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1. TRIGONOMETRY was originally considered to be the doctrine of triangles, but in its present improved state it has a much more extensive signification, which we shall hereafter shew even in this rudimentary treatise.

2. In estimating angular measures, we suppose the right angle to be the primary one, and to be divided into 90 equal parts, each of which is called a degree; each degree is supposed to be divided into 60 equal parts, each of which is called a minute ; each minute is supposed to be divided into 60 equal parts, each of which is called a second, and so on to thirds, fourths, &c. Here one degree is considered as the angular unit.

3. Modern French writers, instead of using the sexagesimal division, use the centesimal; and it is to be regretted that the latter is not universally used, from the great ease with which all calculations are made in that division.

We shall, however, shew how to reduce French into English measures, and vice versa.

If E and F represent the number of English degrees and French grades in the same angle,

. E F E F

90 * 100 or g=10;


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9 4. The circumference of a circle is known to be about 3:14159 times its diameter, or, in other words, the ratio of the circumference to the diameter is represented by 3•14159; for this number writers generally put the Greek letter a.


.: circumference=aD; where D is the diameter, or

2nr, where r is the radius of the circle Hence the length of the arc of a quadrant is **; of a semicircle, or 180°, is ar ; and of 270', or three quadrants, is 37".

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Now if any arc a subtend an angle of A°, then since *** subtends 90°, and that by Euclid vi. 33, angles are proportional to the arcs which subtend them,

4°: 90°:: a : *;

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:: AP=1900. ............. (1). From this expression any one of the quantities may be found when the others are given.

Ex. 1. Find the length of an arc of 45° of a circle whose radius is 10 feet;

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. 5. Most modern writers on Trigonometry take also for the unit of angular measure the number of degrees in an angle, subtended by an arc equal to the radius*. If Uo represent that angle, then by equation (1),

* If ACB be an angle at the centre of a circle, subtended by an arc equal to the radius of the circle, then, since by the 33rd Proposition of the 6th Book of Euclid, the angles at the centre of a circle are to each other as the arcs on which they stand,

Angle ACB : four right angles :: arc AB : circumference, but AB is an arc equal to the radius,

.. Angle ACB : four right angles :: r:27r:: 1:27, on four right angles

-, which, being independent of r, is constant for any circle; it may therefore be used to measure other angles.

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vo – 180°– 100 = 3:14159 = 57°2 Hence, AR = 570-29578 (9) or A°= v* (%) ...... (2). And since Uo=570-29578 is constant,

A® varies asm, i.e. as melius and taking U° as the unit, we have

A°= which is called the circular measure of the angle.

From equation (2) we see that the measuring unit, U', must be multiplied by the fraction to find the angle ; thus if the circular measure of an angle be , then

A = (578-29578) = 28°64789.

If the circular meas

A = (570-29578) = 18 570-29578) = 68°-754986.

Now, suppose we take an angle of 22° 27' 39", then this is put into decimals at once by the centesimal division, without putting down any work on paper, it being 22°.2739; whereas, by the sexagesimal, we must proceed in the following manner:

60) 39
• 60) 27.65

22.4608 If we wish to find how many grades and minutes are contained in this angle, here

E = 22:4608

Ō = 2:4956


=24.9564, which at sight is 248 95' 64".

Find the number of degrees and minutes in 46% 56' 36".

F = 46°5636

10 = 4.65636

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26.064 = 41° 54' 26". (1) If F' and F", E' and E" represent the magnitude of a French and English minute and second respectively, shew that

F 3•3° F 3•33
Ē= 258 E" = 2:58 •

F' = 1 French min.;
.. F'x 100 x 100 = a quadrant,

E' = 1 English min.;
.. E' x60 x 90 = a quadrant,
hence, F'x 10000 = E' x 5400,

F' x 50 = E' x 27,
F 27 3.32

Ē = 50 2:52 •
Also, F" = 1 French second;

F" x 100 x 100 x 100 = a quadrant,

E" = 1 English second;
.. E" x 60 x 60 x 90 = a quadrant,
hence, F" x 1000000 = E" x 324000,

F"'x 250 = E" x 81;
F" 81 3:38

Ē! = 250 = 2:58 • (2) Compare the interior angles of a regular octagon and dodecagon: In a polygon of n sides, A° = 180°– 360°.

In octagon = 180°– 360° = 180° – 45° = 185°.


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