Selected quad for the lemma: reason_n
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A25369
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To hit a mark, as well upon ascents and descents, as upon the plain of the horizon experimentally and mathematically demonstrated / by Robert Anderson.
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Anderson, Robert, fl. 1668-1696.
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1690
(1690)
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Wing A3108; ESTC R113
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25,099
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59
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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 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 D2P L L 3. L 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 LD R R 2 DR DR L L 3 L 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 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 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 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 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
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 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 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