. The Locomotive. he angle DCK. In the same manner DF is the tangent ofthe angle DCI, and DG is the tangent of the angle DCII. If the radius of the circle is 1inch, and the angle DCE is 15 degrees, then the tangent DE will be equal to .268 of aninch. If the angle DCF is 30 degrees, its tangent DF is equal to .577 of an inch, and soon. These dimensions may be proved by measuring the above diagram. If the radius of the circle be made equal to tioo inches^ the tangents would be just twiceas long; if the radius were three inches^ they would be three times as long as they arewhen the radius is one
. The Locomotive. he angle DCK. In the same manner DF is the tangent ofthe angle DCI, and DG is the tangent of the angle DCII. If the radius of the circle is 1inch, and the angle DCE is 15 degrees, then the tangent DE will be equal to .268 of aninch. If the angle DCF is 30 degrees, its tangent DF is equal to .577 of an inch, and soon. These dimensions may be proved by measuring the above diagram. If the radius of the circle be made equal to tioo inches^ the tangents would be just twiceas long; if the radius were three inches^ they would be three times as long as they arewhen the radius is one inch, and so on for any radius. A Table of Natural Tangents is a table where the tangent of every degree and min-ute up to 90 degrees has been calculated for a radius = 1. If we wish to know the tan-gent to a radius equal to 2 or 3 we have only to multiply the tangent found in the tableby 2 or 3, and we have its length at once. Every engineers pocketbook contains such atable, and no mechanic should be without Fig. 2. We are now ready -to lay off our angle. Suppose we have given the line AB, Fig. 2,and wish to lay down another line AC which shall make an angle of 31 degrees and 17minutes with AB. Draw BC perpendicular to AB, and say 4 inches from A to B. Lookin the table for the tangent of 31° 17. This we find to be .6076. Multiply this by 4,and we have inches. Lay oflf BC = and draw AC. Then the angle CAB =31° 17; proceed in the same manner for any other angle. The most convenient length to make the line AB is 10, then the tabular tangent isto be multiplied by 10, which is performed by simply moving the decimal point oneplace to the right. One of the most convenient applications of this method is the division of the pitchcircles of gears having an odd or prime number of teeth. This is generally accomplishedby trial stepping around the pitch circle with a pair of dividers. Any one who has hadanything to do with gears knows the tediousness and uncertainty of thi
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