Distance Off by Similar Triangles

When two triangles have internal angles of the same value they are called similar. One characteristic of similar triangles is that although their sides have different lengths, they are proportionate to each other. The ratio of any two sides in the first triangle equals the ratio of the same sides in the second triangle. If you know the length of any three of these four sides, finding the value of the fourth is straightforward. You would like to know the distance between you and the lighthouse...

The Cross Staff

It consists of a staff, usually about 36 inches (91.5 centimetres) long, and a shorter crosspiece called the transom, which slides up and down the staff to measure altitudes. The earliest known description comes from Persia in the 11th century. The first western description is by Levi ben Gerson in1342 but it was 1485 before the German mathematician and navigator, Martin Behaim, took it to sea when he sailed down the east African coast with the Portuguese explorer, Diego Cao. To make a cross...

Charts for Coastal Passages

Blank charts are fine in the middle of nowhere but when you are approaching the coast, some detail ashore will show you how close your plot puts you to land and whether or not the course you have chosen will bring you to harbour. 19.2 Drawing a Plotting Sheet Step 1 19.3 Drawing a Plotting Sheet Step 2 19.4 Drawing a Plotting Sheet Step 3 19.5 Drawing a Plotting Sheet Step 4 There is a lot of information you can add to your chart. You can begin by inserting your waypoints. Next, you can try...

The Astrolabe

Marine astrolabes were about four inches in diameter with a scale marked in degrees, and heavy to make them steady. They were horribly inaccurate, partly because accuracy is a function of size but also because it took three men to use it one person held it, another took the sight, and a third read it. When Vasco da Gama, on his way towards the Cape of Good Hope, reached St Helena Bay, he went ashore with a large wooden astrolabe because he had been unable to take a sight on the voyage south....

The Traverse Board

It is important to keep a record of course and speed at intervals during each watch. For centuries seamen used a traverse board (see Chapter 5, Figure 5.8). The upper half of a traverse board is a compass rose marked out in 32 points. Eight holes radiate out from the centre and are drilled along each point. In the centre are eight pegs, each on a length of string. Starting from the centre, each hole represents half an hour's sailing and the eight holes on each point made up one four-hour watch....

North by Your Watch

If you have a watch with hour and minute hands and you are between latitudes 35-55 , you can always find an approximation of north, as shown in Figures 7.5 and 7.5b. 7.5 North by your watch in the northern hemisphere 7.5b North by your watch in the southern hemisphere 7.5b North by your watch in the southern hemisphere

Estimated Positions

An estimated position (EP) is your best position short of a fix. To turn a DR into an EP it is necessary to make allowances for tidal stream, currents, leeway, and the pricking of your thumbs, as shown in Figure 11.5. Working out the expected rate and direction of the tidal stream should be part of your normal passage planning and it is a good idea to note this down rather than store it electronically. The actual tidal stream may be slightly different from that predicted, but an estimate based...

Longitude by Time

The kindest way of describing finding longitude using the difference between local and Greenwich is 'approximate'. Small errors in time can produce surprisingly large errors in longitude. Still, it may be better than no longitude at all. In 24 hours the earth turns through 360 . This means that in every hour it turns through 15 or that a difference of one degree of longitude equals four minutes of time (see Figure 17.2). Before leaving port, a ship's chronometer would have been set to GMT. Each...

Distance Off by Echo Ranging

In fog, in inshore, cliffy waters you can try echo ranging for the distance off. This uses the time delay between the production of the sound and the arrival of its echo. This assumes that the speed of sound is a constant. It actually varies with the type of gas The sound travels from the boat, hits the cliff and the echo returns. Remember this sound is omnidirectional. If you are in a fiord you will hear an echo from both sides, probably at slightly different times. The sound travels from the...

Celestial Coordinates

Astronomical Declination Diagram

Just like the earth, the celestial sphere has its system of coordinates that define position. The north-south position of anything on the celestial sphere is called its declination, but as the North and South Celestial Poles are directly above the earth's poles and the celestial equator parallels the earth's equator, (see Figure 8.1) the declination of any celestial body is the same as its latitude if it were on earth, (see Figure 8.3). The starting point for measuring the sun's east-west...

Distance Off by Winking

Eye Thumb Distance

You can make a fair estimate of your distance off using the Wink Method (see Figure 12.6). This needs no instruments and little arithmetic and there is no maths using the appearance of features in judging distance off which can be surprisingly accurate (check what to look for in Figure 12.7). Hold your arm out. Place the edge of your finger at one end of your known distance. Close one eye. Open your closed eye. . J Your finger appears to 'jump' sideways and in this case jumps to almost the...

Clearing Distance by Similar Triangles

You can use similar triangles for a clearing distance. Suppose you wish to stay one mile off a light 41 metres high. AC, which is your hand to eye distance 0.57 metre. AE, which is the distance off 1 nautical mile 1852 metres. DE, which is the height of the feature 41 metres. You want to know BC, which is the height on your ruler. By re-arranging the equation BC DE x AC AE BC 41 x .57 1852 Clearing Distance BC 0.013 0.013m 1.3 centimetres. Now you know that if the reading is less than1.3...

Making a Magnetic Compass

With the materials found on most boats, it is possible to magnetise a needle and float it on a dish of water to discover magnetic north (see Figures 6.1 to 6.3.) If you feel that one needle is not enough to swing the straw then use two or three needles. Be sure that all the north poles point in the same direction (see Figure 6.4). A fair estimate of variation can be found by pointing the bows at Polaris and noting where the compass needle points, or by holding a thin wooden or plastic rod,...

Home Made Compass Card

Celestial Bodies Human Body

The compass rose cut out from a chart makes a compass card. Tape it to a board, extend its radii and place it under your compass bowl. If you do not have a spare compass rose, or a protractor to draw one, fold a square of paper (see Figure 6.5 and Figure 6.5b), to get eight compass points. Insert If you have no protractor you can make a compass card by folding a square of paper to find these eight principal points. All the other points have to be inserted by eye. Fortunately the human eye is...

Map Navigators

Map navigators see the world around them as a whole. Their base is still used for finding direction and describing position but they also see features in relation to each other and frequently use a feature or the sun, stars, or the wind to orient themselves, even though their mental map remains orientated to their base. If they look at the midday sun and say 'There's north,' this is immediately followed by 'and that's the way Route navigators know which way to turn at a junction and which exit...

Plane sailing

Plane Sailing and Passage Planning Traverse Tables Courses For centuries, navigators thumbed through traverse tables using their course and distance sailed to find their latitude and longitude. They called this their 'daily work' (see Figure 18.1). This is 'plane' or 'raverse' sailing, and is based on the angles of a plane, right-angled triangle. Back in the 14th century someone, knowing that trigonometry is not a seaman's strongest suit, worked out all the possible answers to all possible...

The Compass Rose

For centuries sailors found direction by the wind and often named winds by destination. Pliny wrote of sailing the 50 miles from 'Carpathus to Rhodes with Africus', Africus being the wind. The four cardinal winds were divided into 32 half and quarter winds, giving a course to within one point (11.25 ). In the early 13th century, wind roses appeared on Portolan charts, with rhumb lines springing from wind roses in the centre, or around the edge of the chart. Using winds for naming direction is...

Geographic and Nominal Range of Lights

Using the range of a light as a means of judging distance off depends on which of its four ranges you are using (see Figure 12.2). A light's geographical range is the distance it can be seen in conditions of perfect visibility and usually equates to its dipping distance. Its visual range is the distance it can be made out against its background. This varies with atmospheric conditions and sometimes is increased by using binoculars. A light's luminous range is the distance it can be seen...

The Quadrant

Do not confuse this with the forerunner of the sextant. It is a 90 protractor with a plumb line hanging from its vertex. Polaris is sighted along one edge and the angle where the line cuts the scale is read. As sailors were more interested in the latitude of a place rather than latitude as a number, some early quadrants replaced the degree markings with the names of ports. The advantage of the quadrant is that you do not need to see the horizon. Most marine quadrants used a radius of 10-12...

The Star Compass

The Polynesians developed a star compass. Look round your horizon and note the position where stars rise. These positions become your compass points. With luck you will be able to find a convoy of stars popping up one after the other at each compass point so that as one star climbs too high to use for direction, the next comes over the horizon. Knowing the declination of stars we can produce a star compass in minutes. Take Betelgeuse, a bright star in the constellation of Orion. It has a...

The Sun Shadow Board

The Vikings used a sun shadow board to find latitude. It was nothing more than a small wooden disc with a number of concentric circles representing different latitudes and a gnomon whose height could be adjusted for the time of year. At noon the sun shadow board would be placed in a tub of water with the gnomon set to the correct date. If the shadow it cast at noon was beyond the desired latitude they were too far north, and if it did not reach it, then they were too far south. SUSPENSION POINT...

Waves in Soundings

On a continental shelf wavelength shortens and the waves steepen. Closer to land the colour of the sea may change and rubbish may appear in the water. If you are approaching a steep-to coast with no beach to absorb the energy of breaking waves, then when a swell hits the coast it is reflected back out to sea (see Figure 20.3 and Figure 20.4). This creates a cross swell which may be detectable some distance offshore. Before acting on this you must be sure that there is no other possible source...

Making a Nocturnal

They first appeared in the 13th century, and remained in use into the 19th century. Nocturnals came in three parts a pointer, and separate date and hour discs rotating round a common centre. Using one involved sighting Polaris through a hole in the rivet or bolt holding the components together. A competent navigator would expect to find the time to within 15 minutes. A simple nocturnal can be made from two discs. One is a clock face with the months round its edge...

S

Sailing Directions 14 32 Scale 3 19 20 Scale, Charts 162 Scale, Cross Staff 123 Scale, Quadrant 121 Scale, Astrolabe 114 Semi-Diameter 128 130 136 Sextant 2 3 25 104 107 120 136 147 Sextant Corrections 128 Shadow Astrolabe 126 130 Shadow Astrolabe 114 116 fig 126 128 130 Timed Log Line 94 Tobias Mayer 147 Traverse Board 25 33 fig 97 Traverse Tables 13 25 152 et seq Universal Coordinated Time 78 Universal Time 78 Utima Thule 11 Variation 13 36 71 80 Vasco De Gama 11 114 Vega 63 Viking Sun...

Time from the Sun The Equation of Time

Solar time is based upon the apparent solar day, which is the time between two successive local noons at the same longitude. The length of the solar day varies, partly because the earth's orbit is an ellipse, and also because the earth's axis is tilted to the celestial equator. The mean solar day is fixed at 24 hours, but the actual solar day can be nearly 22 seconds shorter or almost 29 seconds longer. These differences can build up to around 17 minutes early or 14 minutes late, (see Figure...

Half Latitude Rule

The half-latitude rule is another way of finding east or west. Stars with a northern declination will rise north of due east and set north of due west. Stars with a southern declination will rise south of due east and set south of due west. When a star passes through your zenith it is at a height equal to your latitude and has moved away from due west or east by an angle equal to half your latitude. Outside the polar regions this is accurate to around five degrees. One advantage of the...

Latitude from Circumpolar Stars

Circumpolar stars are stars that stay above your horizon and they can be used to find your latitude (shown in Figure 16.3). Polaris is behind the clouds all night. (And Dubhe is not ) Add altitudes together and divide by two for an answer which is a shade over 55 but there are a couple of catches. Sights must be 12 hours apart. Possible in the tropics but not in temperate latitudes during summer. Secondly, during that time you mill have changed position so any answer will be approximate even...

Deliberate Error

Direct reckoning (DR) positions are almost certainly wrong, but how wrong When approaching a featureless coast, or a hazard such as a reef, introduce an error of your own choosing and steer a course which avoids the danger, or tells you which way to turn when you sight land. The Dutch coastline is mostly featureless dyke. Sailing from Lowestoft or Harwich towards Ijmuiden you could be almost anywhere, until you spot the harbour piers. Select a landfall a few miles south or north, so that when...

Combining the Corrections

Celestial Body Zenith Distance

Starting with your measured angle, usually called Hs, which stands for Height Sextant, applying these corrections will give your observed or corrected altitude, usually called Height Observed or Ho. Ho Hs dip refraction + -semi-diameter (if using the sun). A meridian sight has a lot going for it. First, as long as you can see it and know its declination you can use any celestial body. Secondly, although it helps to know the time, this is not absolutely essential. You can estimate when a body is...

The Kamai

Making Kamal Latitude

Arab navigators used the Kamal (meaning Guide) which works on the same principles as the latitude hook. It was a small wooden board with a hole through its centre. A knotted line was passed through the hole. The spacing of the knots corresponded to the known latitude of different ports. The navigator put the knot for his destination between his teeth, and held the Kamal out with the bottom edge on the horizon. When Polaris touched the top of the board he was on the latitude of his destination....

Homemade sextants

Hands and fingers (see Figure 13.1) are one answer. Normally the angle subtended by a finger at an arm's length is taken as one degree but it varies from person to person. The Chinese called this measurement a chih and there were 224 chihs (1 chih 1.6 ) in a circle. Arabs called it an issabah. Nature places a limit on the size of angle you can measure au naturel. Unless you have abnormally large hands it is around 30-40 . A word of warning people go blind...

Greater than the Whole

Although this book deals with topics separately, the trick is in putting them together. The sum of the parts is greater than the whole. Sometimes an insignificant, almost overlooked and apparently irrelevant detail in the distant outfield completes the picture. The Crash Bag Navigator is a ravenous and omnivorous collector of data. Once cavemen developed a navigational methodology it was not long before this methodology became formalised with certificates of competence, and a range of gadgets...

Dipping Distance

A dipping distance is when you just catch sight of the very top of a feature. If you know the height of the feature then you can calculate the distance from it to its horizon, add on the distance to your horizon, and you have the dipping distance (see Figure 12.1). Dipping distances are a good example of when you may be working with the height of your eye in feet, and the height of the feature in metres.

Plane Sailing and Passage Planning Course to Steer

Parallel Sailing

Without a course to steer all your effort in working out direction from the stars, sun and moon, and how to steer by the seat of your pants, is wasted. You need to know what direction to aim the bows rather than pointing towards the horizon and declaring, 'Thataway' If you know the latitude and longitude of both your start point and your destination then working out the course to steer is easy (see Figure 18.2). DIFFERENCE IN LONGITUDE LonaitUd< e Sine A BC AC which means f If names instead...

An excellent art

Navigation is that excellent Art, which demonstrateth by infallible conclusions, how a sufficient Ship may be conducted the shortest good way from place to place, by Table and Travers. John Davis, The Seaman's Secrets, 1594 Electronics took its time killing traditional navigation. The first hint of its intentions was in 1906 when the Italians Bellini and Tosi found how to determine the direction from which radio signals were transmitted. By the end of the 1940s radio navigation had grown to...

Horizon Sights

The refraction for 0 is included in the above table in a situation where you have an accurate watch, tables and an almanac, but no sextant. A horizon sight is when a celestial body just touches the horizon. If you use the sun you take the sight when either the upper or lower limb reaches the horizon. A horizon sight has an apparent altitude of 0 and is worked out just like any other. Be warned. Horizon sights are famous for their inaccuracy, mostly because of the problems of allowing for...

Semi Diameter

The altitude of the sun should be measured from its centre, but it is difficult to decide exactly where its centre lies. Measure either from the bottom (lower limb) or top (upper limb) of the sun and apply a small correction. The sun's semi-diameter varies from a minimum of 15' 46 to a maximum of 16' 18 with a mean of 15' 59.6. So add 16' when you measure to the lower limb and subtract 16' if you measure to the upper limb. Semi-diameter applies only to sun sights but not sun sights taken by a...

Dead Reckoning Position

A dead reckoning (DR) position is calculated by using the course steered and speed through the water as shown in Figure 11.1. We usually either over or underestimate our boat speed. Over a few hours an error of a fraction of a knot puts your position out by miles. If you steer a good course and get the tides and leeway right then you will be on track but ahead or behind your estimated position (EP). You still pick up your landfall, albeit either earlier or later than expected. It helps to have...

Wavelength and Speed

The speed at which a wave travels depends on whether it is in deep or shallow water. Deep water is water deeper than half the wavelength. 20.1 Wave Height and Wind Force, Duration and Fetch 20.1 Wave Height and Wind Force, Duration and Fetch For deep-water waves the equation is WAVELENGTH 1.56 x wave period squared. The speed of a wave is found by WAVE SPEED 1.25 x square root of wavelength in metres. In shallow water, both wavelength and speed decrease, and height and steepness (wavelength...

Finding Polaris

To find Polaris, look for the Big Dipper and find the Dipper's ladle. Extend the line joining the two stars that make up the ladle across the sky for a distance equivalent to five times the depth of the ladle and you have Polaris (Figure 8.6). When you face Polaris you are facing true north. If Polaris is obscured, but you can see the Big Dipper and its Pointers, then hold a stick at arm's length and mark on it the distance between the Pointers. Mark this distance off five times from the tip of...

North from Polaris

Right now Polaris is less than 1 from the true pole and circles the Pole every 24 hours. Anything that points north to within one degree is fine for helming, but you will need to find the true pole if you are using Polaris for latitude, (see Chapter 16). 8.7 Finding Polaris in a Cloudy Sky Precession of the Equinoxes Because the earth's axis wobbles, Polaris has not always pointed north. Project this wobble out into space and every 28 500 years it makes one revolution. Stars on or very close to...

Corrections for Polaris

Right now Polaris has a declination of about 89 12'. This means that it circles the true pole at a radius of 48 nautical miles. As it is so far away this circle is invisible to the naked eye but it does mean that if you measure the altitude of Polaris and take it as your latitude, you could be almost one degree out. To find your true latitude by Polaris you must know where Polaris is on its daily circle. This is its hour angle and if you have no tables it can be found from the star clock. A...

Amplitudes

Amplitude Celestial

An amplitude is the bearing of a celestial body as it rises or sets, illustrated in Figure 7.2 and Figure 7. 2b. To calculate a body's amplitude you need to know Lacking tables you need a calculator that can handle simple trigonometry. The general formula is Amplitude sin (sin declination of body cos your latitude). Even if the declination and your latitude have different names, ie, one is north and the other is south, all the figures are taken as positive. Once you have the answer you will...

The Latitude Hook

The latitude hook was a favourite of the Polynesians. One piece of bamboo was twisted into a small loop. Another formed a crosspiece, sometimes called the pointer. The distance between the loop and the crosspiece varied with latitude. When held at arm's length with the crosspiece on the horizon and the Pole star in the centre of the loop you were on the latitude of your destination. There were different latitude hooks for different destinations. Making a latitude hook is easy. A piece of wire...

Calibrating the Cross Staff

Cross staffs came with three or four transoms of varying lengths, each measuring a different range of angles. The scale for each transom would be engraved along its own side of the staff. Calibrate your cross staff using the figures in Figure 13.10. If you have no means of measuring inches or centimetres, use five coins taped together as your base unit. Calibrating a cross staff makes it obvious that it is easiest to use for altitudes of between 30-60 . Line up the top edge of the cross staff...

Making a Sundial

A sundial is not sufficiently precise for navigation but it may be useful for watch keeping and similar time checks. There are many kinds of sundial. The simplest to make is an equatorial sundial instructions are in Figure 10.3 . The gnomon is set at an angle equal to your latitude. The hour lines are spaced every 15 round the face of the dial, which is at right angles to the gnomon and so in the plane of the equator. You will need A piece of stiff card about 25 cm long and slightly wider than...