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longer lines, and the shorter lines again divide these into halftenths, or to 5 hundredths. 24 of these smaller parts are set off on the Vernier, and divided into 25 equal parts, each of which is 24 × .05 therefore = =.048 inch, and is shorter than a division. 25

of the scale by .050.048.002, or two thousandths of an inch, a twenty-fifth part of a division on the scale, to which minuteness the Vernier can therefore read. The reading in the figure is £0.686, (30.65 by the scale and .036 by the Vernier), the dotted line marked D showing where the coincidence takes place.

(349) Circle divided into degrees. The following illustrations apply to the measurements of angles, the circle being variously divided. In this article, the circle is supposed to be divided into degrees.

If 6 spaces on the Vernier are found to be equal to 5 on the circle, the Vernier can read to one-sixth of a space on the circle, i. e. to 10'.

If 10 spaces on the Vernier are equal to 9 on the circle, the Vernier can read to one-tenth of a space on the circle, i. e. to 6'.

If 12 spaces on the Vernier are equal to 11 on the circle, the Vernier can read to one-twelfth of a space on the circle, i. e. to 5'. Fig. 229.

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The above figure shows such an arrangement. The index, or zero, of the Vernier is at a point beyond 358°, a certain distance, which the coincidence of the third line of the Vernier (as indicated

by the dotted and crossed line) shows to be 15'. The whole reading is therefore 358° 15'.

If 20 spaces on the Vernier are equal to 19 on the circle, the Vernier can read to one-twentieth of a division on the circle, i. e. to 3'. English compasses, or "Circumferentors," are sometimes thus arranged.

If 60 spaces on the Vernier are equal to 59 on the circle, the Vernier can read to one-sixtieth of a division on the circle, i. e. to 1'.

(350) Circle divided to 3. Such a graduation is a very common one. The Vernier may be variously constructed. Suppose 30 spaces on the Vernier to be equal to 29 on the 29 × 30'

circle. Each space on the Vernier will be =

30

=

29',

and will therefore be less than a space of the circle by 1', to which the Vernier will then read.

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The above figure shows this arrangement. The reading is 0°, or 360°.

In the following figure, the dotted and crossed line shows what divisions coincide, and the reading is 20° 10'; the Vernier being the same as in the preceding figure, and its zero being at a point of the circle 10' beyond 20°.

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In the following figure, the reading is 20° 40', the index being at a point beyond 20° 30', and the additional space being shown by the Vernier to be 10'.

Fig. 232.

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Sometimes 30 spaces on the Vernier are equal to 31 on the circle.

Each space on the Vernier will therefore be

=

31 × 30'
30

=31', and

will be longer than a space on the circle by 1', to which it will therefore read, as in the last case, but the Vernier will be "retrograde." This is the Vernier of the compass, Fig. 148. The peculiar manner in which it is there applied is shown in Fig. 239. If 15 spaces on the Vernier are equal to 16 on the circle, each space on the Vernier will be

will therefore read to 2'.

=

(351) Circle divided to 20'.

16 x 30'
15

= 32', and the Vernier

If 20 spaces on the Vernier are equal to 19 on the circle, each space of the latter will be =

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19', and the Vernier will read to 20'19' 1'.

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If 40 spaces on the Vernier are equal to 41 on the circle, each

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grade. In the following figure the reading is 360°, or 0°; and it will be seen that the 40 spaces on the Vernier (numbered to whole minutes) are equal to 13° 40' on the limb, i. e. to 41 spaces, each of 20'.

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If 60 spaces on the Vernier are equal to 59 on the circle, each

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59 × 20'

60

19′ 40′′, and the Vernier

will therefore read to 20'-19' 40"-20". The following figure shows such an arrangement. The reading in that position would be 40° 46' 20".

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(352) Circle divided to 15'. If 60 spaces on the Vernier are equal to 59 on the circle, each space on the Vernier will be = 14' 45", and the Vernier will read to 15". In the

59 x 15'

60

=

following figure the reading is 10° 20′ 45′′, the index pointing to 10° 15', and something more, which the Vernier shows to be 5' 45".

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