THE M SCALE IN POLISH SKALA SLIDE RULES Angel Crrso (Roger) Trnslted from Spnish y: Alvro González (Teruteru34) nd José Griel F dez -Rventós (e-lento).
To Mrtin, she s lmost here nd we re witing her imptiently. Mdrid, My 0
ACKNOWLEDGEMENTS I wnt to epress my sinere grtitude to my ollegues Álvro González nd José Griel Fernández, of the Spnish Assoition Amigos de ls Regls de Cálulo (ARC), for their detiled review of the mnusript, providing lots of suggestions nd omments, tht for sure hve enrihed the doument, nd for their kind trnsltion into English. Also I wnt to thnk Gonzlo Mrtín, for mking his wonderful olletion of Frenh rules ville to everyody. In prtiulr, I hve used the slide rules from Grphople rnd ville in his we www.photolul.om, to omplete the different models omprison. And, of ourse, I wish to thnk Jorge Fáregs, the true responsile for ll this, s he reted our Assoition, sed in his we www.reglsdelulo.om.
CONTENT.- INTRODUCTION 5.- SCALE DESCRIPTION 6 3.- SCALE FOUNDATIONS 6 4.- SCALE PLACEMENT IN THE SLIDE RULE 8 5.- SCALE OPERATION 6.- EXAMPLES 7.- SOLUTIONS WITH OTHER SLIDE RULES 8 8.- CONCLUSIONS 4 9.- BIBLIOGRAPHY 5
The M sle in SKALA Polish slide rules.- INTRODUCTION The solution for right tringle hs mny prtil uses in ny tehnil sope, s the trnsformtion of rtesin into polr oordintes or omple numers representtion in their polr form, defining its modulus nd rgument from the rel nd imginry prts respetively. The smll sle mrked with n M letter, ppering in models ELEKTRO SLE nd SLPP from polish Brnd SKALA, llow esy nd fst solutions to right tringles. It dditionlly gives etter ury thn the one otined with trditionl trigonometri sles usge. SKALA ELEKTRO SLE SKALA SLPP In this rtile it is desried the sle M in detiled form, it s eplined its usge nd desried the right tringle solution proedures with other types of slide rules through stndrd trigonometri sles, pplying sines theorem. In the neessrily redued spes of slide rules mnuls is not possile to eplin in detil the theoretil foundtions tht llow the solution of the different mthemtil opertions, giving only in its ple series of reipes. 5
The M sle in SKALA Polish slide rules We intend to eplin those foundtions in the elief tht it s esier to rememer the movements nd redings sequenes needed on the slide rules sles when the orresponding theoretil se is lredy known..- SCALE DESCRIPTION The smll sle M sits on the slide, etween the sles C nd CI, nd its length is of out m. The vlues of the M sle spn from 0 to nd inrese from right to left. The vlue 0 of M sle is just over the 5 of C sle, nd the vlue of M is over the 4,4 of C. 3.- SCALE FOUNDATIONS In right tringle where its legs re nd, the vlue of the hypotenuse is esily otined through the diret pplition of the well known Pythgors theorem, s: Tht sme reltionship n e epressed in the following form: Being: M 6
The M sle in SKALA Polish slide rules 7 Where: M And In this quotient, s is the smller of oth legs, it is nmely lwys true: The proof tht oth epressions re relly equivlent is very simple, it s enough to ehnge the vlue of in M nd lter the M vlue in. Following the proedure would e: M And Operting: Sustituting this vlue of Δ, in the first eqution eomes: Nmely And finly
The M sle in SKALA Polish slide rules Whih demonstrtes the equlity of the two equtions proposed. By pplying this proedure nd using the M sle, the vlue of Δ is otined diretly nd, y simple ddition, the hypotenuse of the tringle. 4.- SCALE PLACEMENT IN THE SLIDE RULE A reltively interesting spet whih deserves some ttention is the position of the sle M in the rule. First, this is not onventionl full-length sle, ut develops from out 4.4 to 5 vlues of the C sle, hving, s lredy mentioned, length of m. As outlined elow, this position is onditioned y the minimum nd mimum vlues tht M n tke s funtion of the vlue of. The vlue of is the rtio etween the smllest nd lrgest leg or wht is the sme, the vlue of the tngent of the ngle etween the hypotenuse nd the higher leg, whih we will ll hereinfter, nd therefore n only tke vlues etween "0" nd "", sine the vlue of the ngle shll not eeed 45 degrees. The vlue = orresponds to the se where nd re equl, ie when the tringle, dditionlly to eing right is lso isoseles. In this se it s esy to see tht M vlue is: M 0.44 Wht is not so ovious is tht when the vlue of is infinitely smll, prtilly lmost zero, M = 0.5. To eplin this we need to dip into the infinitesiml lulus nd the theory of limits. Indeed, to lulte the vlue tken y M when is infinitely smll, it is neessry to lulte the limit of the funtion M, where hs vlue lose to zero, ie: lim 0 8
The M sle in SKALA Polish slide rules In relity this sitution would never e met nd would orrespond to the theoretil se in whih = 0. In this se it s ovious the tringle n t e onstruted, ut for very smll vlues of versus, we would otin vlues of very lose to zero. If =0 in the funtion M we otin wht in the theory of limits is known s type 0/0 indeterminy. Suh unertinties in the theory of limits, re solved y pplying the L'Hôpitl rule. This rule is nmed fter the 7th entury Frenh mthemtiin Guillume Frnçois Antoine, Mrquis de L'Hôpitl (66-704), who mde the rule known in his "Anlyse des infinement petits pour l'intelligene des lignes oures" in 69. The wording of the rule of L'Hopitl is essentilly s follows: If f() nd g() re two differentile funtions (ie, the derivtive funtion eists) in the neighorhood of point p in whih: f (p) = g (p) = 0 And dditionlly it is true tht the derivtive funtion g() t point p is not null, ie g (p) 0 9
The M sle in SKALA Polish slide rules Then, if it eists It lso eists f '( ) lim p g'( ) lim p f ( ) g ( ) And they re the sme, ie f '( ) lim p g'( ) = lim p f ( ) g ( ) Applying the L Hôpitl rule to funtion M, hving in this se p=0, it is: f ( ) g ( ) Their derivtive funtions re: f '( ) g' ( ) Then, dividing oth numertor nd denomintor y, finlly it omes to: f '( ) lim g'( ) 0 0.5 In summry, the etreme vlues of the funtion M re: 0,44 for = nd 0.5 for = 0, nd this eplins the position of this sle in the rule, diretly onfronted with the vlues etween 4.4 nd 5.00, respetively, of the C sle. The et length of the sle M, for rule with sles of 50 mm is: L M = (log 5.00 log 4.4)*50 = 0.49 mm 0
The M sle in SKALA Polish slide rules 5.- SCALE OPERATION This setion desries in detil the proedure to e followed for the resolution of right tringle y using the sle M. In the first ple nd using sles C nd D, lulte the vlue of = tn( ), tking into ount tht orresponds to the smllest of the legs. This is done esily y pling the vlue of, on the C sle over the vlue of, on the sle D. Under the of the C sle, you n red the vlue of tn( ). The orresponding ngle α is otined diretly on the T sle, ering in mind tht these rules hve ngulr sles in minutes rther thn tenths of degrees. C D T tn If to otin the vlue of, the sle M is pled outside the rnge of the D sle, with the help of the ursor the slide should e moved ehnging the left of the C sle for the 0 to the right of the sme sle, s in ny other lultions with the rule. From here the proedure is the sme. Using the ursor nd without moving the slide, move to the vlue otined for on the sle M. On the D sle will then red the result of the produt M M Note tht in relity the funtion vlue of M is red on the C sle nd not on the M sle itself. C M tg M D M.
The M sle in SKALA Polish slide rules To finlly otin the vlue of, is still needed to multiply the ove result gin y. It is more onvenient to use the sle CI, ie to divide y the inverse of. To do this, without moving the ursor, move the slide to red under its min line the vlue of on the reiprol sle CI. The vlue of my e red diretly on the D sle, under the of C sle. D CI C Now we just dd to this vlue the one of, to otin the hypotenuse. When the vlue of is very smll nd therefore diffiult to ple on the M sle with some preision (sine in this re the divisions of the sle re very lose together), it n e more effiient to pproimte the vlue of Δ with the epression: Sine, s mentioned ove, the vlue of the M funtion is lose to / where is lose to zero. Furthermore, this ltter term is used, in ny se, to determine the deiml position on the vlue of, following the known rules of the estimtive lultions, s seen in the following emples. 6.- EXAMPLES Emple. To solve the tringle whose legs re: = 7,35 nd = 8,65 Solution: In the first ple we lulte the vlue of = tn α:
The M sle in SKALA Polish slide rules On T sle we n red the vlue of ngle α, eing in this se: α = 40º Trnsferring the result of, on the sle M, we n red on the C sle, the vlue of M, in this se: M = 0,43 On the D sle we n e red, ut only hs edutionl interest, the intermedite produt: M M 3
The M sle in SKALA Polish slide rules But to finlly get the vlue of, we still need to multiply the ove result gin y. It is esier, however, to divide y the inverse of, for whih, without moving the ursor, vlue is mthed, on the CI sle, with the min line of the ursor. Under the of the C sle, the vlue of Δ n e red on D sle. In this se: Δ =,7 The deiml position hs een otined y the following proimtion lultion: 7 0,8,8 As seen, it is not neessry to red the intermedite results of opertions for otining the vlues of M or M. To get the vlue of the hypotenuse it is only needed to dd the vlue of Δ to the longest leg, tht is: = Δ + =,70 + 8,65 =,35 (et vlue: =.35) 4
The M sle in SKALA Polish slide rules Emple. To solve the tringle whose legs re: = 7,0 y =,85 Solution: As in the previous se, we lulte the vlue of = tn α: On T sle we n red the vlue of ngle α, in this se: α = 9º 5 In this se the sle M is outside the rnge of C sle nd therefore it is neessry to trnspose the slide. To do this, without moving the ursor (whih is lredy pled on the of C sle), the slide is fully moved to mth the 0 of this sle with the min line of the ursor. 5
The M sle in SKALA Polish slide rules From here the proedure is etly the sme s in the previous emple, otining the following results: M = 0,466 Δ =,88 Where the deiml position hs een otined y following pproimte lultion: 7 0,5,75 And finlly: = Δ + =,88 +,85 = 4,73 (et vlue = 4.73) Emple 3. To solve the tringle whose legs re: =,86 y = 9,0 6
The M sle in SKALA Polish slide rules Solution: Now the vlue of = tn α is: On the T sle we n red the vlue of ngle α: α = 8º 8 Here, where the vlue of is very smll, it is diffiult to urtely ple it on the sle M. Then it is esier nd suffiiently pproimtion to lulte:,86 0,49 0,3 In this se the deimls ome diretly from this lst lultion, nd it is not neessry ny pproimte estimtion to find the deiml point position. And finlly: = Δ + = 0,3 + 9,0 = 9,43 (et vlue = 9.4) 7
The M sle in SKALA Polish slide rules 7.- SOLUTIONS WITH OTHER RULES The resolution of tringle with slide rule n e rried out esily y pplying rtios of sines theorem. sen sen sen Where, nd re the sides of the tringle nd, side respetively. nd re the ngles opposite eh For right tringle, with the legs nd, nd the hypotenuse, then: And so: = 90º sin = In ddition nd ngles re omplementry, ie the two of them dd up to 90 º, nd thus: sin os = os = sin Given the ove, the lw of sines for right tringle, n lso e written s follows: sen os Multiplying ll terms y sin : tn sin Although the theory disussed ove is the sme, the prtil proedure, s disussed elow, depends on where the trigonometri sles S nd T re pled in the slide rule. The resolution of tringles, s eplined so fr, is not possile when the vlues of the sle S re red ginst the sle of squres A/B insted of on the sles C/D. In some ses it will e neessry to trnspose the slide, hnging with the help of the ursor, the left inde (the of C sle) for the right one (0 of the sme sle) or vievers. 8
The M sle in SKALA Polish slide rules S nd T sles on the ody of the rule This rrngement is quite ommon in Europen rules, ut rules from the Frenh rnd Grphople nd some Aristo models hve these sles loted on the slide. FABER-CASTELL 5/8 ARISTO 0968 STUDIO To mke use of proportions in this type of slide rule, it is more interesting to epress the ove equtions ording to the inverse of, nd, whih would finlly give: tg sen / / / The operting proedure with the slide rule is s follows: We ple the vlue of (rememer tht is the shorter leg) on the sle CI, ove the 0 on the sle D. Move the ursor to the vlue of on the CI sle. On D sle we red the vlue of tn α nd on the T sle we n red the vlue of the ngle α. T D CI tn 9
The M sle in SKALA Polish slide rules In some ses it is neessry to ompletely disple the slide to reple the 0 y the on D sle. Without moving the slide, move the ursor to red the sme ngle α on the S sle, llowing to red on the CI sle the vlue of the hypotenuse. S CI S nd T sles on the slide This rrngement is quite ommon in Amerin-mde rules nd in the Frenh rnd Grphople. PICKETT N-500-ES LOG LOG KEUFFEL & ESSER 68-00 DECI-LON In this se it is more onvenient to rememer tht the sines lw for right tringle n e epressed lso in its inverse form s: sin os 0
The M sle in SKALA Polish slide rules From this, the only eqution with prtil vlue for the resolution with slide rule is: sin Furthermore, dividing this epression y os, it gives tn os And similrly, here only the first equlity is of interest: tg To solve the tringle y suh epressions we ple the 0 of the C sle over the vlue of ( is the vlue of the longer leg) on D sle. Over the vlue of in the D sle, it n e red the vlue of tn on the C sle, nd on T sle (whih is now on the slide) the vlue of the ngle. T C tg D Without moving the ursor, move the slide to ple the vlue of from S sle elow the enter line thereof. Then under the of the C sle, we n red the vlue of the hypotenuse in D sle. In some ses it is neessry to ompletely disple the slide to reple the 0 y the of C sle. S C D S nd T sles on the k of the slide This set of sles is ommon on simple rules, in whih the only sles on the k re the trigonometri sles rrnged in the slide.
The M sle in SKALA Polish slide rules ARISTO 99 RIETZ GRAPHOPLEX 60 In these ses it is more prtil to write the ove reltions s follows: And tn sin With the equtions presented in this wy, the opertion proedure with the rule would e s follows: We ple the vlue of the smller leg, from the C sle, over the vlue of the longer leg, from D sle. Then, over the of the D sle, we n red the vlue of tn D C tg In this position we n e red on the k of the rule, on the T sle of the slide, the vlue of ngle. T
The M sle in SKALA Polish slide rules We move the slide to red the vlue of on the S sle. S Keeping this position of the slide nd turning round the rule, on the front nd elow the vlue of from C sle, you n red the vlue of the hypotenuse, on the D sle. D C 3
The M sle in SKALA Polish slide rules 8.- CONCLUSIONS In order to stlish riteri so tht to ompre the different sle lyouts, summry tle is inluded with the numer of movements of the ursor nd the slide needed to solve eh one of the emples shown. This tle is to evlute the effiieny of eh of the different sle lyouts presented. For ounting the movements, the folded sles hve not een onsidered, s these re not ville in ll slide rules. This type of sles, in some ses, mkes the hnge from left to right end of the sle unneessry, reduing the numer of movements. In those rules with sles in the k of the rule, models with reding windows t oth ends hve een used for the ounting of movements, so tht the reding is ville either t the right or the left. On the ontrry, some other hnge from left to right sle ends might e needed, dding to the totl numer of movements required. EXAMPLE = 7.35 =8.65 EXAMPLE = 7.0 =.85 EXAMPLE 3 =.86 = 9.0 MOVEMENTS CURSOR SLIDE TOTAL SKALA 4 6 S, T in the ody 4 6 S, T in the slide front 3 5 S, T in the slide k 4 3 7 SKALA 4 3 7 S, T in the ody 4 6 S, T in the slide front 3 5 S, T in the slide k 3 5 SKALA 3 5 S, T in the ody 4 6 S, T in the slide front 3 5 S, T in the slide k 3 5 As n e ppreited in the preeding tle, there re no ig differenes in the numer of movements from one sle lyout to the other. The slight differenes re due to the need to hnge of slide ends in some ses. This differene depends on the prolem initil dt rther thn on the speifi sle lyout. No sle lyout hs een found where the hnge of slide ends hd not een needed t some time. Regrding the preision of the result, s in ll slide rules, it is silly dependnt on the length of the sles nd not on their lyout. The igger the length, the greter is the distne etween sle divisions nd thus the respetive redings re esier. 4
The M sle in SKALA Polish slide rules All models used hve 5 m length sles (eept for M sle, oviously), nd so this does not ring ny differene in regrds to results preision. In reltion with the lultion of ngle, the proedure is the sme no mtter the sle lyout of the slide rule, tht is, first the vlue of tg is found nd, fterwrds, the vlue of the ngle is found y mens of T sle. Therefore, the preision in the lultion of rules nd is not dependnt on the sle lyout. ngle is etly the sme in ll slide The sustntil differene ppers in the lultion of the hypotenuse. In the rules without M sle, independently from the sle lyout, the net step is to trnsfer the vlue of to the S sle. The preision of the result depends on how this djustment is done. It is esy to see, in this type of slide rules, tht the preision inreses with the derese of the vlue of the ngle, s the divisions in the S sle re lrger for smll ngles. For vlues of over 5º (gin, it is onvenient to rememer tht is never greter thn 45º) it is not esy to position the vlue of the ngle in the S sle with enough preision. If the ngle hs deimls of degree, s it is usul, the diffiulty is gretly inresed. When the M sle is used, the vlue to use is not ut tg, to e pled on M sle. The preision hieved with the use of this sle is muh greter, espeilly for ig ngle vlues. For vlues of smller thn 0º the preision otined is similr oth with the trditionl trigonometri sles nd with the M sle. Also, s it hs een shown in emple numer 3, even in the ses of very low tg vlues, where it is somehow diffiult to preisely djust the M sle, the simplified method presented provides vlues very proimte to the et vlue. To sum up, the retngle tringle solving with SKALA slide rules where the M sle is inluded, the vlue otined for the ngle hs the sme preision s when otined with ny other rule, ut the differene in the preision for the lultion of the hypotenuse is igger s igger is the vlue of. 9.- BIBLIOGRAPHY.- Additionl M sle on polish SKALA SLE nd SKALA SLPP 0 inhes slide rules. Adm Entrnt Pplinski. Deemer 00..- The Slide Rule. A Complete Mnul. Alfred L. Slter. Los Angeles City College. Holt, Rinehrt nd Winston, In. 967 3.- Aristo, Fer-Cstell nd Grphople user mnuls. 5