AXIOMS. I. Things which are equal to the same thing, are equal to one another. II. If equals be added to equals, the wholes are equal. III. If equals be taken from equals, the remainders are equal. IV. If equals be added to unequals, the wholes are unequal. V. If equals be taken from unequals, the remainders are unequal, VI. Things which are double of the same, are equal to one another. VII. Things which are halves of the same, are equal to one another. VIII. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. IX. The whole is greater than its part. X. Two straight lines cannot inclose a space. XI. All right angles are equal to one another. XII. "If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines being continually produced, shall at length meet upon that side on which are the angles which are less than two right angles." EXPLANATION OF TERMS AND USEFUL NUMBERS IN ARITHMETICAL CALCULATION. In accordance with the design of this book, terms have been introduced which are not in ordinary use in works of this kind, and which require a little explanation. I trust the following will be deemed sufficient to answer every purpose : Pitch of a Screw.-It is the vertical distance between two threads of a screw, or rather the distance between two consecutive threads. A tolerable idea of a screw may be formed by cutting an inclined plane out of paper and wrapping it round a cylinder whose length is equal to the height of the plane, and whose circumference of base is equal to the base of the inclined plane. The height of the inclined plane is the pitch of the screw. Angle and helix of screw.-The angle at the base of the inclined plane is the angle of the screw, and it is the same as the angle D B C in diagram, page 24. The hypothenuse of the inclined plane is the helix of the screw, and corresponds with the line BD, in diagram, page 24. Screws in general use are formed by numerous convolutions of the inclined plane, or by repeating the process of wrapping the inclined plane round the cylinder, increased in length any convenient number of times. In this manner the screw has a length, which is equal to the vertical distance between the extremities of the helix. The length of the screws, however, employed in propelling vessels is usually about one-sixth of their pitch, and therefore the helix does not form a complete convolution. The pitch of such screws is an abstract magnitude, and can only be measured by means of its relation to the diameter, length, and angle of the screw. The screw propeller-blade, properly speaking, is a portion of a conoidal surface, which is produced by a straight line A (always perpendicular to B), moving uniformly round B, while its extremity moves uniformly in the direction of B. Positive and negative slip of screw.-The slip of a screw is explained in Problem 9, page 25. When the vessel is moving slower than the screw, then the slip is positive, and it is negative when the vessel is moving faster than the screw. The fact of the vessel going faster than the screw which propels it, is deemed by many an anomaly in screw propulsion, and there are those who believe that such a circumstance is impossible, and contrary to the received laws of mechanics. I cannot coincide with this view, as there are several well authenticated facts which prove the contrary, and the reason for the negative slip is not difficult to assign when it is remembered, that the screw revolves in water at the stern of the vessel which moves in the direction of the vessel's motion. When screw propulsion was first introduced, the slip of the screw claimed considerable attention, and it was thought by many that a diminution of this element would be attended by an increase in the speed of the vessel. Hence, the determination of the pitch, diameter, and length of a screw, so as to produce a minimum slip, was a great desideratum. Experience, however, was not slow in correcting this hasty impression. Screws were obtained which produced little or no slip, still the speed of the vessel was not improved: the reason of this result is by no means difficult to understand, as the slip depends on two elements, viz., the speed of the vessel and the rotatory motion of the screw. Metacentre. This term was first introduced by Bougier, and is the ultimate intersection of a perpendicular to the curve, which is the locus of the centre of buoyancy, with the line passing through the centre of gravity of the ship and the centre of gravity of the part immersed when the ship is not inclined. Displacement of a ship is the quantity of water, in cubic feet, or in tons, displaced by the ship when it floats. Solids and Surfaces of Revolution. A sphere is formed by the revolution of a semicircle about its diameter. A cone is produced by the revolution of a right-angled triangle about one of the sides which contains the right angle. A paraboloid is generated by the revolution of a parabola about its abscissa. See Problem 20, page 7. Prolate spheroid is produced by the revolution of an ellipse about its transverse, or longest diameter. Oblate spheroid is produced by the revolution of an ellipse about its conjugate, or shortest diameter. This is the figure which the earth would assume, supposing it to be originally composed of a homogeneous fluid mass, every particle attracting every other particle with an accelerating force, proportional to the mass of the attracting particle directly, and the square of the distance of the attracted particle inversely. The whole fluid mass revolving about an axis in 23 h. 56 m. 4 s. (See Airy's Tracts, page 136.) Hyperboloid is formed by the revolution of a hyperbola about its transverse axis. Ellipsoid is a solid bounded by a surface, the equation of = 1, the constants a, b, c are the x2 y2 which is + 1/2 + 22 a2 C2 semi-diameters, and x, y, z the distances of a point on the surface from the centre, measured in the direction of the diameters respectively. This surface may be generated by a variable ellipse moving upwards parallel to itself, with its centre on a line perpendicular to the plane of the ellipse. |