Before calculators

[J HancockParticipant @jhancock95746]

The actual question was part of my 1961 4th form tech drawing papers , now consigned to the bin.

Show how you would make a square of exactly six square inches.

[SillyOldDufferModerator @sillyoldduffer]

Posted by duncan webster on 15/04/2022 13:37:28:

Either SOD has done it wrong, or QCad isn't as good as it thinks it is…

QCAD is innocent. I drew a 7 unit diameter circle and messed up positioning the centre point. Drawing a circle by snapping on the two line ends gets it right. I finished with a mirror of Calum's image, but same answer.

Now I know how, it's obvious. Doh!

Dave

[Mike PooleParticipant @mikepoole82104]

While at school I used log tables which were actually a book of four figure tables including roots and reciprocals and the trig functions plus various other tables, I started my apprenticeship in September 1972 having just turned 16 and my fathers Faber Castell slide rule found a new lease of life, 1973 saw me build a Sinclair Scientific calculator and be initiated into Reverse Polish Notation (RPN), the trig functions were pretty rough but close enough for jazz. To this day I remember the log of pi was 0.4972 or when I got 5 figure tables 0.49715, it’s funny how these things stick in your mind.

Mike

Edited By Mike Poole on 16/04/2022 13:09:31

[Mike PooleParticipant @mikepoole82104]

I am not sure my slide rule would be good for 1000 decimal places even with an eye loupe, 3 may have been optimistic. I don’t think I have ever owned a calculator that would do more than 10. Machining to 0.001” or 0.01mm is pushing my skills and equipment.

Mike

[An OtherParticipant @another21905]

I think Dr R.V Jones, during World War 2, used to get quite irate about making calculations to unmeasurable decimal places!.

[Robert DoddsParticipant @robertdodds43397]

Hi,
Nobody has mentioned the Theorem of Intersecting Chords so far.

"The product of the line segments of two intersecting chords are equal."

So in this case 6 x 1 = 2.45 x 2.45 which can be expressed as Sq. root of 6 = 2.45 depending on how accurate you want to measure it.
The square root bit only applies in the case of one chord being a diameter as this then bisects the other chord to make two equal halves so forming a square.
Try putting the vertical chord at 4 + 3 so you get 4 x 3 and the chord length is 3.46 i.e. sq root 12 = 3.46

Bob D

[Nicholas FarrParticipant @nicholasfarr14254]

Hi, the best guess that I could read, the sq root of 6 on my late brothers slide rule is 2.4495, which when squared = 6.00005025 according to my calculator, I then looked it up in my copy of F. Castle's five figure logarithmic & other tables, which I used during my Tech college work and his answer is 2.4495.

Below is a photo of the slide rule.

Regards Nick.

Edited By Nicholas Farr on 15/04/2022 21:52:32

[Michael GilliganParticipant @michaelgilligan61133]

Posted by J Hancock on 15/04/2022 14:42:19:

The actual question was part of my 1961 4th form tech drawing papers , now consigned to the bin.

Show how you would make a square of exactly six square inches.

.

Out of curiosity, I searched for that question … but failed to find it

In the process, however, l stumbled across this related problem:

**LINK**

It was good to see the number 5 appear out of his Pythagorean juggling.

MichaelG.

[John Doe 2Participant @johndoe2]

Posted by Gary Wooding on 15/04/2022 11:09:13:

…………who can remember how to divide a line into any number of equal parts using just a straightedge and pencil?

 

I know how to divide a board equally with just a rule/ruler – you simply put the ruler at an angle across the piece you want to divide and mark off convenient whole number divisions.

For example to divide a board 11.3 units wide into 6 equal parts; place the ruler with the zero on one edge and angle it so the 12 mark is on the other edge and mark off every 2nd whole number division. You get a line of dots – albeit at an angle across the board – but which are all equally spaced between the edges of the board, and no difficult maths or fractions required.

Not accurate enough for precision machining obviously, but perfectly adequate for many jobs. Much easier than using a calculator to divide 11.3 by 5 then having to read the ruler down to fractions of a division and adding that to the next division and so on.

 

But how do you do it without a measuring device of any sort?

 

 

 

Edited By John Doe 2 on 16/04/2022 10:39:42

[Clive FosterParticipant @clivefoster55965]

Posted by John Doe 2 on 16/04/2022 10:35:45:

But how do you do it without a measuring device of any sort?

Basically pins and string or equivalent.

Divide by 2 is easy. Fix one end of the string, find the length to the other side and double back to the original end. Loop back position gives you half.

Repeat as needed for all the even divisions.

For thirds you set the string in a right angled triangle with the long side 1 1/2 units long. Without measuring equipment best way is probably to set up a rectangle of with one pair of sides one unit long and the diagonals 1 1/2 long. Equal diagonals give right angled corners as close as you care to go.

Same trick for all the other prime divisions.

Tedious but not hard.

Makes you realise why common units (imperial for us) wer always done by dividinng down from something sensible.

Clive

[Nicholas FarrParticipant @nicholasfarr14254]

Hi, in answer to Gary Wooding's question.

To divide line A-B equally, draw a line A-C at any angle to line A-B, mark off line A-C by placing a straight edge square to line A-C and starting at point A, then draw a line from C-B followed by lines from the other marks on line A-C to line A-B and parallel to line C-B.

Regards Nick.

Edited By Nicholas Farr on 16/04/2022 12:03:57

[John Doe 2Participant @johndoe2]

Sorry, I am missing something in your explanation.

Gary says just a pencil and a straight edge, How do you get the equal spacing of the points on line A-C with only those?

How do you ensure a straight edge is exactly square to anything?

With only a pencil and a straight edge, how do you ensure that all the lines between A-C and A-B are truly parallel – for any spacing?

Apologies for being dim !

[Nicholas FarrParticipant @nicholasfarr14254]

Hi John Doe 2, I did the above with just a my Moore & Wright No. 315 straight edge and a pencil, and as it happens the line at an angle was just right to get all the division lines just the right distance apart to allow each one to be taken from the previous one, but I don't claim it to be absolutely accurate. These next four photos show how it can be done to good effect. First draw lines A-B and A-C, then you can place the straight edge at about square to the angled line with one edge touching point A and draw a mark across line A-C, notice I've extended the lines upwards, that way you can line the back edge to each line and mark the next and so on, this way all the points on the line A-C should be equal.

Once you have all the points required, you next join point C to B with the first line. In the case of this drawing, my straight edge is wider the the divisions, but putting the back edge to the line C-B, two short guide lines can be marked on either side of lines A-B & AC and the straight edge can be placed on the next point and adjusted by eye to get an even gap between the guide lines and the straight edge.

It is a bit long winded and you need to be fussy with the points and lines and the finest pencil will help. I'm still not claiming that it is absolutely accurate, but the rule in the photo below shows that it works very well and I have only used this straight edge and a pencil.

Below is a scan of the drawing without the rule and when I drew the division line marked D, drew the other two lines at the top and bottom the get better alignment for the guide lines for drawing the last line, as the division line D was a bit short.

 

Regards Nick.

Edited By Nicholas Farr on 16/04/2022 23:35:19

[Howard LewisParticipant @howardlewis46836]

Still have the Japanes bamboo slide rule that my father gave me, and the One I used at tech, plus the Thorton Pickard from working days.

Superceded by calculators etc.

But what is new?

The Egyptians used Tablets

Clay with styli to enter the data!

Howard

[DC31kParticipant @dc31k]

Posted by John Doe 2 on 16/04/2022 12:57:28:

Gary says just a pencil and a straight edge…

Gary is misremembering. It was a long time since the Greeks first wrote down these methods, so perhaps his memory can be forgiven.

The classic Greek geometry probelms were 'compass and straight edge'. The straight edge is unmarked (and in theory of unlimited length). The pair of compasses can, inter alia, be used to step off equal unit distances.

The dividing into parts in principle can be done with these two tools (you have to construct a series of parallel lines through given points), but is much easier if you use a 'sliding set square' method.

Other interesting subsets of Greek geometry are the angles you can construct using these tools, and closely related, the regular polygons that can be constructed. Maybe our resident CAD jockeys can produce a series of pictures for the case of 17 sides.

[SillyOldDufferModerator @sillyoldduffer]

Posted by John Doe 2 on 16/04/2022 12:57:28:

…

How do you ensure a straight edge is exactly square to anything?

With only a pencil and a straight edge, how do you ensure that all the lines between A-C and A-B are truly parallel – for any spacing?

Apologies for being dim !

Not dim at all! People had been about for a couple of hundred thousand years before the Ancient Greeks made progress on this stuff, and their efforts were spread over a millennia!

Constructions like this help, and they can be done on a wet sandy beach with a length of string and some pointy sticks.

First draw the straight white line by stretching the string between two stakes. The string can be twanged to mark the sand, Modern builders still mark lines with chalky string in the same way.

Next, bang in another stake near one end off the white line and use a fixed length of string to scribe a circle around it. Put a second stake at the intersection between the circle and white line, and draw a second circle of the same size. Repeat by scribing another pair of circles at the other end of the white line.

Then

• The green line drawn between circle intersections is at a right angle to the white line.
• The red line between circle intersections is parallel to the white line.

Dave

[Nigel Graham 2Participant @nigelgraham2]

Sums and I always agreed to differ! However, I do recall some of those methods, and when one night I could find neither calculator nor slide-rule for some problem just a bit too awkward for brain aided by pencil-and-paper, the log. tables did the trick. (With a few minute's revision of the method!)

Oddly, although I used logarithms while at school, I never really understood them until I was in my 50s and working in acoustics, using the deciBel scales for both sound and electrical signal levels.

.

That line-division method also found a place in those transparent plastic rules made for measuring more finely than could be accommodated by simple edge-graduations alone. Let alone drawn by pencil.

'

Whilst these old geometrical constructions allowed my making a template for setting out two large hexagonal frame about 2 feet A/F for the side-cheeks / handles of a winch. The size and roughness of the very pre-loved hot-rolled steel bar (old miniature-railway rails) precluded any combination-square accuracy and precision.

So I marked a big sheet of thick plywood with line at 120º inside-angle to a manufacturer's edge, by carpenter's beam-compass and pencil, using the familiar division of a circle by chords of length = radius, though using only an arc.

This enabled clamping the bars in pairs at a time, aligned along the line and edge, for tacking together, so spreading the errors vaguely evenly around the hexagon.

.

The slide-rule sometimes has an advantage when you are repeating one calculation several times with a single variable.

Slide-rules are still made, though not the traditional arithmetical type. Looking at the Blundell-Harling catalogue shows a big range of special liner and rotary slide-rules for all sort of special purposes – along with drawing-boards of various types. Blundell-Harling, based still in Weymouth, recognised how things were changing back in the 1960s and diversified swiftly into these other areas while their rivals disappeared. It also made office furniture for a time but seems to have dropped that to concentrate on slide-rules and drawing-boards.

My only connection to the firm is as a customer. I own one of their basic A3 boards with very simple parallel-motion, and might still have my school-issue 6" slide-rule. The latter were made apparently especially for selling to schools. I'd investigated the firm several months ago while trying to identify that elusive roundy-roundy thing with its strange numbers and 16X table.

'

If you need use one of these new-fangled electronic things though, the keyboard holds a neat co-incidence for circular-areas to 4 decimal places, if all you are given, or measure, is the diameter. The area of a circle of diameter D = 0.7854 D^2; and those 4 digits' buttons cluster in very elegant order in the top left corner!

' ' '

That beam-compass is a thing of beauty itself, and very careful examination of its brass fittings and it being stamped "H J Sandford 1920 " suggest it was largely hand-made (with some turned components) by that person, possibly as an apprentice-piece. I inherited it, so this Mr. Sandford would have been the retired cabinet-maker who was our next-door neighbour back in 1960, and passed on some of his tools to our Dad. The dates match for a 40 years career. I like to think he'd have appreciated I still care for it and even sometimes use it, 100+ years after he made it!

[Mick B1Participant @mickb1]

Posted by Peter G. Shaw on 15/04/2022 12:39:37:

Philip,

I think that Sinclair's first calculator was the Science of Cambridge retailing at £39-95. When it came out, Practical Electronics was running a series on building your own calculator for £70-£80. PE admitted they could no longer compete, but they were going to complete the series for information whilst accepting that no-one would be building it.

….

Cheers

Peter G. Shaw

I can't remember how much they cost in the mid – '70s, but it was a significant sum to a machinist on normal wages at the time.

What I do remember is that nearly all of those I knew who bought them, either ready-assembled or as kits, lost the top or bottom line of the LED display within weeks or even days.

What a waste of hard-earned.

I only didn't get caught because I couldn't quite afford one, so I kept my 4-figure tables and bought one of the early simple LCD calculators instead. It didn't develop faults and the battery lasted many times longer than in LED calcs.

As for Curta mechanical calculators, take a look at the Bay to see the sums these are fetching with collectors now!

[Speedy Builder5Participant @speedybuilder5]

The advantage of a slide rule was that when you made a calculation, you could see at a glance the next whole number either side of the calculated result. No tapping of keys, clearing accumulator etc etc. However, not so good for addition and subtraction !

[Michael GilliganParticipant @michaelgilligan61133]

Posted by DC31k on 17/04/2022 12:40:29:

Posted by John Doe 2 on 16/04/2022 12:57:28:

Gary says just a pencil and a straight edge…

Gary is misremembering. […]
The straight edge is unmarked (and in theory of unlimited length). […]

.

… and also, of course, singular [i.e not two parallel edges]

MichaelG.

[NealebParticipant @nealeb]

In the past, I've used a "manual" method of calculating square roots. There's a description here. Pretty simple application of differential calculus – surprised no-one has mentioned it!

Back in the seventies when my wife (then girlfriend) were at university, she had a pile of statistics to analyse. This involved many square root calculations and she had a four-function calculator. I was able to derive the Newton-Raphson formula from first principles (ah, the days of understanding that stuff!) and it all went quite quickly. Essentially, you guess an answer, plug it into a simple calculation, and it gives a better answer. Repeat until bored! Or, at least, until successive answers are close enough that you know you are there. It actually gets to a decent approximation fairly quickly.

As you can do the sums on a four-function calculator, clearly you can also do it with pencil and paper but the formula does involve long division which slows things down.

I've just tried finding sqrt(6) by this method. Daft initial guess (sqrt(6)=1) gives us 2.4495 after 4 iterations. Better guess (by inspection, answer is between 2 and 3 so assume 2.5) gives us 2.4495 after 2 iterations. Well, it's quicker than drawing and can get to any arbitrary accuracy. And some of us like doing sums… But just as pointless as the drawing method given today's technology-in-the-pocket devices!

[Martin KyteParticipant @martinkyte99762]

Ready Reckoner.

Zeus Book.

Martin

[Mike PooleParticipant @mikepoole82104]

Just ask Alexa.

Mike

[JAParticipant @ja]

On my desk I have a Casio Scientific Calculator, which for some reason works in fractions, Tubal Cain's "Model Engineer's Handbook", which contains four figure tables including areas of circles, and a slide rule. I have already said which I prefer.

All will give me the square root of 40 but none will give me the square root of -40!

JA

[NealebParticipant @nealeb]

Posted by JA on 19/04/2022 18:04:18:

All will give me the square root of 40 but none will give me the square root of -40!

JA

Well, just imagine that!

[Author]

Posts