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Circumference : Diameter :: 3.14159:1; and therefore Circumference = 3.14159 Diameter.

* As this number, 3.14159, which expresses the ratio of the circumference to the diameter, frequently occurs in mathematical formulæ, it is convenient to represent it by a single letter; that which is invariably used is the Greek letter 7. Substituting this in the last equation, and 2r for the diameter,

we have

Circumference = 21.

(2) From this equation may be derived the number which represents a right angle. The arc which subtends a right angle is equal to a fourth part of the circumference, or jar, by equation (2). Dividing this by r, we obtain, by equation (1),

The numerical value of a right angle = $1. The number , or 1.57079, which expresses the value of a right angle, referred to this particular angular unit, signifies that this unit

, i. e. the angle ACU (fig. 1), is contained once, and the decimal part 0.57079, of ACU over, in a right angle. The number 7, or 3.14159, which expresses two right angles, signifies that ACÚ is contained in two right angles, three times, and the decimal part 0.14159 of ACU over.

3. Division of the circle.-As the angular unit defined in section 1 is not a submultiple of four right angles, and is, moreover, too large for practical purposes, such as astronomical observations and surveying, another mode of measuring angles, and representing them numerically, has been devised, which may be thus explained :

The circumference of the circle is divided into 360 equal parts; each of these parts subtends at the centre an angle equal to the 36oth part of four right angles, this angle is called a degree. The degree is subdivided into 60 equal parts, called minutes ; and the minute into 60 equal parts, called seconds. Degrees, minutes, and seconds are represented by the symbols ""; for example, 43° 25' 18", signifies 43 degrees, 25 minutes, 18 seconds.

In order to reduce an angle expressed in degrees, minutes, and seconds, to its value, referred to the angular unit, and vice versá, it is necessary to find the number of seconds which are contained in this unit: Number of seconds in angular units : number of se

conds in four right angles :: arc AU : circumfe



Number of seconds in angular unit: 360 x 60 x 60

:: 1 : 271.
From which it follows, since 7 = 3.14159, that

Number of seconds in angular unit = 206265".

If N" be the number of seconds in an angle which is subtended by an arc whose length is a,

N": 206265":: a : AU (= n) ;
from which it follows that

N" = 206265" *
" x

(3) By equation (3), in which are connected together the radius, the number of seconds in the angle at the

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centre, and the length of the arc which subtends it, any two of these quantities being given, the third

may be found.


1. In a circle of 100 feet radius, calculate the angle in degrees, minutes, and seconds, which is subtended by an arc whose length is 9 feet.

By equation (3), the number of seconds in the angle is 18563".85, which, reduced to degrees, minutes, and seconds, gives

Ans. 5° 9' 23".85. 2. A person standing in the centre of a sphere observes that a line on its surface, which he knows to be 6 feet in length, subtends an angle of 3' 28", calculate the radius of the sphere.

Ans. I mile, 223.31 yds. 3. The diameter of the earth is 7926 miles, its distance from the moon is 237,638 miles, calculate the angle which the earth's diameter subtends at the moon.

Ans. 1° 54' 39".6. 4. The angle which the moon subtends at the earth is observed to be 31' 7", calculate her diameter in miles.

Ans. 2151 miles nearly. 5. It has been ascertained that the angle which the diameter of the earth subtends at the centre of the sun is 17".2. Calculate the sun's distance.

Ans. 95,049,790 miles. 6. The sun's diameter subtends at the earth an angle of 32' 3". Calculate its diameter.

Ans. 886,145 miles.



1. Definition of Trigonometrical Magnitudes.—2. Relations of

Trigonometrical Magnitudes.-3. Complemental and Supplemental Angles.-4. Trigonometrical Tables. 1. Definition of trigonometrical magnitudes. - There are certain magnitudes connected with angles, and entering into every trigonometrical computation, which are defined as follows:

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Let the right lines (fig. 2) AA' and DD', be drawn intersecting at right angles in the centre of a circle whose radius we shall suppose to be the linear unit ; let CB be drawn making an angle ACB with AC; let also BP be drawn perpendicular to AC, and AT and DS tangents to the circle at A and D respectively, and cutting the line CB produced in T and S, then The line BP is the sine of the angle ACB.

CP cosine


secant DS

cotangent OS

cosecant AP versed sine If the numerical value of the angle ACB be denoted by A, these quantities are usually written, for brevity, thus : sin A, cos A, tan A, sec A, cotA, cosec A, versin A.

2. Relations of trigonometrical magnitudes. -The relations which these magnitudes have to each other are derived from geometrical principles as follows. Because the triangles BCP, ACT, and CDS are right-angled, it follows (Euclid, Book I. Prop. xLvII.) that

CB? = BP2 + CP2,
CT2 = CA? + AT”,

CS2 = CD2 + DSP. As the radius of the circle is supposed to be unity, CB2 = CAP = CD2 = 1. Substituting this, and their trigonometrical values for BP, CP, CT, &c., we have the following equations:

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I =


sin? A + cos2A, sec?A = 1 + tan’A, cosec’A = I + cot’A.

(3) Because the triangles CBP, CTA, and DCS are equi-angular, the following proportions follow from (Euclid, Book VI. Prop. IV.)

AT: AC :: BP: CP,
DS : DC :: CP: PB,
CT: CA :: CB: CP,
DS: DC :: CA: AT,

CS : CD :: CB : BP. Substituting, as before, unity for the radius, and trigonometrical values for the several lines, these proportions may be written thus:

tan A:1:: sin A :cos A,
cot A:1:: cos A : sin A,
sec A:1::1:cos A,
cot A:1::1: tan A,

cosec A :1::1: sin A.
From these proportions follow the equations

sin A tan A

cos A'


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