When 80 SCIENTIFIC AMERICAN MAY 2006 Slide Rules Ruled Before electronic calculators, the mechanical slide rule dominated scientifi c and engineering computation By Cliff Stoll wo generations ago a standard uniform identifi ed engi- neers: white shirt, narrow tie, pocket protector and slide Trule. The shirt and tie evolved into a T-shirt sporting some software advertisement. The pocket protector has been re- placed by a cell phone holster. And the slide rule has become an electronic calculator. Take another look at that slide rule. Pull it out of the drawer you stashed it in 30 years ago or make one of your own [see box on next page]. You’ll see why it was once so valuable. Before the 1970s the slide rule, or slipstick, was as common as the typewriter or the mimeograph machine. A few seconds of fi ddling let scientists and engineers multiply, divide and fi nd square and cube roots. With a bit more effort, techies could also compute ratios, inverses, sines, cosines and tangents. Inscribed with a dozen or more function scales, the slide rule symbolized the mysteries of arcane science. Truth is, though, two scales did most of the work, as many technical jobs boiled down to multiplication and division. A pianist might play most of the ivories on the keyboard, but rarely did any engineer use all the scales on his (almost never her) slide rule. Some engineers, perhaps bucking for promotion, wielded slide rules made of exotic mahogany and boxwood; others sported rules fashioned from ivory, aluminum or fi berglass. Cheapskates—including this author—carried plastic ones. From the fi nest to the humblest, however, all slide rules were based on logarithms [see box on page 85]. Birth of the Slide Rule john napier, a Scottish mathematician, physicist and as- tronomer, invented logarithms in 1614. His Canon of Loga- rithms begins: “Seeing there is nothing that is so troublesome CECCO E to mathematical practice, nor doth more molest and hinder cal- culators, than the multiplications, divisions, square and cubical D BRAD SCIENTIFIC AMERICAN 81 82 DO-IT-YOURSELF SLIDE RULE SLIDE DO-IT-YOURSELF You can build a working slide rule from paper and cellophane reasonably robust calculating instrument. These construction plans are also available at www.sciam.com/ontheweb at available also are plans construction These instrument. calculating robust reasonably SCIENTIFIC AMERICAN AMERICAN SCIENTIFIC x D 1 1.5 2 3 π 4 5 6 87 9 1 a . Cut here L 0 1 2 3 4 5 6 7 8 9 1 0 lgx sin S 5.5 6 87 9 10 15 20 30 40 50 60 70 90 84.5 80 75 70 60 50 40 30 20 0 cos CHRIS HAMANN AND NANCY SHAW ASSEMBLY INSTRUCTIONS ASSEMBLY 1 3 2 5 4 end to complete the cursor. the complete to end the adhesive sides together. sides adhesive the one Make transparent tape. then remove excess ( excessremove then Slide cursor onto the rule. the onto cursor Slide section the length of the the of length the section use the guides to the left to to left the to guides the use measure two pieces of piecestwo measure 6 for sizing. Use the adhesive adhesive the Use sizing. for length of the red line. Place line. red the of length the other the and line black between parts A and B ( B and A parts between inscribed with a hairline), hairline), inscribeda with Cut out the entire white white entire the out Cut panel ( panel Slip part B into the folded folded the into B part Slip partA. c dotted lines. dotted Fold part A along the the along A part Fold Hairline To make the cursor (the sliding window that is is that window sliding marker in the middle.themarker in Draw a line with a fi ne fi a with line a Draw around the slide rule slide the around Wrap the folded tape tape folded the Wrap Cut here Cut cursor Completed a ). Cut along line line along Cut ). c ). ). b ), ), tape. Photocopyingtape. plansthese Part A 84.5 80 75 70 65 60 55 50 45 cot 5.5 6 7 8 9 10 15 20 25 30 35 40 45 T tan K 123456789 123456 7 8 9 1 2 3456789 1 x³ 9 1 2 3π 4 765 98 1 2 876543 91 A x² b Cut here onto thicker paper yields a yields paper thicker onto B 9 1 2 π 9876543 1 2 876543 91 x² 1 CI 1 98 7 6 5 4 π 3 2 1.5 1 –x C 1 1.5 2 3 π 4 5 6 87 91x Left-hand index Right-hand index Part B (Slider) MAY 2006 MAY extractions of great numbers, which besides the tedious ex- pense of time are for the most part subject to many slippery HOW TO USE A SLIDE RULE errors, I began therefore to consider in my mind by what cer- tain and ready art I might remove those hindrances.” First, get your hands on a slide rule. The top stationary scale Yes, logarithms—that horror of high school algebra—were usually has the A scale; the B and C scales reside on the central slider. The D scale sits on the bottom stationary scale. The left- actually created to make our lives easier. In a few generations, hand index is on the slider—it is the farthest left digit 1 on the C people will be equally shocked to learn that computers were scale. At the extreme right of the slider, you will fi nd another also invented to make our lives easier. number 1 on the C scale—that is the right-hand index. Finally, So how did Napier’s logarithms work? Listen to the inven- the mobile cursor contains the hairline. tor: “Cast away from the work itself even the very numbers To multiply two numbers, move the slider until the left-hand themselves that are to be multiplied, divided, and resolved index points to the fi rst number on the D scale. Now slide the into roots, and putteth other numbers in their place which cursor hairline over so it points to the second number on the C perform much as they can do, only by addition and subtrac- scale. The answer will appear under the hairline on the D scale. So tion, division by two or division by three.” to multiply 2 times 4, adjust the C scale until the left-hand index Which is to say that by using logs, multiplication simplifi es points to 2 on the D scale. Move the hairline to rest over the 4 on into sums, division becomes subtraction, fi nding a square the C scale. You’ll fi nd the answer, 8, right under the hairline on root turns into dividing by two, and fi guring a cube root be- the D scale . comes dividing by three. For example, to multiply 3.8 by 6.61, If your calculation extends off the end of the slide rule, use you look up the logarithms of those numbers in a table. There the right-hand index. So to multiply 7 times 6, set the right-hand you will fi nd 0.58 and 0.82. Add these together to get 1.4. index over 7 on the D scale and the hairline over 6 on the C scale. Read 4.2 on the D scale and then remember that the slippery Now go back to the table and fi nd the number whose log is decimal point must be shifted one place to the right to give the 1.4 to get a close approximation of the answer: 25.12. Begone correct answer, 42. ye slippery errors! To divide, set the hairline over the dividend on the D scale. Napier’s invention revolutionized mathematics—mathe- Then shift the slider until the divisor lies under the hairline (and maticians adopted it immediately to speed their calculations. right next to the dividend). The quotient will be under the index. German astronomer Johannes Kepler used these modern log- For example, let’s divide 47 by 33. Move the cursor so that the arithms to calculate the orbit of Mars at the start of the 17th hairline points to 4.7 on the D scale. Move the slider until 3.3 on century. Without their assistance, he might nev- the C scale rests under the hairline. Now the left-hand index sits er have discovered his three laws of celes- adjacent to the answer, 1.42. tial mechanics. Henry Briggs, Eng- Want to fi nd the square of a number? You will not need to land’s most eminent mathemati- move the slider. Just place the hairline over a number on the D scale. Look up at the A scale, where the hairline points to the cian of the time, traveled to square. So, right above 7 on the D scale, you will fi nd 4.9 on the A Scotland just to meet Na- scale. Slip the decimal point to the right to get the answer, 49. pier. He introduced him- To determine square roots, there is no need to move the self with: “My lord, I slider. But notice that the A scale is divided into two parts: the have undertaken this left half runs from 1 to 10, and the right half goes from 10 to long journey purpose- 100. To fi nd the square root of any number between 1 and 10, ly to see your person, place the hairline over the number on the left side of the A scale and to know by what and read out the square root from the D scale.