TRIGONOMETRICAL TABLES.

106. There are two kinds of trigonometrical tables; the first, called the Table of Natural Sines, Cosines, &c., contains the numerical values of the sines, cosines, tangents, &c., that is, of the trigonometrical ratios for each given value of the angle; the second, called the Table of Logarithmic Sines, &c., contains the logarithms of the numbers in the first Table.*

TABLE OF NATURAL SINES, &c.

107. The trigonometrical functions† or ratios are numbers which are capable of being calculated from geometrical principles, and accordingly certain series have been investigated, and certain algebraic expedients devised for the general purpose of determining the trigonometrical ratios. With such aid the sines, cosines, &c., of all angles from o° to 90° (i..., for all values of A, from Ao up to A = 90) have been computed to several places of decimals and arranged in tables called Tables of Natural Sines, Cosines, &c. In some tables the angles succeed each other at intervals of 1", in others at intervals of 10"; but in ordinary tables (as Table XXVI, NORIE) at intervals of 1', and to the last mentioned we shall refer.

108. The statement of the method by which such tables are constructed is unsuitable to the pages of the present work. The mode of using them in computation we shall now proceed to explain.

109. The arrangement of this table will be understood from a simple inspection. It contains the sines, cosines, &c., of angles between zero and 90°, generally for every minute, and the fluctuations of angles containing a number of degrees, minutes, and seconds, have to be found by interpolation similar in their nature to those that are required to be used in tables of logarithms of numbers. This interpolation is based upon the supposition that the differences of the sines, &c., are proportional to the differences of the angles, and this proportion, though theoretically inexact, gives, in general, a sufficient approximation, provided the difference of the angles of the table are sufficiently small.

110. Referring to the Tables (Table XXVI, NORIE) it will be seen that the degrees are given at the top of the column, and the minutes down the left hand side of the page for the sines.

And, for the cosines, the degrees are given at the bottom of the page, and the minutes up the right hand side of the page.

The usual trigonometrical tables are given in conjunction with tables of logarithms, and they more frequently give logarithms only than sines, cosines, &c., themselves. When logarithms were invented they were called artificial numbers, and the originals for which logarithms were computed, were accordingly called natural numbers. Thus, in speaking of a table of sines, to express that it is not the logarithms of the sines which are given, but sines themselves, that table would be called a table of natural sines, and the logarithms of these would be called not logarithms of sines but logarithmic sines, &c.

+ By the functions of angles (sometimes called their trigonometrical or geometrical functions) are meant their sines, tangents, secants, versed sines, and chords; the word function signifying any quantity that is dependent on another changing as it changes.

M

The difference of the trigonometrical ratios for 100" are given at the foot of each column.

III. In using these Tables, we have either to find the sine, cosine, &c., of an angle whose value is given in degrees (°), minutes ('), and seconds ("); or to find the corresponding angle in degrees, minutes, and seconds.

112. If the value of the angle be given in degrees and minutes only, the sine, cosine, &c., is found directly from the Tables, in which are registered the values of the trigonometrical ratios.

All the numbers contained in such Tables as NORIE's Table XXVI must be understood as decimals.

113. As the sines, cosines, &c., pass through all their possible numerical values while the angle varies from o° to 90°, the tables are not extended beyond 90°; such computations would be superfluous, for the sine or cosine of an angle between one and two right angles, viz., of an angle greater than 90° is the same in numerical value as the sine, cosine, &c., of an angle as much below 90°, and is known from the recorded sine or cosine of its supplement.*

If the angle contains seconds, we must proceed by the method of proportional parts, as in the following examples:

RULE XXXIV.

1o. Find from the Table the nat. sine, cosine, &c., which corresponds to the degrees and minutes. (NORIE, Table XXVI.)

2°. Multiply the difference by the seconds, and divide by 100.

NOTE. To divide by 100 we have merely to cut off the two right-hand figures.

3°. If the required quantity be a nat. sine, tangent, or secant, add the result to the last figures obtained in 1°; if it be a cosine, cotangent, or cosecant, subtract. The result will be the required sine, cosine, &c.

NOTE 1.-The reason of this rule is founded on the principle that for a small interval, such as one minute, the increase of the sine is proportional to the increase of the angle.

NOTE 2.-It is necessary to bear in mind that the sine, tangent, and secant (under 90°) for which the tables are constructed increase as the arc increases, whilst the cosine, cotangent, and cosecant decrease as the arc increases. This will require the corrections connected with a sine, a tangent, or a secant to be added, and those connected with a cosine, a cotangent, or a cosecant to be subtracted whether arcs or their functions be sought from the tables.

* Def.-The supplement of an angle is the result when the angle is subtracted from 180°. In other words, an angle and its supplement together make 180°, or two right angles, thus, 23° 19' is the supplement of 156° 41', and 156° 41′ is the supplement of 23° 19′.

115. If the value of the sine, cosine, &c., be given, and it is required to find the angle, we use the following rule:

RULE XXXV.

1o. Find in the Tables the next lower nat. sine, nat. cosine, &c., and note the corresponding degrees and minutes.

20. Subtract this from the given sine, cosine, &c., multiplying the difference by 100; divide by the tabular difference, and consider the result as seconds.

3°. If the given value be that of a sine, tangent, or secant, add these seconds to the degrees and minutes found in 1°; if it be that of a cosine, cotangent, &c., subtract. The result will be the required angle.

NOTE. In taking out the angle for a natural cosine we may take out the next greater natural cosine, and subtract the given natural cosine from it; and having found the seconds ("), as above, they are additive. The trigonometrical ratio corresponding to the next less angle being written down in every case, confusion will be avoided, as the additional seconds will always be additive.

EXAMPLES.

Ex. 1. Given the natural sine 0732156: find the angle.

Given nat. sine 732156

Sine 47° 4'

Tab. diff.

732147 next lower in Table XXVI, NORIE.

327 327)900(3′′ nearly (additional seconds for nat. sine.)
Ans.: 47° 4' 3".

981

The log. 732156 is sought for in Table XXVI, NORIE, but as it cannot be found exactly, the next less is taken which corresponds to 47° 4'. The difference of the logs. is then found, two cyphers added (which is equivalent to multiplying by 100), and the product divided by the tabular difference; the quotient is the additional seconds.

Ex. 2. Given the natural cosine 853267: find the angle.

769388

9. 817726 II. 999000 *215515 12. .6

Given the nat. cosine, to find the angle. I. *448807 3. 726998 5. 514841 7. 2. 948397 4. *702017 6. 914237 8. 974822 IO.

TABLES OF LOGARITHMS OF TRIGONOMETRICAL RATIOS.

116. The Trigonometrical Ratios being numbers, have logarithms that correspond to them. In practice the logarithmic are generally far more useful than the natural sines, &c., though the latter are often necessary, in some simple kinds of calculation, preferable.

or

117. As the sines and cosines of all angles, and the tangents of angles less than 45°, are less than radius or unity, being proper fractions, the logarithms of the value of these quantities, properly, have negative characteristics. In order to avoid the inconvenience of printing negative logarithms, and for other reasons, 10 is added to the characteristic before it is registered in the table of logarithmic sines, &c., so that we find the characteristic 9 instead of ī, 8 instead of 2, &c.

Thus, on referring to the Table of Natural Sines (Table XXVI, NORIE), we find natural sine of 16° 275637. If we calculate the logarithm of 275637, we find its value is T440338; if to this 10 is added we find that

Log. sine 16° 9'440338.

To preserve uniformity, the characteristics of the logarithms of all the other ratios, namely, of the log. tangents, cotangents, secants, and cosecants are increased by 10. In trigonometrical operations this is convenient, but. principally because the extraction of roots very seldom occurs.

It may be observed here that the uniform addition of 10 to the characteristic gives the logarithm of 10000 million times the natural number.

Thus, 9'599327 is the log. of 3979486000, and this latter number is the natural sine corresponding to a radius of 10000 millions, instead of a radius of unity.

118. Usual arrangement of Tables of Logarithmic Sines, Cosines, &c.— The table of logarithmic sines, cosines, tangents, cotangents, secants, and cosecants, contain all arcs from zero (0°) through all magnitudes up to 90°, the log. of radius, as just stated, being 10. At the top of the page is placed the number of degrees, and in the left-hand column each minute of the degree, opposite to which are arranged the numerical values of the log. sine, cosine, &c., of the corresponding angle in those columns, at the top of which those terms are placed. The headings of the columns run along the top, thus, as far as 44°. The degrees from 45° to 90° are placed at the bottom of the page, and the minutes of the degree arranged in a right-hand column, so that the angles read off on the right-hand side are complemental to those read off at the points exactly opposite on the left-hand side, the values of the sines, cosines, tangents, &c., being found in the columns at the bottom of which those terms are found. This arrangement is rendered practicable by the circumstance of every angle between 45° and 90° being the complement of another between 45° and o°, every sine of an angle less than 45° is the cosine of another greater than 45°, every tangent is a cotangent, &c.; the sines, tangents, &c., of angles being respectively equal to the cosines, cotangents, &c., of the complements of the same angle.

The following shows the usual arrangement of such tables :

Besides the columns headed "sine, tangent," &c., are three smaller columns headed "Diff." They contain, in most tables, the differences between the values of the consecutive logarithms in the contiguous columns on either side, but corresponding to a change of 100" in the arc (not the difference corresponding to 60" of arc or angle); and it must be kept in mind that the same difference is common to the sine and cosecant, to the tangent and cotangent, and to the secant and cosine. They are inserted for the convenience of finding the values of the sines and cosines, &c., of angles which are expressed in degrees, minutes, and seconds.

119. The above, as just stated, is the usual arrangement of most tables, but in the earlier editions of NORIE and some other works the arrangement is somewhat different.