This section is from the book "Turning And Mechanical Manipulation", by Charles Holtzapffel. Also available from Amazon: Turning and Mechanical Manipulation.

The divisions of the circle in the preceding table, require the index to he always of unvarying height or length, as it is applied from hole to hole; but other numbers, not aliquot parts of these circles of holes, may be produced from them with the adjusting index, by the method of differences, or interpolation.

In interpolation, the index itself continuously varies in length, its point being moved through a small space, together with the division plate, between every division marked on the work; with the effect, that when the index is moved in the same direction with the division plate the two quantities are added together, and the spaces marked out upon the work become so much the wider and less in number, than the divisions taken by the point in the circle of holes. When the index is moved in the opposite direction to that travelled by the division plate, the amount of its motion is deducted, producing the contrary effect; the divisions then obtained, being less in width and more in number.

The adjusting index fig. 123, is used for interpolation in the following manner. To obtain, for example, thirty one equal divisions of the circle, a reference to the table, gives three holes in 96 for thirty two; it has then to be ascertained by trial, how many divisions of the micrometer of the adjusting index, are equal to the one thirty second part of the 96 circle, which part has to be expunged, by being equally divided among the thirty one divisions required to remain. To determine their number, the mandrel is retained stationary, the point of the index is placed in one of the holes of the 96 circle, and thence, moved over exactly the distance between three holes, by its micrometer screw. This, it may be assumed, requires 124 divisions, which divided by 31, the number to be produced, gives a result of four. Therefore, by every time moving the division plate the three holes as for 32, and the index also 4 divisions of its micrometer, in the same direction for the difference; the one extraneous division, in the course of the entire circuit, will be equally divided among the 31, augmenting each a minute quantity, so that they exactly divide the circle.

It will however be more frequent that the number of divisions of the micrometer, will not divide without leaving a remainder. Supposing their number to have been say, 113, division by 31, would give 3 with a remainder of 20, which must be introduced among the whole, by making every movement of the index 3⅔ When the micrometer is more finely divided, making the number in such case say, 289, every movement would correctly be 9⅓; but 9.9 and 10 divisions taken in continual succession, would probably suffice for many purposes, while it would save the trouble of dealing with fractions. On the same principle 27 may be obtained from 32, by measuring the value of five thirty second parts of the circle, and dividing that by 27; but it would be more judicious to obtain 27 from 28, in the 112 circle, as when the quantity to be dealt with is large, the inequality between the right line and the arc becomes of some consequence. In pursuing this method in the opposite manner, to increase 32 to 33 divisions, the difference measured by the index, is divided by 33 the number to be obtained, and the index and micrometer are moved in opposite directions. To simplify the use of the apparatus, the graduations are so numbered, that when the figures on the division plate and those of the micrometer of the index, both go on increasing, that is, read in the same direction, then the two movements are added together, enlarging the spaces; and on the other hand, when the two series of numbers are used in opposite directions, the motion of the one, deducted from that of the other, contracts the resulting spaces.

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