Part 11

5. The altitude of the curve, at any elevation, is found by this proportion: as

Radius : tangent of angle of elevation :: range : altitude. ----- 4

6. The time of flight at 45° is equal the square root of the range in feet, divided by 4, or more nearly = √(quotient)² of the range in feet, divided by 16.1, or the space passed through in the first second by gravity.

Having the first graze with a given elevation and charge, to determine the charge for any other first graze and elevation, multiply the known charge and elevation into the proposed first graze; also the proposed elevation into the known first graze, and divide the first product by the last, for the charge required.

_Upon inclined Planes, at 45° Elevation._

_Case 1st. Given the charge and inclination of the plane, to find the range._

Multiply the horizontal range with this given charge, (found in the tables of ranges) by the number found opposite the angle of inclination of the plane, in the first column of multipliers, in the table of amplitudes, under the head _Ascents_, if it be inclined above the horizon; and _Descents_, if below the horizon, for the range required.

_Case 2d. Given the range and inclination of the plane, to find the charge._

Multiply the number found in the abovementioned table opposite the angle of inclination of the plane, in the second column of multipliers, under the head _Ascents_, or _Descents_, according as it is above or below the horizon, by the given range; for the range on a horizontal plane at 45°, the charge for which may be found from the tables of ranges.

_Upon Inclined Planes, at any Elevation._

There are always two elevations with which any range, (less than the greatest) may be made; and these elevations are always the complements of each other. The greatest range upon a horizontal plane is at 45°; or when the direction bisects the angle formed by the horizontal and vertical plane; also the greatest range upon any plane is made with that direction which bisects the angle between the plane and the zenith; and all other directions which make equal angles with this direction, (on each side of it) will also make equal ranges on the said plane; for the direction that bisects the angle between any plane and the zenith is the same with respect to that plane as the direction at 45° is with respect to the plane of the horizon.

_Rules._—1st. The elevation which gives the greatest range on a given ascent, is equal to half the sum of 90° added to the ascent.

2d. The elevation which give equal ranges on a given ascent, are the complements of each other added to the ascent.

3d. The elevation which gives the greatest range on a descent, is equal to half the complement of the descent.

If the range and inclination be given, the least charge that will reach the object, may be found as follows: multiply the tangent of the proper elevation into the proposed range, for the horizontal range whose charge is required.

WILLIAMS’S _Gunnery_.

_Table of Amplitudes._

+--------+-----------------+---------------+ |Degrees.| Ascents. | Descents. | | | Multipliers. | Multipliers | +--------+--------+--------+--------+------+ | |1st.cl. | 2d.cl. |1st.cl. |2d.cl.| | 1 | .983 | 1.02 | 1.02 | .983 | | 2 | .966 | 1.03 | 1.04 | .966 | | 3 | .949 | 1.06 | 1.05 | .950 | | 4 | .932 | 1.07 | 1.07 | .932 | | 5 | .916 | 1.09 | 1.09 | .916 | | 6 | .900 | 1.11 | 1.11 | .900 | | 7 | .884 | 1.13 | 1.13 | .884 | | 8 | .868 | 1.15 | 1.15 | .868 | | 9 | .852 | 1.18 | 1.17 | .853 | | 10 | .836 | 1.20 | 1.19 | .836 | | 11 | .821 | 1.22 | 1.22 | .821 | | 12 | .805 | 1.24 | 1.24 | .805 | | 13 | .789 | 1.27 | 1.27 | .789 | | 14 | .774 | 1.29 | 1.29 | .774 | | 15 | .758 | 1.32 | 1.31 | .763 | | 16 | .742 | 1.35 | 1.34 | .745 | | 17 | .726 | 1.38 | 1.37 | .730 | | 18 | .711 | 1.40 | 1.39 | .720 | | 19 | .693 | 1.45 | 1.42 | .704 | | 20 | .677 | 1.48 | 1.45 | .690 | | 21 | .660 | 1.52 | 1.48 | .675 | | 22 | .643 | 1.56 | 1.52 | .662 | | 23 | .625 | 1.60 | 1.55 | .645 | | 24 | .607 | 1.64 | 1.58 | .633 | | 25 | .589 | 1.70 | 1.62 | .617 | | 26 | .570 | 1.76 | 1.66 | .603 | | 27 | .550 | 1.82 | 1.69 | .592 | | 28 | .530 | 1.86 | 1.73 | .578 | | 29 | .510 | 1.96 | 1.78 | .562 | | 30 | .488 | 2.05 | 1.82 | .549 | | 31 | .466 | 2.14 | 1.87 | .534 | | 32 | .442 | 2.26 | 1.92 | .526 | | 33 | .418 | 2.41 | 1.97 | .508 | | 34 | .393 | 2.55 | 2.02 | .495 | | 35 | .366 | 2.73 | 2.08 | .488 | | 36 | .338 | 2.96 | 2.13 | .470 | | 37 | .309 | 3.24 | 2.20 | .455 | | 38 | .278 | 3.60 | 2.26 | .443 | | 39 | .245 | 4.09 | 2.33 | .430 | | 40 | .210 | 4.80 | 2.40 | .417 | | 41 | .173 | 5.78 | 2.48 | .404 | | 42 | .134 | 7.46 | 2.56 | .390 | | 43 | .092 | 10.90 | 2.64 | .380 | | 44 | .045 | 22.22 | 2.73 | .370 | | 45 | .000 |infinite| 2.83 | .360 | +--------+--------+--------+--------+------+

_Table of Natural Sines, Tangents, and Secants._

+--------+------+---------+--------+ |Degrees.|Sines.|Tangents.|Secants.| +--------+------+---------+--------+ | 1 | .018 | .018 | 1.000 | | 2 | .035 | .035 | 1.000 | | 3 | .052 | .052 | 1.001 | | 4 | .070 | .070 | 1.002 | | 5 | .087 | .087 | 1.004 | | 6 | .105 | .105 | 1.006 | | 7 | .122 | .123 | 1.008 | | 8 | .139 | .141 | 1.010 | | 9 | .156 | .158 | 1.012 | | 10 | .174 | .176 | 1.015 | | 11 | .191 | .194 | 1.019 | | 12 | .208 | .213 | 1.022 | | 13 | .225 | .231 | 1.026 | | 14 | .242 | .249 | 1.031 | | 15 | .259 | .268 | 1.035 | | 16 | .276 | .287 | 1.040 | | 17 | .292 | .306 | 1.046 | | 18 | .309 | .325 | 1.051 | | 19 | .326 | .344 | 1.058 | | 20 | .342 | .364 | 1.064 | | 21 | .358 | .384 | 1.071 | | 22 | .375 | .404 | 1.079 | | 23 | .391 | .424 | 1.086 | | 24 | .407 | .445 | 1.095 | | 25 | .423 | .466 | 1.103 | | 26 | .438 | .488 | 1.112 | | 27 | .454 | .510 | 1.122 | | 28 | .469 | .532 | 1.133 | | 29 | .485 | .554 | 1.143 | | 30 | .500 | .577 | 1.155 | | 31 | .515 | .601 | 1.167 | | 32 | .530 | .625 | 1.179 | | 33 | .545 | .649 | 1.192 | | 34 | .559 | .675 | 1.206 | | 35 | .574 | .700 | 1.221 | | 36 | .588 | .727 | 1.236 | | 37 | .602 | .754 | 1.252 | | 38 | .616 | .781 | 1.269 | | 39 | .629 | .810 | 1.287 | | 40 | .643 | .839 | 1.305 | | 41 | .656 | .869 | 1.325 | | 42 | .669 | .900 | 1.346 | | 43 | .682 | .933 | 1.367 | | 44 | .695 | .966 | 1.390 | | 45 | .707 | 1.000 | 1.414 | | 46 | .719 | 1.036 | 1.440 | | 47 | .731 | 1.072 | 1.466 | | 48 | .743 | 1.111 | 1.494 | | 49 | .755 | 1.150 | 1.524 | | 50 | .766 | 1.192 | 1.556 | | 51 | .777 | 1.235 | 1.589 | | 52 | .788 | 1.280 | 1.624 | | 53 | .799 | 1.327 | 1.662 | | 54 | .809 | 1.376 | 1.701 | | 55 | .819 | 1.428 | 1.743 | | 56 | .829 | 1.483 | 1.788 | | 57 | .839 | 1.540 | 1.836 | | 58 | .848 | 1.600 | 1.887 | | 59 | .857 | 1.664 | 1.942 | | 60 | .866 | 1.732 | 2.000 | | 61 | .875 | 1.804 | 2.063 | | 62 | .883 | 1.881 | 2.130 | | 63 | .891 | 1.963 | 2.203 | | 64 | .899 | 2.050 | 2.281 | | 65 | .906 | 2.145 | 2.366 | | 66 | .914 | 2.246 | 2.459 | | 67 | .921 | 2.356 | 2.559 | | 68 | .927 | 2.475 | 2.669 | | 69 | .934 | 2.605 | 2.790 | | 70 | .940 | 2.747 | 2.924 | | 71 | .946 | 2.904 | 3.072 | | 72 | .951 | 3.078 | 3.236 | | 73 | .956 | 3.271 | 3.420 | | 74 | .961 | 3.487 | 3.628 | | 75 | .966 | 3.732 | 3.864 | | 76 | .970 | 4.011 | 4.134 | | 77 | .974 | 4.331 | 4.445 | | 78 | .978 | 4.705 | 4.810 | | 79 | .982 | 5.145 | 5.241 | | 80 | .985 | 5.671 | 5.759 | | 81 | .988 | 6.314 | 6.392 | | 82 | .990 | 7.115 | 7.185 | | 83 | .993 | 8.144 | 8.206 | | 84 | .995 | 9.514 | 9.567 | | 85 | .996 | 11.430 | 11.474 | | 86 | .998 | 14.301 | 14.336 | | 87 | .999 | 19.081 | 19.107 | | 88 | .999 | 28.636 | 28.654 | | 89 | .999 | 57.290 | 57.299 | | 90 | 1.000|infinite.| ———— | +--------+------+---------+--------+

_GUNS.—Calibers of English and Foreign Guns, expressed in English Inches._

+---------+---------+---------+---------+---------+---------+ |English. | French. | Spanish | Dutch. | Russian.| Portug. | +---------+---------+---------+---------+---------+---------+ |Pr. Inch.|Pr. Inch.|Pr. Inch.|Pr. Inch.|Pr. Inch.|Pr. Inch.| |42 |7.018| — | — | — |— | — | — |36 |6.86 |48 |7.49 | |32 |6.41 |36 |6.9 |36 |6.84 |32 |6.4 |30 |6.47 |36 |6.8 | |24 |5.823|24 |6.03 |24 |6.03 |24 |5.92 |24 |6. |24 |5.93 | |18 |5.292|16 |5.26 |18 |5.52 |18 |5.45 |18 |5.45 |18 |5.4 | |12 |4.623|12 |4.78 |12 |4.8 |12 |4.76 |12 |4.76 |12 |4.7 | | 9 |4.200| 8 |4.18 | 9 |4.2 | 8 |4.13 | 8 |4.17 | 9 |4.3 | | 6 |3.668| — | — | — | — | 6 |3.78 | 6 |3.78 | 6 |3.75 | | 4 |3.204| 4 |3.315| — | — | — | — | — | — | — | — | | 3 |3.913| — | — | — | — | — | — | — | — | — | — | | 1 |2.019| — | — | — | — | — | — | — | — | — | — | +---------+---------+---------+---------+---------+---------+

_Length and Width of English Brass Guns._

+-----------------------+-----------------+-------------+ | Nature. | Length in | Weight. | +-----------------------+-----------------+-------------+ | |Calib’rs.|Ft. In.|Ct. qrs. lbs.| | 42 Pounders | 16.244 | 9 6 | 66 — — | | 32 ” | 18.721 |10 0 | 55 2 — | +----+------------------+-----------------+-------------+ | |Heavy | 19.574 | 9 6 | 53 0 9 | | |Medium | 16.483 | 8 0 | 41 3 2 | | 24 |Light | 10.302 | 5 0 | 16 3 13 | | | ” new | 13.000 | 6 3 | 24 0 — | +----+------------------+-----------------+-------------+ | 18 |Light ” | 13.000 | 5 9 | 18 0 — | +----+------------------+-----------------+-------------+ | |Heavy | 24.659 | 9 0 | 31 2 8 | | |Gen. Desagulier’s | 19.468 | 7 6 | 22 1 21 | | 12 |Medium, old | 16.872 | 6 6 | 21 3 — | | |Medium, new‡ | 16.872 | 6 6 | 18 — — | | |Light | 12.978 | 5 0 | 8 3 4 | | | ” new‡ | 13.000 | 5 0 | 12 — — | +----+------------------+-----------------+-------------+ | |Heavy | 26.112 | 8 0 | 19 1 6 | | |Gen. Desag. Med.‡ | 22.876 | 7 0 | 12 — 24 | | 6 |Med. new | 18.500 | 6 0 | 8 3 27 | | | ” reduced | 17.000 | 5 6 | 8 0 22 | | |General Belford’s‡| 16.342 | 5 0 | 5 2 21 | | |Light, common | 14.706 | 4 6 | 5 — 18 | +----+------------------+-----------------+-------------+ | |Heavy | 28.836 | 7 0 | 11 3 19 | | |Gen. Desagulier’s‡| 24.717 | 6 0 | 6 — — | | 3 |Light, common‡ | 14.418 | 3 6 | 2 2 27 | | |Light infantry | 12.358 | 3 0 | 1 3 16 | | |General Pattison’s| 12.358 | 3 0 | 1 2 19 | +----+------------------+-----------------+-------------+ | 1 | Pr. Amuzette | 29.7 | 5 0 | 2 2 12 | | | Do. | 35.6 | 6 0 | 3 0 11 | | | Do. | 41.5 | 7 0 | 3 1 12 | +----+------------------+-----------------+-------------+

NOTE. The guns marked (‡) are the only ones now used on general service.

_Length and Weight of French Brass Guns, in their own Weights and Measures._

+---------------------+-----------------------+------+ | Nature. | Length in |Weight| +---------------------+---------+-------------+------+ | |Calibers.| Ft. In. Li. | lbs. | | 24 Prs. Siege | — | 9 11 5 | 5628 | | 16 ” | — | 9 7 — | 4111 | +---------------------+---------+-------------+------+ | 12 ” Garrison | — | — — — | 3184 | | 8 ” | — | — — — | 2175 | +---------------------+---------+-------------+------+ | 12 ” | 18.0 | 6 6 — | 1808 | | 8 ” Field | 18. | 5 8 — | 1196 | | 4 ” | 18. | 4 6 — | 590 | +---------------------+---------+-------------+------+ | 1 ” | — | — — — | 266 | +---------------------+---------+-------------+------+

_Length and Weight of English Iron Guns._

+----------+-----------------+---------+---------------+ | | | | Proportion | | Nature. | Length in | Weight. | between shot | | | | | and gun.[8] | +----------+---------+-------+---------+---------------+ | |Calibers.|Ft. In.| Ct. Qr.| | | | | | | | | 42 Prs. | 17.098 | 10 — | 67 — | | | | 16.244 | 9 6 | 65 — | 170 | | | | | | | | 32 ” | 18.721 | 10 — | 58 — | | | | 17.725 | 9 6 | 55 — | 193 | | | | | | | | | 20.604 | 10 — | 52 — | | | 24 ” | 19.574 | 9 6 | 49 2 | 231 | | | 18.542 | 9 — | 47 2 | | | | | | | | | 18 ” | 21.542 | 9 6 | 42 — | | | | 20.408 | 9 — | 40 — | 249 | | | | | | | | | 24.659 | 9 6 | 34 — | | | 12 ” | 23.361 | 9 — | 32 — | | | | 22.063 | 8 6 | 31 2 | | | | 19.468 | 7 6 | 29 1 | 294 | | | | | | | | 9 ” | 21.4 | 7 6 | 24 2 | 305 | | | 19.9 | 7 — | 23 — | | | | | | | | | 6 ” | 26.2 | 8 — | 22 — | | | | 19.6 | 6 — | 16 2 | 411 | | | | | | | | 4 ” | 22.4 | 6 — | 22 1 | 343 | | | 20.6 | 5 6 | 11 1 | | | | | | | | | 3 ” | 18.6 | 4 6 | 7 1 | 270 | +----------+---------+-------+---------+---------------+

[8] This column expresses the number of pounds of metal in the guns, to each pound in the shot.

_French Iron Guns, in English Weights, &c._

+---------------+-------------------+------------+ | Nature. | Length in | Weight. | +---------------+----------+--------+------------+ | |Calibers. |Ft. In. |Ct. Qr. lbs.| | | | | | | 36 Pounders | 16.18 | 9 8 | 74 3 — | +---------------+----------+--------+------------+ | 24 ” | 18.18 | 9 1½ | 51 — — | +---------------+----------+--------+------------+ | | 21.01 | 9 7 | 42 — — | | 16 ” | 18.45 | 8 4 | 43 2 — | | | 16.92 | 7 8½ | 35 — — | +---------------+----------+--------+------------+ | | 21.54 | 8 7 | 31 2 — | | 12 ” | 20.5 | 8 2 | 20 3 — | | | 17.14 | 6 10 | 28 — — | +---------------+----------+--------+------------+ | 8 ” | 24.64 | 8 7 | 24 1 — | | | 17.22 | 6 — | 16 — — | +---------------+----------+--------+------------+ | 4 ” | 17.19 | 4 9 | — — — | +---------------+----------+--------+------------+

_Ranges of Brass Guns, with one Shot. 1793._

+----------------------+--------+-------------------------------+ | | Charge |To the first graze of the Shot.| | Nature. | +-----+----+----+----+----+-----+ | | |P. B.| 1° | 2° | 3° | 4° | 5° | +----------------------+--------+-----+----+----+----+----+-----+ | | lb.oz. | Yds.|Yds.|Yds.|Yds.|Yds.|Yds. | +----------------------+--------+-----+----+----+----+----+-----+ | Heavy | 8 0 | 473| 781|1032|1405|1585|1710 | |24 Medium | 8 0 | 488| 757|1103|1425|1557|1745 | | Light | 3 0 | 162| 364| 606| 722| |1390 | +----------------------+--------+-----+----+----+----+----+-----+ | Heavy | | | | | | | | |12 Medium | 4 0 | | 705| 973|1189| | | | Light | 3 0 | | 601| 816|1063| | | +----------------------+--------+-----+----+----+----+----+-----+ | Desagulier’s | 2 0 | | 646| 966|1325| | | | 6 Feet | 2 0 | | 683| 948|1327| | | | 6 5 : 6 Medi. | 2 0 | | 775|1003|1444| | | | 5 : 6 Red’d. | 2 0 | | 642| 976|1150| | | | 5 Feet | 1 8 | | 587| 825| 950| | | | 4 : 6 Feet | 1 8 | | 628| 804| 991| | | +----------------------+--------+-----+----+----+----+----+-----+ | 3 Desagulier’s | 1 0 | | 679| 883| 918| | | | Ammuzette of 5 feet| 0 8 | | 604| 800| | | | | ” of 7 feet| 0 8 | | 656| 830|1000| | | +----------------------+--------+-----+----+----+----+----+-----+

_Ranges from Brass Guns, with Two Shot. 1793._

+-------------------+--------+----------+--------------------+ | Nature. |Charge. |Elevation.| Medium first Graze.| +-------------------+--------+----------+----------+---------+ | |lbs. oz.| 1° 30′ | 1st Shot.| 2d Shot.| | | | | | | |12 Pounder, Medium | 4 — | 1 30 | 607 | 706 | | 6 : Desagulier’s | 2 — | 1 30 | 621 | 739 | | 6 : of 5 feet | 1 8 | 1 30 | 586 | 732 | | 3 : Desagulier’s | 1 — | 1 30 | 523 | 638 | +-------------------+--------+----------+----------+---------+

_Ranges from Brass Field Guns, with small Charges. 1798._

+-------+------+----------------------------------+------------------+ | | | First Graze with | | |Nature.|Charge| different Elevations. | Extreme Range. | +-------+------+---+---+---+---+---+----+----+----+------------------+ | | | 1°| 2°| 3°| 4°| 5°| 6° | 7° | 8° | | +-------+------+---+---+---+---+---+----+----+----+------------------+ |12 Pr. |10 oz.|199|290|390|385|597| 716| 695| 788|From 800 to 1000.| | | 1 lb.|280|416|729|777|966|1090|1054|1295|From 1200 to 1500.| +-------+------+---+---+---+---+---+----+----+----+------------------+ | 6 Pr. | 5 oz.|111|222|376|432|618| 625| 650| 788|From 800 to 1000.| | | 8 oz.|277|401|754|826|925| 980|1103|1100|From 1000 to 1300.| +-------+------+---+---+---+---+---+----+----+----+------------------+

=N. B.= The above was a 12 Pr. Medium, and a 6 Pr. Desagulier’s. The Distances are given in Yards.

_Effects of Case Shot from a Battalion Gun.—Light 6 Pr. Length 5 feet—Weight 5 cwt. 3 qrs. 21 lbs. against a Target 8 feet high, and 90 feet long._

+--------+-----------------+----------+------------+------------+ |Distance| | |Nᵒ. put |Nᵒ. put | | of | | |into the |into the | | Target.|Nature of Charge.|Elevation.|height of 6 |height of 8 | | | | |feet, or the|feet, or the| | | | |height of |height of | | | | |_Infantry_. |_Cavalry_. | +--------+-----------------+----------+------------+------------+ | Yards. | | Deg. | | | | |12 Balls, 8 oz. | 1 | 3 | 3 | | 500 | each, 3 in a | 1½ | 3 | 4 | | | tier, 1¼ lb. | | | | | | powder. | 2 | 3 | 4 | +--------+-----------------+----------+------------+------------+ | | | 1 | 6 | 6 | | 400 |Same Charge. | 1½ | 4 | 5 | | | | 2 | 4 | 5 | +--------+-----------------+----------+------------+------------+ | | | ½ | 6 | 7 | | 300 |Same Charge. | 1 | 3 | 3 | | | | 1½ | 4 | 6 | +--------+-----------------+----------+------------+------------+ | |34 Balls, 3 oz. | P. B. | 10 | 12 | | 400 | each, 7 in a | ½ | 9 | 10 | | | tier, 1¼ lb. | 1 | 6 | 8 | | | powder. | | | | +--------+-----------------+----------+------------+------------+ | | | P. B. | 11 | 13 | | 300 |Same Charge. | ½ | 12 | 15 | | | | 1 | 7 | 9 | +--------+-----------------+----------+------------+------------+

=N. B.= There were three rounds fired at each change, but they were all so nearly alike that it has been thought necessary to put down only one of them.—1802.

_Ranges with Sea Service Iron Guns. 1796._