Part 196

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 change 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 gives 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.

Table of Amplitudes.

+--------+----------------++----------------+ |Degrees.| Ascents. || Descents. | | | Multip’rs. || Multip’rs. | +--------+--------+-------++--------+-------+ | |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 | infi- || 2.88 | .360 | | | | nite. || | | +--------+--------+-------++--------+-------+

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 | 1.80 | 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.--Calibres of European Guns, expressed in 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 weight of English Brass guns.

-------------------+--------------+----------- Kind. | Length in | Weight. -------------------+------+-------+----------- |Calib.|ft. in.|ct. qr. lb. 42 Pounders |16.244| 9 6 |66 -- -- 32 ---- |18.721|10 0 |55 2 -- {Heavy |19.574| 9 6 |53 0 9 24 {Medium |16.483| 8 0 |41 3 2 {Light |10.302| 5 0 |16 3 13 {Do. new |13.000| 6 3 |24 0 -- 18 Light do. |13.000| 5 9 |18 0 -- {Heavy |24.659| 9 0 |31 2 8 {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 * {Do. new |13.000| 5 0 |12 -- -- {Heavy |26.112| 8 0 |19 1 6 * {Desag’s. Med. |22.876| 7 0 |12 -- 24 6 { {new |18.500| 6 0 | 8 3 27 {Med.{reduced |17.000| 5 6 | 8 0 22 * {Gen. 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 * {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 {Gen. 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 -------------------+------+-------+----------

The guns marked (*) are the only ones used by the British since 1795, on general service.

Length and weight of French brass guns, in their old weights and measures.

-----------------+------------------+---- Kind. | Length in |W’t. -----------------+------+-----------+---- |Calib.|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 -----------------+------+-----------+----

The French weights and measures have assumed new names, and are reduced to strict proportions since the revolution. The weights here referred to are the old. For the new French system of weights and measures, see the word WEIGHTS.

Length and weight of English iron guns.

---------+--------------+-------+-------- | | |Pro- | | |portion | | |between | | |shot and Kind. | Length in | Wt. |gun. ---------+------+-------+-------+-------- |Calib.|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 {|21.4 | 7 6 |24 2 | 305 9 -- {|19.9 | 7 -- |23 -- | { |26.2 | 8 -- |22 -- | 6 -- { |19.6 | 6 -- |16 2 | 411 {|22.4 | 6 -- |22 1 | 343 4 -- {|20.6 | 5 6 |11 1 | 3 -- |18.6 | 4 6 | 7 1 | 270 ---------+------+-------+-------+--------

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

French iron guns, in English weights, &c.

------------+----------------+------------ Kind. | Length in | Weight. ------------+------+---------+------------ |Calib.|ft. in. |ct. qr. lbs. 36 Pounders | 16.18| 9 8 | 74 3 -- 24 ---- | 18.18| 9 1¹⁄₂| 51 -- -- {| 21.01| 9 7 | 42 -- -- 16 ---- {| 18.45| 8 4 | 43 2 -- {| 16.92| 7 8¹⁄₂| 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.

------------------+--------+----------------------------- | | To the first graze of | | the shot. Kind. | Charge.| PB | 1° | 2° | 3° | 4° | 5° ------------------+--------+----+----+----+----+----+---- |lbs. 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| |1399 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 Med. | 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| | Amuzette of 5 feet| 0 8 | | 604| 800| | | Do. 7 feet | 0 8 | | 656| 830|1000| | ------------------+--------+----+----+----+----+----+----

Ranges from Brass Guns, with Two Shot. 1793.

+------------------+--------+------+------------------+ | | |Eleva-|Medium first Graze| | Kind. | Charge.| tion.| in yards. | +------------------+--------+------+---------+--------+ | |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.