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I. The Sexagesimal Method.

46. In this method the right angle is divided into 90 equal parts, each of which is called a degree ; each degree is subdivided into 60 equal parts, each of which is called a minute; and each minute is again subdivided into 60 equal parts, each of which is called a second.

Instruments used for measuring angles are subdivided accordingly; and the size of an angle is known when, with such an instrument, it has been observed that the angle contains a certain number of degrees, and a certain number of minutes beyond the number of complete degrees, and a certain number of seconds beyond the number of complete minutes.

Thus an angle might be recorded as containing 79 degrees + 18 minutes + 36.4 seconds.

Degrees, minutes, and seconds are indicated respectively by the symbols', ', ", and the above angle would be written

79o. 18. 36.4".

* II.

The Centesimal or Decimal Method.

47. The other method of subdivision is the Centesimal or Decimal. Here each right angle is divided into 100 equal parts each of which is called a grade; each grade is subdivided into 100 equal parts, each of which is called a minute; and each minute is again subdivided into 100 equal parts, each of which is called a second.

Instruments of observation would be subdivided accordingly, and any observed angle would be recorded as containing so many grades + so many minutes + so many seconds.

* 48.

+

Grades, minutes and seconds are indicated respectively by the symbols ,;". So that an angle of 26 grades + 19 minutes + 342 seconds would be written

368 19' 34.2". It will be observed that this method is simply that of the decimal system of notation.

The above angle for example=16 +100% +1788800 of a right angle.

That is •3619342 of a right angle.
This is equal to 36•19342 of a grade.
Also to 3619.342 of a minute.
Also to 361934.2 of a second.
Example. Express 3028 2° 4:6" as the decimal of a right angle.
302 2

4.6
This angler
100

of a right angle 10000 1000000

=3.0202046 of a right angle. *49. Hence, to express an angle given in grades, minutes and seconds as the deciinal of a right angle, we have only to observe that the first and second decimal places are occupied by the grades, the third and fourth decimal places are occupied by the minutes, and the fifth and sixth decimal places are occupied by the seconds.

The same observation will enable us to express in grades, minutes, and seconds an angle given as the decimal of a right angle.

Example. Express 3.4650023 of a right angle in grades, minutes, and seconds.

3 right angles=300 grades.
46 of a right angle =46 grades.
*00,50 of a right angle=50 minutes.

*00,00,02,3 of a right angle=2:3 seconds. Therefore the angle is 3468 509 2:3".

* 50.

* EXAMPLES. VII.

Express as the decimal of a right angle, (1) 638 21' 18".

(7) 328 4 5.2 (2) 1045 269 99.1".

(8) 182 3.4" (3) 26 18 27".

(9) 69% O 7.1". (4) 38 29' 48.94".

(10) 1198 3 0.45". (5) 62 41".

(11) 10068 18' 1" (6) 1000% 812".

(12) 28 26° 4.84.

Express in grades, minutes and seconds, (13) 367891 of a right angle. (19) 1.001 of a right angle. (14) 1•043021 of a right angle. (20) *0101001 of a right angle. (15) *012003 of a right angle. (21) 6:451 of a right angle. (16) *00102 of a right angle. (22) 023 of a right angle. (17) *0625 of a right angle. (23) *00011 of a right angle. (18) 3.02125 of a right angle. (24) '00001 of a right angle.

51. An angle given in degrees, minutes, and seconds may be expressed as the decimal of a right angle by the usual method. Example. Express 39° 4' 27" as the decimal of a right angle.

60 ) 27 seconds
60 _4:45 minutes
90 )39•07416666 etc. degrees

.43415740740 etc. right angles

Answer. •43415740 of a right angle. 52. An angle given as the decimal of a right angle may be expressed in degrees, minutes, and seconds by the converse of the above.

Example. Express ·43415740 of a right angle in degrees, minutes, and seconds.

•43415740740 etc. right angles

90 39•07416666600 degrees The last two figures would be 66 if we were to write down the recurring part to more figures. This gives 39:07416666666 etc. degrees

60

4.4499999960. minutes that is

4.449 minutes
4.45. minutes

60

27.00 seconds. The result is 390 4' 27".

or

*53. We have seen that an angle expressed as the decimal of a right angle can be at once expressed in grades, minutes, and seconds.

Hence an angle expressed in degrees, minutes, and seconds, can be expressed in grades, etc. by first reducing the angle to the decimal of a right angle.

Example. Express 39° 4' 27" in grades, minutes, and seconds.

This angle is ·43415740 of a right angle, by Art. 51 and this

=438 41' 57.407". *54. An angle given in grades, minutes, and seconds can be expressed in degrees, minutes, and seconds by first expressing the angle as the decimal of a right angle.

Example. Express •432 41' 57•409" in degrees, minutes, and seconds.

This angle is ·43415740 of a right angle, which is 39° 4' 27" from Art. 52.

* EXAMPLES. VIII.

Express each of the following angles (i) as the decimal of a right angle, (ü) in grades, minutes, and seconds ; (1) 8° 15' 27".

(4) 16° 14' 19". (2) 6° 4' 30'.

(5) 132° 6'. (3) 970 5' 15".

(6) 49o.

Express in degrees, minutes and seconds,
(7) 18 37 50".

(10) 248 0 25".
(8) 88 75'.

(11) 188 1' 15". (9) 1708 45' 35". (12) 358.

55. The decimal or centesimal system of subdividing a right angle was proposed by the French at the commence. ment of the present century; but, although it possesses many advantages over the established method, no one has been found willing to undertake the great expense that would have to be incurred in rearranging all tables and all books of reference, and all the records of observations, which would have to be transferred from the old system to the new, before the advantages of the decimal system could be felt. Thus the decimal system of angular measurement has never been used even in France, and in all probability never will be used in practical work.

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