FIG. 61.
Suppose it is required to compare the force exerted by the slay in beating up (say the front 1 inch of its stroke) in two looms, one with a 12-inch crank-arm and 3-inch crank and the other with an 11-inch arm and 4-inch crank. The weight of the slays, the speed of the looms, the tension on the warps, and the timing of the primary movements, the same in each case.
In [Fig. 61] the smaller circle represents the 3-inch crank and the larger one the 4-inch crank. CP = 1-inch, CB = 11-inch arm, and CD = the 12-inch arm. It is obvious that if we can obtain the two angles made by the cranks, viz. ∠ CAB and ∠ CAD, we shall be able to get the time, or fraction of a revolution, occupied in moving the slay from C to P. As we know the three sides of the triangle we can obtain the angle enclosed by any two sides, and what is required in this case is to obtain the angles BAC and DAC. In triangles of this kind where there is no right angle, we can obtain the cosine of the angle as follows:—
| CA2 + AD2 − DC2 | = AQ, the cosine of angle DAC, | |
| 2CA.AD | ||
| and | CA2 + AB2 − BC2 | = AN, the cosine of angle BAC. |
| 2CA.AB |
The proof of this formula is given in Euclid, Book 2.
Having obtained the cosines of the two angles, we can find the angles themselves by referring to a table of sines and cosines.
Then as AP = 15 inches,
CA = 14, AD 3 inches, DC 12 inches, BA 4 inches, BC 11 inches; and reducing the formulæ to figures, we get:
| 142 + 32 − 122 | = | 196 + 16 − 144 | = | 0·7262 cosine, |
| 2 × (14 × 3) | 84 |