An Easy Introduction to the Slide Rule
Asimov introduces the slide rule, a simple pocket computer solving numerical problems like multiplication, division, powers, and roots using logarithms.
In these excerpts from his "An Easy Introduction to the Slide Rule", he sets out to demystify a tool that, while perhaps not as dazzling as a "giant electronic computer", holds its own quiet power and numerous advantages. Imagine, a computer small enough for your pocket, costing just a couple of dollars, immune to breaking down, and capable of solving "almost any numerical problem that you meet up with under ordinary circumstances". And the best part? It's "simple to operate". Asimov, the noted scientist, teacher, and author, promises that even with just "grade-school arithmetic", you can wield this instrument, perhaps not fully understanding why it works at first, but certainly mastering how. With his characteristic enthusiasm, he aims to make the principles both entertaining and understandable, guiding us towards truly knowing "what ybu are doing and why". This understanding, he assures, will make using the slide rule on day-to-day problems simple. Practice, naturally, will be key to becoming expert.
Our story begins with the most fundamental idea: arithmetic by distance. Asimov starts with a concept we all grasp – adding lengths on a ruler. We see how combining a length of 45 inches with a length of 32 inches gives us 77 inches, illustrating that 45 + 32 = 77. A ruler, in this sense, acts as a simple "analog computer", using the properties of lengths, which are analogous to numbers, to solve problems. He even playfully notes that 2 + 3 = 5 might be easier to find on a foot-rule than on a complex Univac.
To make this length manipulation more useful for arithmetic, we move from a single ruler to two, specifically two 10-inch rulers because our number system is based on 10. Placing these rulers edge to edge, with one sliding along the other, creates the fundamental concept of a slide rule. Asimov introduces us to building our own "addition rule" from cardboard, with a "slide" piece moving within a "body". We learn to use terms like the "slide" and "body", and to refer to numbers on them as "S-numbers" and "B-numbers". The left end, previously unmarked on an ordinary ruler, becomes 0 on this addition rule. The core operation is shown: placing the "S-0" index over a number on the body (say, B-5) and finding the sum (5 + 2) under the corresponding slide number (S-2) on the body (B-7).
The narrative quickly faces a challenge: sums exceeding the 10-unit length. Asimov shows how extending the rule helps, but notes the inconvenience. The ingenious solution lies in using the "S-10" index (the right end) as well, allowing sums up to 20 on the 10-unit rule by simply adding 1 to the result read on the body scale. This illustrates the power of the instrument to provide answers beyond its physical markings.
The addition rule isn't limited to whole numbers; it handles fractions and decimals too. We see how 1½ + 2½ = 3½ can be found by placing S-0 over B-1½ and reading under S-2½. This introduces the necessity of reading between the markings, estimating values like 7.33 on the scale. While these estimations aren't perfect, leading to approximate answers, Asimov argues this is often sufficient in practical fields like science and engineering. Furthermore, the slide rule can serve as a crucial check against "arithmetical mistake involved in long division" or other calculations.
Moving beyond addition, subtraction is presented as the reverse process. By using an "inverted" scale (SI-numbers) or simply moving the slide leftward, the addition rule can find differences. This brings us to the conclusion of the "addition rule" phase; while simple, its principles are foundational for the true slide rule. Addition and subtraction are easy enough with pen and paper that a physical addition rule isn't typically manufactured.
The true magic of the slide rule, the book reveals, lies in the world of logarithms. Asimov introduces the concept of logarithms as exponents. The key insight is that adding logarithms corresponds to multiplying the original numbers (antilogarithms), and subtracting logarithms corresponds to dividing the original numbers. He illustrates this with examples using base 2 and base 10 logarithms. The decimal point in a number affects only the "characteristic" of its logarithm, not the fractional part or "mantissa," which remains constant for a given sequence of digits regardless of the decimal point's position. This is a crucial property that the slide rule exploits, leaving the handling of the decimal point to the user.
The true slide rule, then, is essentially an addition rule designed to add and subtract logarithms. Instead of looking up logarithms in a table, the slide rule incorporates them directly onto its scales. The L-scale is presented as a linear scale representing logarithms (or rather, their mantissas) from 0.0 to 1.0. The D-scale, placed in correspondence with the L-scale using the hairline of an "indicator assembly", is a logarithmic scale where the physical distance from the left index corresponds to the logarithm of the number marked. This results in crowded markings on the right end of the D-scale.
With the D-scale representing antilogarithms and the principle of adding logarithms for multiplication in hand, the next step is clear: create a second, identical, sliding scale – the C-scale. Placing the C-scale index (C-1) over a number on the D-scale aligns the scales such that adding lengths on the C-scale (representing log of the multiplier) to lengths on the D-scale (representing log of the multiplicand) results in a total length on the D-scale representing the log of the product.
The journey continues with strategies for dealing with calculations that run off the physical scale. This leads to the introduction of "folded scales" like the CF and DF scales, which are essentially D and C scales "folded" at pi (or another number). These scales allow operations to continue seamlessly without needing to "switch indices" as often, or they provide convenient relationships, such as automatically multiplying or dividing by pi. The concept of inverse scales, like CI, CIF, and DI, printed in a different color (usually red), further simplifies operations, especially division and finding reciprocals. The D(back) and DI scales on the reverse side act as a crucial link between the front and back of the instrument.
Beyond basic multiplication and division, the book introduces scales for powers and roots. The A and B scales cover a range equivalent to two D-scales, allowing squares to be found directly opposite numbers on the D or D(back) scales because their logarithmic spacing is double. Similarly, the K-scale, covering three D-scale ranges, enables the calculation of cubes and cube roots.
Asimov guides us through these scales, emphasizing that despite the increasing complexity, the underlying principle remains the same: manipulating lengths that represent logarithms to perform arithmetic operations. While accuracy has its limits, especially on compressed scales like A and B, the speed and convenience make the slide rule an invaluable tool. The journey through these scales, from the simple addition rule to the intricate logarithmic arrangements, builds a complete picture of how this elegant "computer" in your pocket tackles a wide range of numerical problems. It's a delightful exploration of applied mathematics and a testament to the ingenuity of a tool that, for generations, felt as essential as a doctor's stethoscope or a painter's brush.
