131°25": Taking the lower of these to avoid re-entering angles, we Lastly, AF-AB. COS A-BE. COS (A + B) - EF COS F= EF. Cos 84° : 1645-292. Note. The preceding three examples comprehend all the varieties which can occur in Polygonometry, when all the sides The unexcept one, and all the angles but two, are known. known 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. Given AB 2200 Ext. Ang. A 96° B = 54° BC= 2400 D = 249 E = 18° F= 148°. Here, reasoning from the principle of cor. th. 2, we have, AB. sin 960· BC. sin 150° +CD. sin 170° DE. sin 16600r AB sin 84o+EF. Sin 1480+ BC sin 30° WhenceS DE.Sin14°.cosec10°. +CD.sin 10° AB.sin 84°. cosec 10° BC sin 30°, cosec 100 And DE.sin24°.cosec10°. -CB sin 20° AF= {+E.F.sin42°.cosec10° BA.sin 74°} S DE.S'n140 } 14874.98. 3045-58. 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. 2dly. In the quadrilateral DEFG, where DG and the angles about it are unknown; we have tan EDG= EF.SINE FG.sin(E+F) EF.Sin 36°+FG.sin81° DE+EFCUSE+FG.COS(E+F) DE+EF.COS36°+FG.Cos81 Whence ErG= 41° 14′ 53′′, FGD = 39° 45′ 7′′. DE. COS 41° 14′ 53′′ And DG = +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, CD = 34; 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={9° 14, BPQ=68° 11', CPQ=36° 24,DPQ= 19° 57 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 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. AXIOMS. 13. EVERY body naturally endeavours to continue in its present state, whether it be at rest, or moving uniformly in a right line. 14. The Change or Alteration of Motion, by any external force, is always proportional to that force, and in the direction. of the right line in which it acts. 15. Action and Re-action, between any two bodies, are equal and contrary. That is, by Action and Re-action, equal changes of motion are produced in bodies acting on each other; and these changes are directed towards opposite or contrary parts. GENERAL LAWS OF MOTION, &c. PROPOSITION I 16. The Quantity of Matter, in all bodies, is in the Compound Ratio of their Magnitudes and Densities. THAT is, b is as ind; where b denotes the body or quantity of matter, in its magnitude, and d its density. For, by art. 4, in bodies of equal magnitude, the mass or quantity of matter is as the density. But, the densities remaining, the mass is as the magnitude: that is, a double magnitude contains a double quantity of matter, a triple magnitude a triple quantity, and so on. Therefore the mass is in the compound ratio of the magnitude and density. 17. Corol. 1. In similar bodies, the masses are as the densities and cubes of the diameters, or of any like linear dimensions. For the magnitudes of bodies are as the cubes of the diameters, &c. 18. Corol. 2. The masses are as the magnitudes and specific gravities. For, by art. 4 and 12, the densities of bodies are as the specific gravities. 19. Scholium. Hence, if b denote any body, or the quantity of matter in it, m its magnitude, dits density, g its specific |