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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
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
log .00123 = 1.3001031 1230345
196 -1230541 *1230541
2.5078867 M. of log N, M. of log 32202 ; de D.
M. of log 32202.29;
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
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
Ex. Find log 7994-3726–
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
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
(1.32756) 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
* 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
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,
.:. 2.5078867 = log 03220229.
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".
= 9.9209393 Tab. diff. for 60" or D 836
24 diff. for 24" or d
334 60 :: L sin 56° 28' 24"
Ex. 2. Find L cos 29° 31' 28".
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.
difference or d, Also, 2648 tab. diff. for 60" = D, d
x 60" =