The Scale Arrangements of the Slide Rule: Trigonometric Functions in Electrical Engineering

Andreas Poschinger Munich University of Applied Sciences [email protected]

Final version, 2019/06/04

Contents

1 Introduction 1 1.1 Motivation, methodology and overview ...... 1 1.2 Slide rule basics ...... 1 1.3 The scales and their arrangements ...... 3 1.4 The scale arrangements in the literature ...... 3

2 Comparison of the trigonometric scale arrangements 4 2.1 Definition of the trigonometric scale arrangements ...... 4 2.2 Electrical task ...... 4 2.3 Solution of the task ...... 4 2.4 Evaluation ...... 5

3 Summary and Outlook 6

1 Introduction

1.1 Motivation, methodology and overview The top models of slide rules in the United States and other countries differed in scale arrangements from those used in the Federal Republic of Germany (West) and the ‘German Democratic Republic’ (East). A scan of the contemporary literature is presented but does not provide a sufficient explanation. Hence a more practical method of examination is chosen. The differences in the scale arrangements of trigonometric functions are shown in this contribution by means of a German and American as well as a Chinese model. The different scale arrangements are tested and evaluated on the basis of a specific electrical engineering task and the introduction of calculation costs. The results of the test are presented and shortly discussed. The summary and outlook also contains a deduction on further development of pocket calculators. First however some slide rule basics are introduced as well as scales and their arrangements.

1.2 Slide rule basics Figure 1 shows three exemplary slide rules: the Faber 62/8, made in Germany, (F83, top), the Pickett N3P-ES, made in U.S.A. (PN3, center) and the Flying Fish 1200, made in

1 China (FF1200, bottom), which will also be used later to compare the scale arrangements.. Top model slide rules usually follow the duplex construction. It consists of the two-part

Figure 1: Faber Castell 62/83 (top), Pickett N3P (center), Flying Fish 1200 (bottom) base which is interconnected with end braces, a slide and a cursor, which can be moved on the body. The cursor has one (PN3) or several (F83, FF1200 only rear) hairlines, with the help of values can be set and read. The cursor does not move when the slide is moved. The slide rule can be used to calculate almost all mathematical functions except addition and subtraction. The scales C (lower edge of the slide) and D (upper edge of the lower half of the body) show the multiplication. For the calculation of 1.25 · 4.8 (see F83), set the 1 of C on the 1.25 of D (cursor not needed), then bring the cursor to 4.8 on C, and read the result 6 on D. In a shorthand notation derived from Strubecker, 1956, this is written as a slider setting (before ‘||’) and a value determination with or without cursor usage (after ‘||’); ‘|’ stands for cursor usage, ‘/’ for settings without cursor:

D 1.25 / C 1 || C 4.8 | D: 6 = 1.25 · 4.8

Accordingly, the following steps – all being optional – are performed:

1. Set the cursor to value on D (not needed in example).

2. Set the slide on cursor or value of D.

3. Set the cursor on value on C or other scale.

2 4. Read result at value of C or cursor.

Setting the cursor to value on D is not needed, if either already done in the previous calculation step, or if side by side scales on slide and body are used, such as C and D, and on one of the scales the index is used. In all other cases it is assumed that the cursor movement is needed. Some recipes require a carry from one scale to another that consists of a reading and a value setting. If the second factor on C is outside the range of D, the ‘setback’ must be used. For the calculation of 8 · 0, 8 (see P3N),place the right 1 (hereinafter referred to as 10) of C on the cursor hair line, and then at 8 from C read on D the result 6.4. The decimal places must usually be estimated; the calculations of e.g. 8· 0.8, 80· 0.8 or 8· 0.08 are identical on the slide rule. The division works by reversing the multiplication (e.g., 6 : 4.8 = 1.25, see P83). It is not possible to set the first factor first with the cursor on C (on the slide) and then to move the body relative to the slide, because the cursor would be moved together with the body.

1.3 The scales and their arrangements In addition to the basic scales C and D, most slide rules have scales for calculating the inverse (CI, sometimes also DI), the trigonometric functions (S, ST, T or T1 and T2, P), √ of logarithms (L and Ln), squares and square roots ( , either W and R, and/or A, B and BI – also allowing multiplication and division without setback), cubes and cubic roots √ ( 3 , K, rarely K’) and for calculating arbitrary powers (LL00-LL03 and LL0-LL3). Folded scales (CF, DF and CIF) allow faster calculations of some tasks. Only a few slide rule models have hyperbolic scales (usually Sh and Th, here only on FF1200). Some of these scales are always on the body (e.g. A, D, DI) or the slide (e.g. B, C, CI); for some, the arrangement is not critical (L, Ln). This contribution shows that the arrangement of some scales is decisive, and yet no particular place has generally prevailed. The trigonometric scales S, ST and T, which are the focus of this work, are arranged on most of the German slide rule models on the body (Figure 1: F83), but on most American and international models on the slide (Figure 1: PN3, FF1200).

1.4 The scale arrangements in the literature Occasionally the system Darmstadt (e.g. Faber-Castell, 1965, p. 75) and the system Rietz are mentioned. Slide rules of these two types were produced by several German and inter- national manufacturers with the same scale arrangement. Nevertheless, there is very little literature that deals originally with scale system. Ewert (1965) introduces a classification of types based on the existing scales in the section “new type definition - an urgent task (neue Typdefinition – eine dringende Aufgabe)”, but does not distinguish whether the scales are arranged on the body or the slide. This view is also found in the products of the East German company Reiss, which equipped their type ’‘Rietz Spezial” with trigono- metric scales on the body rather than on the slide like the original Rietz arrangement. This view is further adopted in — (2007). J¨ager(1957) discusses the scale arrangement for the trigonometric functions. He comes to the conclusion that trigonometric scales in general are better located on the body with the exception that chain calculations include more than one trigonometric function.

3 2 Comparison of the trigonometric scale arrangements

2.1 Definition of the trigonometric scale arrangements Essentially, the trigonometric scale systems can be distinguished according to whether the scales (S, ST, T or√ T1 and T2) are on the body or on√ the slide and whether the Pythagorean scale P ( 1 − x2 ) and possibly the scale H ( 1 + x2) are present and on the body or slide. The top model German slide rules like the F83 have the scales S, ST, T1 and T2 as well as P usually on the body. This scale arrangement is referred to below as ExtTrigOnBody. American slide rules such as the P3N usually have the scales S, ST, T1 and T2 on the slide, but have no P scale. This arrangement is called TrigOnSlide. Very rarely there are slide rules which additionally have the scale P on the slide and sometimes also the scale H. This union of the German slide rules (scales on the body including P) and the American slide rules (scales on the slide) first showed up on Chinese slide rules of the mid-fifties and later on the Aristo HyperLog (J¨ager,1970); The FF1200 belongs to the Chinese top models in this tradition; its trigonometric scale arrangement is called ExtTrigOnSlide. These three scale arrangements are compared on the basis of a concrete electrical task.

2.2 Electrical task Given is the voltage of an asynchronous machine with U = 400V , the frequency f = 50Hz, the apparent power with PS = 8kV A, the power factor cos(ϕ) = 0.8, the slip s = 0.15. What is the maximum value of the mechanical power PM ? Which reactive power PB is generated? What is the inductance L of one inductor in a parallel equivalent circuit with three identical inductors (three-phase!) that absorb this reactive power? In this electrical task the apparent power corresponds to the hypotenuse of a right-angled triangle, the active power PW to its longer cathetus and the reactive power PB to its shorter cathetus . The use of results such as the active power for subsequent calculations is typical, and serves to show that slide rule results should preferably appear on scale D (or DF) for chain calculations. The required formulas are:

U 2 PW = PS · cos(ϕ); PM = PW (1 − s); PB = PS · sin(ϕ); L = (1) 3 · π · 2 · f · PB sin(ϕ) can be calculated either by formula (2) or by formula (3): sin(ϕ) = sin(arccos(cos(ϕ))) (2) p sin(ϕ) = (1 − cos2(ϕ)) (3) The challenge is to calculate the solution as efficiently as possible and as precisely as possible. With the slide rule, the most efficient way is to make as few settings as possible and to be able to make the settings quickly. Being as accurate as possible means that as few intermediate readings and transfers of values as possible are needed, as well as using scales that enable accurate work. Another criterion that is difficult to assess is whether the solution can be solved with the simple means of standard multiplications, or whether more complicated recipes are needed.

2.3 Solution of the task The solution of the tasks is written in the shorthand notation introduced in the section 1.2. Empty settings are available from the previous step. The number of cursor settings (C),

4 slider settings (S) as well as value findings and readings (R) are noted and later used in the cost function. Index marks are assumed not to cause value findings. The calculation of the inductance is not considered because it depends on the existence of quadratic and/or root scales that are not in focus because of U 2. However, reading and memorizing of the reactive power is evaluated. With ExtTrigOnBody the solution is as follows:

D10 / C 8 || P 0.8 | C 4.8 = PB[kvar] 3R, 1S, 1C / || D 0.8 | C 6.4 = PW [kW ] 2R, 1C D 6.4 | CI 0.85 || C1 / D 5.44 = PM [kW ] 4R, 1C, 1S Another variant requires standard multiplications only, but requires three slider settings:

P 0.8 / C10 || C 8 | D 4.8 = PB[kvar] 3R, 1S, 1C D 0.8 / C10 || C 8 | D 6.4 = PW [kW ] 2R, 1S, 1C | CI 0.85 || C1 / D 5.44 = PM [kW ] 3R, 1S With TrigOnSlide without existence of the P scale the following recipe is used: D 8 / C10 || C 0.8 | Cos 36.9 = ϕ 3R, 1S, 1C / || Sin 36.9 | D 4.8 = PB[kvar] 2R, 1C / || C 0.8 | D 6.4 = PW [kW ] 1R, 1C | CI 0.85 || C1 / D 5.44 = PM [kW ] 3R, 1S The following recipe can be used with ExtTrigOnSlide:

D 8 / C10 || P 0.8 | D 4.8 = PB[kvar] 3R, 1S, 1C / || C 0.8 | D 6.4 = PW [kW ] 1R, 1C | CI 0.85 || C1 / D 5.44 = PM [kW ] 3R, 1S

2.4 Evaluation Even more complex recipes may appear simple by appropriate exercise; therefor the pos- sibility of calculation with standard multiplications is not evaluated in a quantitative way. Quantitative features are the number of cursor (C) and slider (S) settings as well as read- ings and findings (R). In the following, three different weight sets are used to calculate the total effort, since absolute effort of the different actions is unknown and may differ between different slide rule users. Readings are always calculated as uniform effort E (R = E). In a first weight set also slide and cursor movements are weighted by one uniform effort (‘Uni’), in a second weight set slide movements are weighted by S = 1.5E and cursor movements by C = 0.5E (‘Slide’) and in third weight set slide movements are weighted by S = 1.5E while cursor movements are not taken into consideration (C = 0E, ‘No cursor’). The results are shown in the following table:

Scale Arrangement Actions ‘Uni’ ‘Slide’ ‘No cursor’ ExtTrigOnBody (1) 9R, 3C, 2S 14 13.5 12 ExtTrigOnBody (2) 8R, 2C, 3S 13 13.5 12.5 TrigOnSlide 9R, 3C, 2S 14 13.5 12 ExtTrigOnSlide 7R, 2C, 2S 11 11 10

The costs of one weight set are found in the respective column, in order to compare between the different scale arrangement and recipes. Independent of the chosen weight set, the results of the recipes for ExtTrigOnBody and TrigOnSlide are close together while

5 the cost for ExtTrigOnSlide is considerably lower as for the others. From this evaluation it can be seen that scale systems can not be defined by the mere existence of scales, but their arrangement is also essential. ExtTrigOnBody does not proof to be superior to TrigOnSlide for that specific task. It is confirmed however that the Pythagorean scale is useful, especially when placed on the slide.

3 Summary and Outlook

The slide rule and the standard multiplication as well as a short notation for calculation recipes were introduced. Three different scale arrangements of the trigonometric functions were described on the basis of three slide rule models. These in turn were evaluated on the basis of a concrete electrical task. Costs of the necessary recipes has been calculated by summing up the necessary steps: cursor and slide settings as well as readings. There were significant differences depending on the arrangement. Precisely, it has been shown that for the given task the location of the trigonometric functions on the slide is preferable, including the Pythagorean scale P. In the course of the work, because of the required square function, it showed up that similar comparisons may also be meaningful for square and root scales as well as the arrangement of hyperbolic scales. A deduction might be to challenge the typical electronic pocket calculator for introduction of the P functionality. Touch screens might also allow for entirely new calculator operation concepts; slide rules and their usage may give some hints.

Acknowledgments

This comparison of the scale arrangements would not have been possible without suitable slide rules. I would like to thank Hans-J¨urgen Hildebrandt, who sold me the P3N, as well as Xiongfeng Lin, who bought the FF1200 in China. I would also like to thank Joe Herning, with whom I could discuss the aspects of the scale arrangements.

References

—, 2007. Der Rechenstab REISS Duplex 3227 im Vergleich zu anderen Modellen online. Online; heruntergeladen am 19.04.2019. Available from: http://www.rechenschieber. org/wordpress/wp-content/uploads/2007/08/reiss3227.pdf.

Ewert, A., 1965. REISS-Duplex und die internationale Rechenstabentwicklung. Berlin (Ost): Kammer der Technik.

Faber-Castell, 1965. CASTELL–Rechenstab Lehrbuch. 13th ed. M¨unchen: Lindauer Verlag.

J¨ager,R., 1957. Gedanken zu Skalenanordnungen. Hamburg: Aristo.

J¨ager,R., 1970. Aristo Neuheiten. Hamburg: Aristo.

Strubecker, K., 1956. Einf¨uhrungin die h¨ohere Mathematik. Vol. Band 1: Grundlagen. M¨unchen: Oldenburg Verlag.

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