tan BAD= BC. Sin B+CD. Sin (B+C) ABEC.COS B+CD.COS (B+ c) Whence BAD = 39° 30′ 42'', CDA = BC. sin 40° + CD sin 70° ABRC COS 40°+ CD. cos 72° 32° 29′ 18′′. AB. Cos 39° 30′42′′ And AD = +c.cos 0 29 18 +CD. cos 32 29 18 =6913-292. 2dly. In the quadrilateral DEFG, where DG and the angles about it are unknown; we have EF. Sin EFG. sin (+F) = EF. sin 36° + FG. sin 81° DEEF.CUS 36°EG.COs 81° tan EDG = { 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′′, 75° 50′ 40′′. Hence there results, lastly, ADG IAB CDE = 47° 13′ 7′′ + 60° 53′26′′ + 39. 30′42′′ = 147° 37′15′′, 32° 29′18′′ +70° 50′40′′ +41° 14′53′′ = 149°34′51′′, 39° 45′ 7+ 43° 15'54" +50° 46'63" : = 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 ABCUQP? APQ=89° 14, BPQ=68° 11, cPQ=36 24,DPQ= 1957; 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 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 effects 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 immersed 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 the force 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 md; where b denotes the body or quantity of matter, m 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, d its density, g its specific gravity, gravity, and a its diameter or other dimension; then. α (pronounced or named as) being the mark for general proportion, from this proposition and its corollaries we have these general proportions : 20. The Momentum, or Quantity of Motion, generated by a Single Impulse, or any Momentary Force, is as the Generating Force. THAT is, m is as f; where in denotes the momentum, and f the force. So For every effect is proportional to its adequate cause. that a double force will impress a double quantity of motion: a triple force, a triple motion; and so on. That is, the motion impressed, is as the motive force which produces it. PROPOSITION HII. 21. The Momenta, or Quantities of Motion, in moving Bodies, are in the Compound Ratio of the Mosses and Velocities. That is, m is as bv. For, the motion of any body being made up of the mo tions of all its parts, if the velocities be equal, the momenta will be as the masses; for a double mass will strike with a double force; a triple mass, with a triple force, and so on. Again, when the mass is the same it will require a double force to move it with a double velocity, a triple force with a triple velocity, and so on; that is, the motive force is as the velocity; but the momentum impressed, is as the force which produces it, by prop. 2; and therefore the momentum is as the velocity when the mass is the same. But the momentum was found to be as the mass when the velocity is the same. Consequently |