128 KNOT tv, r o are linked, and yet the three are inseparably fastened together. The rest of Tait s paper deals chiefly with numerical characteristics of knots, such as their " knottiness," " be- knottedness," and " knotfulness." He also shows that any knot, however complex, can be fully represented by three closed plane curves, none of which has double points, and no two of which intersect. It may be stated here that the notion of beknottedness is founded on a remark of Gauss, who in 1833 considered the problem of the number of inter- linkings of two closed circuits, and expressed it by the electro-lynamic measure of the work required to carry a unit magnetic pole round one of the interlinked curves, while a unit electric current is kept circulating in the other. This original suggestion has been developed at considerable length by Boeddicker (Erweiterung der Gauss 1 schen Theorie der Verschlingungen, Stuttgart, 187G). This author treats also of the connexion of knots with Eie- mann s surfaces. It is to be noticed that, although every knot in which the crossings are alternately F g- 6. over and under is irreducible, the converse is not generally true. This is obvious at once from fig. 6, which is merely the three-crossing knot with a doubled string what Listing calls "paradromic." Klein, in the Matliematisclie Annalcn, ix. 478, has proved the remarkable proposition that knots cannot exist in space of four dimensions. SAILORS KNOTS. The knots used by sailors are of many kinds. The following are the most useful : Overhand Knot (fig. 7). Take the end a of the rope round the end b. Reef Knot (figs. 8, 9). Form an overhand knot as above. Then take the end a over the end b and through the bight. If the end a 6 Fig. 8. were taken under the end b a granny would be formed. This knot is so named from being used in tying the reef-points of a sail. Bowline (figs. 10-12). Lay the end a of a rope over the standing part b. Form with b a bight c over a. Take a round behind b and Fig. 11. down through the bight c. This is a most useful knot employed to form a loop which will not slip. Half Hitch (fig. 13). Pass the end a of the rope round the standing part b and through the bight. Clove Hitch (figs. 14, 15). Pass the end a round a spar and cross it over b. Pass it round the spar again and put the end a through the second bight. Blackwall Hitch (fig. 16). Form a bight at the end of a rope, and put the hook of a tackle through the bight so that the end of the rope may be jammed between the standing part and the back of the hook. Timber flitch (fig. 17). Take the end a of a rope round a spar, then round the standing part b, then several times round its own part c. Fisherman s Send (fig. 18). Take two turns round a spar, then a half hitch round the standing part and between the spar and tho turns, lastly a half hitch round the standing part. L8. Carried Bend (fig. 19). Lay the end of one rope over its own standing part so as to form a bight. Put the end of the other rope through this bight, under .the standing part, over the end beyond the bight, under the standing part beyond the bight, and down through the bight over its own standing part. Sheet Bend (fig. 20). Pass the end of one rope through the bight of another, round both parts of the other, and under its own standing part. Fig. 19. Fig. 20. Fig. 21. Single Wall Knot (fig. 21). Unlay the end of a rope, and with the strand a form a bight. Take the next strand b round the end of a. Take the last strand c round the end of b and through the bight made by a. Haul the ends taut. Single Wall Crowned (fig. 22). Form a single wall, and lay one of the ends, a, over the knot. Lay b over a, and c over b and through the bight of a. Haul the ends taut. Fig. 23. Fig. 24. Fig. 25. Double Wall and Double Crown (fig. 23). Form a single wall crowned ; then let the ends follow their own parts round until all the parts appear double. Put the ends down through the knot. Matthew Walker (figs. 24, 25). Unlay the end of a rope. Take the first strand round the rope and through its own bight ; the
