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71 USING A CALCULATOR FOR NAVIGATION
283. DESIRABLE FEATURES WHEN CHOOSING A CALCULATOR FOR NAVIGATIONAL USE
An individual’s choice of calculator will depend on the amount the purchaser wishes to pay and on personal preference as to type of machine. A sophisticated programmable dedicated or advanced scientific calculator could cost many times more than an adequate scientific calculator.
The advantage of a basic machine is simplicity of operation and lower cost. The procedures necessary to perform the various types of calculation are easily learned with a little practice and once understood can be applied to work out any type of equation without the need for a list of keying instructions. The limiting factor in manual operation is not imposed by the calculator but by the ability of the operator to keep track of the calculations when keystrokes become numerous. For this reason key sequences normally have to be kept fairly short and lengthy programmes are best broken down into smaller parts.
With programmable calculators the number of keystrokes required is reduced and a considerable simplification in data entry and output can be achieved with the more sophisticated machines, with less risk of error. Whichever type of machine is chosen the following functions and facilities should be considered important for navigational work.
One of the most important requirements is the ability to handle trigonometric functions. Sine, cosine, tangent, and their inverses are essential to all but the simplest problems in navigation. Furthermore, the calculator intended for marine use should be able to evaluate trigonometric functions for all angles, not just those in the range of O° to 90°. Preferably it should have the powerful feature of polar to rectangular conversion described earlier in this chapter.
A second useful feature is memory, the ability to save several intermediary results for later use, without having to re-enter them. The more data memories a calculator has, the more useful it will be, but we recommend at least 4 for Astro navigation.
Another desirable feature on a scientific calculator to be used for navigation is sexagesimal to decimal conversion. Programmability is also useful the ability to retain and reproduce a series of keystrokes to, solve a particular type of problem. Having keystrokes stored in a calculators program memory is a benefit that can hardly be over-emphasised. Most calculators have a ‘non volatile’ or constant’ memory which draws so little current that a program is retained even when the calculator is switched off. The scientific calculator replaces tables of values. The programmable calculator looks up those values and then does meaningful calculations with them. Thus, the job of evaluating mathematical expressions is replaced with the more important one of evaluating the results of the calculations.
Data retention is important. Not all calculators retain data. The result of an earlier calculation may have to be re-entered. This can be annoying since the main cause of errors is the incorrect entry of data. A calculator with a ‘data entry’ program separate from the ‘computation’ program is highly desirably as it avoids this need to re-enter data. For example, for a set of star sights the GREENWICH DATE, H. of. E., sextant I.E. and the boat’s COURSE and SPEED would be common to all of them and should not have to be entered more than once.
The choice of logic system is largely a matter for personal preference. Navigators new to calculators will probably find the algebraic logic system easiest to use. Algebraic machines also tend to be cheaper than the alternative RPN models. On the other hand, there are many advocates of the Hewlett-Packard RPN logic/stack register system amongst experienced calculator users. This manufacturer has a reputation for high quality products and good after-sales service, important considerations when an expensive purchase is contemplated.
284. MNEMONICS OF FUNCTIONS ON A SCIENTIFIC CALCULATOR
The functions on a scientific calculator can be somewhat intimidating at first. The following explanation of the meaning of the most common mnemonics will take away some of the mystery of the keyboard of a scientific calculator. The numbers are located in a four or five row “numeric pad” which is largely self explanatory. This pad contains the four arithmetic operators, the decimal point key and either an key or an key. The remainder of the keys are for the manipulation of numbers and other mathematical operations, but some of these functions are accessed by pressing a shift key first.
The Table of Calculator Keyboard Mnemonics gives the symbol or mnemonic for commonly used calculator keys, its colloquial name and meaning, and a description of its use for Navigation.
285. DR POSITION AND COURSE AND DISTANCE USING A CALCULATOR
In plotting the run between sights, say Sun-run-Mer. Alt Sun, it will be required to find the DR Position after sailing a certain Course and Distance. This can be done either by formulae or using the P → R function if available. The following example shows the method where no P → R function is available.
4. EX. No.1 A boat in lat 17°20.0N, long 38°41.0W steers 320°T for 54 miles. Find her DR at the end of this run.
To find the d Lat: D Lat = distance x cos Course Here, multiply the distance (54) by the cosine of the Course.
The Course can be entered as either 320 or – 40 (N40°W).
This gives the result: d.Lat = 41′.36
To find Departure: Dep = Distance x sin Course
This is similar to the previous formula and should give the result: Dep = -34′.71
(This negative value indicates that the departure is W.)
To find D Long: d.Long = Departure ÷ cos mean Lat
Mean Lat = initial Lat + ½ d.Lat = 17° 20.0 N + 20′.7 = 17° 40′.7 (= 17° 40′ 42 N)
Here the departure (-34.71) would be divided by the cosine of mean lat converted to decimal degrees (17° 40′.7)
= 17.67833 cos. = 0.95278) to give the result: d.Long = -36′.43
The boat’s DR position after steering 320° T for 54 miles would therefore be
The Course and Distance between two points can be found in a similar manner with a calculator. First the D Lat and D Long between the two points would be determined together with the mean Lat.. The calculator could then be used to find the departure, the course and the distance between the two points as follows:
Departure = d.long. x cos mean Lat
tan Course = dep.
d.lat.
Distance = d.lat.
cos.Course
The above process could be considerably shortened by use of the P → R function as described earlier in this chapter.
It is very important to note that the P → R function and all the above formulae assume that the Earth is flat (which is why these routines are frequently referred to as Plane Sailing). This assumption is acceptable for distances up to about 500-600 miles, but this method should not be used for distances greater than this.
286. SIGHT REDUCTION USING A CALCULATOR
For sight reduction either a dedicated navigational calculator (described earlier) or a programmable scientific calculator is recommended in order to obviate repetitive operations and reduce potential errors resulting from numerous key operations.
One of the prime advantages of the calculator over the tabular methods described earlier in this chapter is that all sights can be worked from one EP or DR position and it is unnecessary to round off the Lat. or L.H.A. to the nearest whole degree.
Calculators differ so much and new models are appearing with such frequency that it is quite impossible for us to suggest a sequence of key strokes common to all of them. For those who wish to program their own calculator we give below easy formulae for sight reduction which should be entered in accordance with the calculator manufacturer’s instruction manual. It is recommended that the calculator is capable of converting degrees, minutes and seconds of arc for the L.H.A., LAT and DEC into the decimal degree format and that these can be stored in separate memory stores. Sight reduction with a calculator is described in greater depth in later chapters of this study.
In the formulae, Z = azimuth angle, Zn = true bearing and Hc. = calculated altitude.
sin Hc = (cos L.H.A. x cos lat x cos Dec) + (sin lat x sin Dec)
cos z = sin Dec – (sin lat x sin Hc)
cos lat x cos Hc
Under this system the azimuth (Z) is always N. and named E or W according to the L.H.A.. If the L.H.A. is less than 180°, Z is named W.. If the L.H.A. lies between 180° and 360°, Z is named E..
For example, if Z = 100° when the L.H.A. is 72°, this would be NI00°W and the bearing Zn would be 360°-100° = 260°T.
If Z = 150° when the L.H.A. is 327°, this would be N150°E and the bearing Zn would be 150°T.
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