Take logs of the upper factors by themselves and of the lower factors by themselves, add each column, and subtract. Ex. 2. A circular plate of lead, 2 inches in thickness and 8 inches in diameter, is converted without loss into spherical shot of the same density, and each of 05 inch radius. How many shot does it make ? As the density of the plate and of the shot is the same, we require to consider the volume of the materials only. EXERCISE XIV. 1. Find the volume of a cone whose altitude is 2 feet, and the diameter of the base 1 foot 6 inches. 2. Find the surface of a sphere which is one yard in diameter. 3. Find the radius of the sphere the volume of which is equal to the sum of the volumes of two spheres, whose radii are 3 feet and 4 feet. 4. The area of the base of a cylinder is 2 square feet and its height 30 inches; find the height of a cylinder the solid content of which is three times as great, but whose diameter is only two thirds of the given one. 5. If the volume of the first of two cylinders is to that of the second as 10 to 8, and the height of the first is to that of the second as 3 to 4, and if the base of the first has an area of 16·5 square feet, what is the area of the base of the second? 6. Determine the volume of the earth, supposing its diameter to be 8,000 miles. How many masses of the size of the earth would make up one of the size of the sun, the diameter of which is 880,000 miles? 7. Two spheres, A and B, have for radii 9 feet and 40 feet; the superficial area of a third sphere C is equal to the sum of the areas of A and B. Calculate the excess in cubic feet of the volume of C over the sum of the volumes of A and B. 8. A cone and hemisphere being supposed to have a common base and to lie at opposite sides of it; required, the ratio of the altitude of the cone to the radius of the hemisphere, in order that the volumes of the two solids should be equal. 9. Determine approximately the length of the radius of the sphere whose volume is 400 cubic feet. 10. A sphere and a cube have an equal amount of surface; what is the ratio of their volumes? 11. If a model is made on the linear scale of 1/40 inch to the foot; what is the scale for surface and for volume? 12. The altitude of a common cone equals the circumference of its base. Calculate the volume and the area of the whole surface of the cone, the radius of the base being 6 inches. 13. The interior of a building is in the form of a cylinder of 30 feet radius and 12 feet altitude, surmounted by a cone whose vertical angle is a right angle. How many cubic feet of air will it contain? 14. The long axis of a spheroid is 10 inches, and each short axis 6 inches. Find its volume. CHAPTER THIRD. KINEMATICAL. SECTION XV.-TIME. ART. 98.-Sidereal Units. The idea of time is fundamental. The general unit is appropriately denoted by T. The standard of time adopted by all civilized nations is the time occupied by the earth in making one rotation about its axis. This interval is marked out by the successive transits of a particular star across the meridian of a place, and it is on that account denominated a sidereal day. The sidereal day is divided into 24 sidereal hours, the sidereal hour into 60 sidereal minutes, and the sidereal minute into 60 sidereal seconds. The sidereal units are used principally by astronomers. ART. 99. Mean Solar Units. By a year is meant the constant interval occupied by the earth in making a revolution round the sun; it is marked out by the sun leaving and returning to a certain position among the stars. The transit of the centre of the sun across the meridian of a place marks out an interval called the apparent solar day. This apparent day is not completely constant; its length goes through a cycle of small changes in the course of a year. Its average length for the course of a year is called the mean solar day. This interval of the mean solar day is measured out by clocks and chronometers, corrected by observation, on the part of astronomers, of sidereal time. The mean solar day is divided into 24 mean solar hours, each of which is divided into 60 mean solar minutes, each of which is divided into 60 mean solar seconds. When the terms "hour," "minute," "second" are used without qualification, "mean solar is understood. In science the "second" is the unit principally used; it is chosen for the unit of time both in the British system of units and in the C.G.S. system of units. 1 mean solar day = 1.00274 sidereal day. 1 mean solar second = 1.00274 sidereal second. ART. 100.-Relation of Mean Solar Day to Year. The interval occupied by the earth in making one rotation has not a simple relation to the interval occupied by the earth in making one circuit round the sun. Hence the equivalence of the year to the mean solar day involves a complex number. It is 1 year = 365-2422 mean solar days. The first approximation in use is 1 year = 365 days; it is called the common year, and is sufficient for all ordinary calculations. But it is not sufficient for the purpose of arranging the calendar, so that a given date of the month shall always fall on the same season of the year. For this purpose a second approximation was introduced under the authority of Julius Cæsar: 4 × 3651 days, 365.25 days. i.e., 4 years 1 year = = This equivalent is called the Julian year. or or A third approximation is 25 × 4 years = 25(4 × 365 + 1) −1 days, 1 year 365-24 days. = A fourth approximation is 4 x 25 x 4 years 4{25(4 × 365 + 1) −1} + 1 days, = 1 year = 365-2425 days. This equivalent is called the Gregorian year, because the approximation was applied to the calendar under the authority of Pope Gregory. The mode in which the Gregorian equivalence is applied is as follows: A A year whose number is divisible by 4 and by 25, but not A year whose number is divisible by 4 and by 25, and by 4 again, contains 366 days. ART. 101.-Epoch and Era. When we come to specify time as an ordinal quantity we require to choose an origin from which to reckon. The civil day is reckoned from mean midnight to mean midnight; the nautical and the astronomical from mean noon to mean noon. The two latter differ in this respect, that the number for a civil day is by the nautical reckoning given to the interval between the preceding noon and the noon of the civil day; whereas by the astronomical reckoning it is given to the interval between the noon of the civil day and the succeeding noon. Astronomers reckon the hours up to 24, and this mode of reckoning is sometimes found more convenient on extensive railway systems. By the epoch of an era is meant the particular year from which the years are numbered, that of the Christian era being the year of the birth of Christ. ART. 102.-Local Time and Standard Time. The civil day is reckoned from mean noon, but each meridian has its own mean noon; hence the same instant is not denoted by the same number at places on widely different meridians. The connection between difference of time and difference of longitude is given by or 24 hours later 1 hour later = 360 degrees of longitude west, In the British Islands the legal origin for the civil day at a place is the mean midnight at the place; but it is found more |