Sommario:
+ Contare
+ Sistemi di numerazione
+ Addizione e Sottrazione
+ Moltiplicazione
+ Divisione
+ Radice Quadrata
+ Elevamento a potenza
+ Logaritmi
- the slide rule
a magical tool
Origins
Double logarithmic
the regular "modern"
Advanced Calculations
few curious detail
a magical tool
I always thought that the slide rule was an instrument of calculation rather limited ... until I found my father's library in this book of 1936 so as to reveal all the secrets:
It took me a few days to understand, but now I can say ... that was truly a wondrous instrument, capable of making calculations of unexpected complexity. ▲
Le origini
Prima di iniziare a parlare del regolo calcolatore, riscrivo la definizione di logaritmo (di cui ho parlato nel capitolo precedente ):
Il logaritmo decimale di un numero è l'esponente a cui elevare la base 10 per ottenere il numero dato.Ecco come si esprime questo concetto in formule per due numeri N1 e N2, ma anche per il loro prodotto:
Fra le proprietà delle potenze (di cui invece ho parlato qui ) c'è quella per cui il prodotto di due potenze di pari base è la stessa cosa di una potenza della stessa base con esponente uguale alla somma dei due esponenti di partenza. Quindi:
Dalle formule qui sopra risulta quindi che il logaritmo del prodotto è uguale alla somma dei logaritmi dei fattori. Il "trucco" alla base del funzionamento del regolo calcolatore è proprio il fatto che siamo riusciti a trasformare un prodotto in una somma!
A seguito degli studi di Nepero sui Logaritmi ci fu subito chi pensò di sfruttare l'idea in modo da velocizzare i calcoli, anche a scapito della precisione. Già nel 1623 Edmund Gunter, professore di astronomia al Gresham College di Londra, sviluppa una scala logaritmica sulla quale, con l'aiuto di un compasso, si possono eseguire graficamente moltiplicazioni e divisioni. Ecco... ma cos'è esattamente una scala logaritmica ?
Si tratta di un righello in cui si riportano tacche a distanze proporzionali ai logaritmi dei numeri da 1 a 10. Nel diagramma sopra specifico che ogni tacca corrisponde al logaritmo del numero, ma la sigla "log" non è assolutamente necessaria:
Notare che a destra si scrive un 1 e non un 10: questa è una pratica utilizzata in tutti i regoli calcolatori. In pratica "si sa" che all'uno di destra corrisponde un 10; inoltre, come we shall see, in some cases the right one is used just as ... 1 and not 10!.
is built on a logarithmic scale! Now suppose you want to multiply by 1.5 to 4 with this scale and a compass, just like the Gunter: just open the compass with an opening corresponding to the logarithm of 1.5 and contain the same opening at 4. We see the process step by step:
The point of the compass is positioned right on the mark of 1.5. The left point to be put on a notch, because remember that this mark is the logarithm of 1, and the logarithm of 1 is 0. In this way the opening of the compass is the difference between the logarithm of 1.5 and a log of the following:
log (1.5) - log (1) = log (1.5) - 0 = log (1.5)
Once you have found the opening of the compass , just passing it to the right:
is that by putting the left point of the compass on the rear of 4 I find myself right on the tip 6 (remember that the logarithm of 1 is 0) :
log (4) + [log (1.5) - log (1)] = log (4) + log (1.5) = log (4 x 1.5) = log (6)
(that is 1.5 x 4 = 6).
With the same exact positions I could have done the inverse calculation: in fact with the same opening corresponding to the number 1.5, I could make the division 6: 4 = 1.5: Whereas the right to put the tip of the compass on the 6, the tip of my left would have given the quotient correctly sought (the difference of the exponents is in fact the division of powers).
log (6) - [log (1.5) - log (1)] = log (6) - log (1.5) = log (6: 1.5) = log (4) ▲
Double logarithmic
In 1630 Edmund Gunter Wingate uses two scales at each other to perform multiplication and division directly, without using the compass. Let the multiplication 1.5 x 3:
The red area in the lower scale, has the same amplitude of the compass that we have seen above. By starting the source (ie the mark of 1) the scale than just the end of the red zone, I see that the right edge of the area falls on the blue 4.5: in fact 1.5 x 3 = 4, 5.
This procedure is also good to calculate numbers with different orders of magnitude: eg. 15 x 300 = 4500; In these cases the zeros more or less, or any movement of the decimal point must be made by hand (in this sense the errors were still there ... it was necessary to be very, very careful and the ideal was more or less understand what was the result before calculating it, and try to adjust only the accuracy of significant digits).
And if you want to calculate 5 x 3? Here comes a problem, because the top three on the ladder is positioned outside the lower scale, so I can not read the result:
In these cases it uses a trick: instead of multiplying by 3 multiply by 0.3:
La zona marcata in azzurro qui sopra indica la differenza fra il logaritmo di 3 e il logaritmo di 10, quindi
log( 3 ) – log( 10 ) = log( 3 : 10 ) = log( 0,3 )
Allora basta posizionare, sul 5 della scala inferiore, non la tacca dell'uno sinistro della scala superiore, ma la tacca dell'uno destro :
In questo modo la moltiplicazione 5 x 0,3 si riesce a fare senza uscire dalle scale, e il risultato si legge sul 1,5 della scala inferiore; ma siccome abbiamo moltiplicato per 0,3 e non per 3, occorre "aggiustare" il risultato multiplying by 10: The final result is 15!
I said that among the properties of powers (and logarithms) is one for which the division of powers is the subtraction of exponents. So I can calculate the Division 6: 4
From the graph above we see that on 5 of lower scale but not the one I put on 4 of the top scale, one of the top scale is then on the left and the red and blue areas are "escape" here is the mark of a higher scale of 1.5 falls on a smaller scale, determining the correct result.
Let us now calculate the "standard" (Which I've actually found this to be just reading the book I mentioned in the caption of the photo): multiplication and division in one fell swoop! In practice the calculation of a proportion: a number multiplied by a fraction of which the numerator and denominator data, such as the calculation of the three means of five (5 x 3 / 2).
aligning two of the top scale with the scale of less than 5 of which have the mark of a left upper scale indicates the ratio 5: 2 = 2.5. The green and blue areas along the damage factor of 3, then it is as if you were multiplying by 3 the ratio 5: 2. In fact, the top three on the scale of 7.5 falls right on the bottom, which is the exact result!
Obviously with the divisions and mixed with the calculations (multiplication and division together) could happen to leave "out of scale", but there is always a way to fix the stairs to get the result, as we saw for multiplication. ▲
the regular "modern"
With subsequent improvements, and the fact that from the mid-nineteenth century, the precision engineering industry has allowed the construction in series, the slide rule has become a tool for calculating substantially standardized and widely available. Slide rules consist of the following elements:
- a body on which the scales are fixed
- sliding rod with escalators
- a slider with one or more reference lines
scales are of various types, conventionally referred to by some letters. In simple steps, such as the Wingate there are always two, a sliding rod (scale C) and the other on the body (D ). Other scales are used to simplify the calculations when you are in the presence of squares, cubes, square and cube roots, trigonometric functions ... etc. .. The scales are usually sorted out between the front and back of the rod and the body of the ruler, in this rule are the steps:
K - Scale the cube (body )
A - Units of squares (body)
B - Scale of squares (auction)
ST - Breasts and Bribes for small angles (Auction) T
- Scale of tangents for angles> 6 ° (pole vault)
S - Scale of the breasts (and cosines) for angles> 6 ° (pole )
C - Scale of numbers (auction)
D - Scale of numbers (the body)
OF - Units of the inverse of the numbers (1 / x) ( body)
the reverse auction, at the top, there are three other scales:
CI - Inverse scale of numbers (1 / X)
CF - scale "bent back", starting from pi-greek instead of 1
CIF - inverse scale, which starts by pi-greek
For greater clarity of the photos that follow, setting out the calculations so that all numbers are in a short stretch of straight edge, and no escape from the stairs, also for the sake of brevity I do not explain how to determine the zeros or decimal point: my aim is only to show what were capable engineers when they were using a straight edge! ▲
Calcoli avanzati
Abbiamo visto che con due semplici scale si riesce a fare un'operazione mista di moltiplicazione-divisione in un colpo solo. E se invece volessi fare una divisione-divisione? Per fare questo bisogna ricondurre l'operazione che vogliamo fare a quella "standard" di moltiplicazione-divisione usando la scala dei reciproci; con questa scala si converte la divisione per B in una moltiplicazione per 1 / B :
In verde indico i numeri di partenza e il risultato reale (calcolato con la calcolatrice); in rosso il risultato ottenuto with the straight edge. In making the calculation of all operands must be traced to numbers between 1 and 10, then 2100 becomes 2.1, 1.7 and 17 becomes 86 becomes 8.6. Let us see how to use the stairs of the slide rule:
It starts by superimposing the value C (17) C value of the scale A (2100) of the D scale, as we have seen, the notch 1 left the auction falls on the quotient A / C. Move the cursor to the value B (86) of the scale we multiply this quotient by means of the inverse B, then divide this quotient still B . The reading of the cursor on the scale Q gives the desired result.
Now we see a variant of the previous calculation, a multiplication-multiplication. Again we must bring the desired operation to the standard case, then proceed two factors multiplying and dividing by the reciprocal of the third, as shown in formulas:
Overlap value B (65) of the stairs to the value A (19) of the D scale A means to divide by the reciprocal of B: To make this alignment using the cursor, because the CI and D scales are not adjacent. Now read the number on the scale corresponding to the value D C (12) means multiplying the C scale again for C here found the result you want!
Another type of calculation: a simple division, but the number is a square root.
Move the cursor to the value A (350) At the scale of the squares (this position is on the normal scale D, the value 18.7 which is precisely the root square 350). If we now match the value B (1.51) the C scale of the cursor on the line, I will have the value of the D scale at the left of the auction is a division of the square root A for B, which is the result.
Finally, a very complex calculation, which requires the use of four different scales. It is the relationship between a cube root and square root, all multiplied by a normal number. To make this calculation must fit the shaft of the slide rule to the contrary, in order to provide the scale of the square rod (scale B).
You place the cursor on the value A (7400) on the scale of the cubes K (this position is on the normal scale D, the value of 19.487 that it is the cube root of 7400). He then moved the rod to align the cursor value B (290) Scale B. At this point the value of the D scale at the left of the auction is a division of the cube root of A the square root of B , but if I try the value C (1.3) on the scale C, the corresponding value on the scale D will be the product of the quotient previous C, which is the value sought. ▲
few curious detail
There are endless variations of these types of computation that can involve trigonometric functions, logarithmic, exponential ... I now understand how to design engineers did what they were able to build over the last two centuries before the advent of electronics (at least until the early 70s of last century).
Among the nineteenth and twentieth century were built of massive proportions even slide rules: he had climbed two meters in length, and a microscope mounted on the slider. With these tools monumental could appreciate up to 6 significant digits, both in the operands that results!
last interesting note: when the astronauts took to the Moon (1969), electronic calculators had not yet been invented, in fact the first scientific calculator (the HP35) is only 1972. Then the astronauts went to the moon ... carrying a slide rule! The manufacturer of this rule "extraterrestrial" in fact written on the packaging of its rules: "5 moon flights, five flights to the Moon! ▲
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