EXAMPLE.

Required the amount of the principal and interest of 355 dollars, let at 6 per cent. compound interest, for 7 years.

Adding 6 to 100 gives 106; whose logarithm, rejecting

2 in the index, is

Multiplied by

Product....

Principal, 355 dollars...

Sum gives the logarithm of 533.83.....

Therefore the amount of principal and interest is 533 dollars and 83 cents.

To find the logarithm of the sine, tangent, or secant, corresponding to any number of degrees and minutes, by Table XXVII.

The given number of degrees must be found at the bottom of the page when between 45° and 135°, otherwise at the top; the minutes being found in the column marked M, which stands on the side of the page on which the degrees are marked; thus, if the degrees are less than 45, the minutes are to be found in the left-hand column, &c.; and it must be noted that if the degrees are found at the top, the names of hour, sine, cosine, tangent, &c., must also be found at the top; and if the degrees are found at the bottom, the names sine, cosine, &c., must also be found at the bottom. Then opposite to the number of the minutes will be found the log. sine, log. secant, &c. in the columns marked sine, secant, &c. respectively.

EXAMPLE I.

Required the log. sine of 28° 37'.

Find 28° at the top of the page, directly below which, in the left-hand column, find 37'; against which, in the column marked sine, is 9.68029, the log. sine of the given number of degrees; and in the same manner the tangents, &c. are found.

EXAMPLE 11.

Required the log. secant of 126° 20′.

Find 126° at the bottom of the page, directly above which, in the left-hand column, find 20'; against which, in the column marked secant, is 10.22732 required.

To find the logarithm of the sine, cosine, &c. for degrees, minutes, and seconds, by Table XXVII.

Find the numbers corresponding to the even minutes next above and below the given degrees and minutes, and take their difference, D; then say, As 60" is to the number of seconds in the proposed number, so is that difference, D, to a correction, d, to be applied to the number corresponding to the least number of degrees and minutes; additive if it is the least of the two numbers taken from the table, otherwise subtractive.

If the given seconds be,,,, or, or any other even parts of a minute, the like parts may be taken of the difference of the logarithms, and added or subtracted as above, which may be frequently done by inspection. These proportional parts may also be found very nearly by means of the three columns of differences for seconds, given, for the first time, in the ninth edition of this work. The first column of differences, which is to be used with the two columns marked A, A, is placed between

these columns. The second column of differences, which is to be used with the two colun.ns B, B, is placed between these two columns. In like manner, the third column of differences, between the columns C, C, is to be used with them. The correction of the tabular logarithms in any of the columns A, B, C, for any number of seconds, is found by entering the left-hand column of the table, marked S' at the top, and finding the number of seconds; opposite to this, in the column of differences, will be found the corresponding correction. Thus, in the table, page 215, which contains the log. sines, tangents, &c., for 30°, the corrections corresponding to 25", are 9 for the columns A, A, 12 for the columns B, B, 3 for the columns C, C; so that, if it were required to find the sine, tangent, or secant of 30° 12′ 25′′, we must add these corrections respectively to the numbers corresponding to 30° 12'; thus,

these corrections being all added, because the logarithms increase in proceeding from 30° 12′ to 30° 13'. Instead of taking out the logarithms for 30° 12', and adding the correction for 25", we may take out the logarithms for 30° 13′, and subtract the correction for 60". - 25", or 35", found in the margin S'; thus, Logs. for 30° 13'....Sine 9.70180

Corr. for 35t in col. S'; }

or in

....

13

Logs. for 30° 12′ 25′′.... 9.70167

Tangent.... 9.76522

Secant.... 10.06342

4

10.06338

The corrections are in this case subtracted, because the logarithms decrease in proceeding backward 35" from 30° 13', to attain 30° 12′ 25′′. The tangents and secants, in this example, are the same by both methods; the sines differ by one unit, in the last decimal place, and this will frequently happen, because the difference of the logarithms for I', sometimes differ one or two units from the mean values which are used in the three columns of differences. The error arising from this cause is generally diminished by using the smallest angle* S', when the seconds of the proposed angle are smaller than 30'; or the greatest angle G', when the number of seconds are greater than 30". Thus, in the above example, where the angle S' 30° 12', and the angle G'30° 13', it is best to use the angle S' when the given angle is less than 30° 12′ 30", but the angle G' when it exceeds 30° 12′ 30′′. Thus, if it be required to find the sine of 30° 12′ 51′′, it is best to use the angle G'30° 13', and find the correction by entering the margin marked S', with the difference 60"-51"-9", opposite to which, in the column of differences, is 3, to be subtracted from log. sine of 30° 13′ = 9.70180, to get the log. sine of 30° 12′ 51′′ = 9.70177. To save the trouble of subtracting the seconds from 60", we may use the right-hand margin, marked G', and the correction may then be found by the following rules:RULE 1. When the smallest angle S' is used, find the seconds in the column S', and take out the corresponding correction, which is to be applied to the logarithm corresponding to S'; by adding, if the log. of G' be greater than the log. of S'; otherwise, by subtracting.

RULE 2. When the greater angle G' is used, find the seconds in the column Gʻ, and take out the corresponding correction, which is to be applied to the logarithm corresponding to G'; by adding, if the log. of S' be greater than the log. of G'; otherwise, by subtracting; so that, in all cases, the required logarithm may fall between the two logarithms corresponding to the angles S' and G'.

The correctness of these rules will evidently appear by comparing them with the preceding examples; and by the inverse process we may find the angle corresponding to a given logarithm, as in the next article.

We have given at the bottom of the page, in this table, a small table for finding the proportional parts for the odd seconds of time, corresponding to the column of Hours A. M. or P. M.; to facilitate the process of finding the log. sine, cosine, &c,, corresponding to the nearest second of time in the column of hours, or, on the contrary, to find the nearest second of time corresponding to any given log. sine, cosine, &c. Thus, in the preceding examples, where the angle S-30° 12′, and the

If we neglect the seconds in any proposed angle whose sine, &c. is required, we get the angle denoted above by S, and this angle increased by 1, is represented by G'; so that the proposed angle falls between S'and G; S being a smaller, and G' a greater angle than that whose log. sine, &c., is required; the letters S' and G', accented for minutes, being used because they are easily remembered

angle G'30° 13'; the times corresponding in the column of Hours P. M., are S4 1 36; G4 1 44'; and if we wish to find the log. sine, cosine, &c., corresponding to any intermediate time, as, for example, 4 1m 39', which differs 3 from the angle S', we must find the tabular logarithm corresponding to S', and apply the correction for 3*, given by the table at the bottom of the page, as in the following examples:Logs. for S'4b 1m 360 Correction for +35

C.

A.
Sine 9.70159

B.
Tangent 9.76493

Secant 10.06335

+8

+11

+3

are obtained by using the angle G', in the manner we

These corrections must be applied by addition or subtraction, according to the directions, given above, so as to make the required logarithm fall between those which correspond to the times S' and G'.

The inverse process will give the time corresponding to any logarithm. Thus, if the log. sine 9.70167 be given, the difference between this and 9.70159, corresponding to S'4 1m 36', is 8; seeking this in the column A, in the second line of the table at the bottom of the page, it is found to correspond to 3; adding this to the time S'4h 1m 36, we get 4h 1m 39 for the required time. We may proceed in the same manner with the logarithms in the columns B, C; using the numbers corresponding, marked B, C, respectively, in the table at the bottom of the page. To find the degrees, minutes, and seconds, corresponding to any given logarithm sine, cosine, &c. by Table XXVII.

Find the two nearest numbers to the given log. sine, cosine, &c., in the column marked sine, cosine, &c., respectively, one being greater, and the other less, and take their difference, D; take also the difference, d, between the given logarithm and the logarithm corresponding to the smallest number of degrees and minutes; then say, As the first found difference is to the second found difference, so is 60" to a number of seconds to be annexed to the smallest number of degrees and minutes before found. The three columns of differences may also be used, by an inverse operation to that which we have explained in the preceding article.

Then say, As 29:18::60": 38", nearly; which, annexed to 24° 16', give 24° 16′ 38′′, answering to log. sine 9.61400. Subtracting 24° 16′ 38′′ from 180°, there remain 155° 43′ 22, the log. sine of which is also 9.61400. The quantity 38" may also be found by inspection in the side column S' of the page opposite d=18, in the column of differences between the two columns, A, A. If we use the angle G', we shall have d' equal to 11, the difference of the logarithms 9.61411 and 9.61400, and the corresponding number of seconds in column G', is 37", making 24° 16′ 37′′.

To find the arithmetical complement of any logarithm.

The arithmetical complement of any logarithm is what it wants of 10.00000, and is used to avoid subtraction. For, when working any proportion by logarithms, you may add the arithmetical complement of the logarithin of the first term, instead of subtracting the logarithm itself, observing to neglect 10 in the index of the sum of the logarithms. The arithmetical complement of any logarithm is thus found:-Begin at the index, and write down what each figure wants of 9, except the last significant figure, which take from 10.* Thus, the arithmetical complement of 9.62595 is 0.37405; that of 1.86567 is 8.13433; and that of 10.33133 is 89.66867, or 9.66867.

When the index of the given logarithm is greater than 10, as in some of the numbers of Table XXVII, the left-hand figure of it must be neglected; and when there are any ciphers to the right hand of the last significant figure, you may place the same number of ciphers to the right hand of the other figures of the arithmetical complement.

PLANE TRIGONOMETRY.

- PLANE TRIGONOMETRY is the science which shows how to find the measures of the sides and angles of plane triangles, some of them being already known. It is divided into two parts, right-angled and oblique-angled; in the former case, one of the angles is a right angle, or 90°; in the latter, they are all oblique.

In every plane triangle there are six parts, viz. three sides and three angles; any three of which being given (except the three angles), the other three may be found by various methods, viz. by Gunter's scale, by the sliding rule, by the sector, by geometrical construction, or by arithmetical calculation. We shall explain each of these methods; but the latter is by far the most accurate; it is performed by the help of a few theorems, and a trigonometrical canon, exhibiting the natural or the logarithmic sines, tangents, and secants, to every degree and minute of the quadrant.† The theorems alluded to are the following:

THEOREM 1.

In any right-angled triangle, if the hypotenuse be made radius, one side will be the sine of the opposite angle, and the other its cosine; but if either of the legs be made radius, the other leg will be the tangent of the opposite angle, and the hypotenuse will be the secant of the same angle.

Ist. If, in the right-angled plane triangle ACB (fig. 1), we make the hypotenuse AB radius, and upon the centre, A, describe the arc BE, to meet AC produced in E, then it is evident that BC is the sine of the arc BE (or the sine of the angle BAC), and that AC is the cosine of the same angle; and if the arc AD be described about the centre B (fig. 2), AC will be the sine of the angle ABC, and BC its cosine.

2dly. If the leg AC (fig. 3) be made radius, and the arc CD be described about the centre A, CB will be the tangent of that arc, or the tangent of the angle CAB; and AB will be its secant.

3dly. If the leg BC (fig. 4) be made radius, and the arc CD be described about the centre B, CA will be the tangent of that arc, or the tangent of the angle B, and AB will be its secant.

Now, it has been already demonstrated (in Art. 55, Geometry) that the sine, tangent, secant, &c. of any arc in one circle is to the sine, tangent, secant, &c. of a similar arc in another circle as the radius of the former circle to the radius of the latter. And since in any right-angled triangle there are given either two sides, or the angles and one side, to find the rest, we may, if we wish to find a side, make any side radius; then say, As the tabular number of the same name as the given side is to the given side of the triangle, so is the tabular number of the same name as the required side, to the required side of the triangle. If we wish to find an angle, one of the given sides must be made radius; then say, As the side of the triangle made radius is to the tabular

* It will not be necessary to add any further description of the uses of the sector or sliding rule; for what we have already given will be sufficient for any one tolerably well versed in the use of Gunter's scale.

radius, so is the other given side to the tabular sine, tangent, secant, &c. by it represented; which, being sought for in the table of sines, &c., will correspond to the degrees and minutes of the required angle.

THEOREM II.

In all plane triangles, the sides are in direct proportion to the sines of their opposite angles (by Art. 58, Geometry).

Hence, to find a side, we must say, As the sine of an angle is to its opposite side, so is the sine of either of the other angles to the side opposite thereto. But if we wish to find an angle, we must say, As any given side is to the sine of its opposite angle, so is either of the other sides to the sine of its opposite angle.

THEOREM III.

In every plane triangle, it will be, as the sum of any two sides is to their difference, so is the tangent of half the sum of the two opposite angles to the tangent of half their difference (by Art. 59, Geometry).

THEOREM IV.

As the base of any plane triangle is to the sum of the two sides, so is the difference of the two sides to twice the distance of a perpendicular (let fall upon the base from the opposite angle) from the middle of the base (by Art. 60, Geometry).

THEOREM V.

In any plane triangle, as the rectangle contained by any two sides including a sought angle, is to the rectangle contained by the half sum of the three sides and the same half sum decreased by the other side, so is the square of radius to the square of the cosine of half the sought angle (by Art. 61, Geometry).

In addition to these theorems, it will not be amiss for the learner to recall to mind the following articles:

1. In every triangle, the greatest side is opposite to the greatest angle, and the greatest angle opposite to the greatest side.

2. In every triangle equal sides subtend equal angles. (Art. 39, Geometry.)

3. The three angles of any plane triangle are equal to 180°. (Art. 35, Geometry.) 4. If one angle of a triangle be obtuse, the rest are acute; and if one angle be right, the other two together make a right angle, or 90°; therefore, if one of the acute angles of a right-angled triangle be known, the other is found by subtracting the known angle from 90°. If one angle of any triangle be known, the sum of the other two is found by subtracting the given angle from 180°; and if two of the angles be known, the third is found by subtracting their sums from 180°.

5. The complement of an angle is what it wants of 90°, and the supplement of an angle is what it wants of 180°.

In the two following tables we have collected all the rules necessary. for solving the various cases of Right-angled and Oblique-angled Trigonometry.