Somebody has said that "if you can not derive a formula or if you can not understand how it is derived by others, never use it" do you agree with this?
No, we don't need to know how to actually derive a formula in order to use it, but to use it intelligently we do need to have an understanding of where it came from and what limitations should be observed in applying it. When you calculate a result using the formula, you should be able to look at the result and make a decision about whether or not the answer is reasonable and makes sense. Many young engineers that I have taught seem to lack the ability to question the result that they calculate with a given formula, even if it is off by several orders of magnitude. Did they enter each quantity in the formula with the proper units? Did they make a mathematical error in the calculation? Are the units that they are using compatible the units assumed in the derivation of the formula? By failing to consider these potential mistakes it reveals a deeper problem - that they lack insight in the problem that they are attempting to model.
Rick, consider a right triangle with the sides that border the 90 degree angle labeled A and B, and the hypotenuse labeled C. Then, if the angle between the hypotenuse and side B is Theta, we can write
sin(Theta) = A/C
and
cos(Theta) = B/C
You can use these relationships to derive the Pythagorian Theorem. Using the trigonometric identity
[sin(Theata)]^2 + [cos(Theata)]^2 = 1
we find
(A/C)^2 + (B/C)^2 = 1
Multiplying both sides by C^2 we find
A^2 + B^2 = C^2
This is the Pythagorian Theorem. For an actual derivation, look at the content in the following link:
Rick, it wasn't intended as a proof. It was meant to show how the relationship could be arrived at from a simple trigonometric principle and some geomtrical arguments. Yes, you can easily show that the reverse works as well - the trig relationship can be derived from the Pythagorean Theorem. The actual proofs, based on first principles, are contained in the links listed below:
Maui,
now you have to show/derive your trig relationship.... (pretty soon we'll be back onto Russel).. so at what point do you stop re-creating the entire history of maths, science and engineering?
Everything depends on something else.
maui, that's pretty complicated...
Try this: draw the triangle with lengths a, b and c (a and b form the 90 deg angle). draw the square with sides c and draw 3 other same size triangles on the other sides of the square. The area of the square = c^2 and it equals the area of the 4 triangles plus the smaller square, or 2*a*b + b^2. Solve for c^2: c^2 = a^2 + b^2.
Problem with that approach is that it is only feasible for integer sides.
I must admit I burst out laughing when Rick first mentioned it, it is such a neat example of a useful formula we use every day, yet I'm betting that only one in a thousand of us could derive it without looking it up first (cue 400 proofs of Pythagoras). I've never even read the full proof.
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
"epoisses approach" should work for any length sides. And what is an "integer side" exactly ? Wouldn't that depend on an arbitrary choice of unit ? As far as I know, the only objection to that solution is that it employs algebra, unlike the classic solution given by Euclid.
It works for any length sides, I learnt it at high school from a maths teacher in his last year before retirement, who was no less than God to me at the time at least when it came to maths. So don't break my heart and burn down his proof please!
epoisses proof is shown on maui's link,
and as englishmuffin says it is not restricted to integers. i think the "integer" thought comes from it is easy to make a 90degree corner if you can make a 3unit length, a 4unit, and a 5 unit.
Actually, there is an even more elegant "proof" - requiring the addition of only one construction line - which can be found, among other places, in Roger Penrose's new book "The Road to Reality" (about the current state of the art of knowledge in physics and mathematics. Excellent book by the way.
I guess what you mean is draw a line from the 90 deg corner that lands at a 90 deg angle on the hypothenusa, then express the area of the large triangle as the sum of the areas of the smaller two expressed as ratios:
1/2*a*b = 1/2*a*b*b^2/c^2 + 1/2*a*b*a^2/c^2
This does not sound like a current state of art / cutting edge recent mathematical invention, if a dumb @$$ like me can repeat it after a simple hint?
PS ok I saw my proof back in one of the links, it appears that some Babelonian did it already long before Pythagore, that reassures me.
Something else: when I had worked for about 2 months as a chem eng, one of the operators asked for a simple formula to calculate the content of a horizontal drum as a function of the level height. I was glad to be able to use some math again and gave him an excel sheet with a few sines and cosines. My fellow engineers feel off their chairs with surprise. Then I wrote down an integral and worked it out to find another formula that did essentially the same but was more impressive. They fell off their chairs once again and stayed on the floor for about half an hour with astonishement.
Half a year later I found that formula back and was unable to tell how on earth I had done it. Right now (9 years later) I don't even try to imagine where my university calculus has gone, I wonder if I might be able to pick it up again.
I started graduate school 12 years after finishing under graguate. My most common comment was "I can remember having known that". It came back reasonably quickly.
can we leave pythagoras and agree that there are many ways to "skin a cat" (the trouble, of course, is finding one that will sit still long enough)?
can we return to the OP ? I personally think it is nice to be able to derive the equations i use. i think being able to allows me to understand the equations (and the assumptions) better. like the previous poster, david, there are many things i'm sure i use in daily practice that i've forgotten the derivation (lets take the parallel axis theorem, why does one axis have to be thru a CG ?) but i'm sure i could sit down with a book and understand the details (again). i don't think it's elitist to be able to do this, possibly others have an inferiority complex if they can't ??
A very interesting thread - Pythagarean's Theorum notwithstanding. I am a geotecthnical engineer. Did I ever go through the graphical and proof analysis of Bishop for slope stability by slices; yes, I remember seeing it done. Could I do it now. No. Would I want to - No. But I do know that it is well established. Similarly with bearing capacity computations. I understand well the overall development but the intricacies of the Nq and Ng and Nc derivations and differences between those of Terzaghi, Meyerhoff, Hansen are, well, outside my current need (desire?) to know. Again, it has been well established as to which is the most reasonable. Same for Nq values in pile design.
Now, on the application. As I said in other forums, I am not against computer programmes. I undestand that for many other disciplines, they are an absolute necessity. For my discipline, well, I can live without them (if I don't care about the time element). I do most of my calcs as back of the envelope and develop further if needed. I do most calcs, anyway, by hand for it is quicker than hunting for a computer programme that I don't have. Too many spend more time finding "a" programme for which they have no idea of the limitations and accuracies of such. For those with a good background in hand methods, they can be reasonably sure of preventing GIGO complications; but for those who are new to an analysis or may be stepping beyond their field and experience, it can be fraught with danger.
In the end - if you use time-honoured established methods, understand the assumptions and caveats on a particular method; have a experience to know that that your answer is not garbage, then it isn't necessary that you can derive your equation from first principals - each and every time!
Sorry, what I meant by integer sides is that you have to be able to prove that the area is always the square of the length of the side, which can only be done physically if you can fit (say) 25 unit squares into a 5x5 square. Otherwise you are assuming the analytical solution.
So can anybody prove that the area of a square is always L^2?
Cheers
Greg Locock
Please see FAQ731-376 for tips on how to make the best use of Eng-Tips.
![[cheers]](/data/assets/smilies/cheers.gif "[cheers] [cheers]")
