both kinds, viz. as well upon Descents as Ascents: As also, with the Line of Impulse and without it, that the difference may be seen: But this is to be sure, that, with the Line of Impulse, will come nearest the Mark.

In these Examples I have made use of the Ascent of Edinburgh Castle, which is the highest Ascent, if below, or lowest Descent if above, of any Garrison in this part of the World.

If Mortar-Pieces were all similis, and their requisits of Powder as the Cube of the Diameter of their Bores, and their Granadoes, Bombs, Carcasses, and Fire-Balls were similis, as they ought to be, their Ranges upon the plain of the Horizon under the same deg. of Elevation would be equal, comparing like with like: So one Piece being well proved, that is, the Range of the Granada, Bomb, Carcase, and Fire-Ball being found to any degree of Elevation, the whole Work of a Mortar-Piece would become very easie and exact.

Considering they are not similis, there is required the Range of the Piece, at any convenient deg. of Elevation, with its requisite of Powder, then proceed by the Tables.

A Table of Horizontal-Distances.

de. [...]i nu. dif. de. m [...] nu. dif. de. mi. nu. dif. de mi. nu. dif. de. mi. nu. dif. de. mi. nu. dif.
      521       145       82       4       89       151
  30 521     30 5406     30 8851     30 9996     30 8523     30 4808  
      190       144       79       7       92       152
    711   16   5550       8930   46   9989   61   8431   76   4656  
      181       142       76       10       94       154
  30 892     30 169 [...]     30 9006     30 9979     30 8337     30 4502  
      177       149       73       13       97       155
2   1069   17   5832   32   9079   47   9966   62   8240   77   4347  
      175       138       71       16       99       157
  30 1244     30 5970     30 9150     30 9950     30 8141     30 4190  
      173       132       69       19       101       157
    1417   18   6107       9219   48   9931   63   8040   78   4033  
      172       135       65       22       103       159
  30 1589     3 [...] 6242     30 9284     30 9909     30 7937     30 4874  
      170       133       62       25       107       160
4   1759   19   375   34   9346   49   9884   64   7830   79   3714  
      170       131       61       28       110       161
  30 2929     30 6506     30 9407     30 9856     30 7720     30 3553  
      168       129       57       31       111       162
5   2097   20   663 [...]   35   9464   50   9825   65   7609   80   3391  
      16 [...]       128       54       34       113       16 [...]
  30 2264     30 6763     30 9518     30 9791     30 7496     30 3228  
      167       126       51       36       116       165
6   2431   21   6889   36   9569   51   9755   66   7380   81   3063  
      166       123       49       40       118       165
  30 2597     30 7012     30 9618     30 9715     30 7262     30 2898  
      166       122       46       42       120       166
7   2763   22   7134   37   9664   52   9673   67   7142   82   2732  
      164       119       43       45       121       166
  30 2927     30 7253     30 9707     30 9628     30 7021     30 2566  
      163       117       40       49       125       168
8   3090   23   7370   38   9747   53   9579   58   6896   83   2398  
      193       115       37       51       126       168
  30 3253     30 [...]485     30 9784     30 9528     30 6770     30 2230  
      162       113       34       53       128       16 [...]
9   3415   24   [...]598   39   9818   54   9475   69   6642   84   2061  
      160       111       32       58       131       17 [...]
  30 3575     30 [...]709     30 9850     30 9417     30 6511     30 1891  
      159       108       28       59       132       17 [...]
10   3734   25   [...]817   40   9878   55   [...]358   70   6379   85   1721  
      158       106       26       6 [...]       134       17 [...]
  30 3892     30 [...]923     30 9904     30 9295     30 6245     30 1550  
      158       104       22       65       136       171
11   4050   26   8027   41   9926   56   9230   71   6109   86   1379  
      156       102       20       6 [...]       138       17 [...]
  30 [...]206     30 8129     30 9946     30 [...]162     80 5971     30 1208  
      155       98       16       71       139       172
12   1361   27   8227   42   9962   57   9091   72   5832   87   1036  
      153       96       14       73       141       17 [...]
  30 [...]514     30 8323     30 9976     30 9018     30 5691     30 864  
      151       95       11       76       144       173
13   [...]665   28   8418   43   9987   58   8942   73   5547   88   691  
      151       92       8       7 [...]       144       17 [...]
  30 [...]816     30 8510     30 9995     30 8864     30 5403     30 519  
      150       89       4       8 [...]       146       17 [...]
14   [...]966   29   8599   44   9999   59   8782   [...]4   5257   89   346  
      148       87       2       8 [...]       148       17 [...]
  30 5114     30 8686     30 10001     30 8698     30 5109     30 173  
      147       83       1       8 [...]       150       17 [...]
15   5261   30   8769   45   10000   60   861 [...]   [...]5   4959   [...]   [...]000  

The Use Table of Horizontal Distances.

I. Any degree of Elevation under 45 being given, to find at what degree above 45 will hit the same horizontal distance; suppose 12 degrees, I look against 12 degrees in the Table, and find 4 [...]61, which I look for beyond 45 degrees, and find it against 76 deg. 57 min. so I conclude, a Piece charged with the same quantity of the same Powder, and the same Ball put to 12 deg. and 76 deg. 57. min. of Elevation, will Range the Shot to the same Horizontal distance.

Here note, Suppose a Piece be charged with 1, 2, 3, and 4 parts of Powder and the same Ball, and put to those degrees of Elevation, if the upper and lower Ranges be equal, there is no sensible resistance of the Medium.

II. Iune the 5th. 1 [...]77. on Wimbleton Heath I charged the Mortar Piece with 4 ounces of Powder, and put it to 15 deg. of Elevation, it ranged the Ball to the Horizontal distance of 659 paces; with that, I would hit a Mark with the same Piece, Ball, and quantity of Powder, at the Horizontal distance of a 1000 paces. Then as 659, is to a 1000, so is 5261 the tabular Number of 15 deg. to 7983, which gives in the Table 25 deg. 47 min. and 63 deg. 16 min. to hit a Mark at the Horizontal distance of a 1000 paces.

III. Feb. the 12th 1677/8. on Wimbleton Heath, a Piece whose Length of its Chase is 18 Inches and Diameter of Bore 3 Inches, charged with 8 ounces of Powder, and laid to 10 deg. of Elevation, Ranged its shot to the Hori­zontal distance of 1805 paces; with that I would hit a Mark at the Horizon­tal distance of 2112 paces, that is 2 English Miles: Then as 805, is to 2112, so is 3734 the Tabular Number at 10 deg. to 9797, which gives in the Table 38 deg. 41 min. and 50 deg. 25 min. to hit a Mark at the Horizontal distance of 2112 paces, viz. 2 English Miles.

The Making of the Table.

In Fig. to the III. Proposition, the Angle MRF being given, RZ, FV and OI being found; that is RI the Horizontal distance to that degree of Elevation; and then reduced, as in the Table.

A Table of the Requisites of Powder of like Mortar-Pieces, from 6 Inches to 20 Inches Dia­meter; and they are as the Cubes of the Diameters of their Bores.

In. Dec. Pou. Ou. In. Dec. Pou. Ou.
A B A B
6     13 13   8 09
6 5 1 01 13 5 9 10
7   1 05 14   10 11½
7 5 1 10 14 5 11 14
8   2 00 15   13 03
8 5 2 06 15 5 14 09
9   2 14 16   16 00
9 5 3 06 16 5 17 09
10   3 14½ 17   19 03
10 5 4 08 17 5 20 15
11   5 03 18   22 12½
11 5 5 15 18 5 24 11
12   6 12 19   26 13
12 5 7 10 19 5 28 14
        20   31 04

The Use of the Table.

The Diameter of a Mortar Piece 9 Inches and ½ being given. To find the Requisite of Powder.

Look in Column A, and there find 9 Inches and 5 Tenths, and against it in Column B there is 3 Pound 6 Ounces, its Requisite of Powder.

The like in the rest.

OF GRANADOES, CARCASSES, And FIRE-BALLS.

[Page]

[figure]

OF GRANADOES.

AS a Granarium keeps Corn for the Preservation of the Life of Man, so these Granariums (corruptly called Granadoes) are filled with Corns of Fire for the De­struction of Mankind.

A Granade is a hallow Sphere of Iron (as we may so call it) fill'd with Corn-Powder, with a Fusee to fire the dry Pow­der to break the Shell when it arrives to the designed Object (as in Fig. A) the Fusee B, the Vents R, ZR equal to two Thirds of the Diameter of the Granado.

If the Granado be too little for the Chase of the Mortar, marle it with slack twisted Thred, brushing it all over with hot Pitch, throwing Brick-dust thereon, till it be fit for the Piece, by its Gauge C, charge the Fusees with one of these Com­positions following,

1. Of Powder 1 ℥ Of Saltpetre 1 ℥ Of Sulphur 1 ℥
2. Of Powder 3 ℥ Of Saltpetre 2 ℥ Of Sulphur 1 ℥
3. Of Powder 4 ℥ Of Saltpetre 3 ℥ Of Sulphur 2 ℥
4. Of Powder 4 ℥ Of Saltpetre 3 ℥ Of Sulphur 1 ℥

Casimire Simienowiez, Page 206.

Drive it into the Granade to the Shoulder Z, being well glued.

Mortars usually Range the Granads with their due Requisit of Powder about 1250 paces upon the plain of the Horizon; the best place generally to work a Mortar, is from 70 to 80 deg. [Page 28] of Elevation, the greatest Altitude of a Projection at 75 deg. of Elevation whose Horizontal distance is 1250 paces, is about 583 paces, and the time of the Flight of the Granad will be about 27 Seconds of time, which time the Fusee ought to con­tinue burning before it fires the Granad; but if upon an Ascent or Descent, less or more time is to be allowed, according as the Object is situated. And this in the general.

Example.

The greatest Range in the Parabola
1250
A [...]/4 part equal to ND in Fig. I
312.5

In the Right Angled Triangle NDL.

As Radius   90 10.000000
is to ND 312.5 2.494850
so is Tang. of the Angle DNL 75 10.571948
to DL 1166.2 3.066798
the half DQ 583.1  

Then,

As 20.2 Paces 1.305351
is to the square of 2″. 30‴ that is 150‴ that is 22500 4.352183
so is DQ 583.1 2.765743
to the square of the Time   5.812575
The Root 805.9 i. e. 13.43 2.906287
The Double, that is near 27″, 26.86  

To give a Fusee its just time of burning, viz. 27 Seconds.

Take a Bullet of any convenient weight, fasten a String to it, measure from the Centre of the Bullet, 39 Inces 2 Tenths, make a Loop there, hang it upon a Pin, move the Bullet with your Hand to a convenient distance out of its Perpendicularity, then let it loose, these Swings or Vibrations will be Seconds of Time sufficiently near the truth.

[Page 29]Drive your Fusee with one of the former Compositions, then fire it, if it continues burning 27 Seconds of Time, you have hit right, if the Fusee continued not burning the full Time, give it a slower Composition, or the contrary.

Few Trials make Perfectness.

A Bomb has the same Office as a Granade, only it is in the form of a Spheriod, that is, longer one way than the other, therefore its Range will be more uncertain.

There are also Granads to be cast with the hand; their Office is the same as the former; they are of Iron, Brass, Glass, and other mixt Metals that are brittle.

Thomas Malthus, an Engineer of England, was the first that exercised the Mortar-Piece, and taught the Use of Granadoes in the 17 Provinces in Germany and Poland.

The French King hearing of his good success, sent for him, to instruct the French as he had done the Dutch, and imployed him in several Sieges, and at the Siege of Graveline, where he was slain by a Musquet-shot in the Head.

William Eldred, 60 Years a Gunner, sometime Master-Gun­ner of Dover-Castle, an honest industrious Person, gives us many good Experiments of Long Guns, for the space of thirty Years; a good Work: At the latter end of his days he eat the Charity of charitable Sutton, and not the Bounty of his Prince; for he dyed in the Charter-House.

OF CARCASSES.

TAke two Armillas of Iron of a convenient breadth, a little less than the Diameter of the Mortar-Piece to which they belong, put them at right Angles, rivet them fast at each Pole, at a and b in Fig. D.

Rivet a Bason of Iron at the bottom, as c, a and d, to bear the Impulse of Powder and fall of the Carcass, a lesser Bason at the top, as gbh, in which there ought to be Holes to fire the Car­cass, take an Armilla fit to the Diameter of the Piece, and place it equidistant from each Pole, as ef.

Fill it with this Composition.

  • 12 ℥ of Pitch
  • 12 ℥ of Colophone.
  • 4 ℥ of Oyl of Turpentine.
  • 1 ℥ of Hemp cut short.

Or,

  • 2 lb of Pitch.
  • 4 ℥ of Oyl of Turpentine.
  • 1 ℥ of Hemp cut short.

All these melted together, take fit Marlin, and from the two Shoulders d and c marle it fast, fill it as you marel, till you come to gh, fasten the Marlin to the Armillas with fine Marlin, if need be.

[Page 31]Coat it with this Receipt.

  • Pitch 4 ℥
  • Rosin 3 ℥
  • Wax 2 ℥
  • Turpentine 1 ℥

Fit it to its Gauge C, make Holes in the top of the Car­cass about 3 Inches deep, more or less, according to the mag­nitude of the Carcass.

Use for the Priming the same Composition as for Fusees of Granadoes, with quick Match moderately droven into the Holes. If it be design'd to place Petards in the Carcass, let there be Holes made in the Armillas to fix them in, or they may be placed in the Marlin, as the Carcass is a filling: These are things of Common Sense.

There is another kind of Carcass, namely, in form of an Egg, as E, whose length is a Diameter and half of the Piece to which it belongs, composed of two Hoops of Iron and two Basons, as in the former, with a Zone of Iron in the middle, as also a Trunk of Iron E, with two Shoulders, as Z and X, to be placed in the Carcass, as AB, with Holes to place the ends of the Petards in, the Trunk filled with a slow Compo­sition to fire the Petards as the Carcass confumes; these Petards, as F, are to be charged with dry Powder and Bullets, to de­stroy those that come to quench the Carcass. The Range of this Carcass will be very uncertain, by reason of the irregular form.

OF FIRE-BALLS,

IN Fig. G, Take the Trunchion H, wind Marlin about it in form of a Sphere, as AB, leaving the first end of the said Marlin out, as I, about this Sphere of Marlin put Canvas several thicknesses, dipping the Sphere of Marlin in Glue be­twixt every thickness of Cloth.

Then coat it with the Coating before cited, till it fits its Gauge C, pluck out the Truncheon H, then draw the Marlin I till it all comes out, there will remain a Concave Sphere AB, which being filled with this Composition, viz.

Powder-dust
12 ℥
Saltpetre
6 ℥
Sulpher
3 ℥
Colophone
2 ℥ more or less, as necessity requires.

Driving it hard as you fill, it being finished, it's fit for use.

Lastly, place a Bason of Lead at E, as CED, to bear the blast of the Piece, and carry that side of the Ball E foreright, that the mouth of the Ball may be uppermost, when it arrives to its assigned place.

Fill the Concave Sphere AB with either of the Compositions made for the Carcasses, and that will be another sort of Fire-Ball.

Much more might be said of this Subject, as well as of the whole, but this may suffice at the present.

To HIT a MARK WITH Long Guns.

PROPOSITION I.

THE Length of the Chase, Diameter of the Bore, and Requisite of Powder of any Piece; with the Length of the Chase, Diameter of the Bore of any other Piece being given. To find its Requisite of Powder.

[figure]

For if, as AE:BD::AF:BL, then as the Cylinder GE, is to the Cylinders ID; so is the Requisite of Powder in GF, to the Requisite of Powder in IL. But as quantity is to quantity, so is weight to weight being of the same kind. Therefore as the square of AG in AE is to the square of BI in BD; so is the weight of the Powder GF, to the weight of Powder in IL. Here we compare the Basilisk with the Culverin, EAG the Basilisk, and DBI the Culverin.

  • L the Length of the Culverin 11 Foot.
  • D the Diameter of the Bore 5 Inch.
  • P the Requisite of Powder 9.977
  • R the greatest Range 4.837
  • T the Time of the Flight of the shot.
  • F the comparative Force.
  • L the Length of the Basilisk 23.5 Foot.
  • D the Diameter of the Bore 46 Inch.
  • P the Requisite of Powder.
  • R the greatest Range.
  • T the Time of the Flight of the shot.
  • F the comparative Force.

[Page 34]

1 the Diameter of the Bore of the Culverin BI 5 Inches 0.698970
2 the Square of the first     1.397940
3 the Length of the Chase of the Culverin BD 11 Foot 1.041393
4 the Log of the second and third     2.439333
5 the Diameter of the Bore of the Basilisk AG 4.6 0.662758
6 the Square of the fifth     1.325516
7 the length of the Chase of the Basilisk AE 23.5 1.371068
8 the Log of the sixth and seventh     2.696584
9 the Requisite of Powder of the Culverin   9 l. 977 3.998999
10 Log of the 8 and 9     6.695583
11 the Log of the 4 subst.     2.439333
12 the Requisite of Powder of the Basilisk AGFH 18 l. 041 4.256250

That is, LD 2: LD2::P: P, That is, LD 2 P=LD2 P, so any five of these six being given, the sixth is also given.

  • 1. LD 2: LD2::P: P
  • 2. D 2 P:D2P:: L:L
  • 3. L P:LP:: D2:D 2
  • 4. LD2:LD 2:: P:P
  • 5. DP:D 2 P::L: L
  • 6. LP:L P::D 2: D2

PROP. II.

THE Length of the Chase, the Diameter of the Bore and greatest Range of any Piece, with the Length of the Chase and Diameter of the Bore of any other Piece being given, to find the greatest Range.

Let XY=ZR; then as AE to BD, so is the Range of the Cylinder HV, to the Range of the Cylinder KN, but as XC is to XY; so the Range of the Cylin­der KN to the Range of the Cylinder KM. That is as AE in XC, is to BD in ZR, so is the Range of the Cylinder HV, to the Range of the Cylinder KM. But a Sphere is two Thirds of its circumscribed Cylinder, therefore the Ranges of the Bullets are in the same proportion.

1 the length of the Culverin BD 11 Foot 1.041393
2 the Diameter of the Bore of the Basilisk   45 Inches 0.662758
3 the Log of the 1 and 2     1.704151
4 the length of the Basilisk AE 235 Foot 1.371068
5 the Diameter of the Bore of the Culverin BI 5 Inches 0.698970
[...] the Log of the 4 and 5     2.070038
7 the greatest Range of the Culverin   4837 3.684576
8 the Log of the 6 and 7     5.754614
9 the 3 subst.     1.704151
10 the greatest Range of the Basilisk   11.232 4.050403
  • 1 L D:LD::R: R
  • 2 DR:DR:: L:L
  • 3 L R:LR::D: D
  • 4 LD:L D:: R:R
  • 5 DR: DR::L: L
  • 6 LR:L R::D:D

PROP. III.

THE length of the Chase, Diameter of the Bore and force of any Piece; with the length of the Chase, and Diameter of the Bore of any other Piece being given. To find the force.

Let AB be the greatest Range of the Culverin, and CD the great­est

[figure]

Range of the Basilisk. Let the natural falling of the Ball from A to B, be as the Cube of its Diameter, viz. 125 the force of the Cul­verin. The natural falling of the Ball from C to Z be as the Cube of the Diameter of the Ball of the Basilisk 97.336; the falling of the Ball from C to D be the force of the Basilisk, to be found; as the square Root of AB=CZ, is to the square Root of CD so is the force at Z, to the force at D. Torricel. de Motu Proj. Lib. II. Prop. XXII.

1 the greatest Range of the Culverin AB 4.837 3.684576
2 the square Root of the 1     1.842288
3 the greatest Range of the Basilisk   11.232 4.050457
4 the square Root of the 3     2.025228
5 the Diameter of the Bore of the Basilisk   4.6 0.662758
6 the Cube of the 5     1.988274
7 the Log of the 4 and 6     4.013502
8 the 2 subst.     1.842288
9 the force of the Basilisk   148 2.171214
  • 1 L D:LD:: D6: F2
  • 2 DF2:D D6:: L:L
  • 3 L F2: LD6::D: D
  • 4 LD:L D::F2: D6
  • 5 D D6: DF2::L: L
  • 6 LD6:L F2:: D:D

PROP. IV.

THE Length of the Chase, Diameter of the Bore of any Piece, with the Time of the flight of the shot; with the length of the Chase, Diameter of the Bore of any other Piece being given. To find the Time or Duration of the shot in its flight.

Let AB be the greatest Range in the Culverin, CD=AF the greatest Range of the Basilisk, the time of the falling of the Ball from A to B, is to the time of the falling of the Ball from A to F; as the square Root of AB, is to the square Root AF. Or as AB to AF, so the square of the time of AB, to the square of the time AF. Torricel. de Motu Proj. Lib. II. Prop. XIX.

Iune 24. 1686. I made an Experiment of the falling of heavy Bodies from the top of Cripplegate-Steeple, which is 101 Foot higher than the place where they fell, viz. Three Iron Balls, one of 5 Inches Diameter, another of 3 Inches, and another of 2 Inches and an half Diameter, which constantly passed that space in 2′ 30‴ of Time, with such exactness, that not any difference could be discerned; Mr. Leake, Mr. Tompion, our famous Watchmaker, who kept time with a Watch that moved ¼ seconds, Mr. Norris, Mr. Morden, with several other Persons there present.

If any heavy Body fall 101 foot, or 20.2 paces in 150 thirds of time, what time shall it require to fall 4837 paces, the greatest Range of the Culverin. As 20.2 is to the squarè of 150, so is 4837 to the square of the time.

1 the Time 150 thirds 2.176091
2 the square of the 1   4.352182
3 the greatest Range of the Culverin 4837 3.684576
4 the Log of the 2 and 3   8.036758
5 the first space 20.2 subst. 1.305351
6 the Log of the square of the time   6.731407
7 the time 2321‴, that is, 38″ 41‴   3.365703

The time of the flight of the shot of any Piece at 45 degrees of Elevation, is the time of the falling of an heavy Body the greatest Range of the same Piece.

1 the Log of the square of the time of the Culverin
6.731407
2 the greatest Range of the Basilisk 11.232
4.050463
3 the Log of the 1 and 2
10.781870
4 the greatest Range of the Culverin Log. subst. 4837
3 684576
5 the Log of the square of the time of the Basilisk
7.097294
6 the time of the Basilisk 3537‴ that is 58″ 57‴
3.548647

Or,

  • 1 L D:LD::T 2: T2
  • 2 L T2: LT 2::D: D
  • 3 DT2:DT 2:: L:L
  • 4 LD:L D::T2:T 2
  • 5 DT 2: DT2::L: L
  • 6 LT 2:L T2:: D:D

PROP. V.

THE greatest Range in the Parabola, the Line of Impulse, the Angle of Elevation being given. To find the Range upon the plain of the Horizon

[figure]

The greatest Range in the Parabola 1657.2 DE, the Line of Impulse AD, the Angle KAD 45 deg. of Elevation. To find AB the Range upon the plain of the Horizon.

As Rad. 90 10.000000
is to AD 37.48 1.573800
so is the sine of the Angle KAD 45 deg. 9.849485
to KD or AK 26.5 1.423285

DAK being 45 deg. therefore ADK is 45 deg. therefore AK is equal to KD, and by the same reason, GP and DG are equal, therefore GH is equal to half DG.

HG 414.3  
DK 26.5  
HN 440.8  
As HG 414.3 2.617315
is to HN 440.8 2.644242
so is the square of DG or GE 8286 5.836690
to the square of NB   5.863617
ND 854.7 2.931808
NK 828.6  
AK 26.5  
AB 1709.8  

Here note, AK and FB differs but 4 Tenths of a Unit, therefore in the following Work, I take the double of AK for them both.

PROP. VI.

TO make a Table of Ranges, of any Gun; for the plain of the Horizon to every 30 Minutes and single degree of Elevation.

And here remember, that the Ranges upon the plain of the Horizon, are as the sines of the double of the Angle of Elevation.

For Example sake, the Cannon Royal whose greatest Range is 3298 paces.

As AB 1709.8 3.232945
is to AD 37.48 1.573800
so is the greatest Range of the Cannon 3298 3.518251
to the Line of Impulse   72 paces 1.859106

If we make use of the last Diagram, and suppose AB to be the Horizon­tal Range at 45 deg. of Elevation of the Cannon Royal, and AD the Line of Impulse 72 paces. Then,

As the Rad.   90 10.000000
is to the sine of the Angle ADK 45 deg. 9.849485
so is the Line of Impulse AD 72 1.859106
to AK 51 1.708591
The greatest Range of the Cannon Royal
AB 3298
subst.
FB+AK=102
The greatest Range in the Parabola
DE 3196

[Page 39]

As the Rad.   90 subst. 10.000000
is to   DE 3196   3.504607
so is the sine of   1 deg.   8.241855
to   56   1.746462
L:P 72 2 deg.   8.542819
  56 112   2.047426
30 128 3 deg.   8.718800
  112 167   2.223407
1 184 4   8.843585
  167 223   2.348192
30 239 &c.    
  223      
2 295 &c.    

These Calculations may suffice, till such time as we have more Experiments to confirm these; or to make Tables of Ranges more exactly.

The greatest Ranges of Long Guns, are deduced from many Experiments made by Eldred in his Gunners Glass, especially those in Pag. 74. 1611. Iuly 2. and Pag. 75. 1636. August 30. The manner how they are deduced may be seen in my Genuine Use of the Gun, Prop. 1, 2, 3, 4. Printed for Robert Morden, at the Atlas in Cornhil, near the Royal Exchange.

These Ranges for single degrees, are deduced from those Ranges, and what was done on Wimbleton-Heath, Septemb. 17. 1677.

A Table of the Names, Diameters of the Bores, and Length of the Chases of Ten several Pieces of Canon; with their Requisites of Powder, greatest Ranges, Comparative Force, with their Ranges to 8 deg. of Elevation, Experimentally and Mathe­matically demonstrated.

The Names of each Piece. Lengths of the Chases. Diamet. of the Bores. Requisits of Powder. Greatest Ranges. Compa­rative Force.
  Feet. Inches. lb ℥ Paces.  
A Rabbinet 1 3 1.75 5 3769 38
A Falconet 2 4 2 9 4398 61
A Falcon 3 6 2.75 1 10 4797 166
A Minion 4 8 3 2 10 5864 238
A Saker 5 9 3.5 4 5654 371
A Demi Culverin 6 10 4.5 7 5 4886 733
A Culverin 7 11 5 10 4837 1000
A Demi Cannon 8 11 6 14 6 4031 1575
A Whole Cannon 9 12 7 21 5 3769 2422
A Cannon Royal 10 12 8 27 14 3298 3382

The USE of these Ranges.

Suppose an Engineer be commanded to batter a Bastion, or a Curtain, at the Horizontal distance of 900 paces, with a whole Cannon, a Demi Cannon, and a Culverin; I look in the Table, and find 900 against the Demi Cannon, and at the head 6°, for the whole Cannon 6° 27′ and for the Culverin 4° 52′ to batter the said Bastion or Curtain, taking the proportional parts; thus, in the Culverin, 80:30′:60:22′, whole Cannon 63:30′:58:27′.

  0′   30′   1°   30′   2°   30′   3°   30′   4°  
L.P. D. R. D. R. D. R. D. R. D. R. D. R. D. R. D. R. D.
1 83 64 147 63 210 64 274 64 338 63 401 64 465 63 528 63 591 63
2 96 74 170 75 245 74 219 74 393 74 467 74 541 74 615 74 689 74
3 105 81 186 81 267 81 348 81 429 81 510 81 591 80 671 81 752 80
4 128 99 227 99 326 99 425 93 524 99 623 99 722 98 820 99 919 98
5 124 96 220 95 315 96 411 95 506 95 601 9 697 95 792 94 886 95
6 107 83 190 82 272 83 355 82 437 83 520 82 602 82 644 82 766 82
7 106 83 189 81 270 82 352 81 433 82 515 81 596 82 678 80 758 82
8 88 68 156 68 224 68 292 69 361 68 429 67 496 68 564 68 632 67
9 83 64 147 63 210 64 274 64 338 69 401 64 465 63 528 63 591 63
10 72 56 128 56 184 55 239 56 295 56 351 55 406 55 461 56 517 55

[Page 42]Here note well also; The Object that the Ball is to hit, ought to be as high above the Horizon, as the Mouth of the Piece; if not, the Ranges at 0, 1, 2, and 3 degrees of Elevation will sensibly differ, those Ranges at 6, 7, and 8 degrees of Eleva­tion not so much as the former.

  30′   5°   30′   6°   30′   7°   30′   8°
  R. D. R. D. R. D. R. D. R. D R. D. R. D. R.
1 654 3 717 63 780 62 842 63 905 62 967 61 1028 62 1090
2 763 73 836 73 909 73 982 73 1055 72 1127 72 1199 72 1271
3 832 80 912 80 992 79 1071 80 1 [...]51 78 1229 79 1308 78 1386
4 1017 98 1115 97 1212 97 1309 97 1406 97 1503 96 1599 95 1694
5 981 94 1075 94 1169 94 1263 93 135 93 1449 93 1542 92 1634
6 848 81 929 81 [...]010 81 1091 81 1172 80 1252 80 1332 80 1412
7 840 80 920 81 1001 79 1080 81 1161 79 1240 80 1320 78 1398
8 699 67 766 67 833 67 900 67 967 66 1033 66 1099 66 1165
9 654 63 717 63 780 62 842 63 905 62 967 61 1028 62 1090
10 572 55 627 55 682 54 736 55 791 54 845 54 899 54 953

Warlike Musick ILLUSTRATED, In several Consorts of Phrygian Flutes. Clearly demonstrated by Principles of Musick and Mathematicks.

PROP. I.

THE length of the Chase 12 foot of a Cannon Royal being given. To find the length of the Chases of seven lesser Pieces of Cannon that shall be in Musical Proportion.

The greater tone 9/8 their Log. 95424251
90308999
The difference of their Log.   5115 [...]52
The lesser tone 10/9 their Log. 100000000
95424251
The difference of their Log.   4575749
The half Note 16/15 their Log. 1.20411998
1.1 [...]6 [...]9126
The difference of their Log.   2802872

The perfect Eight 2/1 12 1.07918125
      4575749 10/9
The middle Seventh 9/5 10.8 1.03342376
      5115252 9/8
The lesser Sixth 8/5 9.6 0.98227124
      2802872 16/15
The perfect-Fifth 3/2 9 0.95424252
      5115252 9/8
The perfect Fourth 4/3 8 0.90309000
      4575749 10/9
The lesser Third 6/5 7.2 0.85733251
      2802872 16/15
The greater Second 9/8 6.75 0.82930379
      5115252 9/8
The Unisound 1/1 6 0.77815127

PROP. II.

THE Diameter of the Bore of the Cannon Royal being 8 Inches. To find the Diameter of the Bore of seven lesser Pieces of Cannon, which shall be in Musical Proportion: By the first.

PROP. III.

THE Requisite of Powder of the Cannon Royal being 28 Pounds. To find the Requisite of Powder of seven lesser Pieces of Cannon in Musical Proportion: By the first.

PROP. IV.

THE Force of the Cannon Royal being taken to be as the Cube of the Diameter of the Bore, viz. 512. To find the Force of seven lesser of Pieces of Cannon which shall be in Musical Proportion: By the first.

PROP. V.

THE Time of the Flight of the shot of the Cannon Royal 1000. To find the time of the flight of the shot of seven lesser Pieces of Cannon which shall be in Musical Proportion: By the first.

PROP. VI.

THE Length of the Chase, Diameter of the Bore of the Cannon Royal, as also the Length of the Chase, and Force of any of the aforesaid Pieces being given. To find the Diameter of its Bore.

PROP. VII.

THE Length of the Chase, Diameter of the Bore, and the Time of the Flight of the shot of the Cannon Royal, as also the Length of the Chase, and time of any of the aforesaid Pieces being given. To find the Diameter of its Bore.

PROP. VIII.

THE Length of the Chase 12 Foot, the Diameter of the Bore 8 Inches, the Requi­site of Powder 28 lb of a Cannon Royal being given. To find the Length of the Chase, Diameter of the Bore, and Requisite of Powder of seven lesser Pieces of Cannon, which shall be in Geometrical Proportion.

The Length of the Chase 12 Foot   Log. 1.07918125
Its half 6 Foot   Log. 0.77815127
The difference of the     Log. 30102998
The 7th. part is, add       4300428
Add to the length of the least   1 6 foot 0.77815127
    2 6.624 0.82115555
    3 7.314 0.86415983
    4 8.076 0.90718411
    5 8.916 0.95016839
    6 9.844 0.99317267
    7 10.869 1.03617655
    8 12. 1.07918123

The like for the Diameters.

The Requisite of Powder 28 lb   Log. 1.44715803
The eighth part thereof is 3.5   Log. 0.54406804
The difference of the     Log. 0.90308999
The 7th part is       0.12901285
    1 3.5 0.54406804
    2 4.71 0.67308089
    3 6.34 0.80209374
    4 8.53 0.93110659
    5 11.48 1.06011944
    6 15.46 1.18913229
    7 20.80 1.31814514
    8 28. 1.44715809

[Page 46]I. By the first Proposition for the length of the Chase, and second Proposition for the Diameter of the Bore; we compose the first Consort of Phrygian Flutes, whose visible shapes are in Musical Proportion, the Ranges and Time upon the same degree of Elevation are equal, their Force and Requisite of Powder as the Cube of the Diameters of their Bore.

II. By the first Proposition for the Length of the Chase, and third Propo­sition of the Requisite of Powder, we compose the second sort of Phrygian Flutes, whose Sounds or Reports and Ranges shall be in Musical Proportion, the Diameter of their Bores equal.

III. By the first Proposition for the Length of the Chase, and the fourth Proposition for the Force, and Proposition the 5th. for the Diameter of the Bore, we compose a third sort of Phrygian Flutes whose Force shall be in Musical Proportion.

IV. By the first Proposition for the Length of the Chase, and 6th. Proposi­tion for the Time of the Flight of the shot, and Proposition the 7th. for the Diameter of the Bore, we compose a fourth sort of Warlike Flutes, whose Times shall be in Musical Proportion.

V. By the eighth Proposition we find the Length of the Chase, Diameter of the Bore, Requisite of Powder of Eight Pieces of Cannon in Geometri­cal Proportion, their Ranges upon the same degree of Elevation equal.

So then, here is nothing wanting in these things that the Heart of Man can in reason desire; for in the first, the Eye is satisfied, their shapes are in Musical Proportion. In the second, the sense of Hearing is delighted, their Sounds are in Musical Proportion. In the third the sense of Feeling may be terrified, the Smart of the Blow in Musical Proportion. In the fourth, that precious thing called Time is employed in delightful Musical Proportion.

What can be presented to a Prince more delightful, or what can a Prince more delight in, than to conquer his Enemies with Musick and Delight?

[Page 47]FRom what has or may be said concerning Guns in Musical and Geometrical Proportion, and what may be said concerning Guns under other Qualifica­tions, it may manifestly appear, that the business of Gun-founding lyes as open for Improvements in these days, as their Uses did when Galileus first appeared in the World: For,

An Engineer draws the Draught of a Gun, and gives it a certain Fortifi­cation, the Founder founds it according to the Draught or Pattern, a Proof-Master proves it, and gives it its Requisite of Powder. Quere, Whether it be the Engineer's or Draught-Man's business to give it its Requisite of Powder, and consequently its Range, or the Proof-Master's?

Then,

[figure]

I. In the Diagram, Let K, G, B, and P, be any Piece of Cannon. It being demanded to increase the Mettal to I, H, A, and Q; or decrease the Mettal to L, F, C, and O; so as the strength of the Mettal K, G, B, and P, be to the strength of the Mettal I, H, A, and Q; as 3 to 7; and to L, F, C, and O, (being of the same Mettal) as 7 to 3.

The Requisite of Powder, and Range at 45 degrees of Elevation of the Piece I, H, A, and Q; and the Piece L, F, C, and O, are required.

II. If the Lengths of the Chases of two Pieces, and the Diameters of the Bores of the same two Pieces be as 8 to 9. To give them such Mettal that the strength of one Piece may be to the strength of the other, as 3 to 7.

Their Requisites of Powder and greatest Ranges are required.

III. If the Lengths of the Chases of two Pieces be as 9 to 10, and the Diameters of their Bores be as 4 to 5. To give them such Metal, that their strengths may be as 3 to 7.

Their Requisites of Powder, and greatest Ranges are required.

To Conclude.

HERE I might produce a Jury of Mathematicians, famous for their Learning, for the confirmation of that of the Parabola: But it is so evi­dent in it self, that he that is ignorant of, and denies that, viz. The Flight of the Bullet to be in the Curve of a Parabola, deserves not the Name of a Mathematician, Philosopher, Engineer, or Fire-Master, no, not of a private Gunner.

FINIS.

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