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Hence, "Find the logarithm of the given number, divide it by the number indicating the root.—this is the logarithm of the required root-the corresponding number is the root itself."
All this has to be divided by 7. This will be effected most easily by adding and subtracting such a multiple of 10 as shall make the negative part 70.
If we had to extract the 6th root of the above number, we must of course take log. .00972 57.9876663 — 60. And again, to extract the cube root, we must take log. 0097257.9876663 - 30.
11. The student must exercise himself in working several examples, like each of those above given. When he has done so, he may then, for practice, work some more complicated examples, such as the following:
N.B. To find the logarithm of a mixed number reduce it to an improper fraction. To find the logarithm of a vulgar fraction use the formula. If N=
log. N.= log. a+ Ar. Com. log. b-10.
precisely as in division, excepting that there is no occasion to find N itself.
1. On representing Lines and Angles by Numbers. If we have a line of any length, we can represent it numerically by the number of times it contains a given line, which we take to represent unity. Thus if we take a line a foot long to be the unit of length, a line seven feet long can be represented by 7. Of course, the same holds good of any other line. And so when we speak of a line 8, 5, or whatever the number may be, we mean that the line in question contains 8 or 5 of the given unit, as 8 feet, or 5 feet. And of course, if we can represent lines by numbers, we can generalize the numbers by letters, and thus we can represent lines by algebraical symbols; so that a b c, x y z, &c. may be understood to represent lines. In the same manner as before, if we speak of a line a, we mean a line containing as many units of length (e.g. feet) as a contains units of number.
On the same principle we may express angles by numbers or by letters. This is done by dividing the right angle into 90 equal parts, each of which is called a degree, and dividing the degree into 60 equal parts, each called a minute, and the minute into 60 equal parts, each called a second. An angle is then expressed as being so many degrees, with odd: minutes and seconds, e.g. 36 degrees, 57 minutes, 31 seconds, (which is usually written 36° 57' 31"), in the same manner as a line is expressed by so many yards, with odd feet and inches.
Of course, as we can thus represent angles by numbers, we may also represent them by letters, and may have angles ABC; where the angle A (for instance) means that the angle contains as many degrees and parts of a degree as A contains units and parts of a unit. *
In the same manner as we may measure lines either by feet or yards, or miles, so we might take, as the unit of angular measure, any other part of the right angle than the oth; and in fact at the end of last century, when the decimal notation was intro- ! duced into France, it was proposed by certain French mathematicians, to make the degree the tooth part of the right angle. The proposition was at no time extensively accepted, and is now quite abandoned.
2. Definition of the Science of Trigonometry. We are thus enabled to express lines and angles by numbers; and this is the first step towards making calculations in which lines and angles are the data. However, before these calculations can be performed, it is necessary that the relations which exist between straight lines and angles should be investigated. It is the object of the science of Trigonometry to make these investigations.
The object of the science will, perhaps, be more clearly stated, if we limit the definition so as to make it correspond more closely to its derivational meaning, by saying that the science of Trigonometry has for its object the investigation of the relations
* It is usual to denote angles either by Roman capital letters, ABC; or else by Greek small letters, a, b, y... 0, 0,4 ... while generally the small Roman a b c denote lines. This is, of course, only a conventional arrangement.
which exist between the sides and angles of triangles and the algebraical expression of those relations.
The immediate application of the science is to the calculation of certain parts of a
viz., the measurement of triangles.
triangle from certain given parts; e. g., having given the sides BA, AC, and the angle BAC of the triangle ABC, we can calculate the magnitude of the side BC.
The science has, however, very many other uses besides the one from which its name is taken,
3. The Circular Measure of an Angle.
The measures above given enable us to compare arithmetically one straight line with another, and one angle with another. But it is to be observed, that an angle and a line are heterogeneous magnitudes; and therefore, if we would perform algebraical operations in which lines and angles enter, we must devise some plan of measuring angles that shall express them by means of lines, or of the ratios of lines.
In fact, when we speak of an angle (of 57° suppose) it tells us what the angle is, but does not at once give us the means of comparing that angle with given lines.
The measure of the angle adopted for the purpose of such calculations, is called the circular measure.
It is founded on the two well-known geometrical propositions.
(a) That in circles of the same radius the angle is proportional to the arc which
(b) And that for the same angle, in circles of different radii, the arc varies as the radius.
ace when r is constant.
ar when @ is constant.
.. are when both vary.
or the angle is measured by the ratio of the arc to the radius.
If we take the unit of angle to be the angle which is subtended by an arc of the same length as the radius, then
In this case, the angle being measured by the ratio of two lines, it can enter a calculation in which we are dealing with lines.
N.B.-We can easily find the number of degrees in the angle which is the unit of circular measure.