. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. 100 MECHANICAL RECREATIONS [CH. V to take one instance, the sail to be fixed, that is, suppose a to be a constant. Then v is a maximum if 8 + a = \tt, that is, if 0 is equal to the complement of a. In this case we have v = u cosec a, and therefore v is greater than u. Hence, if the wind makes the same angle a abaft the beam that the sail makes with the keel, the velocity of the boat will be greater than the velocity of the wind. Next, suppose that the boat is r
. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. 100 MECHANICAL RECREATIONS [CH. V to take one instance, the sail to be fixed, that is, suppose a to be a constant. Then v is a maximum if 8 + a = \tt, that is, if 0 is equal to the complement of a. In this case we have v = u cosec a, and therefore v is greater than u. Hence, if the wind makes the same angle a abaft the beam that the sail makes with the keel, the velocity of the boat will be greater than the velocity of the wind. Next, suppose that the boat is running close to the wind, so that the wind is before the beam (see figure below), then in the same way as before we have v sin a = u sin (6 + a), or v sin a = u sin <p, where $ = angle WAS = ir— 0 — eu Hence v = u sin cosec Let to be the component velocity of the boat in the teeth of the wind, that is, in the direction AW. Then we have w = v cos BA W = v cos (a + <p) = u sin $ cosec 2 cos (a + <£). If a is constant, this is a maximum when = \tt — Ja; and, if has this value, then w = \u (cosec a — 1). This formula shows that w is greater than u, if sin a < £. Thus, if the sails can be set so that a is less than sin-1£, that is, rather less than 19° 29', and if the wind has the direction above assigned, then the component velocity of the boat in the face of the wind is greater than the velocity of the wind. The above theory is curious, but it must be remembered that in practice considerable allowance has to be made for the fact that no boat for use on water can be constructed in which the resistance to motion in the direction of the keel can be wholly neglected, or which would not drift slightly to leeward if the wind was not dead astern Still this makes less. Please note that these images are extracted from scanned page images that may have been digitally enhanced for readability - coloration and appearance of these illustrations may not perfectly res
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Keywords: ., bookcentury1900, bookdecade1920, booksubjectgeometry, bookyear192
