Knot

Overhand Knot (fig. 1).—Used at the end of ropes to prevent their unreeving and as the commencement of other knots. Take the end a round the end b.

Figure-of-Eight Knot (fig. 2).—Used only to prevent ropes from unreeving; it forms a large knob.

Fig. 3.	Fig. 4.

Reef Knot (figs. 3, 4).—Form an overhand knot as above. Then take the end a over the end b and through the bight. If the end a 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.

Fig. 5.	Fig. 6.	Fig. 7.

Bowline (figs. 5-7).—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 down through the bight c. This is a most useful knot employed to form a loop which will not slip. Running bowlines are formed by making a bowline round its own standing part above b. It is the most common and convenient temporary running noose.

Fig. 8.	Fig. 9.	Fig. 10.

Bowline on a Bight (figs. 8, 9).—The first part is made similar to the above with the double part of the rope; then the bight a is pulled through sufficiently to allow it to be bent over past d and come up in the position shown in fig. 9. It makes a more comfortable sling for a man than a single bight.

Half-Hitch (fig. 10).—Pass the end a of the rope round the standing part b and through the bight.

Two Half-Hitches (fig. 11).—The half-hitch repeated; this is commonly used, and is capable of resisting to the full strength of the rope. A stop from a to the standing part will prevent it jamming.

Fig. 11.	Fig. 12.	Fig. 13.

Clove Hitch (figs. 12, 13).—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. 14).—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.

Double Blackwall Hitch (fig. 15).—Pass the end a twice round the hook and under the standing part b at the last cross.

Fig. 14.	Fig. 15.	Fig. 16.	Fig. 17.

Cat’s-paw (fig. 16).—Twist up two parts of a lanyard in opposite directions and hook the tackle in the eyes i, i. A piece of wood should be placed between the parts at g. A large lanyard should be clove-hitched round a large toggle and a strap passed round it below the toggle.

Marling-spike Hitch (fig. 17).—Lay the end a over c; fold the loop over on the standing part b; then pass the marline-spike through, over both parts of the bight and under the part b. Used for tightening each turn of a seizing.

Fig. 18.	Fig. 19.	Fig. 20.

Fisherman’s Bend (fig. 18).—Take two turns round a spar, then a half-hitch round the standing part and between the spar and the turns, lastly a half-hitch round the standing part.

Studding-sail Halyard Bend (fig. 19).—Similar to the above, except that the end is tucked under the first round turn; this is more snug. A magnus hitch has two round turns and one on the other side of the standing part with the end through the bight.

Timber Hitch (fig. 20).—Take the end a of a rope round a spar, then round the standing part b, then several times round its own part c, against the lay of the rope.

Fig. 21.

Carrick Bend (fig. 21).—Lay the end of one hawser over its own part to form a bight as e´, b; pass the end of another hawser up through that bight near b, going out over the first end at c, crossing under the first long part and over its end at d, then under both long parts, forming the loops, and above the first short part at b, terminating at the end e´´, in the opposite direction vertically and horizontally to the other end. The ends should be securely stopped to their respective standing parts, and also a stop put on the becket or extreme end to prevent it catching a pipe or chock; in that form this is the best quick means of uniting two large hawsers, since they cannot jam. When large hawsers have to work through small pipes, good security may be obtained either by passing ten or twelve taut racking turns with a suitable strand and securing each end to a standing part of the hawser, or by taking half as many round turns taut, crossing the ends between the hawsers over the seizing and reef-knotting the ends. This should be repeated in three places and the extreme ends well stopped. Connecting hawsers by bowline knots is very objectionable, as the bend is large and the knots jam.

Sheet Bend (fig. 22).—Pass the end of one rope through the bight of another, round both parts of the other, and under its own standing part. Used for bending small sheets to the clews of sails, which present bights ready for the hitch. An ordinary net is composed of a series of sheet bends. A weaver’s knot is made like a sheet bend.

Single Wall Knot (fig. 23).—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. 24).—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. 22.	Fig. 23.	Fig. 24.

Double Wall and Double Crown (fig. 25).—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. 26, 27).—Unlay the end of a rope. Take the first strand round the rope and through its own bight; the second strand round the rope, through the bight of the first, and through its own bight; the third through all three bights. Haul the ends taut.

Fig. 25.	Fig. 26.	Fig. 27.	Fig. 28.

Inside Clinch (fig. 28).—The end is bent close round the standing part till it forms a circle and a half, when it is securely seized at a, b and c, thus making a running eye; when taut round anything it jams the end. It is used for securing hemp cables to anchors, the standing parts of topsail sheets, and for many other purposes. If the eye were formed outside the bight an outside clinch would be made, depending entirely on the seizings, but more ready for slipping.

Midshipman’s Hitch (fig. 29).—Take two round turns inside the bight, the same as a half-hitch repeated; stop up the end or let another half-hitch be taken or held by hand. Used for hooking a tackle for a temporary purpose.

Fig. 29.	Fig. 30.	Fig. 31.	Fig. 32.

Turk’s Head (fig. 30).—With fine line (very dry) make a clove hitch round the rope; cross the bights twice, passing an end the reverse way (up or down) each time; then keeping the whole spread flat, let each end follow its own part round and round till it is too tight to receive any more. Used as an ornament variously on side-ropes and foot-ropes of jibbooms. It may also be made with three ends, two formed by the same piece of line secured through the rope and one single piece. Form with them a diamond knot; then each end crossed over its neighbour follows its own part as above.

Spanish Windlass (fig. 31).—An iron bar and two marling-spikes are taken; two parts of a seizing are twisted like a cat’s-paw (fig. 16), passed round the bar, and hove round till sufficiently taut. In heaving shrouds together to form an eye two round turns are taken with a strand and the two ends hove upon. When a lever is placed between the parts of a long lashing or frapping and hove round, we have what is also called a Spanish windlass.

Slings (fig. 32).—This is simply the bight of a rope turned up over its own part; it is frequently made of chain, when a shackle (bow up) takes the place of the bight at s and another at y, connecting the two ends with the part which goes round the mast-head. Used to sling lower yards. For boat’s yards it should be a grummet with a thimble seized in at y. As the tendency of all yards is to cant forward with the weight of the sail, the part marked by an arrow should be the fore-side—easily illustrated by a round ruler and a piece of twine.

Sprit-Sail Sheet Knot (fig. 33).—This knot consists of a double wall and double crown made by the two ends, consequently with six strands, with the ends turned down. Used formerly in the clews of sails, now as an excellent stopper, a lashing or shackle being placed at s and a lanyard round the head at l.

Fig. 33.	Fig. 34.	Fig. 35.

Turning in a Dead-Eye Cutter-Stay fashion (fig. 34).—A bend is made in the stay or shroud round its own part and hove together with a bar and strand; two or three seizings diminishing in size (one round and one or two either round or flat) are hove on taut and snug, the end being at the side of the fellow part. The dead-eye is put in and the eye driven down with a commander.

Turning in a Dead-Eye end up (fig. 35).—The shroud is measured round the dead-eye and marked where a throat-seizing is hove on; the dead-eye is then forced into its place, or it may be put in first. The end beyond a is taken up taut and secured with a round seizing; higher still the end is secured by another seizing. As it is important that the lay should always be kept in the rope as much as possible, these eyes should be formed conformably, either right-handed or left-handed. It is easily seen which way a rope would naturally kink by putting a little extra twist into it. A shroud whose dead-eye is turned in end up will bear a fairer strain, but is more dependent on the seizings; the under turns of the throat are the first to break and the others the first to slip. With the cutter-stay fashion the standing part of the shroud gives way under the nip of the eye. A rope will afford the greatest resistance to strain when secured round large thimbles with a straight end and a sufficient number of flat or racking seizings. To splice shrouds round dead-eyes is objectionable on account of opening the strands and admitting water, thus hastening decay. In small vessels, especially yachts, it is admissible on the score of neatness; in that case a round seizing is placed between the dead-eye and the splice. The dead-eyes should be in diameter 1½ times the circumference of a hemp shroud and thrice that of wire; the lanyard should be half the nominal size of hemp and the same size as wire: thus, hemp-shroud 12 in., wire 6 in., dead-eye 18 in., lanyard 6 in.

Fig. 36.

Short Splice (fig. 36).—The most common description of splice is when a rope is lengthened by another of the same size, or nearly so. Fig. 36 represents a splice of this kind: the strands have been unlaid, married and passed through with the assistance of a marling-spike, over one strand and under the next, twice each way. The ends are then cut off close. To render the splice neater the strands should have been halved before turning them in a second time, the upper half of each strand only being turned in; then all are cut off smooth. Eye Splice.—Unlay the strands and place them upon the same rope spread at such a distance as to give the size of the eye; enter the centre strand (unlaid) under a strand of the rope (as above), and the other two in a similar manner on their respective sides of the first; taper each end and pass them through again. If neatness is desired, reduce the ends and pass them through once more; cut off smooth and serve the part disturbed tightly with suitable hard line. Uses too numerous to mention. Cut Splice.—Made in a similar manner to an eye splice, but of two pieces of rope, therefore with two splices. Used for mast-head pendants, jib-guys, breast backstays, and even odd shrouds, to keep the eyes of the rigging lower by one part. It is not so strong as two separate eyes. Horseshoe Splice.—Made similar to the above, but one part much shorter than the other, or another piece of rope is spliced across an eye, forming a horseshoe with two long legs. Used for back-ropes on dolphin striker, back stays (one on each side) and cutter’s runner pendants. Long Splice.—The strands must be unlaid about three times as much as for a short splice and married—care being taken to preserve the lay or shape of each. Unlay one of the strands still further and follow up the vacant space with the corresponding strand of the other part, fitting it firmly into the rope till only a few inches remain. Treat the other side in a similar manner. There will then appear two long strands in the centre and a long and a short one on each side. The splice is practically divided into three distinct parts; at each the strands are divided and the corresponding halves knotted (as shown on the top of fig. 38) and turned in twice. The half strand may, if desired, be still further reduced before the halves are turned in for the second time. This and all other splices should be well stretched and hammered into shape before the ends are cut off. The long splice alone is adapted to running ropes.

Fig. 37.

Shroud Knot (fig. 37).—Pass a stop at such distance from each end of the broken shroud as to afford sufficient length of strands, when it is unlaid, to form a single wall knot on each side after the parts have been married; it will then appear as represented in the figure, the strands having been well tarred and hove taut separately. The part a provides the knot on the opposite side and the ends b, b; the part c provides the knot and the ends d, d. After the knot has been well stretched the ends are tapered, laid smoothly between the strands of the shroud, and firmly served over. This knot is used when shrouds or stays are broken. French Shroud Knot.—Marry the parts with a similar amount of and as before; stop one set of strands taut up on the shroud (to keep the parts together), and turn the ends back on their own part, forming bights. Make a single wall knot with the other three strands round the said bights and shroud; haul the knot taut first and stretch the whole; then heave down the bights close: it will look like the ordinary shroud knot. It is very liable to slip. If the ends by which the wall knot is made after being hove were passed through the bights, it would make the knot stronger. The ends would be tapered and served.

Fig. 38.

Flemish Eye (fig. 38).—Secure a spar or toggle twice the circumference of the rope intended to be rove through the eye; unlay the rope which is to form the eye about three times its circumference, at which part place a strong whipping. Point the rope vertically under the eye, and bind it taut up by the core if it is four-stranded rope, otherwise by a few yarns. While doing so arrange six or twelve pieces of spun-yarn at equal distances on the wood and exactly halve the number of yarns that have been unlaid. If it is a small rope, select two or three yarns from each side near the centre; cross them over the top at a, and half-knot them tightly. So continue till all are expended and drawn down tightly on the opposite side to that from which they came, being thoroughly intermixed. Tie the pieces of spun-yarn which were placed under the eye tightly round various parts, to keep the eye in shape when taken off the spar, till they are replaced by turns of marline hove on as taut as possible, the hitches forming a central line outside the eye. Heave on a good seizing of spun-yarn close below the spar, and another between six and twelve inches below the first; it may then be parcelled and served; the eye is served over twice, and well tarred each time. As large ropes are composed of so many yarns, a greater number must be knotted over the toggle each time; a 4-in. rope has 132 yarns, which would require 22 knottings of six each time; a 10-in. rope has 834 yarns, therefore, if ten are taken from each side every time, about twice that number of hitches will be required; sometimes only half the yarns are hitched, the others being merely passed over. The chief use of these eyes has been to form the collars of stays, the whole stay in each case having to be rove through it—a very inconvenient device. It is almost superseded for that purpose by a leg spliced in the stay and lashing eyes abaft the mast, for which it is commonly used at present. This eye is not always called by the same name, but the weight of evidence is in favour of calling it a Flemish eye. Ropemaker’s Eye, which also has alternative names, is formed by taking out of a rope one strand longer by 6 in. or a foot than the required eye, then placing the ends of the two strands a similar distance below the disturbance of the one strand, that is, at the size of the eye; the single strand is led back through the vacant space it left till it arrives at the neck of the eye, with a similar length of spare end to the other two strands. They are all seized together, scraped, tapered, marled and served. The principal merit is neatness. Mouse on a Stay.—Formed by turns of coarse spun-yarn hove taut round the stay, over parcelling at the requisite distance from the eye to form the collar; assistance is given by a padding of short yarns distributed equally round the rope, which, after being firmly secured, especially at what is to be the under part, are turned back over the first layer and seized down again, thus making a shoulder; sometimes it is formed with parcelling only. In either case it is finished by marling, followed by serving or grafting. The use is to prevent the Flemish eye in the end of the stay from slipping up any farther.

Rolling Hitch (fig. 39).—Two round turns are taken round a spar or large rope in the direction in which it is to be hauled and one half-hitch on the other side of the hauling part. This is very useful, as it can be put on and off quickly.

Fig. 39.	Fig. 40.

Round Seizing (fig. 40).—So named when the rope it secures does not cross another and there are three sets of turns. The size of the seizing line is about one-sixth (nominal) that of the ropes to be secured, but varies according to the number of turns to be taken. An eye is spliced in the line and the end rove through it, embracing both parts. If either part is to be spread open, commence farthest from that part; place tarred canvas under the seizing; pass the line round as many times (with much slack) as it is intended to have under-turns; and pass the end back through them all and through the eye. Secure the eye from rendering round by the ends of its splice; heave the turns on with a marling-spike (see fig. 17), perhaps seven or nine; haul the end through taut, and commence again the riding turns in the hollows of the first. If the end is not taken back through the eye, but pushed up between the last two turns (as is sometimes recommended), the riders must be passed the opposite way in order to follow the direction of the under-turns, which are always one more in number than the riders. When the riders are complete, the end is forced between the last lower turns and two cross turns are taken, the end coming up where it went down, when a wall knot is made with the strands and the ends cut close; or the end may be taken once round the shroud. Throat Seizing.—Two ropes or parts of ropes are laid on each other parallel and receive a seizing similar to that shown in figure 35—that is with upper and riding but no cross turns. As the two parts of rope are intended to turn up at right angles to the direction in which they were secured, the seizing should be of stouter line and short, not exceeding seven lower and six riding turns. The end is better secured with a turn round the standing part. Used for turning in dead-eyes and variously. Flat Seizing.—Commenced similarly to the above, but it has neither riding nor cross turns.

Fig. 41.	Fig. 42.	Fig. 43.

Racking Seizing (fig. 41).—A running eye having been spliced round one part of the rope, the line is passed entirely round the other part, crossed back round the first part, and so on for ten to twenty turns, according to the expected strain, every turn being hove as tight as possible; after which round turns are passed to fill the spaces at the back of each rope, by taking the end a over both parts into the hollow at b, returning at c, and going over to d. When it reaches e a turn may be taken round that rope only, the end rove under it, and a half-hitch taken, which will form a clove-hitch; knot the end and cut it close. When the shrouds are wire (which is half the size of hemp) and the end turned up round a dead-eye of any kind, wire seizings are preferable. It appears very undesirable to have wire rigging combined with plates or screws for setting it up, as in case of accident—such as that of the mast going over the side, a shot or collision breaking the ironwork—the seamen are powerless.

Diamond Knot (figs. 42, 43).—The rope must be unlaid as far as the centre if the knot is required there, and the strands handled with great care to keep the lay in them. Three bights are turned up as in fig. 42, and the end of a is taken over b and up the bight c. The end of b is taken over c and up through a. The end c is taken over a and through b. When hauled taut and the strands are laid up again it will appear as in fig. 43. Any number of knots may be made on the same rope. They were used on man-ropes, the foot-ropes on the jibboom, and similar places, where it was necessary to give a good hold for the hands or feet. Turk’s heads are now generally used. Double Diamond.—Made by the ends of a single diamond following their own part till the knot is repeated. Used at the upper end of a side rope as an ornamental stopper-knot.

Stropping-Blocks.—There are various modes of securing blocks to ropes; the most simple is to splice an eye at the end of the rope a little longer than the block and pass a round seizing to keep it in place; such is the case with jib-pendants. As a general rule, the parts of a strop combined should possess greater strength than the parts of the fall which act against it. The shell of an ordinary block should be about three times the circumference of the rope which is to reeve through it, as a 9-in. block for a 3-in. rope; but small ropes require larger blocks in proportion, as a 4-in. block for a 1-in. rope. When the work to be done is very important the blocks are much larger: brace-blocks are more than five times the nominal size of the brace. Leading-blocks and sheaves in racks are generally smaller than the blocks through which the ropes pass farther away, which appears to be a mistake, as more power is lost by friction. A clump-block should be double the nominal size of the rope. A single strop may be made by joining the ends of a rope of sufficient length to go round the block and thimble by a common short splice, which rests on the crown of the block (the opposite end to the thimble) and is stretched into place by a jigger; a strand is then passed twice round the space between the block and the thimble and hove taut by a Spanish windlass to cramp the parts together ready for the reception of a small round seizing. The cramping or pinching into shape is sometimes done by machinery invented by a rigger in Portsmouth dockyard. The strop may be made the required length by a long splice, but it would not possess any advantage.

Fig. 44.	Fig. 45.

Grummet-Strop (fig. 44).—Made by unlaying a piece of rope of the desired size about a foot more than three times the length required for the strop. Place the centre of the rope round the block and thimble; mark with chalk where the parts cross; take one strand out of the rope; bring the two chalk marks together; and cross the strand in the lay on both sides, continuing round and round till the two ends meet the third time; they are then halved, and the upper halves half-knotted and passed over and under the next strands, exactly as one part of a long splice. A piece of worn or well-stretched rope will better retain its shape, upon which success entirely depends. The object is neatness, and if three or multiples of three strops are to be made it is economical.

Double Strop (fig. 45).—Made with one piece of rope, the splice being brought as usual to the crown of the block t, the bights fitting into scores some inches apart, converging to the upper part, above which the thimble receives the bights a, a; and the four parts of the strop are secured at s, s by a round seizing doubly crossed. If the block be not then on the right slew (the shell horizontal or vertical) a union thimble is used with another strop, which produces the desired effect; thus the fore and main brace-blocks, being very large and thin, are required (for appearance) to lie horizontally; a single strop round the yard vertically has a union thimble between it and the double strop round the block. The double strop is used for large blocks; it gives more support to the shell than the single strop and admits of smaller rope being used. Wire rope is much used for block-strops; the fitting is similar. Metal blocks are also used in fixed positions; durability is their chief recommendation. Great care should be taken that they do not chafe the ropes which pass by them as well as those which reeve through.

Selvagee Strop.—Twine, rope-yarn or rope is warped round two or more pegs placed at the desired distance apart, till it assumes the requisite size and strength; the two ends are then knotted or spliced. Temporary firm seizings are applied in several places to bind the parts together before the rope or twine is removed from the pegs, after which it is marled with suitable material. A large strop should be warped round four or six pegs in order to give it the shape in which it is to be used. This description of strop is much stronger and more supple than rope of similar size. Twine strops (covered with duck) are used for boats’ blocks and in similar places requiring neatness. Rope-yarn and spun-yarn strops are used for attaching luff-tackles to shrouds and for many similar purposes. To bring to a shroud or hawser, the centre of the strop is passed round the rope and each part crossed three or four times before hooking the “luff”; a spun-yarn stop above the centre will prevent slipping and is very necessary with wire rope. As an instance of a large selvagee block-strop being used—when the “Melville” was hove down at Chusan (China), the main-purchase-block was double stropped with a selvagee containing 28 parts of 3-in. rope; that would produce 112 parts in the neck, equal to a breaking strain of 280 tons, which is more than four parts of a 19-in cable. The estimated strain it bore was 80 tons.

Stoppers for ordinary running ropes are made by splicing a piece of rope to a bolt or to a hook and thimble, unlaying 3 or 4 ft., tapering it by cutting away some of the yarns, and marling it down securely, with a good whipping also on the end. It is used by taking a half-hitch round the rope which is to be hauled upon, dogging the end up in the lay and holding it by hand. The rope can come through it when hauled, but cannot go back.

Whipping and Pointing.—The end of every working rope should at least be whipped to prevent it fagging out; in ships of war and yachts they are invariably pointed. Whipping is done by placing the end of a piece of twine or knittle-stuff on a rope about an inch from the end, taking three or four turns taut over it (working towards the end); the twine is then laid on the rope again lengthways contrary to the first, leaving a slack bight of twine; and taut turns are repeatedly passed round the rope, over the first end and over the bight, till there are in all six to ten turns; then haul the bight taut through between the turns and cut it close. To point a rope, place a good whipping a few inches from the end, according to size; open out the end entirely; select all the outer yarns and twist them into knittles either singly or two or three together; scrape down and taper the central part, marling it firmly. Turn every alternate knittle and secure the remainder down by a turn of twine or a smooth yarn hitched close up, which acts as the weft in weaving. The knittles are then reversed and another turn of the weft taken, and this is continued till far enough to look well. At the last turn the ends of the knittles which are laid back are led forward over and under the weft and hauled through tightly, making it present a circle of small bights, level with which the core is cut off smoothly. Hawsers and large ropes have a becket formed in their ends during the process of pointing. A piece of 1 to 1½ in. rope about 1½ to 2 ft. long is spliced into the core by each end while it is open: from four to seven yarns (equal to a strand) are taken at a time and twisted up; open the ends of the becket only sufficient to marry them close in; turn in the twisted yarns between the strands (as splicing) three times, and stop it above and below. Both ends are treated alike; when the pointing is completed a loop a few inches in length will protrude from the end of the rope, which is very useful for reeving it. A hauling line or reeving line should only be rove through the becket as a fair lead. Grafting is very similar to pointing, and frequently done the whole length of a rope, as a side-rope. Pieces of white line more than double the length of the rope, sufficient in number to encircle it, are made up in hanks called foxes; the centre of each is made fast by twine and the weaving process continued as in pointing. Block-strops are sometimes so covered; but, as it causes decay, a small wove mat which can be taken off occasionally is preferable.

Fig. 46.

Sheep-Shank (fig. 46).—Formed by making a long bight in a topgallant back-stay, or any rope which it is desirable to shorten, and taking a half-hitch near each bend, as at a, a. Rope-yarn stops at b, b are desirable to keep it in place till the strain is brought on it. Wire rope cannot be so treated, and it is injurious to hemp rope that is large and stiff.

Knotting Yarns (fig. 47).—This operation becomes necessary when, a comparatively short piece of junk is to be made into spun-yarn, or large rope into small, which is called twice laid. The end of each yarn is divided, rubbed smooth and married (as for splicing). Two of the divided parts, as c, c and d, d, are passed in opposite directions round all the other parts and knotted. The ends e and f remain passive. The figure is drawn open, but the forks of A and B should be pressed close together, the knot hauled taut and the ends cut off.

Fig. 47.	Fig. 48.

Butt Slings (fig. 48).—Made of 4-in. rope, each pair being 26 ft. in length, with an eye spliced in one end, through which the other is rove before being placed over one end of the cask; the rope is then passed round the opposite side of the cask and two half-hitches made with the end, forming another running eye, both of which are beaten down taut as the tackle receives the weight. Slings for smaller casks requiring care should be of this description, though of smaller rope, as the cask cannot possibly slip out. Bale Slings are made by splicing the ends of about 3 fathoms of 3-in. rope together, which then looks like a long strop, similar to the double strop represented in fig. 45—the bights t being placed under the cask or bale and one of the bights a, a rove through the other and attached to the whip or tackle.

For a complete treatise on the subject the reader may be referred to The Book of Knots, being a Complete Treatise on the Art of Cordage, illustrated by 172 Diagrams, showing the Manner of making every Knot, Tie and Splice, by Tom Bowling (London, 1890).

Tait starts with the almost self-evident proposition that, if any plane closed curve have double points only, in passing continuously along the curve from one of these to the same again an even number of double points has been passed through. Hence the crossings may be taken alternately over and under. On this he bases a scheme for the representation of knots of every kind, and employs it to find all the distinct forms of knots which have, in their simplest projections, 3, 4, 5, 6 and 7 crossings only. Their numbers are shown to be 1, 1, 2, 4 and 8. The unique knot of three crossings has been already given as drawn by Listing. The unique knot of four crossings merits a few words, because its properties lead to a very singular conclusion. It can be deformed into any of the four forms—figs. 51 and 52 and their perversions. Knots which can be deformed into their own perversion Tait calls “amphicheiral” (from the Greek ἀμφί, on both sides, around, χείρ, hand), and he has shown that there is at least one knot of this kind for every even number of crossings. He shows also that “links” (in which two endless physical lines are linked together) possess a similar property; and he then points out that there is a third mode of making a complex figure of endless physical lines, without either knotting or linking. This may be called “lacing” or “locking.” Its nature is obvious from fig. 53, in which it will be seen that no one of the three lines is knotted, no two 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,” “beknottedness” 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-dynamic 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 Otto Boeddicker (Erweiterung der Gauss’schen Theorie der Verschlingungen (Stuttgart, 1876). This author treats also of the connexion of knots with Riemann’s surfaces.

Fig. 53.	Fig. 54.

It is to be noticed that, although every knot in which the crossings are alternately over and under is irreducible, the converse is not generally true. This is obvious at once from fig. 54, which is merely the three-crossing knot with a doubled string—what Listing calls “paradromic.”

Christian Felix Klein, in the Mathematische Annalen, ix. 478, has proved the remarkable proposition that knots cannot exist in space of four dimensions.