The type of Astro-navigation so far described in this study has been what is called the Tabular Method, i.e., the use of pre-calculated tables of data such as The N.A. and Sight Reduction Tables, involving no more mathematics than the ability to add or subtract sets of figures. Another type of Astro-navigation is the so-called Direct Method in which the PZX and other spherical triangles on the celestial sphere are solved by a branch of mathematics called Spherical Trigonometry.

Although the use of pocket calculators can greatly simplify the mathematics involved, nevertheless this use quite clearly falls into the ‘Direct’ type of Astro-navigation, and herein lies a danger. To professional mathematicians, scientists, architects, engineers or teachers who use calculators frequently in the course of their daily work the extension of their use for navigation will prove no problem. But for others who use pocket calculators either very little or not at all, their use is not recommended without first studying the mathematics involved (for those interested, direct methods of Astro-navigation are covered in other chapters of the Astro-navigation study).

Although ‘formulae’ for use with calculators are given later in these chapters and can be found in many almanacs, text books, handbooks and similar references for small craft owners and navigators, unless the student fully understands the underlying mathematics involved considerable difficulty is likely to ensue. Using a calculator by rote, i.e., ‘if you press this button you will come up with the answer’ is almost certain to lead to serious if not positively dangerous errors of which the user may not even be aware.

These chapters describes the various types of pocket calculator available, including those types and their functions most suitable for Astro-navigation. This should prove useful introduction to calculators for those unfamiliar with them, at the same time showing those who use them daily how they can be adapted for navigational use.


Problem-solving by graphical or chart plotting methods has been based on procedures designed to avoid or minimise calculation. In cases where an alternative to some form of calculation has not been feasible, pre-calculated tables or similar aides have been devised, for example the Sight Reduction Tables described earlier in this study.

The advent of the electronic calculator has changed all this. The mathematical and trigonometrical formulae that were previously tedious and difficult to manage can now be solved literally at the touch of a few buttons.

There are many different types of pocket calculators ranging from simple ones that just do basic sums to scientific calculators for doing complex mathematical calculations, and dedicated calculators which are specially designed to do specific jobs, for instance, those used by accountants and engineers. Calculators can be broadly divided into four categories, providing:

Simple Arithmetic Functions only

Full range of mathematical functions (the ‘Scientific Calculator’)

The capability of being programmed (the ‘Programmable Calculator’)Specialised programming (the ‘Dedicated Calculator’)

Arithmetical calculators, although inexpensive are not ideally suited to the tasks of a navigator – the savings made in time being minimal. They are useful for performing the arithmetic parts of calculations, and are always handy for dealing with interpolations.

A simple arithmetical calculator is illustrated in fig. 69-1, which shows the jobs done by the principal keys.  Each job is called a function: adding, multiplying, squaring, etc., are all functions. Calculators vary enormously in the functions they provide and the instruction booklet which is sold with the calculator should always be consulted for detail.

Calculators of type (2), Scientific Calculators, are described in some detail in the following sub-section of this Study.

A Programmable Calculator (type (3)) is one which can remember a sequence of keystrokes, repeating it on a command without a mistake. In other words, it can store the instructions for performing long, complex calculations (i.e., a ‘formula’) and use these instructions over and over again. A set of instructions is called a program. You can work out your own programs or formulae and give them to the calculator by pressing keys in sequence, or (with certain types of programmable calculator) you can buy ready made programs in small boxes called modules which slot into the back of the calculator.

Calculators of type (4), Dedicated Calculators have specialised programs built into their circuitry to do specific jobs. For instance, there are special Engineering Calculators programmed to perform specific functions such as testing the strength of a bridge. There are special Financial Calculators with functions and programs for working out interest on investments, depreciation, profits and losses, etc. of particular interest to accountants and businessmen, and there are special navigational Calculators (such as the Tamaya NC-77 & 88 illustrated in fig. 69-2) specifically designed and programmed to solve navigational problems, some even incorporating Astronomical data for many years ahead.

Specialist celestial navigation calculators such as these can be grouped into those providing almanac information for the Sun and Aries (hence suitable for stars) and those that include, in addition, ephemeris for the Planets and the Moon which are more complex and require extra calculating power.

The first celestial navigation calculator suitable for small craft was the Tamaya NC77 fig. 69-2(a) which, together with the programmed Sharp EL-512 calculator (MERLIN) and the programmed Sharp PC246 pocket computer (TERN) include the Sun and Aries almanacs.  The Tamaya NC88 together with the plug-in modules for the Sharp PCSOOA and the Hewlett Packard 4CV pocket calculators provide the full Sun, Aries, Moon and Planet almanacs.

Guidance on the use of a dedicated Navigational Calculator will be given later in this Manual. However, such calculators are sufficiently expensive to be mainly of interest to the professional navigator since they can not readily be employed for general use. They have the drawback that they cannot be programmed by the operator, and provide only a limited range of functions for use in the manual mode. An amateur yachtsman is not going to make heavy use of a calculator, but it would be quite possible to couple its navigational use with some other in order to justify the expense of purchasing a good one for navigational use.

Common sense must apply as to which type of calculation lends itself to solution by calculator and which is best solved by conventional means.  Undoubtedly calculators are best at solving the more complex problems.  Calculators are, after all, nothing but another aid to navigation but, used intelligently, they can become a very valuable aid.

The aim of the first part of this chapter is to provide the navigator with the means to develop his or her own methods of working using either a relatively simple scientific or programmable calculator.  When using calculators, it should be borne in mind that, although these are remarkably reliable, failures do occur. Arrangements should be made, therefore, to provide a ‘back-up’ system, not merely by carrying a duplicate calculator.

We urge that under no circumstances should you go to sea without almanac, nautical tables, reduction tables, and so on. and the knowledge of how to use them.


Today, calculators having all the functions of a slide-rule are available for only a moderate price. These are the scientific calculators. Not only do they perform elementary arithmetic without error, but compute trigonometric, logarithmic, and many other useful functions

For navigational work a scientific calculator is required. The term scientific is applied to machines which provide a range of functions such as the trigonometrical ratios (sine, cosine and tangent), logarithms, decimal degrees, etc.

Scientific calculators range from basic or manual machines to sophisticated programmable models.  With a basic machine the operator performs calculations by depressing keys in the order required to enter data, obtain functions and execute the arithmetic necessary for the solution of a particular problem.

Programmable calculators will perform calculation sequences automatically in addition to manual operations once the appropriate instructions – The Program have been entered.

In the scientific calculator, the mariner has an instrument capable of rendering a speedy solution of even highly complex problems, and with a degree of accuracy not attainable by the use of most tables.

In Astro-navigation, for example, the calculator permits the navigator to reduce a series of sights from a dead reckoning or E.P., rather than having to employ a series of assumed positions, thus not only saving time in plotting, but also doing away with the errors arising from long intercepts sometimes caused by the use of assumed positions.

The keys and labels on scientific calculators vary a great deal and yours may not look exactly like the one shown in fig. 69-3. You may need to check how the keys on your calculator work in your instruction booklet.

In most scientific calculators the memory can store numbers for weeks or months, even when the calculator is switched off.  This facility is called Constant Memory and can be seen in the calculator illustrated in fig. 69-3.

Many of the keys on a scientific calculator perform more than one function. You will notice that these keys have symbols printed above them in addition to those printed on them. The symbols printed above the keys are second functions. Keys may even have a third function, the symbol for which would be in a different colour from the second. To perform these functions you have to press a key marked ‘2nd ‘.  ‘3rd ‘, or ‘INV’ (short for inverse). The label ‘inverse’ is used on some calculators because the second function is often the opposite of the first. For example, squaring a number is the opposite of finding a square root, so these two functions are on the same key.


Whereas scientific calculators are very similar in regard to the types of calculations they can perform, they often differ in the actual keystrokes used to effect those calculations. The method by which data are received and processed by a calculator is called the logic system and this determines the key sequence necessary to perform calculations. For scientific calculators three systems are common: simple algebraic, Full Hierarchy (priority algebraic). and RPN (Reverse Polish Notation).

A simple algebraic machine executes a chain of calculations progressively, the depression of any of the operation keys (+, -, x, ÷ or =) resulting in the sub-total for the calculation being displayed. With Full Algebraic Hierarchy (priority algebraic) logic, the input data and instructions are initially stored in pending registers known as a stack.  The calculations are performed in accordance with the normal mathematical rules, so that multiplication and division are completed first followed by addition and subtraction.

Expressions or equations involving a series of mixed arithmetical operations must always be evaluated in a specific sequence to conform to the normal rules of arithmetic

The priority of calculations is:

functions (e.g. powers, roots, trigonometrical ratios logarithms, etc)

multiplication and division ( x and ÷ Or

addition and subtraction (+ and – )

For example:

3  x  5  +  6  =  15  +  6  =  21              (not 3  x  11  =  33)

8  +  6 /  2  =  8  +  3  =  11                  (not 14  /  2  =  7)

5  x  3 –  12  /  4  =  15    3  =  12       (not 3  /  4)

3  x  62  = 3  x  36  =  108                   (not 182  =  324)

Parts of an equation in parentheses (between brackets) receive priority over the parts outside.  For example:

3  x  (5  +  6)  =  3  x  11  =  33

Using the parentheses for implied multiplication such as 7(3 + 5) is invalid.

Fig: 69 4 Operation of an RPN Calculator

The calculator requires the ‘x’ symbol whenever multiplication is intended, thus:

7  x  3  +  5  =  26   7  x  (3  +  5)  =  56

Likewise the expression sin 24° sin 147° is unacceptable. This must be expressed as sin 24° x sin 147°.

Your calculator may have an Equation Operating System (EOS) which enables you to enter numbers along with operations into the calculator in a simple straightforward sequence from left to right. The EOS decides whether an operation is to be completed or temporarily delayed based on the priorities described above.  When the calculator reaches the end of the expression, it performs any remaining delayed operations to arrive at the solution.

RPN calculators have no   =   key.  With this type of logic, the function key is depressed after both the values, upon which that operation is to be performed, have been entered.  RPN logic is used exclusively in the calculators manufactured by Hewlett-Packard.  Input data is stored in a stack of four registers X, Y, Z and T, the X register being displayed.  The procedure is best explained by means of an example.

Consider the expression:

The keying sequence and display registers for this calculation are shown in fig. 69-4. Depressing the function keys causes the value in the X register to be added to, subtracted from, multiplied by or divided into the value in the Y register, the result being displayed in the X register and the stack dropping. The first entry has to be moved into the Y Register by the ENTER key, but the stack rises automatically by an entry following the depression of a function key.

An algebraic calculator waits until the entire expression has been keyed in to evaluate it, whereas the RPN calculator evaluates small portions of the expression as each operation is performed.

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