the sine of twice the angle s A F; half the impetus being radius.

for

Whence, at the directions of 15° or 75°, the amplitude is equal to the impetus; from what has been said, half the impetus being radins, a fourth part of the amplitude is the sine of twice the angle of elevation; but the sine of twice 15°, that is, the sine of 30°, is always equal to half the radius; or in this case a fourth part of the impetus is equal to a fourth part of the amplitude. From this and the preceding proposition there are two easy practical methods for finding the impetus of any piece of ordnance. The fourth part of the amplitude is a mean proportional between the impetus at the curve's principal vertex and its altitude. For MN: Ns:: Ns: NA=sF=v D.

The altitudes are as the versed sines of double the angles of elevation, the impetus remaining the same. For making half the impetus radius, AN the altitude is the versed sine of the angle AC s = twice angle SAF. And also, radius: tangent angle elevation :: one-fourth amplitude: altitude; that is, R: tangent angle s Aƒ:: Aƒ:ƒs =

Dv.

Projections on Ascents and Descents, fig. 8, 9. If the mark can be hit only with one direction A G, the impetus in ascents will be equal to the sum of half the inclined plane and half the perpendicular height, and in descents it will be equal to their difference; but if the mark can be reached with two directions, the impetus will be greater than that sum or difference. For when AG is the line of direction, the angle g GA being MAG=GAg; Gg=Ag, and g z added to or subtracted from both makes Gz half the impetus equal to the sum or difference of Ag a fourth part of the inclined plane, and g z a fourth part of the perpendicular height. In any other direction F P is greater than Fo=A F; and Ff added to or subtracted from both, makes fP half the impetus greater than the sum or difference of AF a fourth part of the inclined plane, and Ff a fourth part of the perpendicular height. Whence if in ascents the impetus be equal to the sum of half the inclined plane and half the perpendicular height, or if in descents it be equal to their difference, the mark can be reached only with one direction; if the impetus is greater than that sum or difference, it may be hit with two directions; and if the impetus is less, the mark can be hit with none at all.

Prob. II. The angles of elevation, the

horizontal distance, and perpendicular height being given, to find the impetus. Fig. 8, 9.

From these data you have the angle of obliquity, and length of the inclined plane; then as

As AM S. angle A M 8: S. angle As M :: S. angle s A F: S. angle M A F, and AF: As::S. angle M As: S. angle M AF; whence by the ratio of equality, AF: A M:: S. angle s AFX S. angle M As: S. angle MAFX S. angle M A F, which gives

this rule.

garithmic sine of the angle MAF; from Add the logarithm of A F to twice the lotheir sum subtract the logarithmic sines of the angles s AF and MA s, and the reremainder will give the logarithm of A M the impetus.

When the impetus and angles of elevation are given, and the length of the inclined plane is required, this is the rule. sines of the angles s AF and MAs; from Add the logarithm of A M to the logarithmic sine of angle MA F, and the remainder will their sum subtract twice the logarithmic give the logarithm of A F the fourth part of the length of the inclined plane.

If the angle of elevation t A H and its amplitude A B (fig. 11,) and any other angle of elevation t A H is given; to find the amplitude Ab for that other angle, the impetus A M and angle of obliquity DAH remaining the same.

Describe the circle AG M, take A Fa fourth part of A B, and Aƒ a fourth part of Ab: from the points F, f, draw the lines Fs and fp parallel to A M, and cutting the circle in the points s,p; then AF: AM:: S. augle s A F× S. angle M As: S. angle M A F × S. angle MAF; and A M: Aƒ:: S. angle MAF S. angle MAF: S. angle p Afx S. angle p A M; whence by the ratio of equality.

A FAƒ:: S. angle & A F x S. angle M As: S. angle p A ƒ × S. angle p A M, which gives this rule.

Add the logarithm of A F to the logafrom their sum subtract the logarithmic rithmic sines of the angles pAƒ, pAM; sines of the angles s AF, s A M, and the remainder will give the logarithm of Aƒ, a fourth part of the amplitude required.

Prob. III. To find the force or velocity of a ball or projectile at any point of the curve, having the perpendicular height of that point, and the impetus at the point of

projection given. From these two data find out the impetus at that point; then 2 x 16 feet 1 inch is the velocity acquired by the descent of a body in a second of time; the square of which (4 × the square of 16 feet 1 inch) is to the square of the velocity required, as 16 feet 1 inch is to the impetus at the point given; wherefore multiplying that impetus by four times the square of 16 feet 1 inch, and dividing the product by 16 feet 1 inch, the quotient will be the square of the required velocity; whence this rule. Multiply the impetus by four times 16 feet 1 inch, or 64 feet, and the square root of the product is the velocity.

Thus suppose the impetus at the point of projection to be 3,000. and the perpendicular height of the other point 100; the impetus at that point will be 2,900. Then 2,900 feet multiplied by 64 feet gives 186,566 feet, the square of 432 nearly, the space which a body would run through in one second, if it moved uniformly.

And to determine the impetus or height, from which a body must descend, so as at the end of the descent it may acquire a given velocity, this is the rule:

Divide the square of the given velocity (expressed in feet run through in a second) by 64 feet, and the quotient will be the impetus.

The duration of a projection made perpendicularly upwards, is to that of a projection in any other direction whose impetus is the same, as the sine complement of the inclination of the plane of projection (which in horizontal projections is radius) is to the sine of the angle contained between the line of direction and that plane.

Draw out At (fig. 8,) till it meets m B continued in E, the body will reach the mark B in the same time it would have moved uniformly through the line A E; but the time of its fall through MA the impetus, is to the time of its uniform motion through A E, as twice the impetus is to A E.

And therefore the duration of the perpendicular projection being double the time of its fall, will be to the time of its uniform motion through A E; as four times the impetus is to A E; or as A E is to EB; that is, as A t is to t D; which is as the sine of the angle t DA (or M A B its complement to a semicircle) is the sine of the angle t A D.

Hence the time a projection will take to arrive at any point in the curve, may be

found from the following data, viz. the im petus, the angle of direction, and the inclination of the plane of projection, which in this case is the angle the horizon makes with a line drawn from the point of projec tion to that point.

Hence also in horizontal cases, the durations of projections in different directions with the same impetus, are as the sines of the angles of elevation. But in ascents or descents their durations are as the sines of the angles which the lines of direction make with the inclined plane. Thus, suppose the impetus of any projection were 4,500 feet; then 16 feet 1 inch : 1": : 4,500 feet: 275" the square of the time a body will take to fall perpendicularly through 4,500 feet, the square root of which is 16'' nearly, and that doubled gives 32" the duration of the projection made perpendicularly upwards. Then to find the duration of a horizontal projection at any elevation, as 20°; say K: S. angle 20°:: 32′′: duration of a projection at that elevation with the impetus 4,500. Or if with the same impetus a body at the direction of 35° was projected on a plane inclined to the horizon 17o, say as sine 73°: sine 18°;: 32": duration required.

The tables in the next leaf, at one view, give all the necessary cases as well for shooting at objects on the plane of the ho rizon, with proportions for their solutions, as for shooting on ascents and descents. We shall in this place mention some of the more important maxims laid down by Mr. Robins, as of use in practice. 1. In any piece of artillery, the greater quantity of powder with which it is charged, the greater will be the velocity of the bullet. 2. If two pieces of the same bore, but of different lengths, are fired with the same charge of powder, the longer will impel the bullet with a greater celerity than the shorter. 3. The ranges of pieces at a given elevation, are no just measures of the velocity of the shot: for the same piece fired successively at an invariable elevation, with the powder, bullet, and every other circumstance as nearly the same as possible, will yet range to very different distances, 5. The greatest part of the uncertainty in the ranges of pieces, arises from the resistance of the air. 6. The resistance of the air acts upon projectiles by opposing their motion, and diminishing celerity; and it also diverts them from the regular track which they would otherwise follow. 7. If the same piece of cannon be successively fired at an invariable

elevation,but with various charges of powder, the greatest charge being the whole weight of the ball in powder, and the least not less than the fifth part of that weight; then, if the elevation be not less than eight or ten degrees, it will be found that some ranges with the least charge, will exceed some of those with the greatest. 8. If two pieces of cannon of the same bore, but of different lengths, are successively fired at the same elevation, with the same charge of powder, then it will frequently happen

that some of the ranges with the shorter piece will exceed some of those with the longer. 9. Whatever operations are performed with artillery, the least charges of powder with which they can be effected, are always to be preferred. 10. No fieldpiece ought at any time to be loaded withi more than one-sixth, or at most one-fifth of the weight of its bullet in powder, nor should the charge of any battering-piece exceed one-third of the weight of its bullet.

Cases.

GUNPOWDER, a composition of nitre, sulphur, and charcoal, mixed together, and usually granulated. This easily takes fire, and when fired it rarefies or expands with great vehemence, by means of its elastic force. It is to this powder that we owe all the effect and action of guns, and ordnance of all sorts, so that fortification, with the modern military art, &c. in a great measure depends upon it.

The invention of gunpowder is ascribed by Polydore Virgil to a chemist, who having accidentally put some of his composition in a mortar, and covered it with a stone, it happened to take fire, and blew up the stone. Thevet says, that the person here spoken of was a monk of Fribourg, named Constantine Anelzen; but Belleforet, and other authors, with more probability, hold it to be Bartholdus Schwartz, or the black, who discovered it, as some say, about the year 1320; and the first use of it is ascribed to the Venetians in the year 1380, during the war with the

Am: Bm::R: T. angle B Am, half of which added to 45°, gives angle GA 2. AM:AB:: Ac: AC=CG. T. angle G Az: R::Gz: Az, and A z — Aƒ= f=PG.

CG: PG::R: V. S. of S G, half of which added to, or taken from GA z, gives the higher or lower direction required.

Log. of A M= Log. of AF x2
Log. S. angle MAF-Log. S.
angle s A F
Log. S. angle
MA s.

Genoese. But there are earlier accounts of its use, after the accident of Schwartz, as well as before it; for Peter Mexia, in his "Various Readings," mentions that the Moors being besieged, in 1343, by Alphonsus the Eleventh, King of Castile, discharged a kind of iron mortars upon them, which made a noise like thunder: and this is seconded by what is related by Don Pedro, Bishop of Leon, in his Chronicle of King Alphonsus, who reduced Toledo, viz. that in a sea combat between the King of Tunis, and the moorish King of Seville, about that time, those of Tunis had certain iron tubs or barrels, with which they threw thunderbolts of fire,

Du Cange adds, that there is mention made of gunpowder in the registers of the chambers of accounts in France, as early as the year 1338. But it appears that Roger Bacon knew of gunpowder near one hundred years before Schwartz was born; and M. Dutens carries the antiquity of gunpowder still much higher, and refers to

the writings of the ancients themselves for the proof of it. It appears too, from many authors and many circumstances, that this composition has been known to the Chinese and Indians for thousands of years.

For some time after the invention of artillery, gunpowder was of a much weaker composition than that now in use, or that described by Marcus Græcus, which was chiefly owing to the weakness of their first pieces. Of twenty-three different compositions, used at different times, and mentioned by Tartaglia in his “Ques, and Inv. lib. 3, ques. 5;" the first, which was the oldest, contained equal parts of the three ingredients. But when guns of modern structure were introduced, gunpowder of the same composition as the present came into use. In the time of Tartaglia the cannon powder was made of four parts of nitre, one of sulphur, and one of charcoal; and the musket-powder of forty-eight parts of nitre, seven parts of sulphur, and eight parts of charcoal; or of eighteen parts of nitre, two parts of sulphur, and three parts of charcoal. But the modern composition is six parts of nitre, to one of each of the other two ingredients: though Mr. Napier says, he finds the strength commonly to be greatest when the proportions are, nitre three pounds, charcoal about nine ounces, and sulphur about three ounces. See his paper on gunpowder in the Transactions of the Royal Irish Academy, vol. ii. The cannon-powder was in meal, and the musket-powder grained; and it is certain, that the graining of powder, which is a very considerable advantage, is a modern improvement.

To make gunpowder duly, regard is to be had to the purity or goodness of the ingredients, as well as the proportions of them, for the strength of the powder depends much on that circumstance, and also on the due working or mixing of them together. See NITRE.

These three ingredients in their purest state being procured, long experience has shown that they are then to be mixed together in the proportion before-mentioned, to have the best effect, viz. three quarters of the composition to be nitre, and the other quarter made up of equal parts of the other two ingredients, or, which is the same thing, six-parts nitre, one part sulphur, and one part charcoal.

But it is not the due proportion of the materials only, which is necessary to the making of good powder, another circum

stance, not less essential, is the mixing them well together; if this be not effectually done, some parts of the composition will have too much nitre in them, and others too little; and in either case there will be a defect of strength in the powder.

After the materials have been reduced to fine dust, they are mixed together, and moistened with water, or vinegar, or urine, or spirit of wine, &c. and then beaten toge ther for twenty-four hours, either by hand or by mills, and afterwards pressed into a hard, firm, solid cake. When dry, it is grained or corned, which is done by breaking the cake of powder into small pieces, and so running it through a sieve; by which means the grains may have any size given them, according to the nature of the sieve employed, either finer or coarser; and thus also the dust is separated from the grains, and again mixed with other manufacturing powder, or worked up into cakes again.

Powder is smoothed or glazed, as it is called, for small arms, by the following operation: a hollow cylinder or cask is mounted on an axis, turned by a wheel; this cask is half filled with powder, and turned for six hours, and thus by the mutual friction of the grains of powder it is smoothed or glazed. The fine mealy part, thus separated or worn off from the rest, is again granulated.

The velocity of expansion of the flame of gunpowder, when fired in a piece of artil lery, without either bullet or other body before it, is prodigiously great, viz. seven thousand feet per second, or upwards, as appears from the experiments of Mr. Robins. But M. Bernoulli and M. Euler suspect it is still much greater; and Dr. Hutton supposes it may not be less, at the moment of explosion, than four times as much.

It is this prodigious celerity of expansion of the flame of gunpowder which is its peculiar excellence, and the circumstance in which it so eminently surpasses all other inventions, either ancient or modern; for as to the momentum of these projectiles only, many of the warlike machines of the ancients produced this in a degree far surpassing that of our heaviest cannon shot or shells; but the great celerity given to these bodies, cannot be in the least approached by any other means but the flame of powder.

To prove gunpowder. There are several ways of doing this. 1. By sight; thus if it be too black, it is a sign that it is moist, or