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28 THE Marine REVIEW the performance of the second, minute and hour hands of a watch or clock. While the second hand ticks off 60 seconds (making a com- plete revolution) the- minute hand moves forward or passes over but 1-60 of its circumference, or in other words, describes an arc of 1 minute, and during the time that the minute hand is making a -com- plete revolution through 60 minutes of space the hour hand advances through 1 hour of space, or what is equivalent to 60 minutes; there- fore the second hand ticks 60 times while the minute hand ticks 1, and while the minute hand ticks 60 the hour hand ticks 1; therefore, 60 second ticks equals 1 minute; 60 minute ticks equals 1 hour, or 60 minutes. During the time that the minute hand makes a complete revolution, the second hand makes 60 revolutions, or 60 X 60 seconds, equals 3,600 seconds; therefore, 3,600 seconds equals 60 minutes. Now, supposing that there was another hand on a watch or clock that would indicate the next denomination lower than seconds, what would its performance be. In the first place it would be called the "third" hand, meaning not that there were three hands, but thirds, 60 of which make a second. 'This hand in indicating thirds would have to tick 60 times, or make one complete revolution, while the second hand ticked one, or moved through one second of space. The construction of such a piece of mechanism would be a most difficult task, though were it possible to construct it, the above would be the principle upon which it would operate. Now, remember that every circle, no matter how large or how small, contains 360°, that is, the circumference of all circles is divided into 360 equal parts, or graduations, called degrees, and all angles are measured by these. A single glance at the accompanying diagram will illustrate this. The angles at a do not increase in size' because A QUARTER CIRCLE, their boundary lines are prolonged. If these lines were prolonged till they reached the apparent sky the angle at their juncture (where they meet) would be the same size-- 45°. A degree, therefore, is 1-360 of the circumference of any circle, no matter what size. Do not forget this im- portant matter. Now, each degree is again divided into 60 equal parts called minutes and each minute into sixty equal parts called seconds, and each second into 60 equal parts called thirds of are, Or: Of circle. _ Every circle, great or small (the compass card is a circle), is also divided into 32 equal parts called points (compass points). The circumference then of every cir- cle, no matter what its size, is divided into 32 equal parts called points, and all angles can likewise be méasured by them. ° . Now, you will readily see that it would take a circle of very large circumference in order to put in all the subdivisions, or graduations, belonging to the degree, but, they are there, nevetheless; even though the space between the degree graduations are not large enough to contain these smaller subdivisions, such as the ', ",'" of the de- gree. Practically speaking, they are not there, that is, they are not there so they can be seen, but theoceti- cally, they are. . Now, if there are 360° to every circle how many minutes (') will there be in 360°? 360 X 60' = 21,600' How many seconds then to a circle? 360 X 60 X 60" = 1,296,000" How many thirds then to a circle? 360 X 60 X 60 X 60"" = 77,760,000"" You will see from this that there isn't any difference between 360° and 21,600" or 1,296,000" or 77,760,000"". ~ Now, if you have a full understanding of what has al- ready been said you will see that 360° and 32 compass points are practically the same thing. Then, 180°, or a semi-circle (one-half of a circle) contains 16 points, one- half of 360° being 180° and one-half of 32 is 16. Every circle is divided into four equal parts call quadrants, each containing 90°, or 8 points (%4 of 360° is 90° and % of 32 is 8). One-eighth of a circle is 45°, or 4 points (1% of 360° is 45°, and % of 32 is 4, or % of go° is 45°, and % of 8 points is 4 points). One-sixteenth of a circle is 22° 30', equals two compass points (360 + 16), or %4 of 45° is 2212", or 22°- 30°; there- fore, 1-32 of 360° is 114° = 11° 15', or % of 2214° is 114°, or 11° 18', all of which will be explained in another lesson. Note.--If it bothers you to remember how many minutes of arc in a degree of arc; how many seconds of are in a minute of arc, etc., try this: Let ° stand for hours, ' for minutes of time, and " for seconds of time. COMPOUND DENOMINATE NUMBERS. A compound denominate number expresses two or more denominations of the same kind. 4o° 15' is a compound denominate number, 40° is a simple denominate number. 17 yards 1 foot 9 inches is a compound denominate num- ber, because it expresses.two or more denominations of the same kind. REDUCTION. Reduction is the process of changing a denominate num- ber from one denomination to another without altering its value. REDUCTION DESCENDING. The changing of a denominate number to an equivalent denominte number of a lower denomination is called re- duction descending. Reduction descending is to reduce from a higher to a lower denomination; as.5° = 300'; 300' =. 18,000"; 18,000" = 1,080,000', or thus, 5 X 60' = 300'; 300 X 60" = 18,000'; 18,000 X 60""' = 1,080,000'. That is to say, 5° equals that number of minutes, seconds, thirds. 5° in each case is reduced to a lower denomination, but its value is not altered in the least. | Explanation:--Since there are 60' in 1°, in 5° there are 5 times 60' = 300'; again, since there are 60" in 1', in 300' there are 300 times 60" = 18,000"; again, since there are 60" in 1", in 18,000" there are 18,000 times 60'" = 1,080,- 000'"" Rule--Multiply the number of the highest denomina- tion given, by the number indicating how many units of the next lower denomination are equal to one of the high- er, and to the product add the number given of this lower denomination. Proceed in like manner with this and each successive result thus obtained, until the number is re- duced to the required denomination. Example:--Change 40° 15' to ' (minute's). ay We multiply by 60 because it. takes 60' to make 60 a degree. 2400' is the same thing as 40°, only es this expresses it in a lower denomination. We 2400" add in the 15' because they are of the same de- ris nomination; hence, 2415' is the same as 40° ---- 15% 24t5. |b. Change, or reduce 5 yards 9 feet to feet. 5 3 teet to the yard, hence 3 X 5° = 15 +9 = X3 ft | 24. 24 feet or 5 yards 9 feet are one and' the same. 15, ft Change, or reduce: or tt a5 14 30 to ~ (seconds). Ans. 54870". 15 * -- | 60 + 14' X 60" + 30" = 54870", because 60' = 24 ft | 1°; 60" =1', etc. 60 is always the basis in degree, minute and second work. This degree work is on the same principle as yards, feet and inches, or gallons, quarts and pints. Change 15 gallons to pints. Ans. 120 pints.