Logarithmic Tables |
Popular passages
Page 3 - The logarithm of a number is the exponent of that power to which another number, the base, must be raised to give the number first named.
Page 3 - ... and an endless decimal, the mantissa. It is to be observed that the common logarithms of all numbers expressed by the same figures in the same order with the decimal point in different positions have different characteristics but the same mantissa. To illustrate: — if the decimal point stand after the first figure of a number, counting from the left, the characteristic is 0; if after two figures, it is 1; if after three figures, it is 2, and so forth. If the decimal point stand before...
Page 8 - Thus in page 69, the sine of 5° 30' is .09585. The cosine of 5° 30' „ .99540. Sine of 40° 25' (page 73) „ .64834. Cosine of 40° 25' „ .76135. When the angle exceeds 45°, the degrees are found at the bottom of the page, and the minutes are counted upwards in the right hand column of the page, as in the table of logarithmic sines. Thus, sine of 84° 20
Page 3 - Since most numbers are incommensurable powers of ten, a common logarithm, in general, consists of an integer which is called the characteristic and an endless decimal, the mantissa. It is to be observed that the common logarithms of all numbers expressed by the same figures in the same order with the decimal point in different positions have different characteristics but the same mantissa. To illustrate: — if the decimal point stand after the first figure of a number, counting from the left, the...
Page 11 - A distinction is always made between the "probable error of a single observation" and "the probable error of the general mean" (ie of the adjusted value). The two errors are supposed to be closely related, but they are essentially distinct ; the determination of the second depends on a knowledge of the first. Let us therefore consider first the probable error of a single observation.
Page 10 - The logarithm of the cotangent of a small angle i found by subtracting the modified logarithm of the tangent of the angle from 10 that of the tangent of an angle near 90°, by subtracting the modified logarithm о the tangent of the complementary small angle from 10.
Page 9 - To find the trigonometric functions corresponding to an angle between 45° and '90°, we take the degrees at the bottom of the page and the minutes in the right.hand column.
Page 10 - ... its sine and tangent are also very small; but their logarithms are negative and very large, and they change rapidly and at rapidly varying rates. Such logarithms, therefore, are not convenient for use where interpolation is necessary, and in their stead the logarithms given below may be used ; they are based on the following considerations : An angle whose bounding arc is just as long as a radius is a radian; it is equal to 57° 17'44.8", ie to 206264.8", and the number of seconds in an angle...
Page 10 - ... is a very little larger, than unity. These ratios approach unity closer and closer as the angle grows smaller. If the angle be expressed in seconds, the ratio sin A"/A is a very little smaller than the reciprocal of 206264.8. and the ratio tanA"/A is a very little larger than this reciprocal. These ratios change very slowly, and hence interpolation is always possible ; the table below gives their logarithms as far as 5°. Angle. log(sinA"/A).
Page 10 - ... by subtracting the modified logarithm of the tangent of the complementary small angle from 10. TO TAKE OUT THE SINE OR TANGENT OF A SMALL ANGLE. Take out the logarithm that corresponds to the number of degrees and minutes ; and add the logarithm of the whole number of seconds in the angle. Let A be the number of seconds in an angle ; their.' sin A"= (sin A "/A) . A, .'. log-sin A" = log (sin A"/A) +log A ; and v tan A" = (tan A "/A ) . A, .-. log-tan A" = log (tan A "/A) + log A. Eg log-sin 10'...