Pythagorean (P) Scale on slide rules

[@738polarbear]

I can’t thank you enough for this series of videos about a small but powerful analogue instrument . When you think about all the magnificent buildings ,bridges,dams,s etc that were designed and built just using these it is amazing.I just bought a Logarex 27602_ 2 the scales are marked as functions and NOT as letters . It is very powerful and was dirt cheap . Only 15 USD.

[@terryfisher8872]

Professor Herning, this tutorial really makes the grade. Had the 2/82N Faber Castell with the P scale and did not have a clue how to operate. This really clears things up. Using the K&E 4081 book to switch out problems similar to your demo and it works everytime!

[@ProfessorHerning]

Thanks! This is one of the videos Of like to improve a bit though.

[@jadenephrite]

Thank you demonstrating the P scale’s accuracy in finding cosines of acute angles. Regarding 12:02, I was able to read the cosine of 7 degrees up to five decimal places using my larger slide rules. On my Lafayette Model 99-7055 slide rule 10 inch P scale, I am able to read the cosine of 7 degrees as 0.99255. Similarly on my Scientific Instruments Model 1610-D five inch diameter circular slide rule, I am also able to read the cosine of 7 degrees as 0.99255. My Hemmi 260 Slide Rule also includes the P scale.

[@terryfisher8872]

Professor Herning, you are a smash hit on Slide Rule Fan Club on facebook. Thought you should know, in particular for revealing the mystery of the notorious P-scale.

[@ProfessorHerning]

Thanks! This is not even one of my better videos, though, in my opinion.

[@sandcat-maurice]

Extremely helpful and great explanation, thank you! I was stuck on how to properly use the P scale. The manuals are not particularly helpful in explaining how/why the P-scale works. Also good to know how to use the P-scale to get better readout on large angles.

[@thomaswetmore7905]

I just purchased a Flying Fish 1003 slide rule. It has the P scale ( labelled H0' [0 is a subscript, the ' a superscript], since the scale is the inverse of the H0 scale). Unlike the Logarex, the 1003 has P on the slide. This makes solving triangles considerable simpler than the procedure when the P is on the body. Step 1: Normalize the triangle put placing an index of C over the hypotenuse on D. Step 2: Move hairline to length of known side on D, so the normalized value of that side is on C. Step 3: If the S scale is on the slide, the base angle of the known side can be read directly off S under the hairline. Step 4. Because the P scale is on the slide, the normalized length of the unknown side is also under the hairline on the P! Step4. Because the C index is already over the hypotenuse length, the actual length of the unknown side can be read (without moving the slide) by putting the hairline over the normalized length on C or CF and reading the length on the D or DF. Note the in this full procedure the slide was moved ONCE, and the hairline TWICE. Summary: The only side movement normalizes all sides of triangle. The first hairline move finds both the base angle and the normalized length of the unknown side. The second hairline move "unnormalizes" the unknown side by multiplying the normalized length by the original hypotenuse. (I realize these comments are seven months out of date, but I just found your videos and have been enjoying them.)

[@ProfessorHerning]

Thanks Thomas! I plan on improving some of the videos including this one over the summer. I've learned much more about the use of slide rules since making them. I haven't seen any rules with P on the slide, so that's intriguing. I've still been wondering about the real advantage of the P scale since the angle and the other side are determined with the traditional S scale solution to this problem (align hypotenuse on D with index of C, hairline to shorter side length on D, read angle on S, hairline to complement on S, other side length is on D). Perhaps the advantage is accuracy in certain cirumstances and avoidance of index-swaps through use of CF/DF (when available)?

[@ProfessorHerning]

Also, since you seem to have experience with slide rule trigonometric solutions, I'm wondering if you have a solution to the following problem: Traditional Darmstadt rules have no ST scale. Of course, sines and tangents of small angles may be computed with C/D, but this is very annoying when attempting to use the Law of Sines. For example, if the two known angles are 11 deg and 3 deg, then I could switch to ST when applying the Law of Sines on a Rietz slide rule. Even on a Mannheim-with-trig slide rule, I could swap the index of S with ~5.73 on C and finish the solution using C as S. Do you know of an elegant solution to such a problem on a Darmstadt rule? I have a sneaking suspicion this is why ST is added eventually to most simplex slide rules having have trig. on the base.

[@thomaswetmore7905]

I think your videos are the best available on the web. Thanks for making them. I look forward to any others you may do. I'm a retired software engineer. My BS was in '71 and PhD in '82. My university years spanned the demise of the slide rule and the ascent of the calculator -- I started college with a K&E DeciLon in '67 and switched to an HP-21 in '75; I still have both. The calculator I use today is the HP-21 simulation on my iPhone! I view the slide rule as one of the greatest inventions of humankind, and hope that we interested few can keep knowlege of it alive in case our civilation collapses and we must boostrap engineering once again. I have a modest collection, as it seems you must also. I had been trying to find an affordable Flying Fish for years, and one finally came my way. The 1003 certainly rivals the very best of K&E, Hemmi, Dietzgen and Faber-Castell, IMHO. It was seeing the P scale on the slide, and doing some examples that led me to writing my comment above. I have a PDF of an English version of the 1003's manual, and it describes a few examples using the P scale. It's all by rote, with no explanations about the whys, but it provides enough to skull it out. My main interest in slide rules has evolved into wanting to attain a full understanding of their mathematical principles and the construction and layout of all the scales. So in my collecting I am trying to find examples of all the scales that have been used on the more or less standard slide rules. I've worked up a lot of material and may submit an article to the Oughtred Society Journal some day.

[@ProfessorHerning]

Thomas, I am a mathematics professor, so I actually probably know more about the mathematics than the actual use of the slide rules -- especially since I'm not old enough to have used them in school. I initially preferred the more complex layouts, but I've come recently to appreciate some of the simpler ones (e.g. the Rietz layout) for their simplicity as well. It's amazing that the basic Mannheim-with-trig slide rules can approximate an answer to basically all common numerical operations.

[@Ensign_Cthulhu]

Nice! A neat trick if you want to set up the cosine of a small angle for a subsequent multiplication (say), and you have a slide rule with sin and tan on the slide (as opposed to the frame) is to remember that cos = sin / tan. So for A cos B deg., everything to base state, cursor to sin B, pull tan B under the cursor, left index of C scale is now at cos B; advance to the A multiplier on the C scale and read the answer on D.

[@ProfessorHerning]

Interesting trick but what about the larger angles! I would simply use cos(B) = sin(90-B). This is why most nice rules have cosine marked on the S scale. Use the following procedure: find A on D first, pull index of S/C there (can avoid cursor use here if you like that sort of thing). Then find B as a complement on the S scale with the cursor, result is on D. This moves the slide only once. Dividing a cosine is tricky on simple rules but is possible if you have a C scale on the slide as you seem to have. Using two slide movements, you could close the rule (1st move) then bring the cursor to the complement on the S scale. The cosine then is reading on the D scale and you could do a second operation (2nd move) however you like to multiply/divide and get the result on D -- assuming again that you have that C scale.

[@ProfessorHerning]

On another note, you the first procedure is great for vector components. In that scenario you often want Asin(B) and Acos(B). Ignoring the off-scale possibility, you can find both with one movement of the slide using the above procedure. Align the C/S index with A on D (essentially setting the hypotenuse to A). Then moving the cursor to B and 90-B on S computes the two components on D.

[@johnsavard7583]

While the P scale is a standard feature of the Darmstadt layout, and Darmstadt slide rules sometimes sneak into North America, there they are almost nonexistent on duplex slide rules - although found on the Hemmi 260 and some European duplex slide rules.

[@thorhilda]

Unwittingly, the P scale is using what will become Wildberger's rational trigonometry : en.wikipedia.org/wiki/Rational_trigonometry

[@robinj.9329]

Yup! I've never worked with "The P scale" before. But you do have my attention!

[@mgmartin51]

I love me my slide rules. Got us to the moon and back. Oh, wait. NASA faked the moon landing, I forgot. Let the kids learn their math with a calculator.

[@robinj.9329]

Mike Martin When someone confronted "Buzz Aldrin" in a bar, with that "Fake moon landing" 💩 He punched the guy right in the nose!! And later on, when that dufous charged Astronaut Aldrin with assault and took him to court? THE JUDGE SIDED WITH BUZZ!!!!!!! 😉

[@marklammas2465]

I will shortly be getting a slide rule with a P scale, when it arrives. Better get a triangle or two to try it with, then :)