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CONSTRUCTION AND DESCRIPTION OF THE TABLES.
XXVII. If the radius of a circle is taken equal to 1, and the lengths of the lines representing the sines, cosines, tangents, cotangents, &c. for every minute of the quadrant be calculated, and written in a table, this would be a table of natural sines, cosines, &c.
XXVIII. If such a table were known, it would be easy to calculate a table of sines, &c. to any other radius; since, in different circles, the sines, cosines, &c. of arcs containing the same number of degrees, are to each other as their radii.
XXIX. If the trigonometrical lines themselves were used, it would be necessary, in the calculations, to perform the operations of multiplication and division. To avoid so tedious a method of calculation, we use the logarithms of the sines, cosines, &c.; so that the tables in common use show the values of the logarithms of the sines, cosines, tangents, cotangents, &c. for each degree and minute of the quadrant, calculated to a given radius. This radius is 10,000,000,000, and consequently its logarithm is 10.
XXX. Let us glance for a moment at one of the methods of calculating a table of natural sines.
The radius of a circle being 1, the semi-circumference is known to be 3.14159265358979. This being divided successively, by 180 and 60, or at once by 10800, gives .0002908882086657, for the arc of 1 minute. Of so small an arc the sine, chord, and arc, differ almost imperceptibly from the ratio of equality; so that the first ten of the preceding figures, that is, .0002908882 may be regarded as the sine of 1'; and in fact the sine given in the tables which run to seven places of figures is .0002909. By Art. XVI. we have for any arc, cos=√(1--sin3). This theorem gives, in the present case, cos l'=.9999999577. Then by Art. XXII. we shall have
2 cos l'x sin l'-sin 0'-sin 2'=.0005817764
Thus may the work be continued to any extent, the whole difficulty consisting in the multiplication of each successive result by the quantity 2 cos 1'1.9999999154.
Or, the sines of 1' and 2' being determined, the work might be continued thus (Art. XXI.):
sin 1'; sin 2'-sin 1':: sin 2' + sin l' : sin 3'
In like manner, the computer might proceed for the sines of degrees, &c. thus:
sin 1° : sin 2°-sin 1° : : sin 2° + sin 1° : sin 3o
Above 45° the process may be considerably simplified by the theorem for the tangents of the sums and differences of arcs. For, when the radius is unity, the tangent of 45° is also unity, and tan (a+b) will be denoted thus:
tan (45°+b)−1 + tan b
And this, again, may be still further simplified in practice. The secants and cosecants may be found from the cosines and sines.
TABLE OF LOGARITHMS.
XXXI. If the logarithms of all the numbers between 1 and any given number, be calculated and arranged in a tabular form, such table is called a table of logarithms. The table annexed shows the logarithms of all numbers between 1 and 10,000.
The first column, on the left of each page of the table, is the column of numbers, and is designated by the letter N ; the decimal part of the logarithms of these numbers is placed directly opposite them, and on the same horizontal line.
The characteristic of the logarithm, or the part which stands to the left of the decimal point, is always known, being 1 less than the places of integer figures in the given number, and therefore it is not written in the table of logarithms. Thus, for all numbers between 1 and 10, the characteristic is 0: for numbers between 10 and 100 it is 1, between 100 and 1000 it is 2, &c.
To find from the table the logarithm of any number.
When the number is less than 100.
Look on the first page of the table of logarithms, along the columns of numbers under N, until the number is found; the number directly opposite it, in the column designated Log., is the logarithm sought.
When the number is greater than 100, and less than 10,000.
Find, in the column of numbers, the three first figures of the given number. Then, pass across the page, in a horizontal line, into the columns marked 0, 1, 2, 3, 4, &c., until you come to the column which is designated by the fourth figure of the given number: to the four figures so found, two figures taken from the column marked 0, are to be prefixed. If the four figures found, stand opposite to a row of six figures in the column marked 0, the two figures from this column, which are to be prefixed to the four before found, are the first two on the left hand; but, if the four figures stand opposite a line of only four figures, you are then to ascend the column, till you come to the line of six figures: the two figures at the left hand are to be prefixed, and then the decimal part of the logarithm is obtained. To this, the characteristic of the logarithm is to be prefixed, which is always one less than the places of integer figures in the given number. Thus, the logarithm of 1122 is 3.049993.
In several of the columns, designated 0, 1, 2, 3, &c., small dots are found. Where this occurs, cipher must be written for each of these dots, and the two figures which are to be prefixed, from the first column, are then found in the horizontal line directly below. Thus, the log. of 2188 is 3.340047, the two dots being changed into two ciphers, and the 34 from the column 0, prefixed. The two figures from the colum 0, must also be taken from the line below, if any dots shall have been passed over, in passing along the horizontal line: thus, the logarithm of 3098 is 3.491081, the 49 from the column 0 being taken from the line 310.
When the number exceeds 10,000, or consists of five or more places of figures.
Consider all the figures after the fourth from the left hand, as ciphers. Find, from the table, the logarithm of the first four places, and prefix a characteristic which shall be one less than the number of places including the ciphers. Take from the last column on the right of the page, marked D, the number on the same horizontal line with the logarithm, and multiply this number by the numbers that have been considered as ciphers: then, cut off from the right hand as many places for decimals as there are figures in the multiplier, and add the product, so obtained, to the first logarithm: this sum will be the logarithm sought.
Let it be required to find the logarithm of 672887. The log. of 672800 is found, on the 11th page of the table, to be 5.827886, after prefixing the characteristic 5. The corresponding number in the column D is 65, which being multiplied by 87, the figures regarded as ciphers, gives 5655; then, pointing off two places for decimals, the number to be added is 56.55. This number being added to 5.827886, gives 5.827942 for the logarithm of 672887; the decimal part .55, being omitted.
This method of finding the logarithms of numbers, from the table, supposes that the logarithms are proportional to their respective numbers, which is not rigorously true. In the example, the logarithm of 672800 is 5.827886; the logarithm of 672900, a number greater by 100, 5.827951: the difference of the logarithms is 65. Now, as 100, the difference of the numbers, is to 65, the difference of their logarithms, so is 87, the difference between the given number and the least of the numbers used, to the difference of their logarithms, which is 56.55: this difference being added to 5.827886, the logarithm of the less number, gives 5.827942 for the logarithm of 672887. The use of the column of differences is therefore manifest.
• When, however, the decimal part which is to be omitted exceeds .5, we come nearer to the true result by increasing the next figure to the left by 1; and this will be done in all the calculations which follow. Thus, the difference to be added, was nearer 57 than 56; hence it would have been more exact to have added the former number.
The logarithm of a vulgar fraction is equal to the logarithm of the numerator, minus the logarithm of the denom
inator. The logarithm of a decimal fraction is found, by considering it as a whole number, and then prefixing to the decimal part of its logarithm a negative characteristic, greater by unity than the number of ciphers between the decimal point and the first significant place of figures. Thus, the logarithm of .0412. is
To find from the table, a number answering to a given logarithm.
XXXII Search, in the column of logarithms, for the decimal part of the given logarithm, and if it be exactly found, set down the corresponding number. Then, if the characteristic of the given logarithm be positive, point off, from the left of the number found, one place more for whole numbers than there are units in the characteristic of the given logarithm, and treat the other places as decimals; this will give the number sought.
If the characteristic of the given logarithm be 0, there will be one place of whole numbers; if it be -1, the number will be entirely decimal; if it be -2, there will be one cipher between the decimal point and the first significant figure; if it be -3, there will be two, &c. The number whose logarithm is 1.492481 is found in page 5, and is 31.08.
But if the decimal part of the logarithm cannot be exactly found in the table, take the number answering to the nearest 'less logarithm; take also from the table the corresponding difference in the column D: then, subtract this less logarithm from the given logarithm; and having annexed a sufficient number of ciphers to the remainder, divide it by the difference taken from the column D, and annex the quotient to the number answering to the less logarithm: this gives the required number, nearly. This rule, like the one for finding the logarithm of a number when the places exceed four, supposes the numbers to be proportional to their corresponding logarithms.
Ex. 1. Find the number answering to the logarithm 1.532708. Here,
The given logarithm, is
Next less logarithm of 34,09, is
Their difference is
And the tabular difference is 128: hence
128) 81.00 (63
which being annexed to 34,09, gives 34.0963 for the number answering to the logarithm 1.532708.