Here, 1 taken from 1, gives 2 for a result. The subtraction, as in this case, is always to be performed in the algebraic sense.

3. Divide 37.149 by 523.76.

Ans. 0.0709274.

The operation of division, particularly when combined with that of multiplication, can often be simplified by using the principle of

THE ARITHMETICAL COMPLEMENT.

17. The ARITHMETICAL COMPLEMENT of a logarithm is the result obtained by subtracting it from 10. Thus, 8.130456 is the arithmetical complement of 1.869544. The arithmetical complement of a logarithm may be written out by commencing at the left hand and subtracting each figure from 9, until the last significant figure is reached, which must be taken from 10. The arithmetical complement is denoted by the symbol (a. c.)

Let a and b represent any two logarithms whatever, and ab their difference. Since we may add 10 to,

and subtract it from, a- b, without altering its value, we have,

a − b = a + (106) — 10.

(10.)

--

But 10b is, by definition, the arithmetical complement of b: hence, Equation (10) shows that the difference between two logarithms is equal to the first, plus the arithmetical complement of the second, minus 10.

Hence, to divide one number by another by means of the arithmetical complement, we have the following

RULE. Find the logarithm of the dividend, and the arithmetical complement of the logarithm of the divisor, add them together, and diminish the sum by 10; the number corresponding to the resulting logarithm will be the quotient required.

3. Divide the product of 358884 and 5672, by the product of 89721 and 42.056.

20 is here subtracted, as (a. c.) has been twice used.

8.376182

2.731978 .. 539.48, result.

4. Solve the proportion,

3976 : 7952 :: 5903 : X.

Applying logarithms, the logarithm of the 4th term is equal to the sum of the logarithms of the 2d and 3d terms, minus the logarithm of the 1st: Or, the arithmetical complement of the logarithm of the 1st term, plus the logarithm of the 2d term, plus the logarithm of the 3d term, minus 10, is equal to the logarithm of the 4th term.

RULE.-Find the logarithm of the number, and multiply it by the exponent of the power; then find the number corresponding to the resulting logarithm, and it will be the power required.

EXTRACTING ROOTS BY MEANS OF LOGARITHMS.

19. From the principle proved in Art. 8, we have the following

RULE.-Find the logarithm of the number, and divide it by the index of the root; then find the number corresponding to the resulting logarithm, and it will be the root required.

Examples.

1. Find the cube root of 4096.

The logarithm of 4096 is 3.612360, and one third of this is 1.204120. The corresponding number is 16, which is the root sought.

If the characteristic of the logarithm of the given number is negative and not exactly divisible by the index of the root, add to it such negative quantity as shall make it exactly divisible, and add also to the mantissa a numerically equal positive quantity.

2. Find the 4th root of .00000081.

The logarithm of .00000081 is 7.908485, which is equal to 8+1.908485, and one fourth of this is 2.477121. The number corresponding to this logarithm is .03; hence, .03 is the root required.

PLANE TRIGONOMETRY.

20. PLANE TRIGONOMETRY is that branch of Mathematics which treats of the solution of plane triangles.

In every plane triangle there are six parts: three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation. The operation of finding the unknown parts is called the solution of the triangle.

21. A plane angle is measured by the arc of a circle. included between its sides, the centre of the circle being at the vertex, and its radius being equal to 1.

Thus, if the vertex A is taken as a centre, and the radius AB is equal to 1,

the intercepted arc BC measures the angle

A (B. III., P. XVII., S.).

B

Let ABCD represent a circle whose radius is equal to 1, and AC, BD, two diameters perpendicu

B

lar to each other. These diameters

divide the circumference into four equal

parts, called quadrants; and because

A

each of the angles at the centre is a right angle, it follows that a right

angle is measured by a quadrant. An

acute angle is measured by an arc less than a quadrant, and an obtuse angle, by an arc greater than a quadrant.