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Appendix.

4. The dodecaedron, is a solid bounded by twelve equal

pentagons

5. The cosaedron, is a solid, bounded by twenty equal equilateral triangles.

6. The regular solids may easily be made of pasteboard. Draw the figures of the regular solids accurately on paste board, and then cut through the bounding lines: this will give figures of pasteboard similar to the diagrams. Then, cut the other lines half through the pasteboard, after which, turn up the parts, and glue them together, and you will form the bodies which have been described.

ELEMENTS OF TRIGONOMETRY.

INTRODUCTION.

SECTION I.

OF LOGARITHMS.

1. The logarithm of a number is the exponent of the power to which it is necessary to raise a fixed number, in order to produce the first number.

This fixed number is called the base of the system, and may be number except 1: in the common system 10 is assumed as the base.

any

2. If we form those powers of 10, which are denoted by entire exponents, we shall have

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From the above table, it is plain, that 0, 1, 2, 3, 4, &c., are respectively the logarithms of 1, 10, 100, 1000, 10000, &c.; we also see that the logarithm of any number between 1 and 10 is greater than Ond less than 1: thus

Log 2 0.301030

Of Logarithms.

The logarithm of any number greater than 10, and less than 100, is greater than 1 and less than 2: thus

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The logarithm of any number greater than 100, and less than 1000, is greater than 2 and less than 3: thus

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If the above principles be extended to other numbers, it will appear, that the logarithm of any number, not an exact power of ten, is made up of two parts, an entire and a decimal part. The entire part is called the characteristic of the logarithm, and is always one less than the number of places of figures in the given number.

3. The principal use of logarithms, is to abridge numerical computations.

Let M denote any number, and let its logarithm be denoted by m; also let N denote a second number whose logarithm is n; then from the definition we shall have

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Multiplying equations (1), and (2), member by member, we

have

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10 = MXN or, m+n = log MX N: hence, The sum of the logarithms of any two numbers is equal to the logarithm of their product.

Dividing equation (1) by equation (2), member by member, we have

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The logarithm of the quotient of two numbers, is equal to the logarithm of the dividend diminished by the logarithm of the divisor.

Of Logarithms.

4. Since the logarithm of 10 is 1, the logarithm of the product of any number by 10, will be greater by 1 than the logarithm of that number; also, the logarithm of any number divided by 10, will be less by 1 than the logarithm of that number.

Similarly, it may be shown that the logarithm of any number multiplied by a hundred, is greater by 2 than the logarithm of that number, and the logarithm of any number divided by 100 is less by 2, than the logarithm of that number, and so on.

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From the above examples, we see, that in a number composed of an entire and decimal part, we may change the place of the decimal point without changing the decimal part of the logarithm, but the characteristic is diminished by 1 for every place that the decimal point is removed to the left.

In the logarithm of a decimal, the characteristic becomes nega tive, and is numerically 1 greater than the number of ciphers im mediately after the decimal point. The negative sign extends only to the characteristic, and is written over it as in the examples given above.

TABLE OF LOGARITHMS.

5. A table of logarithms, is a table in which are written the logarithms of all numbers between 1 and some given number. The logarithms of all numbers between 1 and 10,000 are given

Of Logarithms.

in the annexed table. Since rules have been given for determining the characteristics of logarithms by simple inspection, it has not been deemed necessary to write them in the table, the decimal part only being given. The characteristic, however, is given for all numbers less than 100.

The left hand column of each page of the table, is the column of numbers, and is designated by the letter N; the logarithms of these numbers are placed opposite them on the same horizontal line. The last column on each page, headed D, shows the difference between the logarithms of two consecutive numbers. This difference is found by subtracting the logarithm under the column headed 4, from the one in the column headed 5 in the same horizontal line, and is nearly a mean of the differences of any two consecutive logarithms on the line.

6. To find from the table the logarithm of any number.

If the number is less than 100, look on the first page of the table, in the column of numbers under N, until the number is found the number opposite is the logarithm sought: Thus

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7. When the number is greater than 100 and less than 10000. Find in the column of numbers, the first three figures of the given number. Then pass across the page along a horizontal line until you come into the column under the fourth figure of the given number: at this place, there are four figures of the required logarithm, to which two figures taken from the column marked 0, are to be prefixed.

If the four figures already found stand opposite a row of six figures in the column marked 0, the two left hand figures of the six, are the two to be prefixed; but if they stand opposite

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