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Older readers will remember using a slide rule to carry out multiplication and division in the days before calculators. There was also a graphical tool called a nomogram, which was used for evaluating certain functions. Although such devices have little practical use nowadays, I was interested in investigating which classes of functions with two independent variables could be calculated using this type of device.Figure 1 shows how a slide rule was used to multiply 3 by 4 (or equivalently to divide 12 by 4). To do this, you align the 1 on the upper scale with the 3 on the lower scale and then read off the value below the 4 on the upper scale to get 12 (or, for division, you line up the 4 above the 12 and read off the answer 3 on the lower scale).

As an undergraduate I was taught to multiply two numbers with the help of log tables, using the formulaHaving graduated to teach calculus to Engineers, I learned that log tables were to be replaced by slide rules. It was then that Imade the fateful decision that there was no need for me to learn how to use this tedious device, as I could always rely on the students to perform the necessary computations. In the course of time, slide rules were replaced by pocket calculators and personal computers, but I stuck to my original decision.My computer phobia did not prevent me from taking an interest in the theoretical question: what is a computation? This question goes back to Hilbert's 10th problem (see Browder [1976]), which asks for an algorithm, or computational procedure, to decide whether any given polynomial equation is solvable in integers. It quickly leads to the related question: which numerical functionsf: ℕn→ ℕ are computable?While Hilbert's 10th problem was only resolved in 1970, this related question had some earlier answers, of which I shall single out the following three:(1)f is recursive(Gödel, Kleene),(2)f is computable on an abstract machine(Turing, Post),(3)f is definable in the untyped λ-calculus(Church, Kleene).These tentative answers were shown to be equivalent by Church [1936] and Turing [1936–7].I shall discuss here some of the more recent developments of these notions of computability and their relevance to linguistics and logic. I hope to be forgiven for dwelling on some of the work I have been involved with personally, with greater emphasis than is justified by its historical significance.

I should like at the outset to disclaim any intention of laying down the law dogmatically. The opinions expressed are stated definitely, but they are submitted with deference for your consideration.

The procedures of dead reckoning require a number of different steps, amongst which are distance measurement, angle measurement, construction of straight-line tracks, the solution of the vector triangle and the process of multiplication and division. The navigator's plotting and dead reckoning instruments are designed to meet these needs. The ruler serves both as a distance-measuring scale and as a means of drawing a straight line track; the protractor measures angles; the dividers measure distances and the Dalton computer serves both as a method of solving the vector triangle and as a circular slide rule for multiplication and division.

The fundamental advantage of two-dimensional photoelasticity is its great simplicity. Neither the frozen-stress three-dimensional method nor the photoelastic layer technique share this simplicity as yet. Whereas the frozen-stress and layer techniques are most suitable for final investigation into the strength of a finished design, the simple plane photoelasticity can be used as a tool by the designer at every stage of his design, as a specialised slide rule, so to say, that gives sometimes faster and frequently more accurate answers than complicated stress calculations. The consequences of given changes in the outlines of a component can often be studied more conveniently and cheaply by this method than by almost any other.

Throughout the history of aircraft design, a trend that has remained quite constant up to the present time is the steady increase of wing loading with successive types. This feature with its consequent raising of stalling speed, and therefore landing and take-off distances, has resulted in techniques rather more scientific than those used in the early days of civil flying. With take-off distances of anything up to 6000 feet, the pilot in deciding whether a particular airfield is adequate for a given aircraft is no longer the sole arbiter; the rule of thumb must give way to the slide rule.The pilot is, of course, still responsible for the safety of a particular operation, and to help him in this respect, performance information for his particular type of aircraft is supplied in graphic form with the safety factor already built in. It is the purpose of this paper to explain why and how this is done.

A Logarithm as generally understood and used by the ordinary student is the expression of a number as a power of 10. In order to multiply two numbers together he adds the logarithms of these numbers and takes as the result the anti-logarithm of this sum. Similarly for division he takes the anti-logarithm of the difference of the logarithms of the two numbers.

Simultaneous equations of three or more variables are notoriously trouble some to solve numerically and, moreover, are frequently sensitive for certain relative values of the variables. This usually precludes the use of the slide rule and in consequence resort has to be had to accurate working to an increasing number of decimal places involving much labour even with a calculating machine.A method is here given which is applicable to certain classes of equations of frequent occurrence in mathematical physics and engineering science, which method is a combination of what is knovwi as the Iteration process and the means of applying this principle to end moment distribution devised by Professor Hardy Cross.

The calculator is a device in slide rule form for assisting in the design and stressing of flat rectangular panels in shear.In the Incomplete Tension Field theory, the ratio q/qb becomes an important factor, whereq applied shear stress (lb./in.2)qb shear stress at which panel first buckles (lb./in.2).This ratio is normally evaluated by use of the formulaqb = KE(t/b)2where E Young's modulus (lb./in.2)t thickness of panel (in.)a length of longer side of panel (in.)b length of shorter side of panel (in.)and K depends on the value of b/a and the edge conditions, and is normally obtained from R.Ae.S. Data Sheet 02.03.01.This method becomes laborious when dealing with large numbers of panels.By the use of the Calculator, q/qb may be found quickly, with no unnecessary paper work.

In a sense Stirling’s formula can be adapted to slide-rule computation of x! when a; is a fraction and tables are not at hand.