Or, 116° + c = S 149°23'26", {,34803834"! -AB. sin 540 -BC. sin 116° DE Sin(E+F) + EF . sin F. The second of these will give for c, a re-entering angle, the first will give exterior angle c = 33° 23′ 26′′, and then will D = 14°36'34". Lastly, AB. COS 54° +BC. COS 64° AF +CD.cos 30°36′34′′ =3885.905. +DE. COS 44° -EF. Cos 72° Ex. 3. In a hexagon ABCDEF, are known, all the sides except AF, and all the angles except в and E; to find the rest. Given AB 1200 Exterior angles A = 64° Suppose the diagonal BE drawn, dividing the figure into two trapeziums. Then, in the trapezium BCDE the sides, except BE, and the angles except в and E, will be known; and these may be determined as in exam. 1. Again, in a trapezium ABEF, there will be known the sides except AF, and the angles except the adjacent ones в and E. Hence, first for BCDE: (cor. 3 th. 2), tan CBE = CD.sin C+DE.Sin (c+D) CD sin 720+DE. sin 33° BC+CD cos 72°+DE.COS 147° BC+CD.COS 72° - DE. cos 33°° Whence CBE 79° 2' 1"; and therefore DEB =67°57′59′′. BC. cos 79° 2' 1"'' +CD.cos 7° 2' 1" Secondly, in the trapezium ABEF, } =2548.581. AB. Sin A+BE. sin (A + B) = EF. Sin F: whence Taking the lower of these, to avoid re-entering angles, we have B (exterior ang.)=95°4′6′′; ABE 84°55′ 54′′; FEB = 63°4'6": therefore ABC 163°57'55"; and FED = 131 2'5": and consequently the exterior angles at в and E are 16° 2′5′′ and 48° 57' 55" respectively. Lastly, AF-AB. COS A-BE. COS (A + B) — EF COS —— AB. COS 64° + BE. cos 20° 55' 54"-EF. cos 84°1645.292. Note. The preceding three examples comprehend ail the varieties which can occur in Polygonometry, when all the sides except one, and all the angles but two, are known. The unknown angles may be about the unknown side;, or they may be adjacent to each other, though distant from the unknown side; and they may be remote from each other, as well as from the unknown side. Ex. 4. In a hexagon ABCDEF, are known all the angles, and all the sides except AF and CD: to find those sides. Here, reasoning from the principle of cor. th. 2, we have AB. sin 96°· +BC. sin 150° +CD.sin 170° S DE.Sin 149 DE. sin 166° or AB. sin 84° +EF. Sin 148°. +BC. sin 30° +CD sin 10°EF.Sin 32°. Whence DE. sin 14°. cosec 10°- AB sin 84° cosec 10° And S DE. sin 24°. cosec 10°-CB sin 20°2 AF= =14874.98. Ex. 5. In the nonagon ABCDEFGHI, all the sides are known, and all the angles except A, D, G: it is required to find those angles. Suppose diagonals drawn to join the unknown angles, and dividing the polygon into three trapeziums and a triangle; as in the marginal figure. Then, G 1st. In the trapezium ABCD, where AD and the angles about it are unknown; we have (cor. 3 th. 2) C = 329 F 45° H = 48° I = 50°. E I A tan tan BAD BC. SinB +CD.sin (B+c). BC. sin 40°+CD sin 70° AB+BC.COSB+CD.COS(B+C) AB+BC.CUS40°+CD.cos720* Whence BAD 39°30′42, CDA = 32°29′18′′. 2dly. In the quadrilateral DEFG, where DG and the angles about it are unknown; we have Whence EDG = 41° 14′ 53′′, FGD = 39° 45′ 7′′. And DG = { DE. COS 41° 14′ 53′′ + EF. COS 5° 14' 53" + FG. COS 39° 45' 7" } 8812.803. 3dly. In the trapezium GHIA, an exactly similar process gives HGA 50° 46' 53", IAG = 47° 13'7', and AG 9780-591. 4thly. In the triangle ADG, the three sides are now known, to find the angles: viz, DAG=60° 53′ 26′′, AGD=43° 15′ 54′′, ADG 75° 50′ 40". Hence there results, lastly, IAB 47° 13'7''+60° 53′ 26!+39° 30′ 42" 147° 37'15", CDE 32°29'18"+70°50′ 40"+41° 14′ 53′′ 149° 34′ 51", FGH=39°45′ 7′′+43°15′ 54′′+50° 46′53′′ = 133° 47′ 54′′. Consequently, the required exterior angles are A-32° 22′ 45′′, D= 30°25' 9", G = 46° 12′ 6“. = Ex. 6. Required the area of the hexagon in ex. 1. Ans. 16530191. Ex. 7. In a quadrilateral ABCD, are given AB=24, BC = = 30, CD34; angle ABC = 92° 18', BCD = 97° 23'. Required the side AD, and the area. Ex. 8. In prob. 1, suppose PQ = 2538 links, and the angles as below; what is the area of the field ABCDQP? APQ=89° 14, BPQ=68°11′,CPQ=36° 24',DPQ= 19057; AQP=25° 18', BQP=69° 24', cqP=94° 6′, DQP=121° 18'. OF OF MOTION, FORCES, &c. DEFINITIONS. Art. 1. BODY is the mass, or quantity of matter, in any material substance; and it is always proportional to its weight or gravity, whatever its figure may be. 2. Body is either Hard, Soft, or Elastic. A Hard Body is that whose parts do not yield to any stroke or percussion, but retains its figure unaltered. A Soft Body is that whose parts yield to any stroke or impression, without restoring themselves again; the figure of the body remaining altered. And an Elastic Body is that whose parts yield to any stroke, but which presently restore themselves again, and the body regains the same figure as before the stroke. We know of no bodies that are absolutely, or perfectly, either hard, soft, or elastic; but all partaking these properties, more or less, in some intermediate degree. 3. Bodies are also either Solid or Fluid. A Solid Body, is that whose parts are not easily moved among one another, and which retains any figure given to it. But a Fluid Body is that whose parts yield to the slightest impression, being easily moved among one another; and its surface, when left to itself, is always observed to settle in a smooth plane at the top. 4. Density is the proportional weight or quantity of matter in any body. So, in two spheres, or cubes, &c, of equal size or magnitude; if the one weigh only one pound, but the other two pounds; then the density of the latter is double the density of the former; if it weigh 3 pounds, its density is triple; and so on. 5. Motion is a continual and successive change of place. If the body move equally, or pass over equal spaces in equal times, it is called Equable or Uniform Motion. But if it increase or decrease, it is Variable Motion; and it is called Accelerated Motion in the former case, and Retarded Motion in the latter. Also, when the moving body is considered with respect to some other body at rest, it is said to be Absolute Motion. But when compared with others in motion, it is called Relative Motion. 6. Velocity, or Celerity, is an affection of motion, by which a body passes over a certain space in a certain time. Thus, if a body in motion pass uniformly over 40 feet in ⚫ 4 seconds of time, it is said to move with the velocity of 10 feet per second; and so on. 7 Momentum, or Quantity of Motion, is the power or force in moving bodies, by which they continually tend from their present places, or with which they strike any obstacle that opposes their motion. 8. Force is a power exerted on a body to move it, or to stop it. If the force act constantly, or incessantly, it is a Permanent Force: like pressure or the force of gravity. But if it act instantaneously, or but for an imperceptibly small time, it is called Impulse, or Percussion like the smart blow of a hammer. 9. Forces are also distinguished into Motive, and Accelerative or Retarding. A Motive or Moving Force, is the power of an agent to produce motion; and it is equal or proportional to the momentum it will generate in any body, when acting, either by percussion, or for a certain time as a permanent force. 10. Accelerative, or Retardive Force, is commonly understood to be that which affects the velocity only; or it is that by which the velocity is accelerated or retarded; and it is equal or proportional to the motive force directly, and to the mass or body moved inversely.-So, if a body of 2 pounds weight, be acted on by a motive force of 40; then the accelerating force is 20. But if the same force of 40 act on another body of 4 pounds weight; then the accelerating force in this latter case is only 10; and so is but half the former, and will produce only half the velocity. 11. Gravity, or Weight, is that force by which a body endeavours to fall downwards. It is called Absolute Gravity, when the body is in empty space; and Relative Gravity, when emersed in a fluid. 12. Specific Gravity is the proportion of the weights of different bodies of equal magnitude; and so is proportional to the density of the body. AXIOMS. |