ABBREVIATIONS.

+Increased by.

Diminished by.

× Multiplied by. Divided by. Equal to.

Since, or seeing that.

.. Hence, or therefore.

: Indicates the quotient of one divided by the other of the quantities it connects, called sometimes the ratio of the quanlities.

:: Indicates an equality of ratios, and connects equal ratios na proportion. Thus, a b c d indicates that a ÷ b = c d; or it may be read, a is to b as c is to d.

( ) Brackets indicate that the operations embraced by them shall first be performed, and the result treated as a single term in the remaining processes required by a formula. Thus, u x b÷ (a + b) requires that the product of a and b shall be divided by their sum. This expression may also be written ab

a + b'

or a X

b
a + b'

α

or b+

a+b

If the brackets be omitted

the expression a × b÷a+b would mean

A2. A small secondary figure annexed thus to an expression is called its exponent. It requires the principal to which it is attached to be used as many times in continued multiplication as there are units in the exponent. Thus, A2 = A × A; A3 = A× A × A, which is called the cube, or third power, of A.

This is called the square root sign: it signifies that the square root of the quantity covered by it is to be taken.

If preceded by a small secondary figure, called the index, as in the marginal figure, it indicates that the cube root of the quantity covered by it shall be taken; and so on.

A. If an exponent be fractional, as in the marginal figure, it requires that the square root of the third power of the quantity covered shall be taken, the numerator indicating the power and the denominator the root.

B. M. Bench-mark; any fixed reference point for the level,

ix

as outcropping ledge, water-table of building, or other permanent object. Usually a blunt conical seat for the rod, hewn on a buttressed tree-base, having a small nail sometimes driven flush in the top of it, and a blaze opposite, on which the elevation is marked with kiel.

T. P. Turning-point: usually marked O in the field-book. P. I. Point of intersection: as of tangents, which are to be connected by a curve.

A. D. Apex distance: i.e., the distance from the P. I. to the point where a curve merges in the tangent.

P. C. Point of curve: the stake-mark at the beginning of

a curve.

P. T. Point of tangent: the stake-mark at the end of a

curve.

P. C. C. Point of compound curvature: the stake-mark where a curve merges in another of different curvature, turning in the same direction.

P. R. C. Point of reverse curvature: the stake-mark where a curve merges in another turning in the opposite direction. B. S. Backsight, in transit work; or the reading of the rod to ascertain the instrument height in levelling.

F. S. Foresight, in transit work; or the reading of the rod to ascertain elevations in levelling.

H. I. Height of instrument: elevation of the level above the datum or zero plane.

H. W. High water.
L. W. Low water.

LOGARITHMS.

I.-II.

LOGARITHMS.

I.

DEFINITIONS AND PRINCIPLES.

1. THE logarithm of a number is the exponent of the power to which it is necessary to raise a fixed number to produce the given number; that is to say, it represents the number of times a fixed number must be multiplied by itself in order to produce any given number.

The fixed number is called the base of the system. In the common system, this base is 10.

It follows from the above, that the logarithm of any power of 10 is equal to the exponent of that power. If, therefore, a number is an exact power of 10, its logarithm is a whole number.

If a number is not an exact power of 10, its logarithm will not be a whole number, but will be made up of an entire part plus a fractional part, which is generally expressed decimally. The entire part of the logarithm is called the characteristic ; the decimal part is called the mantissa.

2. The characteristic of the logarithm of a whole number is positive, and numerically 1 less than the number of places of figures in the given number.

Thus, if a number lies between 1 and 10, its logarithm lies between 0 and 1; that is, it is equal to 0 plus a decimal. If a number lies between 10 and 100, its logarithm is equal to 1 plus a decimal; and so on.

3. The characteristic of the logarithm of a decimal fraction is negative, and numerically 1 greater than the number of O's that immediately follow the decimal point.

The characteristic alone, in this case, is negative, the man

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