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the line which meets the circle, the line which meets the cirle shall touch it.

Let any point D be taken without the circle ABC, and from it let two straight lines, DCA, DB, be drawn, of which DCA cuts the circle, and DB meets it; and let the rectangle AD, DC be equal to the square on DB.

Then DB shall touch the circle. Draw DE CONSTRUCTION.-Draw the straight line DE, touching the touching the circle. circle ABC (III. 17);

Find F the centre (III. 1' and join FB, D

FD, FE

PROOF.-Then FED is a right angle (III. 18).

And because DE touches the circle ABC, BA

5 and DCA cuts it, the reetangle AD, DC is equal to the square on DE (III. 36).

But the rectangle AD, DC is equal to the square on DB (Hyp.);

Therefore the square on DE is equal to

the square on DB (Ax. 1); Then Therefore the straight line DE is equal to the straight line DE = DB DB.

And EF is equal to BF (I. Def. 15);

Therefore the two sides DE, EF are equal to the two

sides DB, BF, each to each; And tri- And the base DF is common to the two triangles DEF, angles DBF and DBF; DEF are equal in

Therefore the angle DEF is equal to the angle DBF (1.8). every te

But DEF is a right angle (Const.); spect.

Therefore also DBF is a right angle (Ax. 1). :: DBF is And BF, if produced, is a diameter; and the straight line a right angle;

which is drawn at right angles to a diameter, from the ex

tremity of it, touches the circle (III. 16, Cor.); and there Therefore DB touches the circle ABC. fore DB touches Therefore, if from a point, &c. Q.E.D. the cir le

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log .00123 = 1.3001031 1230345

196 -1230541 *1230541

-4922164

2.5078867 M. of log N, M. of log 32202 ; de D.

=

M. of log 32202.29;
log .03220229.
the number required.

rical Tables.

tables much in the same way Sarithms of numbers.

natural sines, cosines, &c., and nes, &c. It is with the latter gh many of our remarks apply

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ural sines and cosines of all we (Art. 11) less than unity, it

logarithms are negative. To am in a negative form, and for add 10 to their real value, and must allow for this. The same Hause of logarithmic tangents, conts.

true logarithmic sine by log sin, sine by L sin. A = L sin A

= L cos A. &c. in using the tables that, although ud tangent of an angle increase as

10,

10,

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Ex. Find log 7994-3726–
M. of log 79943

.9027804
Diff. for 7

381 2

11 6

32 :: M. of log 79943726 .9027843

Hence, log 7994:3726 3.9027843. 28. Having given the logarithm of a number to find the number.

After the explanations of Art. 26, the method of working the following examples will be easily understood :

Ex. 1. Find the number whose logarithm is 1-9030173. Taking from the tables the mantissæ next above and below, we have

•9030194 M. of log 79987
•9030140 M. of log 79986 . .(1).

54 Again, 9030173 = M. of log N............... ..(2). Hence, subtracting (1) from (7)

33 d, the difference between the logarithms of the required number and the next lower.

33 Now •61, the difference between the next lower

54 number and the required number. Hence •9030173 M. of log 79986.61 ;

... 1.9030173 = log ·7998661 ; .:. 7998661 is the number required. Ex. 2.* Find the value of

(1.023) (-00123)

(1.32756) We have

3 log 1.023 + 1 log .00123 4 log 1.32756. Now, 3 log 1.023 = 3 x .0098756

0276268 1 log .00123 1 (30899051)

1 (4 + 1.0899051) 1.2724763

log N

* The logarithms used in this example are taken from the tables.

:: adding, 3 log. 1.023 + 4 log •00123 = 1.3001031 Again, M. of log 13275 •1230345 and diff. for

.6

196 .. M. of log 13275.6 = '1230541 . . 4 log 1.32756 = 4 x .1230541

4922164 Then, subtracting, log N

2.5078867 Hence we have, .5078867 M. of log N, and •5078828 M. of log 32202;

39 d, also 135 D,

39
and

•29.
135
.:: -5078867 = M. of log 32202.29;

.:. 2.5078867 = log 03220229.
Hence •03220229 is the number required.

Trigonometrical Tables. 29. We use trigonometrical tables much in the same way as we do tables of ordinary logarithms of numbers.

Tables have been formed of natural sines, cosines, &c., and also of logarithmic sines, cosines, &c. It is with the latter only we shall now deal, though many of our remarks apply equally to the former.

As the values of the natural sines and cosines of all angles between 0° and 90° are (Art. 11) less than unity, it follows (Art. 21) that their logarithms are negative. To avoid, however, printing them in a negative form, and for other reasons,

it is usual to add 10 to their real value, and hence in using them we must allow for this. The same thing is also done in the case of logarithmic tangents, cotangents, secants, and cosecants.

We generally express the true logarithmic sine by log sin, and the tabular logarithmic sine by L sin. Hence, we have, log sin A L sin A - 10,

10, &c. It must be remembered in using the tables that, although (Art. 11) the sine, secant, and tangent of an angle increase as the angle increases from 0° to 90°, yet the cosine, cosecant, and cotangent diminish as the angle increases.

log cos A L cos A

Hence, when any angle is not exactly contained in the tables, we must add the difference in the case of a sine, secant, or tangent; but subtract it in the case of a cosine, cosecant, or cotangent.

And, conversely, when the given logarithm is not contained exactly in the tables, we must in the case of the sine, secant, or tangent take out the next lower tabular logarithm as corresponding to the angle next lower; but in the case of a cosine, cosecant, or cotangent, we must take out the next higher tabular logarithm as corresponding to the angle next lower in the tables.

We shall assume that small differences in the angles are proportional to the corresponding differences of the logarithmic trigonometrical ratios

Ex. 1. Find L sin 56° 28' 24".
Referring to tables, we have-
L sin 56° 28'

= 9.9209393 Tab. diff. for 60" or D 836

24 diff. for 24" or d

334 60 :: L sin 56° 28' 24"

= 9.9209727

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x 836

x 716

Ex. 2. Find L cos 29° 31' 28".
Now L cos 29° 31'

9.9396253 Tab. diff. for 60" or D 716

28 ... diff. for 28"

334 60 .. L cos 29° 31' 28"

9.9395919 Ex. 3. Find the angle A, when L tan A 9.8658585

We have 9.8658585 L tan A.
Next lower, 9.8657702 L tan 36° 17',

883

difference or d, Also, 2648 tab. diff. for 60" = D, d

883 And

20".
D

2648
Hence, 9•8658585 = L tan 36° 17' 20".

x 60"

x 60" =

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