2.

EXAMPLES FOR PRACTICE.

21636 by 36; 6384 by 76; and 31250 by 250.

Divide 6391 by 77: Divide 3600 by 60; 4500 by 9; 3. Divide 1755 by 39; 4. Divide 36808 by 127;

5. Divide 76'4 by 1·6;

6. Divide 2020 by 202;

646 by 34;

3654 by 38; and 58469 by 981.

2160 by 30; and 365.55 by 5'5.

147392 by 440; 72864 by 184; and 16882 by 734. 32·08 by 8·8; 69′52 by 2·4; and 1·728 by 1*2.

2000 by 2000; 87648 by 368; and 147000 by 1470. 7. Divide 135056 by 734; 8746·9 by 36'4; 674·80 by '0763; and 3372 36 by 5'378. Divide 236 by 19'1; 0472 by 3'12; 03755 by 025; and 476·14 by 248. 9. 1972; 19 — '72; 19 ÷ 7·2; and 19÷0072.

IO.

II.

001237 108·46; 287642 ÷ 834°56; 472 ÷322; and 10011'1 — 99'3. 10004572; 112221 ÷ 111; 1÷6'729; and 56262·5 ÷ 52·643. 12. 32567129; 585900124; 72384 ÷ 192; and 132'45 ÷ 385. 13. *00010001; 01001; '01 100; and I'÷'001.

14.

10.

364096·6; 34′56÷0024; 47520003500; and 10000 When it is proposed to find the value of an expression in which both multiplication and division are signified, the sum of the logarithms of the factors of the dividend, diminished by the sum of the logarithms of the factors of the divisor will be the logarithm of the value required.

It is very often expedient to transform the logarithm of a divisor into that of a multiplier, and it is customary, in such calculations, to avoid not only negative logarithms, but negative indices also, by substituting for a subtraction logarithm its arithmetical complement (See page 27). This makes the operation consist of a single addition; only we must diminish the result by subtracting 10 for every arithmetical complement that has been used.

To apply this method to the example above:-Having found in the table the log. of the divisor 287, we may at once transform it into the addition logarithm 7'542118, and similarly for the log. of 2101 we may write 6.677574, and then the calculation will proceed continuously as follows:

pare the result with the product of 8.71979 X 057447 X 0206168.

Trigonometrical magnitudes are numbers capable of being calculated from geometrical principles, and accordingly, tables, called Tables of Natural Sines, have been computed, in which the values of the Sines, Cosines, &c., of every degree and minute of the quadrant are registered. The statement of the method by which such tables are constructed is unsuited to the present treatise. The mode of using them in computation we shall now proceed to explain. 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 being given the value of the angle in degrees, minutes, and seconds, to find the corresponding value of the sine, cosine, &c.

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

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

RULE XV.

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

2o.

Multiply the tabular difference by the seconds, and divide by 100.

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.

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.

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

RULE XVI.

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

2°.

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.

Tab. diff. = 327

732147 next lower in table XXVI.

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

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. Given the natural cosine 853267: find the angle. Given nat. cosine 853267

2.

Cosine 31° 26' =

Tab. diff. 253

129 Ans. 31° 25′ 53′′. EXAMPLES FOR PRACTICE.

TABLES OF LOGARITHMIC SINES, ETC.

In order to apply logarithmic calculations to trigonometrical quantities, it is necessary to construct tables of logarithms of the natural sines, cosines, &c., and the real logarithmic sines, tangents, &c., are just the logarithms of those numbers which are the natural sines, tangents, &c.* In practice the logarithmic are generally far more useful than the natural sines, &c., though the latter are often necessary, or, in some simple kinds of calculation, preferable.

As all the sines and cosines, all the tangents from o° to 45°, and all the cotangents from 45° to 90°, are less than radius or unity; the logarithms of the values of these quantities are decimal fractions and have negative characteristics. In order to avoid the necessity of entering negative numbers, 10 is added to every logarithm before it is registered in the tables of logarithmic sines.

Thus, on referring to the table of natural sines (Table XXVI, Norie), we find that sine 16° 0'275637. If we calculate the logarithm of o 275637, we find that its value is I'440338; if to this 10 is added, we find that

Log. sine 16° 9'440338.

In trigonometrical operations this is convenient, but principally because the extraction of roots very seldom occurs. The same thing is done, for the sake of uniformity, with logarithmic tangents, though only those of angles under 45° would, as just stated, have negative indices.

It may be observed here that the uniform addition of 10 to the index 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.

The table of logarithmic sines, cosines, tangents, cotangents, secants, and cosecants, contain all arcs from 1' of a degree through all magnitudes up to a quadrant or 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

There are independent methods of calculating logarithmic tables, but any investigation of these methods would be out of place in these pages.