,,the ,,slide ,,rule3 ,a ,,practical ,,manual ,,by ,,charles ,n4 ,,pickworth ,,whitworth ,,scholar2 ,,editor ,,of 8,,the ,,mechanical ,,world02 ,,author ,,of 8,,logarithms ,,for ,,beginners02 8,,the ,,indicator3 ,,its ,,construction ,,and ,,application02 8,,the ,,indicator ,,diagram3 ,,its ,,analysis ,,and ,,calculation10 ,,etc4 ,,seventeenth ,,edition ,,manchester3 ,,emmott ,,and ,,co41 ,,limited1 #fe ,,king ,,street2 ,,new ,,york3 ,d4 ,,van ,,nostrand ,,co41 #h ,,warren ,,street4 ,,london3 ,,emmott ,,and ,,co41 ,,limited1 #bj ,,bedford ,,street1 ,w4,c4 ,,and ,,pitman ,,and ,,sons1 ,,limited1 ,,parker ,,st41 ,,kingsway1 ,w4,c4 #b4 ,three ,shillings and ,sixpence net ,all rights reserved4 ,,preface ,,to ,,the ,,fifteenth ,,edition4 ,several new slide rules for special calculations are described in this edition1 and the contents further extended to include a section dealing with screw-cutting gear calculations by the slide rule--an application of the instrument to which attention has been given recently4 ,mention should be made of the fact that some of the special slide rules described in previous editions are no longer obtainable4 ,as1 however1 the descriptive notes may be of service to those possessing the instruments1 and are1 in some measure1 of general interest1 they have been allowed to remain in the present issue4 ,the author tenders his thanks to the many who have evinced their appreciation of his efforts to popularise the subject2 also for the many kind hints and suggestions which he has received from time to time1 and with a continuance of which he trusts to be favoured in the future4 ,c4 ,n4 ,p4 ,,withington1 ,,manchester1 ,november #aiag4 ,,preface ,,to ,,the ,,seventeenth ,,edition4 ,the sustained demand for this very successful work having resulted in the early call for a new edition1 the opportunity has been taken to introduce descriptions of new slide rules and to effect some slight revisions4 ,c4 ,n4 ,p4 ,,withington1 ,,manchester1 ,december #aibj4 ,,contents4 ,,page ,introductory #e ,the ,mathematical ,principle of the ,slide ,rule #f ,notation by ,powers of #aj #h ,the ,mechanical ,principle of the ,slide ,rule #i ,the ,primitive ,slide ,rule #aj ,the ,modern ,slide ,rule #ab ,the ,notation of the ,slide ,rule #ad ,the ,cursor or ,runner #ag ,multiplication #ai ,division #bd ,the ,use of the ,upper ,scales for ,multiplication and ,division #bf ,reciprocals #bg ,continued ,multiplication and ,division #bh ,multiplication and ,division with the ,slide ,inverted #cj ,proportion #ca ,general ,hints on the ,elementary ,uses of the ,slide ,rule #cf ,squares and ,square ,roots #cg ,cubes and ,cube ,roots #dj ,miscellaneous ,powers and ,roots #de ,power and ,roots by ,logarithms #de ,other ,methods of ,obtaining ,powers and ,roots #dg ,combined ,operations #di ,hints on ,evaluating ,expressions #eb ,gauge ,points #ec ,examples in ,technical ,calculations #ef ,trigonometrical ,application #gd ,slide ,rules with ,log-log ,scales #hd ,special ,types of ,slide ,rules #ib ,long-,scale ,slide ,rules #if ,circular ,calculators #aja ,slide ,rules for ,special ,calculations #aji ,construction ,improvements in ,slide ,rules #aaj ,the ,accuracy of ,slide ,rule ,results #aaa ,appendix3-- ,new ,slide ,rules #aac ,the ,solution of ,algebraic ,equations #abb ,screw-,cutting ,gear ,calculations #abd ,gauge ,points and ,signs on ,slide ,rules #abf ,tables and ,data #abh ,slide ,rule ,data ,slips #acc ,,the ,,slide ,,rule4 ,,introductory4 ,the slide rule may be defined as an instrument for mechanically effecting calculations by logarithms4 ,those familiar with logarithms and their use will recognise that the slide rule provides what is in effect a concisely arranged table of logarithms1 together with a simple and convenient means for adding and subtracting any selected values4 ,those1 however1 who have no acquaintance with logarithms will find that only an elementary knowledge of the subject is necessary to enable them to make full use of the slide rule4 ,it is true that for simple slide-rule operations1 as multiplication and division1 a knowledge of logarithms is unnecessary2 indeed1 many who have no conscious understanding of logarithms make good use of the instrument4 ,but this involves a blind reliance upon rules without an appreciation of their origin or limitations1 and this1 in turn1 engenders a want of confidence in the results of any but the simplest operations1 and prevents the fullest use being made of the instrument4 ,for this reason a brief1 but probably sufficient rsum of the principles of logarithmic calculation will be given4 ,those desiring a more detailed explanation are referred to the writer's 8,logarithms for ,beginners40 ,the slide rule enables various arithmetical1 algebraical and trigonometrical processes to be performed with ease and rapidity1 and with sufficient accuracy for most practical purposes4 ,a grasp of the simple fundamental principles which underlie its operation1 together with a little patient practice1 are all that are necessary to acquire facility in using the instrument1 and few who have become proficient in this system of calculating would willingly revert to the laborious arithmetical processes4 ,,the ,,mathematical ,,principle ,,of ,,the ,,slide ,,rule4 ,logarithms may be defined as a series of numbers in arithmetical progression1 as #j1 #a1 #b1 #c1 #d1 etc41 which bear a definite relationship to another series of numbers in geometrical progression1 as #a1 #b1 #d1 #h1 #af1 etc4 ,a more precise definition is3--,the logarithm of a number to any base1 is the index of the power to which the base must be raised to equal the given number4 ,in the logarithms in general use1 known as common logarithms1 and with which we are alone concerned1 #aj is the base selected4 ,the general definition may therefore be stated in the following modified form3--,the common logarithm of a number is the index of the power to which #aj must be raised to equal the given number4 ,applying this rule to a simple case1 as #ajj #ajb1 we see that the base #aj must be squared 7i4e41 raised to the #bnd power7 in order to equal #ajj1 the number selected4 ,therefore1 as #b is the index of the power to which #aj must be raised to equal #ajj1 it follows from our definition that #b is the common logarithm of #ajj4 ,similarly the common logarithm of #ajjj will be #c1 while proceeding in the opposite direction the common log4 of #aj must equal #a4 ,tabulating these results and extending1 we have3-- ,numbers #aj1jjj #ajjj #ajj #aj #a ,logarithms #d #c #b #a #j ,it will now be evident that for numbers between #a and #aj the logs4 will be between #j and #a #aj #ajj #a #b #ajj #ajjj #b #c #ajjj #aj1jjj #c #d ,in other words1 the logarithms of numbers between #a and #aj will be wholly fractional 7i4e41 decimal72 the logs4 of numbers between #aj and #ajj will be #a followed by a decimal quantity2 the logs4 of numbers between #ajj and #ajjj will be #b followed by a decimal quantity1 and so on4 ,these decimal quantities for numbers from #a to #aj 7which are the logarithms of this particular series7 are as follows3-- ,numbers #a #b #c #d #e #f #g #h #i #aj ,logarithms #j #jcja #jdgg #jfjb #jfii #jggh #jhde #jijc #jied #ajjj ,combining the two tables1 we can complete the logarithms4 ,thus for #c multiplied successively by #aj1 we have3-- ,numbers #c #cj #cjj #cjjj #cj1jjj etc4 ,logarithms #jdgg #adgg #bdgg #cdgg #ddgg ,we see from this that for numbers having the same significant figure 7or figures71 #c in this case1 the decimal part or mantissa of the logarithm is the same1 but that the integral part or characteristic is always one less than the number of figures before the decimal point4 ,for numbers less than #a the same plan is followed4 ,thus extending our first table downwards1 we have3-- ,numbers #a #ja #jja #jjja #jjjja etc4 ,logarithms #j #a #b #c #d so that for #c divided successively by #aj1 we have3-- ,numbers #c #jc #jjc #jjjc #jjjjc etc4 ,logarithms #jdgg #adgg #bdgg #cdgg #ddgg ,here again we see that with the same significant figures in the numbers1 the mantissa of the logarithm has always the same 7positive7 value1 but the characteristic is one more than the number of #j's immediately following the decimal point1 and is negative1 as indicated by the minus sign written over it4 ,only the decimal parts of the logarithms of numbers between #a and #aj are given in the usual tables1 for1 as shown above1 the logarithms of all tenfold multiples or submultiples of a number can be obtained at once by modifying the characteristic in accordance with the rules given4 ,an examination of the two rows of figures giving the logarithms of numbers from #a to #aj will reveal some striking peculiarities1 and at the same time serve to illustrate the principle of logarithmic calculation4 ,first1 it will be noticed that the addition of any two of the logarithms gives the logarithm of the product of these two numbers4 ,thus1 the addition of log4 #b and log4 #d #jcja #jfjb #jijc1 and this is seen to be the logarithm of #h1 that is1 of #b #d4 ,conversely1 the difference of the logarithms of two numbers gives the logarithm of the quotient resulting from the division of these two numbers4 ,thus1 log4 #h log4 #b #jijc #jcja #jfjb1 which is the log4 of #d1 or of #h #b4 ,one other important point is to be noted4 ,if the logarithm of any number is multiplied by #b1 #c1 or any other quantity1 whole or fractional1 the result is the logarithm of the original number1 raised to the #bnd1 #crd1 or other power respectively4 ,thus1 multiplying the log4 of #c by #b1 we obtain #jdgg #b #jied1 and this is seen to be the log4 of #i1 that is1 of #c raised to the #bnd power1 or #c squared4 ,again1 log4 #b multiplied by #c #jijc--that is1 the log4 of #h1 or of #b raised to the #crd power1 or #b cubed4 ,conversely1 dividing the logarithm of any original number by any number n1 we obtain the logarithm of the nth root of the original number4 ,thus1 log4 #h #c #jijc #c #jcja1 and is therefore equal to log4 #b or to the log4 of the cube root of #h4 ,only simple logs4 have been taken in these examples1 but the student will understand that the same reasoning applies1 whatever the number4 ,thus for #bjc we prefix the characteristic 7#a in this case7 to log4 #b1 giving #acja4 ,multiplying by #c1 we have #cijc as the resulting logarithm1 and as its characteristic is #c1 we know that it corresponds to the number #hjjj4 ,hence #bjc #hjjj4 ,in this brief explanation is included all that need now be said with regard to the properties of logarithms4 ,the main facts to be borne clearly in mind are3--7#a47 ,that to find the product of two numbers1 the logarithms of the numbers are to be added together1 the result being the logarithm of the product required1 the value of which can then be determined4 7#b47 ,that in finding the quotient resulting from the division of one number by another1 the difference of the logarithms of the numbers gives the logarithm of the quotient1 from which the value of the latter can be ascertained4 7#c47 ,that to find the result of raising a number to the nth power1 we multiply the logarithm of the number by n1 thus obtaining the logarithm1 and hence the value1 of the desired result4 ,and 7#d47 ,that to find the nth root of a number1 we divide the logarithm of the number by n1 this giving the logarithm of the result1 from which its value may be determined4 ,,notation ,,by ,,powers ,,of #aj4 ,a convenient method of representing an arithmetical quantity is to split it up into two factors1 of which the first is the original number1 with the decimal point moved so as to immediately follow the first significant figure1 and the second1 #ajn where n is the number of places the decimal point has been moved1 this index being positive for numbers greater than #a1 and negative for numbers less than #a4a ,in this system1 therefore1 we regard #c1faj1jjj as #cfa #a1jjj1jjj1 and write it as #cfa #ajf4 ,similarly #cfa #cfa x #ajb2 #jjcfa 7 7#cfa7_/7ajj77 #cfa #ajb2 #jjjjjcfa #cfa #aje1 etc4 ,to restore a number to its original form1 we have only to move the decimal point through the number of places indicated by the index1 moving to the right if the index is positive and to the left 7prefixing #j's7 if negative4 ,this method1 which should be cultivated for ordinary arithmetical work1 is substantially that followed in calculating by the slide rule4 ,thus with the slide rule the multiplication of #fc1bjj by #jjjce virtually resolves itself into #fcb #ajd #ce #ajc or #fcb #ce #ajd-c #bbab x #aja #bbab4 ,it will be seen later1 however1 that the result can be arrived at by a more direct1 if less systematic1 method of working4 ,,the ,,mechanical ,,principle ,,of ,,the ,,slide ,,rule4 ,illustration3 ,,fig4 #a4 ,the mechanical principle involved in the slide rule is of a very simple character4 ,in ,fig4 #a1 ,a and ,b represent two rules divided into #aj equal parts1 the division lines being numbered consecutively as shown4 ,if the rule ,b is moved to the right until #j on ,b is opposite #c on ,a1 it is seen that any number on ,a is equal to the coinciding number on ,b1 plus #c4 ,thus opposite #d on ,b is #g on ,a4 ,the reason is obvious4 ,by moving ,b to the right1 we add to a length #jc1 another length #jd1 the result read off on ,a being #g4 ,evidently1 the same result would have been obtained if a length #jd had been added1 by means of a pair of dividers1 to the length #jc on the scale ,a4 ,by means of the slide ,b1 however1 the addition is more readily effected1 and1 what is of much greater importance1 the result of adding #c to any one of the numbers within range1 on the lower scale1 is immediately seen by reading the adjacent number on ,a4 ,of course1 subtraction can be quite as readily performed4 ,thus1 to subtract #d from #g1 we require to deduct from #jg on the ,a scale1 a length #jd on ,b4 ,we do this by placing #d on ,b under #g on ,a1 when over #j on ,b we find #c1 on ,a4 ,it is here evident that the difference of any pair of coinciding numbers on the scales is constantly equal to #c4 ,illustration3 ,,fig4 #b4 ,an important modification results if the slide-scale ,b is inverted as in ,fig4 #b4 ,in this case1 to find the sum of #d and #c we require to place the #d of the ,a scale to #c on the ,b scale1 and the result is read on ,a over #j on ,b4 ,here it will be noted1 the sum of any pair of coinciding numbers on the scales is constant and equal to #g4 ,this case1 therefore1 resembles that of the immediately preceding one1 except that the sum1 instead of the difference1 of any pair of coinciding numbers is constant4 ,to find the difference of two factors1 the converse operation is necessary4 ,thus1 to subtract #d from #g1 #j on ,b is placed opposite #g on ,a1 and over #d on ,b is found #c on ,a4 ,from these examples it will be seen that with the slide inverted the methods of operation are the reverse of those used when the slide is in its normal position4 ,it will be understood that although we have only considered the primary divisions of the scales1 the remarks apply equally to any subdivisions into which the primary spaces of the scales might be divided4 ,further1 we note that the length of scale taken to represent a unit is quite arbitrary4 ,,the ,,primitive ,,slide ,,rule4 ,the application of the foregoing principles to the slide rule can be shown most conveniently by describing the construction of a simple form of slide rule3--,take a strip of card about #aa in4 long and #b in4 wide2 draw a line down the centre of its width1 and mark off two points1 #aj in4 apart4 ,draw cross lines at these points and figure them #a and #aj on each side1 as in ,fig4 #c4 ,next mark off lengths of #cja1 #dgg1 #fjb1 #fii1 #ggh1 #hde1 #ijc and #ied inches1 from the line marked #a4 ,draw cross lines as before1 and figure these lines1 #b1 #c1 #d1 #e1 #f1 #g1 #h and #i4 ,to fill in the intermediate divisions of the scale1 take the logs1 of #aa1 #ab1 #ac1 etc4 7from a table71 multiply each by #aj1 and thus obtain the distances from #a1 at which the several subdivisions are to be placed4 ,mark these #ab1 #ac1 #ad1 etc41 and complete the scale1 making the interpolated division marks shorter to facilitate reading1 as with an ordinary measuring rule4 ,cutting the card cleanly down the centre line1 we have the essentials of the slide rule4 ,illustration3 ,,fig4 #c4 ,the fundamental principle of the slide rule is now evident3--,each scale is graduated in such a manner that the distance of any number from #a is proportional to the logarithm of that number4 ,illustration3 ,,fig4 #d4 8,we know that to find the product of #b #c by logarithms1 we add #jcja1 or log4 #b1 to #jdgg1 the log4 of #c1 obtaining #jggh1 or log4 #f4 ,with our primitive slide rule we place #a on the lower scale to #cja in4 7which we have marked #b7 on the upper scale 7,fig4 #d74 ,then over #dgg in4 on the lower scale 7which we marked #c71 we have #ggh in4 7which we marked #f7 on the upper scale4 ,conversely1 to divide #f by #c1 we place #c on the lower scale in agreement with #f on the upper1 and over #a on the lower scale read #b on the upper scale4 ,this method of adding and subtracting scale lengths will be seen to be identical with that used in the simple case shown in ,fig4 #a4 ,,the ,,modern ,,slide ,,rule4 ,the modern form of slide rule1 variously styled the ,gravt1 the ,tavernier-,gravt1 and the ,mannheim rule1 is frequently made of boxwood1 but all the leading instrument makers now supply rules made of boxwood or mahogany1 and faced with celluloid1 the white surface of which brings out the graduations much more distinctly than lines engraved on a boxwood surface4 ,the celluloid facings should not be polished1 as a dull surface is much less fatiguing to the eyes4 ,the most generally used1 and on the whole the most convenient size of rule1 is about #aj;in4 long1 #a;in4 wide1 and about in4 thick2 but #e in41 #h in41 #ae in41 #bj in41 #bd in4 and #dj in4 rules are also made4 ,in the centre of the stock of the rule a movable slip is fitted1 which constitutes the slide1 and corresponds to the lower of the two rules of our rudimentary examples4 ,illustration3 ,,fig4 #e4 ,from ,fig4 #e1 which is a representation of the face of a ,gravt or ,mannheim slide rule1 it will be seen that four series of logarithmic graduations or scale-lines are employed1 the upper and lower being engraved on the stock or body of the rule1 while the other two are engraved upon the slide4 ,the two upper sets of graduations are exactly alike in every particular1 and the lower sets are also similar4 ,it is usual to identify the two upper scale-lines by the letters ,a and ,b1 and the two lower by the letters ,c and ,d1 as indicated in the figure at the left-hand extremities of the scales4 ,referring to the scales ,c and ,d1 these will each be seen to be a development of the elementary scales of ,fig4 #c1 but in this case each principal space is subdivided1 more or less minutely4 ,the principle1 however1 is exactly the same1 so that by moving the slide 7carrying scale ,c71 multiplication and division can be mechanically performed in the manner described4 ,the upper scale-line ,a consists of two exactly similar scales1 placed end to end1 the first lying between ,,il and ,,ic1 and the second between ,,ic and ,,ir4 ,the first of these scales will be designated the left-hand ,a scale1 and the second the right-hand ,a scale4 ,similarly the coinciding scales on the slide are the left-hand ,b scale and the right-hand ,b scale4 ,each of these four scales is divided 7as finely as convenient7 as in the case of the ,c and ,d scales1 but1 of course1 they are exactly one half the length of the latter4 ,the two end graduations of both the ,c and ,d scales are known as the left- and right-hand indices of these scales4 ,sometimes they are figured #a and #aj respectively2 sometimes both are marked #a4 ,similarly ,,il and ,,ir are the left- and right-hand indices of the ,a and ,b lines1 while ,,ic is the centre index of these scales4 ,other division lines usually found on the face of the rule are one on the left-hand ,a and ,b scales1 indicating the ratio of the circumference of a circle to its diameter1 #cadaf2 and a line on the right-hand ,b scale marking the position of 77_/7#d7 #jghed1 used in calculating the areas of circles4 ,reference will be made hereafter to the scales on the under-side of the slide1 and we need now only add that one of the edges of the rule1 usually bevelled1 is generally graduated in millimetres1 while the other edge has engraved on it a scale of inches divided into eighths or tenths4 ,on the bottom face inside the groove of the rule either one or the other of these scales is continued in such a manner that by drawing the slide out to the right and using the scale inside the rule1 in conjunction with the corresponding scale on the edge1 it is possible to measure #bj inches in the one case1 or nearly #ejj millimetres in the other4 ,on the back of the rule there is usually a collection of data1 for which the slips given at the end of this work may often be substituted with advantage4 ,,the ,,notation ,,of ,,the ,,slide ,,rule4 ,hitherto our attention has been confined to a consideration of the primary divisions of the scales4 ,the same principle of graduation is1 however1 used throughout2 and after what has been said1 this part of the subject need not be further enlarged upon4 ,some explanation of the method of reading the scales is necessary1 as facility in using the instrument depends in a very great measure upon the dexterity of the operator in assigning the correct value to each division on the rule4 ,by reference to ,fig4 #e1 it will be seen that each of the primary spacings in the several scales is invariably subdivided into ten2 but since the lengths of the successive primary divisions rapidly diminish1 it is impossible to subdivide each main space into the same number of parts that the space #a-b can be subdivided4 ,this variable spacing of the scales is at first confusing to the student1 but with a little practice the difficulty is soon overcome4 ,with the ,c or ,d scale1 it will be noticed that the length of the interval #a-b is sufficient to allow each of the #aj subdivisions to be again divided into #aj parts1 so that the whole interval #a-b is divided into #ajj4 ,the shorter main space #b-c1 and the still shorter one #c-d1 only allow of the #aj subdivisions of each being divided into five parts4 ,each of these main spaces is therefore divided into #ej parts4 ,for the remainder of the scale each of the #aj subdivisions of each main space is divided into two parts only2 so that from the main division #d to the end of the scale the primary spaces are divided into #bj parts only4 ,in the upper scales ,a or ,b1 it will be found that--as the space #a-b is of only half the length of the corresponding space on ,c or ,d--the #aj subdivisions of this interval are divided into five parts only4 ,similarly each of the #aj subdivisions of the intervals #b-c1 #c-d1 and #d-e are further divided into two parts only1 while for the remainder of the scale only the #aj subdivisions are possible1 owing to the rapidly diminishing lengths of the primary spacings4 ,the values actually given on the rule run from #a to #aj on the lower scales and from #a to #ajj on the upper scales1 and1 as explained on page #i1 all factors are brought within these ranges of values by multiplying or dividing them by powers of #aj4 ,by following this plan1 we virtually regard each factor as merely a series of significant figures1 and make the necessary modification due to the 8powers of #aj0 when fixing the position of the decimal point in the answer4 ,many1 however1 find it convenient in practice to regard the values on the rule as multiplied or divided by such powers of #aj as may be necessary to suit the factors entering into the calculation4 ,if this plan is adopted1 the values given to each graduation of the scales will depend on that given to the left index figure 7#a7 of the lower scales1 this being any multiple or submultiple of #aj4 ,thus ,,il on the ,d scale may be regarded as #a1 #aj1 #ajj1 #ajjj1 etc41 or as #ja1 #jja1 #jjja1 #jjjja1 etc42 but once the initial value is assigned to the index1 the ratio of value must be maintained throughout the whole scale4 ,for example1 if #a on ,c is taken to represent #aj1 the main divisions #b1 #c1 #d1 etc41 will be read as #bj1 #cj1 #dj1 etc4 ,on the other hand1 if the fourth main division is read as #jjjd1 then the left index figure of the scale will be read as #jjja4 ,the figured subdivisions of the main space #a-b are to be read as #aa1 #ab1 #ac1 #ad1 #ae1 #af1 #ag1 #ah and #ai--;if the index represents #aj1--;and as corresponding multiples for any other value of the index4 ,independently considered1 these remarks apply equally to the ,a or ,b scale1 but in this case the notation is continued through the second half of the scale1 the figures of which are to be read as tenfold values of the corresponding figures in the first half of the scale4 ,the reading of the intermediate divisions will1 of course1 be determined by the values assigned to the main divisions4 ,thus1 if ,,il on ,d is read as #a1 then each of the smallest subdivisions of the space #a-b will be read as #jja1 and each of the smallest subdivisions of the spaces #b-c or #c-d as #jjb1 while for the remainder of the scale the smallest subdivisions are read as #jje4 ,in the ,a or ,b scale the subdivisions of the space #a-b of the first half of the scale are 7if ,,il #a7 read as #jjb1 #jjd1 etc42 for the divisions #b-c1 #c-d1 and #d-e1 the smallest intervals are read as #jje of the primary spaces1 and from #e to the centre index of the scale the divisions represent #ja of each main interval4 ,passing the centre index1 which is1 now read as #aj1 the smallest subdivisions immediately following are read #ajb1 #ajd1 etc41 until #bjj is reached2 then we read #bje1 #baj1 #bae #bbj1 etc41 until the figured main division #e is reached4 ,the remainder of the scale is read #ea1 #eb1 #ec1 etc41 up to #ajj1 the right-hand index4 ,further subdivision of any of the spaces of the rule can be effected by the eye1 and after a little practice the operator will become quite expert in estimating any intermediate value4 ,it affords good practice to set #a on ,c to #ajd1 #aji1 etc4 on ,d1 and to read the values on ,d1 under #d1 #f1 #h1 etc4 on ,c4 ,as the exact results are easily calculated mentally1 the student1 by this means1 will receive better instruction in estimating intermediate results than can be given by any diagram4 ,some rules will be found figured as shown in ,fig4 #e2 in others1 the right-hand upper scales are marked #aj1 #bj1 #cj1 etc4 ,again1 others are marked decimally1 the lower scales and the left-hand upper scales being figured #a1 #aa1 #ab1 #ac 444 #be1 etc4 ,the latter form has advantages from the point of view of the beginner4 ,the method of reading the ,a and ,b scales1 just given1 applies only when these scales are regarded as altogether independent of the lower pair of scales ,c and ,d4 ,some operators prefer to use the ,a and ,b scales1 and some the ,c and ,d scales1 for the ordinary operations of proportion1 multiplication1 and division4 ,each method has its advantages1 as will be shown1 but in the more complex calculations1 as involution and evolution1 etc41 the relation of the upper scales to the lower scales becomes a very important factor4 ,the distance #a-aj on the upper scales is one-half of the distance #a-aj on the lower scales4 ,hence any distance from #a1 taken on the upper scales1 represents twice the logarithm which the same distance represents on the lower scales4 ,in other words1 the length which represents log4 ,n on ,d1 would represent #b log4 ,n on ,a2 and1 conversely1 the length which represents log4 ,n on ,a1 would represent 7log4 ,n7_/7#b7 on ,d4 ,now we have seen 7page #h7 that multiplying the log4 of a number by #b gives the log4 of the square of the number4 ,hence1 above any number on ,d we find its square on ,a1 or1 conversely1 below any number on ,a1 we find its square root on ,d4 ,thus1 above #b we find #d2 under #di1 we find #g and so on4 ,obviously the same relation exists between the ,b and ,c scales4 ,,the ,,cursor ,,or ,,runner4 ,all modern slide rules are now fitted with a cursor or runner1 which usually consists of a light metal frame moving under spring control in grooves in the edges of the stock of the rule4 ,this frame carries a piece of glass1 mica or transparent celluloid1 about #a in4 square1 across the centre of which a fine reference line is drawn exactly at right angles to the line of scales4 ,to 8set the cursor0 to any value on the scales of the rule1 the frame is taken between the thumb and forefinger and adjusted in position until the line falls exactly upon the graduation1 or upon an estimated value1 between a pair of graduations1 as the case may be4 ,having fixed one number in this way1 another value on either of the scales on the slide may be similarly adjusted in reference to the cursor line4 ,the cursor will be found very convenient in making such settings1 especially when either or both of the numbers are located by eye estimation4 ,it also finds a very important use in referring the readings of the upper scale to those of the lower1 or vice versa1 while as an aid in continued multiplication and division and complex calculations generally1 its value is inestimable4 ,multiple ,line ,cursors4--,cursors can be obtained with two lines1 the distance between them being that between #ghed and #aj on the ,a scale4 ,the use of this cursor is explained on page #eg4 ,another multiple line cursor has short lines engraved on it1 corresponding to the main graduations from #ie to #aje on the respective scales4 ,this is useful for adding or deducting small percentages4 ,the ,broken ,line ,cursor4--,to facilitate setting1 broken line cursors are made1 in which the hair-line is not continued across the scales1 but has two gaps1 as shown in ,fig4 #f4 ,the ,pointed ,cursor has an index or pointer1 extending over the bevelled edge of the rule1 on which is a scale of inches4 ,it is useful for summing the lengths of the ordinates of indicator diagrams1 and also for plotting lengths representing the logarithms of numbers1 sometimes required in graphic calculations4 ,the ,goulding ,cursor4--,it has been pointed out that in order to obtain the third or fourth figure of a reading on the #aj in4 slide rule1 it is frequently necessary to depend upon the operator's ability to mentally subdivide the space within which the reading falls4 ,this subdivision can be mechanically effected by the aid of the ,goulding ,cursor 7,fig4 #g71 which consists of a frame fitting into the usual grooves in the rule1 and carrying a metal plate faced with celluloid1 upon which is engraved a triangular scale ,a ,b ,c4 ,the portion carrying the chisel edges ,e is not fixed to the cursor proper1 but slides on the latter1 so that the index marks on the projecting prongs can be moved slightly along the scales of the rule1 this movement being effected by the short end of the bent lever ,f working in the slot as shown4 ,d is a pointer which can be moved along ,f under spring control4 ,as illustrating the method of use1 we will assume that #a on ,c is placed to #aee on ,d1 and that we require to read the value on ,d under #bg on ,c4 ,this is seen to lie between #daej and #dbjj1 so setting the pointer ,d to the line ,b ,c--always the first operation--we move the whole along the rule until the index line on the lower prong agrees with #dbjj4 ,we then move ,f across the scale until the index line agrees with #dajj1 set the pointer ,d to the line ,a ,c1 and move the lever back until the index line agrees with #bg on the slide4 ,it will then be found that the pointer ,d gives #he on ,a ,b as the value of the supplementary figures1 and hence the complete reading is #dahe4 ,illustration3 ,,fig4 #f4 ,illustration3 ,,fig4 #h4 ,illustration3 ,,fig4 #g4 ,illustration3 ,,fig4 #i4 ,magnifying ,cursors are of assistance in reading the scales1 and in a good and direct light are very helpful4 ,in one form an ordinary lens is carried by two light arms hinged to the upper and lower edges of the cursor1 so that it can be folded down to the face of the rule when not in use4 ,a more compact form1 shown in ,fig4 #h1 consists of a strip of plano-convex glass1 on the under-side of which is the hair-line4 ,in a cursor made by ,nestler of ,lahr1 the plano-convex strip is fixed on the ordinary cursor4 ,the magnifying power is about #b1 so that a #e in4 rule1 having the same number of graduations as a #aj in4 rule1 can be read with equal facility1 by the aid of this cursor4 ,the ,digit-registering ,cursor1 supplied by ,mr4 ,a4 ,w4 ,faber1 ,london1 and shown in ,fig4 #i1 has a semicircular scale running from #j at the centre upward to #f and downward to #f4 ,a small finger enables the operator to register the number of digits to be added or subtracted at the end of a lengthy operation1 as explained at page #bh4 ,,multiplication4 ,in the preliminary notes it was shown that by mechanically adding two lengths representing the logarithms of two numbers1 we can obtain the product of these numbers2 while by subtracting one log4 length from another1 the number represented by the latter is divided by the number represented by the former4 ,hence1 using the ,c and ,d scales1 we have the ,,rule ,,for ,,multiplication4--,set the index of the ,c scale to one of the factors on ,d1 and under the other factor on ,c1 find the product on ,d4 ,illustration3 ,,fig4 #aj4 ,thus1 to find the product of #b #d1 the slide is moved to the right until the left index 7#a7 of ,c is brought over #b on ,d1 when under the other factor 7#d7 on ,c1 is found the required product 7#h7 on ,d4 ,following along the slide1 to the right1 we find that beyond #e on ,c 7giving #aj on ,d71 we have no scale below the projecting slide 7,fig4 #aj74 ,if we imagine the ,d scale prolonged to the right1 we should have a repetition of the earlier portion1 but1 as with the two parts of the ,a scales1 the repeated portion would be of tenfold value1 and #aj on ,c would agree with #bj on the prolonged ,d scale4 ,we turn this fact to account by moving the slide to the left until #aj on ,c agrees with #b on ,d1 and we can then read off such results as #b #f #ab2 #b #h #af1 etc41 remembering that as the scale is now of tenfold value1 there will be two figures in the result4 ,hence1 for those who prefer rules1 we have the ,,rule ,,for ,,the ,,number ,,of ,,digits ,,in ,a ,,product4--,if the product is read with the slide projecting to the ,,left1 ,,add ,,the ,,number ,,of ,,the ,,digits ,,in ,,the ,,two ,,factors2 if read with the slide to the ,,right1 deduct #a from this sum4 ,,ex4--#be #gj #agej4 ,the product is found with the slide projecting to the left1 so the number of digits in the product #b #b #d4 ,,ex4--#cf #be #ij4 ,the slide projects to the right1 and the number of digits in the product is therefore #a #b #a #b4 ,,ex4--#jjbe #jg #jjage4 ,the product is obtained with the slide projecting to the left1 and the number of digits is therefore #a #j #a4 ,,ex4--#jjjjahd #jjje #jjjjjjjib4 ,the sum of the number of digits in the two factors #c 7#b7 #e1 but as the slide projects to the right1 the number of digits will be #e #a #f4 ,from the last two examples it will be seen that when the first significant figure of a decimal factor does not immediately follow the decimal point1 the minus sign is to be prefixed to the number of digits1 counting as many digits minus as there are #j's following the decimal point4 ,thus1 #jjc has #a digit1 #jjjce has #b digits1 and so on4 ,some little care is necessary to ensure these minus values being correctly taken into account in determining the number of digits in the answer4 ,for this reason many prefer to treat decimal factors as whole numbers1 and to locate the decimal point according to the usual rules for the multiplication of decimals4 ,thus1 in the last example we take #ahd #e #ibj1 but as by the usual rule the product must contain #f #c #i decimal places1 we prefix six cyphers1 obtaining #jjjjjjjib4 ,when both factors consist of integers as well as decimals1 the number of digits in the product1 and therefore the position of the decimal point1 will be determined by the usual rule for whole numbers4 ,another method of determining the number of digits in a product deserves mention1 which1 not being dependent upon the position of the slide1 is applicable to all calculating instruments4 ,,general ,,rule ,,for ,,number ,,of ,,digits ,,in ,a ,,product4--,when the first significant figure in the product is smaller than in ,,either of the factors1 the number of digits in the product is equal to the ,,sum of the digits in the two factors4 ,when the contrary is the case1 the number of digits is #a ,,less than the sum of the digits in the two factors4 ,when the first figures are the same1 those following must be compared4 ,estimation of the ,figures in a ,product4--,we have given rules for those who prefer to decide the number of figures by this means1 but experience will show that to make the best use of the instrument1 the result1 as read on the rule1 should be regarded merely as the significant figures of the answer1 the position of the decimal point1 if not obvious1 being decided by a very rough mental calculation4 ,in very many instances1 the magnitude of the result will be evident from the conditions of the problem--e4g41 whether the answer should be #jc in41 #c in41 or #cj in42 or #aj tons1 #ja ton1 #ajj tons1 etc4 ,in those cases where the magnitude of the answer cannot be estimated1 and the factors contain many figures1 or have a number of #j's following the decimal point1 the use of notation by powers of #aj 7page #h7 is of considerable assistance2 but more usually it will be found1 that a very rough calculation will settle the point with comparatively little trouble4 ,considerable practice is needed to work rapidly and with certainty1 when using rules4 ,moreover1 the experience thus acquired is confined to slide-rule work4 ,the same time spent in practising the 8rough approximation0 method will enable reliable results to be obtained rapidly1 with the advantage that the method is applicable to calculations generally4 ,however1 the choice of methods is a matter of personal preference4 ,both methods will be given1 but whichever plan is followed1 the student is strongly advised to cultivate the habit of forming an idea of the magnitude of the result4 ,,ex4--#ccf #bcf #gicj4 ,setting #a on ,c to #ccf on ,d1 we read under #bcf on ,d and find #gic on ,d1 as the significant figures of the answer4 ,a rough calculation1 as #cj #bjj #fjjj1 indicates that the result will consist of #d figures1 and is therefore to be read as #gicj4 ,,ex4--#ag1cjj #cghj #fe1djj1jjj4 ,by factorising with powers of #aj #agc #ajd #cgh #ajc #agc #cgh #ajg4 ,setting #a on ,c to #agc on ,d1 we read1 under #cgh on ,c1 the result of the simple multiplication1 as #fed4 ,multiplying by #ajg moves the decimal point #g places to the right1 and the answer is #fe1djj1jjj4 ,if it is required to find a series of products of which one of the factors is constant1 set #a on ,c to the constant factor on ,d and read the several products on ,d1 under the respective variable factors4 ,if the factors are required which will give a constant product 7really a case of division71 set the cursor to the constant product on ,d4 ,then obviously1 as the slide is moved along1 any pair of factors found simultaneously under the cursor line on ,c1 and on ,d under index of ,c1 will give the product4 ,a better method of working will be explained when we deal with the inversion of the slide4 ,it is sometimes useful to remember that although we usually set the slide to the rule1 we can obtain the result equally well by setting the rule to the slide4 ,thus1 bringing #a 7or #aj7 on ,d to #b on ,c1 we find on ,c1 over any other factor1 n on ,d1 the product of #b n4 ,but note that the slide and rule have now changed places1 and if we use rules for the number of digits in the result1 we must now deduct #a from the sum of the digits in the factors1 when the rule projects to the right of the slide4 ,with the ordinary #aj in4 rule it will be found in general that the extent to which the ,c and ,d scales are subdivided is such as to enable not more than three figures in either factor being dealt with4 ,for the same reason it is impossible to directly read more than the first three figures of any product1 although it is often possible--by mentally dividing the smallest space involved in the reading--to correctly determine the fourth figure of a product4 ,necessarily this method is only reliable when used in the earlier parts of the ,c and ,d scales4 ,however1 the last numeral of a three-figure1 and in some cases the last of a four-figure1 product can be readily ascertained by an inspection of the factors4 ,,ex4--#ai #bg #eac4 ,placing the ,l4,h4 index of ,c to #ai on ,d1 we find opposite #bg on ,c1 the product1 which lies between #eaj and #eae4 ,a glance at the factors1 however1 is sufficient to decide that the third figure must be #c1 since the product of #i and #g is #fc1 and the last figure of this product must be the last figure in the answer4 ,,ex4--#gi #ia #gahi4 ,in this case the division line #ia on ,c indicates on ,d that the answer lies between #gahj and #gaij4 ,as the last figure must be #i1 it is at once inferred that the last two figures are #hi4 ,when there are more than three figures in either or both of the factors1 the fourth and following figures to the right must be neglected4 ,it is well to note1 however1 that if the first neglected figure is #e1 or greater than #e1 it will generally be advisable to increase by #a the third figure of the factor employed4 ,generally it will suffice to make this increase in one of the two factors only1 but it is obvious that in some cases greater accuracy will be obtained by increasing both factors in this way4 ,,continued ,,multiplication4--,to find the product of more than two factors1 we make use of the cursor to mark the position of successive products 7the value of which does not concern us7 as the several factors are taken into the calculation4 ,setting the index of ,c to the #ast factor on ,d1 we bring the line of the cursor to the #bnd factor on ,c1 then the index of ,c to the cursor1 the cursor to the #crd factor1 index of ,c to cursor1 and so on1 reading the final product on ,d under the last factor on ,c4 7,note that the #ast factor and the result are read on ,d2 all intermediate readings are taken on ,c47 ,if the rule for the number of digits in a product is used1 it is necessary to note the number of times multiplication is effected with the slide projecting to the right4 ,this number1 deducted from the sum of the digits of the several factors1 gives the number of digits in the product4 ,ingenious devices have been adopted to record the number of times the slide projects to the right1 but some of these are very inconvenient4 ,the author's method is to record each time the slide so projects1 by a minus mark1 thus 4 ,these can be noted down in any convenient manner1 and the sum of the marks so obtained deducted from the sum of the digits in the several factors1 gives the number of digits in the product as before explained4 ,,ex4--#db #ga #ae #jcb #aba #agc1bjj4 ,the product given1 which is that read on the rule1 is obtained as follows3--,set ,r4,h4 index of ,c to #db on ,d1 and bring the cursor to #ga on ,c4 ,next bring the ,l4,h4 index of ,c to the cursor1 and the latter to #ae on ,c4 ,this multiplication is effected with the slide to the right1 and a memorandum of this fact is kept by making a mark 4 ,bring the ,r4,h4 index of ,c to the cursor and the latter to #jcb on ,c4 ,then set the ,l4,h4 index of ,c to the cursor and read the result1 #agcb1 on ,d under #aba on ,c1 while as a slide again projects to the right1 a second memo-mark is recorded4 ,there are #b #b #a #j #c #h digits in the factors1 and as there were #b marks recorded during the operation1 there will be #h #b #f digits in the product1 which will therefore read #agc1bjj 7#agc1aidef74 ,for a very rough evaluation of the result1 we note that #ae #jc is about #je2 hence1 as a clue to the number of figures we have #dj #gj #fj #cjjj #fj #ahj1jjj4 ,,division4 ,the instructions for multiplication having been given in some detail1 a full discussion of the inverse process of division will be unnecessary4 ,,rule ,,for ,,division4--,place the divisor on ,c1 opposite the dividend on ,d1 and read the quotient on ,d under the index of ,c4 ,,ex4--#bbe #ah #abe4 ,bringing #ah on ,c to #bbe on ,d1 we find #abe under the ,l4,h4 index of ,c4 ,as in multiplication1 the factors are treated as whole numbers1 and the position of the decimal point afterwards decided according to the following rule1 which1 as will be seen1 is the reverse of that for multiplication3-- ,,rule ,,for ,,the ,,number ,,of ,,digits ,,in ,a ,,quotient4--,if the quotient is read with the slide projecting to the ,,left1 subtract the number of digits in the divisor from those in the dividend2 but if read with the slide to the ,,right1 ,,add #a to this difference4#b ,in the above example the quotient is read off with the slide to the right1 so the number of digits in the answer #c #b #a #b4 ,,ex4--#jjjjbba #jjag #jjac4 ,here the number of digits in the dividend is #c1 and in the divisor #a4 ,the difference is #b2 but as the result is obtained with the slide to the right1 this result must be increased by #a1 so that the number of digits in the quotient is #b #a #a1 giving the answer as #jjac4 ,if preferred1 the result can be obtained in the manner referred to when considering the multiplication of decimals4 ,thus1 treating the above as whole numbers1 we find that the result of dividing #bba by #ag #ac1 since the difference in the number of digits in the factors1 which is #a1 is1 owing to the position of the slide1 increased by #a1 giving #b as the number of digits in the answer4 ,then by the rules for the division of decimals we know that the number of decimal places in the quotient is equal to #f #c #c1 showing that a cypher is to be prefixed to the result read on the rule4 ,as in multiplication1 so in division1 we have a ,,general ,,rule ,,for ,,number ,,of ,,digits ,,in ,a ,,quotient4--,when the first significant figure in the ,,divisor is greater than that in the ,,dividend1 the number of digits in the quotient is found by subtracting the digits in the divisor from those in the dividend4 ,when the contrary is the case1 #a ,,is ,,to ,,be ,,added to this difference4 ,when the first figures are the same1 those following must be compared4 ,,estimation ,,of ,,the ,,figures ,,in ,a ,,quotient4--,the method of roughly estimating the number of figures in a quotient needs little explanation4 ,,ex4--#cie #ecdj #jjjjgd4 ,setting #ecd on ,c to #cie on ,d we read under the 7,r4,h47 index of ,c1 the significant figures on ,d1 which are #gd4 ,then #ci #e is about #jh and #jh #ajjj gives #jjjjh as a rough estimate4 ,,ex4--#jjjjjjbhe #jjjjaig #jjaddf4 ,regarding this as #bhe #ajf #aig #ajd we divide #bhe by #aig and obtain #addf4 ,dividing the powers of #aj we have #ajf #ajd #ajb1 so the decimal point is to be moved two places to the left and the answer is read as #jjaddf4 ,another method of dividing deserves mention as of special service when dividing a number of quantities by a constant divisor3--,set the index of ,c to the divisor on ,d and over any dividend on ,d1 read the quotient on ,c4 ,for the division of a constant dividend by a variable divisor1 set the cursor to the dividend on ,d and bring the divisor on ,c successively to the cursor1 reading the corresponding quotients on ,d under the index of ,c4 ,another method which avoids moving the slide is explained in the section on 8,multiplication and ,division with the ,slide ,inverted40 ,,continued ,,division1 if we can so call such an expression as 7#cad7_/7ghe #jjjjba #dc #fdd7 #jjfhh1 may be worked by repeating as follows3--,set #ghe on ,c to #cad on ,d1 bring cursor to index of ,c1 #ba on ,c to cursor1 cursor to index1 #dc to cursor1 cursor to index1 #fdd to cursor1 and under index of ,c read #fhh on ,d as the significant figures of the answer4 ,for the number of figures in the result1 we deduct the sum of the number of digits in the several factors and add #a for each time the slide projects to the right1 which in this case occurs once4 ,there are #c 7#c7 #a #b #c denominator digits1 #a numerator digit1 and #a is to be added to the difference4 ,therefore there are #a #c #a #a digits in the answer1 which is therefore #jjfhh4 ,the foregoing method of working may confuse the beginner1 who is apt to fall into the process of continued multiplication4 ,for this reason1 until familiarity with combined methods has been acquired1 the product of the several denominators should be first found by the continued multiplication process1 and the figures in this product determined4 ,then divide the numerator by this product to obtain the result4 ,as the denominator product will be read on ,d1 we may avoid resetting the slide by bringing the numerator on ,c to this product and reading the result on ,c over the index of ,d4 ,the slide and rule have here changed places2 hence if rules are followed for the number of figures in the result1 #a must be added to the difference of digits1 when the rule projects to the right of the slide4 ,the author's method of recording the number of times division is performed with the slide to the right is by vertical memorandum marks1 thus 4 ,the full significance of these memo-marks will appear in the following section4 ,for a rough calculation to fix the decimal point1 in this example we move the decimal points in the factors1 obtaining 7#c7_/7jh #b #d #f7 7#c7_/7dj7 #jjge4 ,,the ,,use ,,of ,,the ,,upper ,,scales ,,for ,,multiplication ,,and ,,division4 ,many prefer to use the upper scales ,a and ,b1 in preference to ,c and ,d4 ,the disadvantage is that as the scales are only one-half the length of ,c or ,d1 the graduation does not permit of the same degree of accuracy being obtained as when working with the lower scales4 ,but the result can always be read directly from the rule without ever having to change the position of the slide after it has been initially set4 ,hence1 it obviates the uncertainty as to the direction in which the slide is to be moved in making a setting4 ,when the ,a and ,b scales are employed1 it is understood that the left-hand pair of scales are to be used in the same manner as ,c and ,d1 and so far the rules relating to the latter are entirely applicable4 ,but in this case the slide is always moved to the right1 so that in multiplication the product is found either upon the left or right scales of ,a4 ,if it is found on the left ,a scale1 the rule for the number of digits in the product is found as for the ,c and ,d scales1 and is equal to the sum of the digits in the two factors1 minus #a2 but if found on the right-hand ,a scale1 the number of digits in the product is equal to the sum of the digits in the two factors4 ,in division1 similar modifications are necessary4 ,if when moving the slide to the right the division can be completely effected by using the ,l4,h4 scale of ,a1 the quotient 7read on ,a above the ,l4,h4 of index ,b7 has a number of digits equal to the number in the dividend1 less the number in the divisor1 plus #a4 ,but if the division necessitates the use of both the ,a scales1 the number of digits in the quotient equals the number in the dividend1 less the number in the divisor4 ,,reciprocals4 ,a special case of division to be considered is the determination of the reciprocal of a number n1 or 7#a7_/7n74 ,following the ordinary rule for division1 it is evident that setting n on ,c to #a on ,d1 gives 7#a7_/7n7 on ,d under #a on ,c4 ,it is more important to observe that by inverting the operation--setting #a 7or #aj7 on ,c to n on ,d--we can read 7#a7_/7n7 on ,c over #a 7or #aj7 on ,d4 ,hence whenever a result is read on ,d under an index of ,c1 we can also read its reciprocal on ,c over whichever index of ,d is available4 ,the ,number of ,digits in a ,reciprocal is obvious when n #aj1 #ajj1 or any power 7p7 of #aj4 ,thus 7#a7_/7aj7 #ja2 7#a7_/7ajj7 #jja2 7#a7_/7ajp7 #a preceded by p #a cyphers4 ,for all other cases we have the rule3--,subtract from #a the number of digits in the number4 ,,ex4--7#a7_/7cci7 #jjjbie4 ,there are #c digits in the number2 hence1 there are #a #c #b digits in the answer4 ,,ex4--7#a7_/7jjjjjaef7 #fd1ajj4 ,there are #d digits in the number2 hence1 there are #a 7#d7 #e digits in the result4 ,,continued ,,multiplication ,,and ,,division4 ,by combining the rules for multiplication and division1 we can readily evaluate expressions of the form 7a7_/7b7 7c7_/7d7 7e7_/7f7 7g7_/7h7 x4 ,the simplest case1 7a c7_/7b7 can be solved by one setting of the slide4#c ,take as an example1 7#adde #fj7_/7he7 #ajb4 ,setting #he on ,c to #adde on ,d1 we can1 if desired1 read #ag on ,d under #a on ,c1 as the quotient4 ,however1 we are not concerned with this1 but require its multiplication by #fj1 and the slide being already set for this operation1 we at once read under #fj on ,c the result1 #ajb1 on ,d4 ,the figures in the answer are obvious4 ,when there are more factors to take into account1 we place the cursor over #ajb on ,d1 bring the next divisor on ,c to the cursor1 move the cursor to the next multiplier on ,c1 bring the next divisor on ,c to the cursor1 and so on1 until all the factors have been dealt with4 ,note that only the first factor and the result are read on ,d2 also that the cursor is moved for multiplying and the slide for dividing4 ,number of ,digits in ,result in ,combined ,multiplication and ,division4--,for those who use rules the author's method of determining the decimal point in combined multiplication and division may be used4 ,each time multiplication is performed with the slide projecting to the right1 make a mark2 each time division is effected with the slide to the right1 make a mark2 but allow the marks to cancel the marks as far as they will4 ,subtract the sum of the digits in the denominator from the sum of digits in the numerator1 and to this difference add any uncancelled memo-marks1 if of character1 or subtract them if of character4 ,,ex4--7#dce #bid #ea #cb7_/7bg #chc #aje #aca7 #adfh4 ,sidenote3 ,set #bg on ,c to #dce on ,d1 and as with this division the slide is to the right1 make the first mark4 ,bring cursor to #bid on ,c1 and as in this multiplication the slide is to the right1 make the first mark1 cancelling as shown4 ,setting #chc on ,c to the cursor1 requires the second mark1 which1 however1 is cancelled in turn by the multiplication by #ea4 ,the division by #aje requires the third mark1 and after multiplying by #cb 7requiring no mark7 the final division by #aca requires the fourth mark4 ,then1 as there are #h numerator digits1 #f denominator1 and #b uncancelled memo-marks 7which1 being #a1 are additive7 we have ,number of digits in result #h #f #b #d4 ,had the uncancelled marks been in character1 the number of digits would have been #h #f #b #j4 ,for quantities less than #ja the digit place numbers will be negative4 ,the troublesome addition of these may be avoided by transferring them to the opposite side and treating them as positive4 #b #d #jjjcef #bga #jjhcge ,thus3-- #bhh #jadbf #ihe #jjjjjb #b #a #a ,the first numerator1 #jjjcef1 has #b digits4 ,note this by placing #b below the lower line as shown4 #bga has #b digits2 place #b over it4 #jjhcge has #a digit2 hence place #a below the lower line4 ,the first denominator has no digits2 the second1 #ihe1 has #a digit2 hence place #a under it4 #jjjjjb has #d digits2 place #d above the upper line4 ,the sum of the top series is #b #d #f2 of the bottom series #b #a #a #d4 ,subtracting the bottom from the top1 we have #f #d #b digits1 to which #a has to be added for an uncancelled memo-mark1 and the result is read as #bhh4 ,moving the decimal point often facilitates matters4 ,thus1 7#cbd #jih #dcb #jjbag7_/7dga #jage #jjjjjjfba #dabjjj7 is much more conveniently dealt with when re-arranged as 7#cbd #ih #dcb #bag7_/7dga #age #fba #dab7 #ada4 ,to determine the number of figures in the result by rough cancelling and mental calculation1 we note that #dga enters #dcb about #ajj times2 #ih enters #age about #b2 #fba into #cbd about #e2 and #bag into #dab about #b4 ,this gives 7#ejj7_/7d7 #abe1 showing that the result contains #c digits4 ,from the slide rule we read #ada1 which is therefore the result sought4 ,the occasional traversing of the slide through the rule1 to interchange the indices--a contingency which the use of the ,c and ,d scales always involves--may often be avoided by a very simple expedient4 ,such an example as 7#fai #cab #dbb7_/7aabj #hhf #b4ji7 #cic is sometimes cited as a particularly difficult case4 ,working through the expression as given1 two traversings of the slide are necessary2 but by taking the factors in the slightly different order1 7#fai #cab #dbb7_/7hhf #bji #aabj71 so that the significant figures of each pair are more nearly alike1 we not only avoid any traversing the slide1 but we also reduce the extent to which the slide is moved to effect the several divisions4 ,such cases as 7a b7_/7c d e f g7 or 7a b c d e7_/7f g7 really resolve themselves into 7a b #a #a #a7_/7;c d e f g7 and 7a b c d e7_/7f g #a #a #a71 but1 of course1 if rules are used to locate the decimal point1 the #a's so 7mentally7 introduced are not to be counted as additional figures in the factors4 ,,multiplication ,,and ,,division ,,with ,,the ,,slide ,,inverted4 ,if the slide be inverted in the rule but with the same face uppermost1 so that the scale lies adjacent to the ,a scale1 and the right and left indices of the slide and rule are placed in coincidence1 we find the product of any number on ,d by the coincident number on 7readily referred to each other by the cursor7 is always #aj4 ,hence1 by reading the numbers on as decimals1 we have over any unit number on ,d1 its reciprocal on 4 ,thus #b on ,d is found opposite #je on 2 #c on ,d opposite to #jccc2 while opposite #h on is #jabe on ,d1 etc4 ,the reason of this is that the sum of the lengths of the slide and rule corresponding to the factors1 is always equal to the length corresponding to the product--in this case1 #aj4 ,it will be seen that if we attempt to apply the ordinary rule for multiplication1 with the slide inverted1 we shall actually be multiplying the one factor taken on ,d by the reciprocal of the other taken on 4 ,but multiplying by the reciprocal of a number is equivalent to dividing by that number1 and dividing a factor by the reciprocal of a number is equivalent to multiplying by that number4 ,it follows that with the slide inverted the operations of multiplication and division are reversed1 as are also the rules for the number of digits in the product and the position of the decimal point4 ,hence1 in multiplying with the slide inverted1 we place 7by the aid of the cursor7 one factor on opposite the other factor on ,d1 and read the result on ,d under either index of 4 ,it follows that with the slide thus set1 any pair of coinciding factors on and ,d will give the same constant product found on ,d under the index of 4 ,one useful application of this fact is found in selecting the scantlings of rectangular sections of given areas or in deciding upon the dimensions of rectangular sheets1 plates1 cisterns1 etc4 ,thus by placing the index of to #gb on ,d1 it is readily seen that a plate having an area of #gb sq4 ft4 may have sides #h by #i ft41 #f by #ab1 #e by #add1 #d by #ah1 #c by #bd1 #b by #cf1 with innumerable intermediate values4 ,many other useful applications of a similar character will suggest themselves4 ,,proportion4 ,with the slide in the ordinary position and with the indices of the ,c and ,d scales in exact agreement1 the ratio of the corresponding divisions of these scales is #a4 ,if the slide is moved so that #a on ,c agrees with #b on ,d1 we know that under any number n on ,c is n #b on ,d1 so that if we read numerators on ,c and denominators on ,d we have ,c #a #ae #b #c #d ,d#a #b #c #d #f #h4 ,in other words1 the numbers on ,d bear to the coinciding numbers on ,c a ratio of #b to #a4 ,obviously the same condition will obtain no matter in what position the slide may be placed4 ,the rule for proportion1 which is apparent from the foregoing1 may be expressed as follows3-- ,,rule ,,for ,,proportion4--,set the first term of a proportion on the ,c scale to the second term on the ,d scale1 and opposite the third term on the ,c scale read the fourth term on the ,d scale4 ,,ex4--,find the #dth term in the proportion of #bj #bg #gj x4 ,set #bj on ,c to #bg on ,d1 and opposite #gj on ,c read #ide on ,d4 ,thus ,c #bj #gj ,d #bg #ide4 ,it will be evident that this is merely a case of combined multiplication and division of the form1 7#bj #gj7_/7bg7 #ide4 ,hence1 given any three terms of a proportion1 we set the #ast to the #bnd1 or the #crd to the #dth1 as the case may be1 and opposite the other given term read the term required4#d ,thus1 in reducing vulgar fractions to decimals1 the decimal equivalent of 7#c7_/7af7 is determined by placing #c on ,c to #af on ,d1 when over the index or #a of ,d we read #jahge on ,c4 ,in this case the terms are #c #af x #a4 ,for the inverse operation--to find a vulgar fraction equivalent to a given decimal--the given decimal fraction on ,c is set to the index of ,d1 and then opposite any denominator on ,d is the corresponding numerator of the fraction on ,c4 ,if the index of ,c be placed to agree with #cadaf on ,d1 it will be clear from what has been said that this ratio exists throughout between the numbers of the two scales4 ,therefore1 against any diameter of a circle on ,c will be found the corresponding circumference on ,d4 ,in the same way1 by setting #a on ,c to the appropriate conversion factor on ,d1 we can convert a series of values in one denomination to their equivalents in another denomination4 ,in this connection the following table of conversion factors will be found of service4 ,if the ,a and ,b scales are used instead of the ,c and ,d scales1 a complete set of conversions will be at once obtained4 ,in this case1 however1 the left-hand ,a and ,b scales should be used for the initial setting1 any values read on the right-hand ,a or ,b scales being read as of tenfold value4 ,with the ,c and ,d scales a portion of the one scale will project beyond the other4 ,to read this portion of the scale1 the cursor or runner is brought to whichever index of the ,c scale falls within the rule1 and the slide moved until the other index of the ,c scale coincides with the cursor1 when the remainder of the equivalent values can then be read off4 ,it must be remembered that if the slide is moved in the direction of notation 7to the right71 the values read thereon have a tenfold greater value2 if the slide is moved to the left1 the readings thereon are decreased in a tenfold degree4 ,although preferred by many1 in the form given1 the case is obviously one of multiplication1 and is so treated in the ,data ,slips at the end of the book4 ,,table ,,of ,,conversion ,,factors4 ,,geometrical ,,equivalents4 ,,scale ,c4 ,,scale ,d4 ,if ,c #a1 ,d ,diameter of circle ,circumference of circle #cadaf ,side of inscribed square #jgjg equal square #jhhf equilateral triangle #acdf ,circum4 of circle inscribed square #jbbe equal square #jbhb ,side of square ,diagonal of square #adad ,square inch ,circular inch #abgc ,area of circle ,area of inscribed square #jfcf ,,measures ,,of ,,length4 ,inches ,millimetres #bedj ,centimetres #bed #hths of an inch ,millimetres #cage #afths #aehg #cbnds #jgid #fdths #jcig ,feet ,metres #jcjdh ,yards #jiadd ,chains #bjaaf ,miles ,kilometres #afji ,,measures ,,of ,,area4 ,square inches ,square centimetres #fdf ,circular #ejfg ,square feet metres #jjibi yards #jhcf miles kilometres #bei ,hectares #beijj ,acres #jdjdf ,,measures ,,of ,,capacity4 ,cubic inches ,cubic centimetres #afch ,imperial gallons #jjjcfj ,u4,s4 gallons #jjjdcb ,litres #jjafch ,cubic feet ,cubic metres #jjbhc ,imperial gallons #fbc ,u4,s4 gallons #gdh ,litres #bhcg yards ,cubic metres #jgfd ,imperial gallons ,litres #ded ,u4,s4 gallons #abjj ,bushels ,cubic metres #jjcfc feet #abhc ,,measures ,,of ,,weight4 ,grains ,grammes #jjfdh ,ounces 7,troy7 #caajc 7,avoird47 #bhce ,kilogrammes #jjbhce ,pounds 7,troy7 #jcgcb 7,avoird47 #jdecf ,hundredweights #ejhjb ,tons #ajafd ,metric tonnes #ajaf ,,compound ,,factors--,,velocities4 ,feet per second ,metres per second #jcjdh minute #ahbhh ,miles per hour #j4fhb minute ,meters per second #jjjejh minute #jcjdh ,miles per hour #jjaacf ,yards per #jjcda ,miles per hour ,metres per minute #bfhb ,knots #cjhh ,miles per hour #aaea ,,compound ,,factors--,,pressures4 ,pounds per sq4 inch ,grammes per sq4 mm4 #jgjca ,kilos4 per sq4 centimetre #jjgjc ,atmospheres #jjfh ,head of water in inches #bgga feet #bcji metres #jgeg ,inches of ,mercury #bjd ,inches of water ,pounds per square inch #jjcfa ,inches of mercury #jjgad ,pounds per square foot #ebj ,inches of mercury ,atmospheres #jjccc ,atmospheres ,metres of water #ajcd ,kilos4 per sq4 cm4 #ajcc ,feet of water ,pounds per square foot #fbce ,atmospheres #jjbid ,inches of mercury #jhhc ,pounds per sq4 foot #jjadag ,kilos4 per square metre #dhhc ,atmospheres #jjjjdgb ,pounds per sq4 yard ,kilos4 per square metre #jedbe ,tons per sq4 inch square mm4 #aege sq4 foot ,tonnes per square metre #ajicf ,,compound ,,factors--,,weights1 ,,capacities1 ,,etc4 ,pounds per lineal ft4 ,kilos4 per lineal metre #adhh per lineal yd4 #jdif per lineal mile ,kilos4 per kilometre #jbhah ,tons ,tonnes #jfcac ,feet ,metres #ahid ,pounds per cubic in4 ,grammes per cubic cm4 #bgfh per cubic ft4 ,kilos4 per cubic metre #afjb per cubic yd4 #jeic ,tons per cubic yard ,tonnes #acbi ,cubic yds4 per pound ,cubic metres per kilo4 #afhe per ton per tonne #jgebe ,cubic inch of water ,weight in pounds #jjcfjh ,cubic feet of water #fbce kilos #bhbc ,imperial gallons #fbce ,u4,s4 gallons #gdh ,litre of water ,cubic inches #fajbe ,gallons of water ,weight in kilos #ded ,pounds of fresh water ,pounds of sea water #ajbf ,grains per gallon ,grammes per litre #jjadbf ,pounds per gallon ,kilos4 per litre #jjiih per ,u4,s4 gal4 #jaae ,,compound ,,factors--,,power ,,units1 ,,etc4 ,british ,ther4 ,units4 ,kilogrammetres4 #ajh ,joules #ajeh ,calories 7,fr4 ,ther4 units7#jbeb per sq4 ft4 per square metre #bgac per pound per kilogramme #jeee ,pounds per sq4 ft4 ,dynes1 per sq4 cm4 #dgi ,foot-pounds ,kilogrammetres #jachb ,joules #acef ,thermal ,units #jjjabi ,calorie #jjjjcbd ,foot-tons ,tonne-metres #jccc ,horse-power ,force decheval 7,fr4,h4,p47 #ajad ,kilowatts #jgdf ,pounds per ,h4,p4 ,kilos4 per cheval #jddg ,square feet per ,h4 ,p4 ,square metres per cheval #jjaif ,cubic ,cubic #jjbgi ,watts ,ther4 ,units per hour #cdd ,foot-pounds per second #jgc per minute #ddbd ,watt-hours ,kilogrammetres #cfg ,joules #cfjj ,kilogrammetres #ihjf ,inverse ,proportion4--,if 8more0 requires 8less10 or 8less0 requires 8more10 the case is one of inverse proportion1 and although it will be seen that this form of proportion is quite readily dealt with by the preceding method1 the working is simplified to some extent by inverting the slide so that the ,c scale is adjacent to the ,a scale4 ,by the aid of the cursor1 the values on the inverted ,c 7or 7 scale1 and on the ,d scale1 can be then read off4 ,these will now constitute a series of inverse ratios4 ,for example1 in the proportion #h #d ,d #ae #c the #d on the scale is brought opposite #c on ,d1 when under #h on is found #ae on ,d4#e ,,general ,,hints ,,on ,,the ,,elementary ,,uses ,,of ,,the ,,slide ,,rule4 ,before the more complex operations of involution1 evolution1 etc41 are considered1 a few general hints on the use of the slide rule for elementary operations may be of service1 especially as these will serve to enforce some of the more important points brought out in the preceding sections4 ,always use the slide rule in as direct a light as possible4 ,study the manner in which the scales are divided4 ,follow the graduations of the ,c and ,d scales from #a to #aj1 noting the values given by each successive graduation and how these values change as we follow along to the right4 ,do the same with the two halves of the ,a and ,b scales and note the difference in the value of the subdivisions1 due to the shorter scale-lengths4 ,practise reading values by setting #a on ,c to some value on ,d and reading under #b1 #c1 #d1 etc41 on ,c1 checking the readings by mental arithmetic4 ,to the same end1 find squares1 square roots1 etc41 comparing the results with the actual values as given in tables4 ,practise setting both slide and cursor to values taken at random4 ,aim at accuracy2 speed will come with practice4 ,when in doubt as to any method of working1 verify by making a simple calculation of the same form4 ,follow the orthodox methods of working until entirely confident in the use of the instrument1 and even then do not readily make a change4 ,if any altered procedure is adopted1 first work a simple case and guard carefully against unconsciously lapsing into the usual method during the operation4 ,unless the calculation is of a straightforward character1 time taken in considering how best to attack it 7rearranging the expression if desirable7 is generally time well spent4 ,in setting two values together1 set the cursor to one of them on the rule1 and bring the other1 on the slide1 to the cursor line4 ,in multiplying factors1 as #eg #jabef1 take the fractional value first4 ,it is easier to set #a on ,c to #abef on ,d and read under #eg on ,c1 than to reverse the procedure4 ,when both values are eye-estimated1 set the cursor to the second factor on ,c and read the result on ,d1 under the cursor line4 ,in continuous operations avoid moving the slide further than necessary1 by taking the factors in that order which will keep the scale readings as close together as possible4 ,,squares ,,and ,,square ,,roots4 ,we have seen that the relation which the upper scales bear to the lower set is such that over any number on ,d is its square on ,a1 and1 conversely1 under any number on ,a is its square root on ,d1 the same remarks applying to the ,c and ,b scales on the slide4 ,taking the values engraved on the rule1 we have on ,d1 numbers lying between #a and #aj1 and on ,a the corresponding squares extending from #a to #ajj4 ,hence the squares of numbers between #a and #aj1 or the roots of numbers between #a and #ajj1 can be read off on the rule by the aid of the cursor4 ,all other cases are brought within these ranges of values by factorising with powers of #aj1 as before explained4 ,the more practical rule is the following3-- ,to ,find the ,square of a ,number1 set the cursor to the number on ,d and read the required square on ,a under the cursor4 ,the rule for ,the ,number of ,digits in a ,square is easily deducible from the rule for multiplication4 ,if the square is read on the left scale of ,a1 it will contain twice the number of digits in the original number less #a2 if it is read on the right scale of ,a1 it will contain twice the number of digits in the original number4 ,,ex4--,find the square of #aad4 ,placing the cursor to #aad on ,d1 it is seen that the coinciding number on ,a is #ac4 ,as the result is read off on the left scale of ,a1 the number of digits will be 7#c #b7 #a #e1 and the answer is read as #ac1jjj4 ,the true result is #ab1iif4 ,,ex4--,find the square of #jjjic4 ,the cursor being placed to #ic on ,d1 the number on ,a is found to be #hfe4 ,the result is read on the right scale of ,a1 so the number of digits #b #b #d1 and the answer is read as #jjjjjhfe #jjjjjhfdi4 ,square ,root4--,the foregoing rules suggest the method of procedure in the inverse operation of extracting the square root of a given number1 which will be found on the ,d scale opposite the number on the ,a scale4 ,it is necessary to observe1 however1 that if the number consists of an odd number of digits1 it is to be taken on the left-hand portion of the ,a scale1 and the number of digits in the root 7,n #a7_/7b71 ,n being the number of digits in the original number4 ,when there is an even number of digits in the number1 it is to be taken on the right-hand portion of the ,a scale1 and the root contains one-half the number of digits in the original number4 ,,ex4--,find the square root of #cf1ejj4 ,as there is an odd number of digits1 placing the cursor to #cfe on the ,l4,h4 ,a scale gives #aia on ,d4 ,by the rule there are 7,n #a7_/7b7 7#e #a7_/7b7 #c digits in the required root1 which is therefore read as #aia #aiaje4 ,,ex4--,find 7#jjjih47 ,placing the cursor to #ih on the right-hand scale of ,a 7since #b is an even number of digits71 it is seen that the coinciding number on ,d is #ii4 ,as the number of digits in the number is #b1 the number of digits in the root will be 7#b7_/7b7 #a4 ,it will therefore be read as #jjii #jjihii4 ,,ex4--,find 7#jjih74 ,the number of digits is #a1 so under #ih on the left scale of ,a1 we find #cac on ,d4 ,by the rule the number in the root will be 7#a #a7_/7b7 #j1 and the root is therefore read as #jcac #jcacjdi4 ,,ex4--,find 7#jadi47 ,as the number of digits 7#j7 is even1 the cursor is set to #adi on the right-hand scale of ,a1 giving #chf on ,d4 ,by the rule1 the number of digits in the root will be 7#j7_/7b7 #j1 and the root will be read as #jchf #jchfje4 ,another method of extracting the square root1 by which more accurate readings may generally be obtained1 is by using the ,c and ,d scales only1 with the slide inverted4 ,if there is an odd number of digits in the number1 the right index1 or if an even number of digits the left index1 of the inverted scale is placed so as to coincide with the number on ,d of which the root is sought4 ,then with the cursor1 the number is found on ,d which coincides with the same number on 1 which number is the root sought4 ,,ex4--,find 7#bbb47 ,placing the left index of to #bbb on ,d1 the two equal coinciding numbers on and ,d are found to be #dga4 ,note that under the cursor line we have the original number1 #bbb1 on ,a1 and from this the number of digits in the root is determined as before4 ,the plan of finding the square of a number by ordinary multiplication is often very convenient4 ,the inverse process of finding a square root by trial division is not to be recommended4 ,to obtain a close value of a root or to verify one found in the usual way1 the author has1 on occasion1 adopted the following plan3--,set #a 7or #aj7 on ,b to the number on the ,a scale 7,l4,h4 or ,r4,h4 as the case may require71 and bring the cursor to the number on ,d4 ,if the root found is correct1 the readings on ,c under the cursor and on ,d under the index of ,c1 will be in exact agreement4 ,if #a on ,b is placed to a number n on the ,l4,h4 ,a scale1 the student will note that while root n is read on ,d under #a on ,c1 the root of #aj n is read on ,d under #aj on ,b4 ,hence1 if preferred1 the number can be taken always on the first scale of ,a and the root read under #a or #aj on ,b1 according to whether there is an odd or even number of digits in the number4 ,obviously the second root is the first multiplied by 7#aj74 ,,cubes ,,and ,,cube ,,roots4 ,in raising a number to the third power1 a combination of the preceding method and ordinary multiplication is employed4 ,,to ,,find ,,the ,,cube ,,of ,a ,,number4--,set the ,l4,h4 or ,r4,h4 index of ,c to the number on ,d1 and opposite the number ,,on ,,the ,,left-,,hand scale of ,b read the cube on the ,l4,h4 or ,r4,h4 scale of ,a4 ,by this rule four scales are brought into requisition4 ,of these1 the ,d scale and the ,l4,h4 ,b scale are always employed1 and are to be read as of equal denomination4 ,the values assigned to the ,l4,h4 and ,r4,h4 scales of ,a will be apparent from the following considerations4 ,commencing with the indices of ,c and ,d coinciding1 and moving the slide to the right1 it will be seen that1 working in accordance with the above rule1 the cubes of numbers from #a to #baed 7 7#aj77 will be found on the first or ,l4,h4 scale of ,a4 ,moving the slide still farther to the right1 we obtain on the ,r4,h4 ,a scale cubes of numbers from #baed to #dfda 7or 7#aj7 to 7#ajj774 ,had we a third repetition of the ,l4,h4 ,a scale1 the ,l4,h4 index of ,c could be still further traversed to the right1 and the cubes of numbers from #dfda to #aj read off on this prolongation of ,a4 ,but the same end can be attained by making use of the ,r4,h4 index of ,c1 when1 traversing the slide to the right as before1 the cubes of numbers from #dfda to #aj on ,d can be read off on the ,l4,h4 ,a scale over the corresponding numbers on the ,l4,h4 ,b scale4 ,hence1 using the ,l4,h4 index of ,c1 the readings on the ,l4,h4 ,a scale may be regarded comparatively as units1 those on the ,r4,h4 ,a scale as tens2 while for the hundreds we again make use of the ,l4,h4 ,a scale in conjunction with the right-hand index of ,c4 ,by keeping these points in view1 the number of digits in the cube 7,n7 of a given number 7n7 are readily deduced4 ,thus1 if the units scale is used1 ,n #cn #b2 if the tens scale1 ,n #cn #a2 while if the hundreds scale be used1 ,n #cn4 ,placed in the form of rules3-- ,n #cn #b when the product is read on the ,l4,h4 scale of ,a with the slide to the right 7units scale74 ,n #cn #a when the product is read on the ,r4,h4 scale of ,a2 slide to the right 7tens scale74 ,n #cn when the product is read on the ,l4,h4 scale of ,a with the slide to the left 7hundreds scale74 ,with decimals the same rule applies1 but1 as before1 the number of digits must be read as #a1 #b1 etc41 when one1 two1 etc41 cyphers follow immediately after the decimal point4 ,,ex4--,find the value of #adc4 ,placing the ,l4,h4 index of ,c to #ad on ,d1 the reading on ,a opposite #ad on the ,l4,h4 scale of ,b is found to be about #bgde #bgdd4 ,,ex4--,find the value of #bfdc4 ,placing the ,l4,h4 index of ,c to #bfd on ,d1 the reading on ,a opposite #bfd on the ,l4,h4 scale of ,b is found to be about #ah1djj #ah1ciigdd4 ,,ex4--,find the value of #gcc4 ,in this case it becomes necessary to use the ,r4,h4 index of ,c1 which is set to #gc on ,d1 when opposite #gc on the ,l4,h4 scale of ,b is read #chi #chijag on ,a4 ,,ex4--,find the value of #jjgcc4 ,from the setting as before it is seen that the number of digits in the number must be multiplied by #c4 ,hence1 as there is #a digit in #jjgc1 there will be #c in the cube1 which is therefore read #jjjjchi4 ,the last two examples serve to illustrate the principle of factorising with powers of #aj4 ,thus #jjgc #gc #ajb2 #jjgcc #gcc 7#ajb7c #chi #ajf #jjjjchi4 ,cube ,root 7,direct ,method74--,one method of extracting the cube root of a number is by an inversion of the foregoing operation4 ,using the same scales1 the slide is moved either to the right or left until under the given number on ,a is found a number on the ,l4,h4 ,b scale1 identical with the number simultaneously found on ,d under the right or left index of ,c4 ,this number is the required cube root4 ,from what has already been said regarding the combined use of these scales in cubing1 it will be evident that in extracting the cube root of a number1 it is necessary1 in order to decide which scales are to be used1 to know the number of figures to be dealt with4 ,we therefore 7as in the arithmetical method of extraction7 point off the given number into sections of three figures each1 commencing at the decimal point1 and proceeding to the left for numbers greater than unity1 and to the right for numbers less than unity4 ,then if the first section of figures on the left consists of-- #a figure1 the number will evidently require to be taken on what we have called the 8units0 scale--i4e41 on the ,l4,h4 scale of ,a1 using the ,l4,h4 index of ,c4 ,if of #b figures1 the number will be taken on the 8tens0 scale--i4e41 on the ,r4,h4 scale of ,a1 using the ,l4,h4 index of ,c4 ,if of #c figures1 the number will be taken on the 8hundreds0 scale--i4e41 on the ,l4,h4 scale of ,a1 using the ,r4,h4 index of ,c4 ,to determine the number of digits in cube roots it is only necessary to note that when the number is pointed off into sections as directed1 there will be one figure in the root for every section into which the number is so divided1 whether the first section consists of #a1 #b1 or #c digits4 ,of numbers wholly decimal1 the cube roots will be decimal1 and for every group of three #js immediately following the decimal point1 one #j will follow the decimal point in the root4 ,if necessary1 #js must be added so as to make up complete multiples of #c figures before proceeding to extract the root4 ,thus #jh is to be regarded as #jhjj1 and #jjjjjh as #jjjjjhj in extracting cube roots4 ,,ex4--,find 7#ad1jjj47 ,pointing the number off in the manner described1 it is seen that there are two figures in the first section--viz41 #ad4 ,setting the cursor to #ad on the ,r4,h4 scale of ,a1 the slide is moved to the right until it is seen that #bda on the ,l4,h4 scale of ,b falls under the cursor1 when #bda on ,d is under the ,l4,h4 index of ,c4 ,pointing #ad1jjj off into sections we have #ad #jjj--that is1 two sections4 ,therefore1 there are two digits in the root1 which in consequence will be read #bda #bdajad4 ,,ex4--,find 7#jafb47 ,as the divisional section consists of three figures1 we use the 8hundreds0 scale4 ,setting the cursor to #jafb on the ,l4,h4 ,a scale1 and using the ,r4,h4 index of ,c1 we move the slide to the left until under the cursor #jede is found on the ,l4,h4 ,b scale1 while the ,r4,h4 index of ,c points to #jede on ,d1 which is therefore the cube root of #jafb4 ,,ex4--,find 7#jjjjb47 ,to make even multiples of #c figures requires the addition of #jj2 we have then #bjj1 the cube root of which is found to be about #ehe4 ,then1 since the first divisional group consists of #js1 one #j will follow the decimal point1 giving 7#jjjjb7 #jjehe #jjehdh4 ,cube ,root 7,inverted ,slide ,method74--,another method of extracting the cube root involves the use of the inverted slide4 ,several methods are used1 but the following is to be preferred3--,set the ,l4,h4 or ,r4,h4 index of the slide to the number on ,a1 and the number on 7i4e41 ,b inverted71 which coincides with the same number on ,d1 is the required root4 ,setting the slide as directed1 and using first the ,l4,h4 index of the slide and then the ,r4,h4 index1 it is always possible to find three pairs of coincident values4 ,to determine which of the three is the required result is best shown by an example4 ,,ex4--,find 7#e17 7#ej17 and 7#ejj47 ,setting the ,r4,h4 index of the slide to #e on ,a1 it is seen that #aga on ,d coincides with #aga on 4 ,then setting the ,l4,h4 index to #e on ,a1 further coincidences are found at #cfh and at #gic1 the three values thus found being the required roots4 ,note that the first root was found on that portion of the ,d scale lying under #a to #e on ,a2 the second root on that portion lying under #e to #ej on ,a2 and the third root on that portion of ,d lying under #ej to #ajj on ,a4 ,in this connection1 therefore1 scale ,a may always be considered to be divided into three sections--viz41 #a to n1 n to #ajn1 and #ajn to #ajj4 ,for all numbers consisting of #a1 #a #c1 #a #f1 #a #i--;i4e41 of #a1 #d1 #g1 #aj1 or #b1 #e1 etc41 figures--the coincidence under the first section is the one required4 ,if the number has #b1 #e1 #h1 or #a1 #d1 #g1 etc41 figures1 the coincidence under the second section is correct1 while if the number has #c1 #f1 #i1 or #j1 #c1 etc41 figures1 the coincidence under the last section is that required4 ,the number of digits in the root is determined by marking off the number into sections1 as already explained4 ,cube ,root 7,pickworth's ,method74--,one of the principal objections to the two methods described is the difficulty of recollecting which scales are to be employed and with which index of the slide they are to be used4 ,with the direct method another objection is that the readings to be compared are often some distance apart1 the maximum distance intervening being two-thirds of the length of the rule4 ,to carry the eye from one to another is troublesome and time-taking4 ,with the inverted scale method the reading of a scale reversed in direction and with the figures inverted is also objectionable4 ,with the author's method these objections are entirely obviated4 ,the same scales and index are always used1 and are read in their normal position4 ,the three roots of n1 #ajn and #ajjn 7n being less than #aj and not less than #a7 are given with one setting and appear in their natural sequence1 no traversing of the slide being needed4 ,the readings to be compared are always close together1 the maximum distance between them being one-sixth of the length of the rule4 ,the setting is always made in the earlier part of the scales where closer readings can be obtained1 and finally1 if desired1 the result may be readily verified on the lower scales by successive multiplication4 ,for this method two gauge points are required on ,c4 ,to conveniently locate these1 set #ec on ,c to #bdf on ,d2 join #a on ,d to #a on ,a with a straight-edge and with a needle point draw a short fine line on ,c4 ,set #bdf on ,c to #ec on ,d1 and repeat the process at the other end of the rule4 ,the gauge points thus obtained 7dividing ,c into three equal parts7 will be at #baed and #dfda1 and should be marked 7#aj7 and 7#ajj7 respectively4#f ,,ex4--,find 7#bhf17 7#bhf7 and 7#bhf74 ,set cursor to #bhf on ,a and drawing the slide to the right find #adb under #a on ,c1 when #adb on ,b is under the cursor4 ,then reading under #a1 7#aj7 and 7#ajj17 we have 7#bhf7 #adb2 7#bhf7 #cjf and 7#bhf7 #fei4 ,it will be seen that factorising with powers of #aj1 we multiply the initial root by 7#aj7 and 7#ajj74 ,obviously the three roots will always be found on ,d1 in their natural order and at intervals of one-third the length of the rule4 ,the number of digits in the roots of numbers which do not lie between #a and #ajjj1 is found as before explained4 ,in any method of extracting cube roots in which the slide has to be adjusted to give equal readings on ,b and ,d1 the author has found it of advantage to adopt the following plan3--,the cursor being set to1 say1 #dh on ,a1 bring a near main division line on ,b1 as #ag1 to the cursor2 then #a on ,c is at #afh on ,d4 ,the difference in the readings is two small divisions on ,d1 and moving the slide forward by one-third the space representing this difference1 we obtain #afhg as the root required4 ,with a little practice it is possible to obtain more accurate results by this method than by comparing the reading on ,d with that on the less finely-graded ,b scale4 ,,miscellaneous ,,powers ,,and ,,roots4 ,in addition to squares and cubes1 certain other powers and roots may be readily obtained with the slide rule4 ,two-thirds ,power4--,the value of ,n is found on ,a over ,n on ,d4 ,the number of digits is decided by the rule for squares1 working from the number of digits in the cube root4 ,it will often be found preferable to treat ,n as ,n ,n1 as in this way the magnitude of the result is much more readily appreciated4 ,three-two ,power4--,n can be obtained by cubing the square root1 deciding the number of digits in each process4 ,for the reason just given1 it is preferable to regard ,n as ,n ,n4 ,fourth ,power4--,for ,n#d set the index of ,c to ,n on ,d and over ,n on ,c read ,n#d on ,a2 or find the square of the square of ,n1 deciding the number of digits at each step4 ,fourth ,root4--,similarly for ,n1 take the square root of the square root4 ,four-third ,power4--,n ,n#acc 7useful in gas-engine diagram calculations7 is best treated as ,n ,n4 ,other powers can be found by repeated multiplication4 ,thus setting #a on ,b to ,n on ,a1 we have on ,a1 ,n#b over ,n2 ,n#c over ,n#b2 ,n#d over ,n#c2 ,n#e over ,n#d1 etc4 ,in the same way1 setting ,n on ,b to ,n on ,d1 we can read such values as ,n1 ,n1 etc4 ,,powers ,,and ,,roots ,,by ,,logarithms4 ,for powers or roots other than those of the simple forms already discussed1 it is necessary to employ the usual logarithmic process4 ,thus to find an x1 we multiply the logarithm of a by n1 and find the number x corresponding to the logarithm so obtained4 ,similarly1 to find a x we divide the logarithm of a by n1 and find the number x corresponding to the resulting logarithm4 ,the ,scale of ,logarithms4--,upon the back of the slide of the ,gravt and similar slide rules there will be found three scales4 ,one of these--usually the centre one--is divided equally throughout its entire length1 and figured from right to left4 ,it is sometimes marked ,l1 indicating that it is a scale giving logarithms4 ,the whole scale is divided primarily into ten equal parts1 and each of these subdivided into #ej equal parts4 ,in the recess or notch in the right-hand end of the rule is a reference mark1 to which any of the divisions of this evenly-divided scale can be set4 ,as this decimally-divided scale is equal in length to the logarithmic scale ,d1 and is figured in the reverse direction1 it results that when the slide is drawn to the right so that the ,l4,h4 index of ,c coincides with any number on ,d1 the reading on the equally-divided scale will give the decimal part of the logarithm of the number taken on ,d4 ,thus if the ,l4,h4 index of ,c is placed to agree with #b on ,d1 the reading of the back scale1 taken at the reference mark1 will be found to be #jcja1 the logarithm of #b4 ,it must be distinctly borne in mind that the number so obtained is the decimal part or mantissa of the logarithm of the number1 and that to this the characteristic must be prefixed in accordance with the usual rule--viz41 ,the integral part1 or characteristic of a logarithm is equal to the number of digits in the number1 minus #a4 ,if the number is wholly decimal1 the characteristic is equal to the number of cyphers following the decimal point1 plus #a4 ,in the latter case the characteristic is negative1 and is so indicated by having the minus sign written over it4 ,to obtain any given power or root of a number1 the operation is as follows3--,set the ,l4,h4 index of ,c to the given number on ,d1 and turning the rule over1 read opposite the mark in the notch at the right-hand end of the rule1 the decimal part of the logarithm of the number4 ,add the characteristic according to the above rule1 and multiply by the exponent of the power1 or divide by the exponent of the root4 ,place the decimal part of the resultant reading1 taken on the scale of equal parts1 opposite the mark in the aperture of the rule1 and read the answer on ,d under the ,l4,h4 index of ,c1 pointing off the number of digits in the answer in accordance with the number of the characteristic of the resultant4 ,,ex4--,evaluate #cfadad4 ,set #a on ,c to #cf on ,d and read the decimal part of log4 #cf on the scale of logarithms on the back of the slide4 ,this value is found to be #jeef4 ,as there are two digits in the number1 the characteristic will be #a2 hence log4 #cf #aeef4 ,multiply by #adad1 using the ,c and ,d scales1 and obtain #bb as the log4 of the result4 ,set the decimal part1 #jb1 on the log4 scale to the mark in the notch at the end of the rule and read #aehe on ,d under #a on ,c4 ,since the log4 of the result has a characteristic #b1 there will be #c digits in the result1 which is therefore read as #aehe4 ,this example will suffice to show the method of obtaining the nth power or the nth root of any number4 ,,other ,,methods ,,of ,,obtaining ,,powers ,,and ,,roots4 ,a simple method of obtaining powers and roots1 which may serve on occasion1 is by scaling off proportional lengths on the ,d scale 7or the ,a scale7 of the ordinary rule4 ,thus1 to determine the value of #abeafg we take the actual length #a-abe on ,d scale1 and increase it by any convenient means in the proportion of #a #afg4 ,then with a pair of dividers we set off this new length from #a1 and obtain #add as the result4 ,one convenient method of obtaining the desired ratio is by a pair of proportional compasses4 ,thus to obtain #aeb1 the compasses would be set in the ratio of #af to #ag1 and the smaller end opened out to include #a-aeb on the ,d scale2 the opening in the large end of the compasses will then be such that setting it off from #a we obtain #aef on ,d as the result sought4 ,illustration3 ,,fig4 #aa4 ,the converse procedure for obtaining the nth root of a number ,n will obviously resolve itself into obtaining 7#a7_/7n7th of the scale length #a-,n1 and need not be further considered4 ,simple geometrical constructions are also used for obtaining scale lengths in the required ratio4 ,a series of parallel lines ruled on transparent celluloid or stout tracing paper may be placed in an inclined position on the face of the rule and adjusted so as to divide the scale as desired4 ,when much work is to be done which requires values to be raised to some constant but comparatively low power1 n1 the author has found the following device of assistance3--,on a piece of thin transparent celluloid a line ,,oc is drawn 7,fig4 #aa7 and in this a point ,b is taken such that 7,,oc7_/7,,ob7 is the desired ratio4 ,it is convenient to make ,,ob #a-aj on the ,a scale1 so that assuming we require a series of values of v#ace1 ,,ob would be #abe cm4 and ,,oc1 #afhge cm4 ,on these lines semi-circles are drawn as shown1 both passing through the point ,o4 ,applying this cursor to the upper scales so that the point ,o is on #a and the semi-circle ,o ,m ,b passes through v on ,a1 the larger semi-circle will give on ,a the value of vn4 ,thus for p vn #cie #diace1 set #a on ,b to #cie on ,a 7,fig4 #ab7 and apply the cursor to the working edge of ,b1 so that ,o agrees with #a and ,o ,m ,b passes through #di on ,b4 ,the larger semi-circle then cuts the edge of the slide on a point1 giving #ccg on ,a as the result required4 ,of course any number of semi-circles may be drawn1 giving different ratios4 ,if a number of evenly-spaced divisions are used as bases1 the device affords a simple means of obtaining a succession of small powers or roots1 while it also finds a use in determining a number of geometric means between two values as is required in arranging the speed gears of machine tools1 etc4 ,the converse operation of finding roots will be evident as will also many other uses for which the device is of service4 ,illustration3 ,,fig4 #ab4 ,the lines should be drawn in ,indian ink with a very sharp pen and on the under side of the celluloid so that the lines lie in close contact with the face of the rule4 ,the ,radial ,cursor1 another device for the same purpose1 is always used in conjunction with the upper scales4 ,as will be seen from ,fig4 #ac1 the body of the cursor ,p carries a graduated bar ,s which can be removed in a direction transverse to the rule1 and adjusted to any desired position4 ,pivoted to the lower end of ,s is a radial arm ,r of transparent celluloid on which a centre line is engraved4 ,a reference to the illustration will show that the principle involved is that of similar triangles1 the width of the slide being used as one of the elements4 ,thus1 to take a simple case1 if #b on ,s is set to the index on ,p1 and #a on ,b is brought to ,n on ,a1 then by swinging the radial arm until its centre line agrees with #a on ,c1 we can read ,n#b on ,a4 ,evidently1 since in the two similar triangles ,a ,o ,n#b and ,n t ,n#b the length of ,a ,o is twice that of ,n t1 it results that ,a ,n#b #b ,a ,n4 ,in general1 then1 to find the nth power of a number1 we set the cursor to #a or #aj on ,a1 bring n on the cross bar ,s to the index on the cursor1 and #a on ,b to ,n on ,a4 ,then to #a on ,c we set the line on the radial arm1 and under the latter read ,nn on ,a4 ,the inverse proceeding for finding the nth root will be obvious4 ,illustration3 ,,fig4 #ac4 ,an advantage offered by this and analogous methods of obtaining powers and roots is that the result is obtained on the ordinary scale of the rule1 and hence it can be taken directly into any further calculation which may be necessary4 ,,combined ,,operations4 ,thus far the various operations have been separately considered1 and we now pass on to a consideration of the methods of working for solving the various formul met with in technical calculations4 ,we propose to explain the methods of dealing with a few of the more generally used expressions1 as this will suffice to suggest the procedure in dealing with other and more intricate calculations4 ,in solving the following problems1 both the upper and lower scales are used1 and the relative value of the several scales must be observed throughout4 ,thus1 in solving such an expression as 77#gde7_/7aeh77 #fhf1 the division is first effected by setting #aeh on ,b to #gde on ,a4 ,from the relation of the two parts of the upper scales 7page #cg7 we know that such values as #gde1 #gde1 etc41 will be taken on the left-hand ,a and ,b scales1 while values as #aeh1 #aehj1 etc41 will be taken on the right-hand ,a and ,b scales4 ,hence1 #aeh on the ,r4,h4 ,b scale is set to #gde on the ,l4,h4 ,a scale1 and the result read on ,d under the index of ,c4 ,had both values been taken on the ,l4,h4 ,a and ,b scales1 or both on the ,r4,h4 ,a and ,b scales1 the results would have corresponded to x 77#gde7_/7aeh77 #bag1 or to x 77#gde7_/7aeh77 #bag1 i4e41 to 7#fhf7_/77aj774 ,hence if a wrong choice of scales has been made1 we can correct the result by multiplying or dividing by 7#aj7 as the case may require4 ,if the result is read on ,d1 set to it the centre index 7#aj7 of ,b and read the corrected result under the index of ,c4 ,to solve a b#b x4 ,set the index of ,c to b on ,d1 and over a on ,b read x on ,a4 ,to solve 7a#b7_/7;b7 x4 ,set b on ,b to a on ,d by using the cursor1 and over index of ,b read x on ,a4 ,to solve 7b7_/7a#b7 x4 ,set a on ,c to b on ,a1 and over #a on ,b read x on ,a4 ,to solve 7a b#b7_/7;c7 x4 ,set c on ,b to b on ,d1 and over a on ,b read x on ,a4 ,to solve 7a b7#b x4 ,set #a on ,c to a on ,d1 and over b on ,c read x on ,a4 ,to solve 77a7_/7b77#b x4 ,set b on ,c to a on ,d1 and over #a on ,c read x on ,a4 ,to solve 7a b7 x4 ,set #a on ,b to a on ,a1 and under b on ,b read x on ,d4 ,to solve 77a7_/7b77 x4 ,set b on ,b to a on ,a1 and under #a on ,c read x on ,d4 ,to solve a 7b7_/7c#b7 x4 ,set b on ,c to c on ,d and over a on ,b read x on ,a4 ,to solve c77a7_/7b77 x4 ,set b on ,b to a on ,a1 and under c on ,c read x on ,d4 ,to solve 7a7_/7b7 x4 ,set b on ,c to a on ,a1 and under #a on ,c read x on ,d4 ,to solve 7a7_/7b7 x4 ,set b on ,b to a on ,d1 and under #a on ,c read x on ,d4 ,to solve ba x4 ,set #a on ,c to b on ,d1 and under a on ,b read x on ,d4 ,to solve 7a#c7 x4 ,treat as aa4 ,to solve a7b#c7 x4 ,treat as ab b4 ,to solve 7a#c7_/7;b7 x4 ,treat as 7a a7_/7b74 ,to solve 77a#c7_/7;b77 x4 ,treat as 7a a7_/7b7 77a7_/7b77 a4 ,to solve 77a b7_/7c77 x4 ,set c on ,b to a on ,a1 and under b on ,b read x on ,d4 ,to solve 7a b7_/7c7 x4 ,set c on ,b to b an ,d1 and under a on ,c read x on ,d4 ,to solve 77a#b b7_/7c77 x4 ,set c on ,b to a on ,d1 and under b on ,b read x on ,d4 ,to solve 7a#b b#b7_/7;c7 x4 ,set c on ,b to a on ,d1 and over b on ,c read x on ,a4 ,to solve 7ab7_/7c7 x4 ,set c on ,c to b on ,a1 and under a on ,c read x on ,d4 ,to solve 77a b7_/7c77#b x4 ,set c on ,c to a on ,d1 and over b on ,b read x on ,a4 ,,hints ,,on ,,evaluating ,,expressions4 ,as a general rule1 the use of cubes and higher powers should be avoided whenever possible4 ,thus1 in the foregoing section1 we recommend treating an expression of the form a7b#c7 as a b b2 the magnitudes of the values thus met with are more easily appreciated by the beginner1 and mistakes in estimating the large numbers involved in cubing are avoided4 ,,ex4--#gc 7#egc7 #cadj4 ,set #a on ,c to #eg on ,d2 bring cursor to #eg on ,b 7,r4,h41 since #eg has an even number of digits72 bring #a on ,c to cursor1 and under #gc on ,c read #cadj on ,d4 ,as a rough estimate we have 7#eg71 about #h2 #h #eg1 about #djj2 #djj #g1 gives #bhjj1 showing the result consists of #d figures4 ,an expression of the form a7b#b71 or a b1 is better dealt with by rearranging as a 7b7_/7b74 ,,ex4--#cfd7dcbb7 #ife4 ,set cursor to #dcb on ,a1 and move the slide until #afc is found simultaneously under the cursor on ,b and on ,d under #a on ,c2 bring cursor to #a on ,c2 #dcb on ,c to cursor1 and over #cfd on ,d read #ife on ,c4 7,note that in this case it is convenient to read the answer on the slide2 see page #bb74 ,from the slide rule we know 7#dcb7 about #af2 this into #dcb is roughly #c2 #cfd #c is about #aj1 showing the answer to be #ife4 ,similarly products of the form a b are best dealt with as a b b4 ,factorising expressions sometimes simplifies matters1 as1 for instance1 in x#d y#d 7x#b y#b77x#b y#b74 ,here1 working with the fourth powers involves large numbers and the troublesome determination of the number of digits in each factor2 but squares are read on the rule at once1 the number of digits is obvious1 and1 in general1 the method should give a more accurate result4 ,take the expression1 ,d#a 77,d#d d#d7_/7;,d77 giving the diameter ,d#a of a solid shaft equal in torsional strength to a hollow shaft whose external and internal diameters are ,d and d respectively4 ,rearranging as ,d#a 777,d#b d#b77;,d#b d#b77_/7;,d77 and taking1 as an example1 ,d #ae in4 and d #g in41 we have ,d#b d#b #bgd and ,d#b d#b #agf2 hence ,d#a 77#bgd #agf7_/7ae77 7#cbaj7 #adge in4 ,reversed ,scale ,notation4--,with expressions of the form #a x1 or #ajj x1 it is often convenient to regard the scales as having their notation reversed1 i4e41 to read the scale backwards4 ,when this is done the ,d scale is read as shown on the lower line-- ,direct ,notation #a #b #c #d #e #f #g #h #i #aj ,d ,scale ,reversed ,notation #i #h #g #f #e #d #c #b #a #j ,the new reading can be found by subtracting the ordinary reading from #a1 #aj1 #ajj1 etc41 according to the value assigned to the ,r4,h4 index1 but actually it is unnecessary to make this calculation1 as with a little practice it is quite an easy matter to read both the main and subdivisions in the reversed order4 ,applications are found in plotting curves1 trigonometrical formul1 etc4 ,,ex4--,find the per cent4 of slip of a screw propeller from #ajj ,s 7#ajacc,v7_/7,,pr7 taking the speed1 ,v1 as #ae knots1 the pitch of the propeller1 ,p1 as #bg ft4 #f in41 and the revolutions per minute1 ,r1 as #fj4 ,set #bge on ,b to #ajacc on ,a 7,n4,b4--,take the setting near the centre index of ,a72 bring the cursor to #ae on ,b and #fj on ,b to cursor4 ,reading the ,l4,h4 ,a scale backwards1 the slip1 ,s1 #h per cent4 is found on ,a over #aj on ,b4 ,percentage ,calculations4--,to increase a quantity by x per cent4 we multiply by #ajj x2 to diminish a quantity by x per cent4 we multiply by #ajj x4 ,hence1 to add x per cent41 set #ajj x on ,c to #a on ,d and read new values on ,d under original values on ,c4 ,to deduct x per cent4 read the ,d scale backwards from #aj and set ,r4,h4 index of ,c to x per cent4 so read4 ,then read as before4 ,,gauge ,,points4 ,special graduations1 marking the position of constant factors which frequently enter into engineering calculations1 are found on most slide rules4 ,usually the values of #cadaf and 77_/7#d7 #jghed--the 8gauge points0 for calculating the circumference and area of a circle--are marked on the upper scales4 ,the first should be given on the lower scales also4 ,marks c and c#a are sometimes found on the lower scales at #aabh 77#d7_/777 and at #cefh 77#dj7_/7774 ,these are useful in calculating the contents of cylinders and are thus derived3--,cubic contents of cylinder of diameter d and length l 77_/7#d7;d#bl2 substituting for 77_/7#d7 its reciprocal 7#d7_/771 the formula becomes 7d#b7_/7abgc l71 and by taking the square root of the fractional part we have 7d7_/7#aabh7b l4 ,this is now in a very convenient form1 since by setting the gauge point c on ,c to d on ,d1 we can read over l on ,b the cubic contents on ,a4 ,this example indicates the principle to be followed in arranging gauge points4 ,successive multiplication is avoided by substituting the reciprocal of the constant1 thus bringing the expression into the form 7a b7_/7c71 which1 as we know1 can be resolved by one setting of the slide4 ,the advantage of dividing d before squaring is also evident4 ,the mark c#a c 7#aj7 is used if it is necessary to draw the slide more than one-half its length to the right4 ,a gauge point1 ,m1 at #cahc 7#ajj7_/77 is found on the upper scales of some rules4 ,setting this point on ,b to the diameter of a cylinder on ,a1 the circumference is read over #a or #ajj on ,b or the area of the curved surface over the length on ,b4 ,as another example of establishing a gauge point1 we will take the formula for the theoretical delivery of pumps4 ,if d is the diameter of the plunger in inches1 l the length of stroke in feet1 and ,q the delivery in gallons1 we have ,q d#b 77_/7#d7 l 7#ab7_/7bgg74 7,n4,b4--#bgg cubic inches #a gallon47 ,multiplying out the constant quantities and taking its reciprocal1 we readily transform the statement into ,q 7d#bl7_/7#bid7 or 77d7_/7#edb77b l4 ,hence set gauge point #edb on ,c to d on ,d and over length of stroke in feet on ,b1 read delivery in gallons per stroke on ,a2 or over piston speed in feet per minute on ,b1 read theoretical delivery in gallons per minute on ,a4 ,several examples of gauge points will be found in the section on calculating the weights of metal 7see pages #ei and #fj74 ,in most cases their derivation will be evident from what has been said above4 ,in the case of the weight of spheres1 we have ,vol4 #jebcf;d#c1 and this multiplied by the weight of #a cubic inch of the material will give the weight ,w in lb4 ,hence for cast-iron1 ,w #jebcf d#c #jbf1 which is conveniently transformed into ,w 7d d#b7_/7gce7 as in the example on page #fj4 ,with these examples no difficulty should be experienced in establishing gauge points for any calculation in which constant factors recur4 ,marking ,gauge ,points4--,the practice of marking gauge points by lines extending to the working edge of the scale is not to be recommended1 as it confuses the ordinary reading of the scales4 ,generally speaking1 gauge points are only required occasionally1 and if they are placed clear of the scale to which they pertain1 but near enough to show the connection1 they can be brought readily into a calculation by means of the cursor4 ,usually there is sufficient margin above the ,a scale and below the ,d scale for various gauge points to be marked4 ,another plan consists in cutting two nicks in the upper and lower edges of the cursor near the centre and about in4 apart4 ,these centre pieces1 when bent out1 form a tongue1 which are in line with the cursor line and run nearly in contact with the square and bevelled edges of the rule respectively4 ,a fine line in the tongue can then be set to gauge points marked on these two edge strips1 the ordinary measuring graduations being removed1 if desired1 by a piece of fine sand-paper4 ,for gauge points marked on the face of the rule1 the author prefers two fine lines drawn at #de--thus1 --and crossing in the exact point which it is required to indicate4 ,with the 8cross0 gauge point the meeting lines facilitate the placing of the cursor1 and an exact setting is readily made4#g ,all lines should be drawn in ,indian ink with a very sharp drawing pen4 ,for a more permanent marking the ,indian ink may be rubbed up in glacial acetic acid or the special ink for celluloid may be used4 ,if any difficulty is found in writing the distinguishing signs against the gauge point1 the inscription may be formed by a succession of small dots made with a sharp pricker4 ,,examples ,,in ,,technical ,,calculations4 ,in order to illustrate the practical value of the slide rule1 we now give a number of examples which will doubtless be sufficient to suggest the methods of working with other formul4 ,a few of the rules give results which are approximate only1 but in all cases the degree of accuracy obtained is well within the possible reading of the scales4 ,in many cases the rules given may be modified1 if desired1 by varying the constants4 ,in most of the examples the particular formula employed will be evident from the solution1 but in a few of the more complicated cases a separate statement has been given4 ,,mensuration1 ,,etc4 ,given the chord c of a circular arc1 and the vertical height h1 to find the diameter d of the circle4 ,set the height h on ,b to half the chord on ,d1 and over #a on ,b read x on ,a4 ,then x h d4 ,,ex4--c #f2 h #b2 find d4 ,set #b on ,b to #c on ,d1 and over #a on ,b read #de on ,a4 ,then #de #b #fe d4 ,given the radius of a circle r1 and the number of degrees n in an arc1 to find the length l of the arc4 ,set r on ,c to #egc on ,d1 and over any number of degrees n on ,d read the 7approximate7 length of the arc on ,c4 ,,ex4--r #bd2 n #cj2 find l4 ,set #bd on ,c to #egc on ,d1 and over #cj on ,d read #abef l on ,c4 ,given the diameter d of a circle in inches1 to find the circumference c in feet4 ,set #aia on ,c to #ej on ,d1 and under any diameter in inches on ,c read circumference c1 in feet on ,d4 ,,ex4--,find the circumference in feet of a pulley #ag in4 in diameter4 ,set #aia on ,c to #ej on ,d1 and under #ag on ,c read #dde ft4 on ,d4 ,given the diameter of a circle1 to find its area4 ,set #jghed on ,b to #aj 7centre index7 on ,a and over any diameter on ,d read area on ,b4 ,when the rule has a special graduation line #jghed1 on the right-hand scale of ,b1 set this line to the ,r4,h4 index of ,a and read off as above4 ,if only is marked1 set this special graduation on ,b to #d on ,a4 ,on the ,c and ,d scales of some rules a gauge point marked c will be found indicating 77#d7_/777 #aabhf4 ,in this case1 therefore1 set #a on ,c to gauge point c on ,d1 and read area on ,a as above4 ,if the gauge point c is used1 divide the result by #aj4 ,or set c on ,c1 to diameter on ,d1 and over index of ,b read area on ,a4 ,cursors are supplied1 having two lines ruled on the glass1 the interval between them being equal to 7#d7_/77 #abgc on the ,a scale4 ,in this case1 if the right hand of the two cursor lines be set to the diameter on ,d1 the area will be read on ,a under the left-hand cursor line4 ,for diameters less than #aaa it is necessary to set the middle index of ,b to the ,l4,h4 index of ,a1 reading the areas on the ,l4,h4 ,b scale4 ,the confusion which in general work is sometimes caused by the use of two cursor lines might be obviated by making the left-hand line in two short lengths1 each only just covering the scales4 ,given diameter of circle d in inches1 to find area a in square feet4 ,set #f on ,b to #aa on ,a1 and over diameter in inches on ,d read area in square feet on ,b4 ,to find the surface in square feet of boiler flues1 condenser tubes1 heating pipes1 etc41 having given the diameter in inches and length in feet4 ,find the circumference in feet as above and multiply by the length in feet4 ,,ex4--,find the heating surface afforded by #afj locomotive boiler tubes #a;in4 in diameter and #ab ft4 long4 ,set #aia on ,c to #ej on ,d2 bring cursor #age on ,c1 ,l4,h4 index of ,c to cursor2 cursor to #ab on ,c2 #a on ,c to cursor2 and under #afj on ,c read #hhj sq4 ft4 of heating surface on ,d4 ,if the dimensions are in the same denomination and the rule has a gauge point ,m at #cahc 7 7#ajj7_/7771 set this mark on ,b to diameter of cylinder on ,a1 and read cylindrical surface on ,a over length on ,b4 ,to find the side s of a square1 equal in area to a given rectangle of length l and breadth b4 ,set ,r4,h4 or ,l4,h4 index of ,b to l on ,a1 and under b on ,b read s on ,d4 ,,ex4--,find the side of a square equal in area to a rectangle in which l #ca ft4 and b #e ft4 ,set the 7,r4,h47 index of ,b to #ca on ,a1 and under #e on ,b read #abde ft4 on ,d4 ,to find various lengths l and breadths b of a rectangle1 to give a constant area a4 ,invert the slide and set the index of to the given area on ,d4 ,then opposite any length l on find the corresponding breadth b on ,d4 ,,ex4--,find the corresponding breadths of rectangular sheets1 #af1 #ah1 #bd1 #cf1 and #fj ft4 long1 to give a constant area of #gb sq4 ft4 ,set the ,r4,h4 index of to #gb on ,d1 and opposite #af1 #ah1 #bd1 #cf1 and #fj on read #de1 #d1 #c1 #b1 and #ab ft41 the corresponding breadths on ,d4 ,to find the contents in cubic feet of a cylinder of diameter d in inches and length l in feet4 ,find area in feet as before1 and multiply by the length4 ,if dimensions are all in inches or feet1 set the mark c 7 #aabh7 on ,c to diameter on ,d and over length on ,b1 read cubic contents on ,a4 ,to find the area of an ellipse4 ,set #bje on ,c to #afa on ,d2 bring cursor to length of major axis on ,c1 #a on ,c to cursor1 and under length of minor axis on ,c read area on ,d4 ,,ex4--,find the area of an ellipse the major and minor axes of which are #af in4 and #ab in4 in length respectively4 ,set #bje on ,c to #afa on ,d2 bring cursor to #af on ,c1 #a on ,c to cursor1 and under #ab on ,c read #aejh in4 on ,d4 ,to find the surface of spheres4 ,set #cadaf on ,b to ,r4,h4 or ,l4,h4 index of ,a1 and over diameter on ,d read by the aid of the cursor1 the convex surface on ,b4 ,to find the cubic contents of spheres4 ,set #aia on ,b to diameter on ,a1 and over diameter on ,c read cubic contents on ,a4 ,,weights ,,of ,,metals4 ,to find the weight in lb4 per lineal foot of square bars of metal4 ,set index of ,b to weight of #ab cubic inches of the metal 7i4e41 one lineal foot1 #a square inch in section7 on ,a1 and over the side of the square in inches on ,c read weight in lb4 on ,a4 ,,ex4--,find the weight per foot length of #d;in4 square wrought-iron bars4 ,set middle index of ,b to #ccc on ,a1 and over #d on ,c read #fge lb4 on ,a4 7,n4,b4--,for other metals use the corresponding constant in column 7#b71 below74 ,to find the weight in lb4 per lineal foot of round bars4 ,set ,r4,h4 or ,l4,h4 index of ,b to weight of #ab cylindrical inches of the metal on ,a 7column 7#d71 below71 and opposite the diameter of the bar in inches on ,c1 read weight in lb4 per lineal foot on ,a4 ,,ex4--,find the weight of #a lineal foot of #b in4 round cast steel4 ,set ,l4,h4 index of ,b to #bfh on ,a1 and over #b on ,c read #ajg lb4 on ,a4 ,to find the weight of flat bars in lb4 per lineal foot4 ,set the breadth in inches on ,c to 7#a7_/7weight of #ab cub4 in47 of the metal 7column 7#c71 below7 on ,d1 and above the thickness on ,d read weight in lb4 per lineal foot on ,c4 ,,ex4--,find the weight per lineal foot of bar steel1 #d;in4 wide and in4 thick4 ,set #de on ,c to #jbid on ,d1 and over #jfbe on ,d read #ief lb4 per lineal foot on ,c4 ,to find the weight per square foot of sheet metal1 set the weight per cubic foot of the metal 7col4 #a7 on ,c to #ab on ,d1 and 7#a7 7#b7 7#c7 7#d7 ,weight in ,weight of 7#a7_/7,wt4 ,weight of ,metals4 lb4 per #ab cubic of #ab cub4 #ab cubic ft4 in4 in47 cylindrical in4 ,wrought iron#dhj #ccc #jcjj #bfb ,cast iron #dej #cabe #jcbj #bde ,cast steel #dij #cdj #jbid #bfh ,copper #eej #chb #jbfb #cjj ,aluminium #afh #aaff #jjhe #jiae ,brass #ebj #cfa #jbgg #bhc ,lead #gaj #dic #jbjc #chg ,tin #dfb #cba #jcab #beb ,zinc 7cast7 #dcj #bih #jcce #bcd 7sheet7 #dej #cabe #jcbj #bde above the thickness of the plate in inches on ,d read weight in lb4 per square foot on ,c4 ,,ex4--,find the weight in lb4 per square foot of aluminium sheet in4 thick4 ,set #afh on ,c to #ab on ,d1 and over #jcge on ,d read #ebe lb4 on ,c4 ,to find the weight of pipes in lb4 per lineal foot4 ,set mean diameter of the pipe in inches 7i4e41 internal diameter plus the thickness1 or external diameter minus the thickness7 on ,c to the constant given below on ,d1 and over the thickness on ,d read weight in lb4 per lineal foot on ,c4 ,metals4 ,constant for ,pipes4 ,constant for ,spheres4 ,wrought iron #jjiee #fhg ,cast iron #jajbj #gce ,steel #jjicf #fgc ,brass #jjhhb #fce ,copper #jjhcd #fjj ,lead #jjfdf #dfe ,,ex4--,find the weight per foot of cast-iron piping #d in4 internal diameter and in4 thick4 ,set #de on ,c to #jajb on ,d1 and over #je on ,d read #bba lb4 on ,c1 the required weight4 ,to find the weight in lb4 of spheres or balls1 given the diameter in inches4 7,w #jebcf;d#c wt4 of #a cub4 in4 of material74 ,set the constant for spheres 7given above7 on ,b to diameter in inches on ,a1 and over diameter on ,c read weight in lb4 on ,a4 ,,ex4--,find the weight of a cast-iron ball #g;in4 in diameter4 ,set #gce on ,b to #ge on ,a1 and over #ge on ,c read #egg lb4 on ,a4 ,to find diameter in inches of a sphere of given weight4 ,set the cursor to the given weight in lb4 on ,a1 and move the slide until the same number is found on ,c under the cursor that is simultaneously found on ,a over the constant for the sphere on ,b4 ,,ex4--,find diameter in inches of a sphere of cast-iron to weigh #glb4 ,setting the cursor to #ge on ,a1 and moving the slide1 it is found that when #ch on ,c falls under the cursor1 #ch on ,a is simultaneously found over #gce on ,b4 ,the required diameter is therefore #ch in4 ,the rules for cubes and cube roots 7page #dj7 should be kept in view in solving the last two examples4 ,,falling ,,bodies4 ,to find velocity in feet per second of a falling body1 given the time of fall in seconds4 ,set index on ,c to time of fall on ,d1 and under #cbb on ,c read velocity in feet per second on ,d4 ,to find velocity in feet per second1 given distance fallen through in feet4 ,set #a on ,c to distance fallen through on ,a1 and under #fdd on ,b read velocity in feet per second on ,d4 ,,ex4--,find velocity acquired by falling through #ad ft4 ,set 7,r4,h47 index of ,c to #ad on ,a1 and under #fdd on ,b read #cj ft4 per second on ,d4 ,to find distance fallen through in feet in a given time4 ,set index of ,c to time in seconds on ,d1 and over #afa on ,b read distance fallen through in feet on ,a4 ,,centrifugal ,,force4 ,to find the centrifugal force of a revolving mass in lb4 ,set #bidj on ,b to revolutions per minute on ,d2 bring cursor to weight in lb4 on ,b2 index of ,b to cursor1 and over radius in feet on ,b read centrifugal force in lb4 on ,a4 ,to find the centrifugal stress in lb4 per square inch1 in rims of revolving wheels of cast iron4 ,set #fac on ,c to the mean diameter of the wheel in feet on ,d1 and over revolutions per minute on ,c read stress per square inch on ,a4 ,,ex4--,find the stress per square inch in a cast-iron fly-wheel rim #h ft4 in diameter and running at #abj revolutions per minute4 ,set #fac on ,c to #h on ,d1 and over #abj on ,c read #bde lb4 per square inch on ,a4 ,,the ,,steam ,,engine4 ,given the stroke and number of revolutions per minute1 to find the piston speed4 ,set stroke in inches on ,c to #f on ,d1 and over number of revolutions on ,d read piston speed in feet per minute on ,c4 ,to find cubic feet of steam in a cylinder at cut-off1 given diameter of cylinder and period of admission in inches4 ,set #bbjj on ,b to cylinder diameter on ,d1 and over period of admission on ,b read cubic feet of steam on ,a4 ,,ex4--,cylinder diameter #bf in41 stroke #dj in41 cut-off at of stroke4 ,find cubic feet of steam used 7theoretically7 per stroke4 ,set #bbjj on ,b to #bf on ,d1 and over #dj or #be in4 on ,b1 read #gfh cub4 ft4 on ,a1 as the number of cubic feet of steam used per stroke4 ,given the diameter of a cylinder in inches1 and the pressure in lb4 per square inch1 to find the load on the piston in tons4 ,set pressure in lb4 per square inch on ,b to #bheb on ,a1 and over cylinder diameter in inches on ,d read load on piston in tons on ,b4 ,,ex4--,steam pressure #ahj lb4 per square inch2 cylinder diameter1 #db in4 ,find load in tons on piston4 ,set #ahj on ,b to #bheb on ,a1 and over #db on ,d read #aaa tons1 the gross load1 on ,b4 ,given admission period and absolute initial pressure of steam in a cylinder1 to find the pressure at various points in the expansion period 7isothermal expansion74 ,invert the slide and set the admission period1 in inches1 on to the initial pressure on ,d2 then under any point in the expansion stroke on find the corresponding pressure on ,d4 ,,ex4--,admission period #ab in41 stroke #db in41 initial pressure #hj lb4 per square inch4 ,find pressure at successive fifths of the expansion period4 ,set #ab on to #hj on ,d1 and opposite #ah1 #bd1 #cj1 #cf and #db in4 of the whole stroke on find the corresponding pressures on ,d3--#ecc1 #dj1 #cb1 #bff and #bbh lb4 per square inch4 ,to find the mean pressure constant for isothermally expanding steam1 given the cut-off as a fraction of the stroke4 ,find the logarithm of the ratio of the expansion r1 by the method previously explained 7page #df74 ,prefix the characteristic and to the number thus obtained1 on ,d1 set #a on ,c4 ,then under #bcjb on ,c read x on ,d4 ,to x #a on ,d set r on ,c1 and under index of ,c read mean pressure constant on ,d4 ,the latter1 multiplied by the initial pressure1 gives the mean forward pressure throughout the stroke4 7,n4,b4--,common log4 #bcjb hyperbolic log47 ,,ex4--,find the mean pressure constant for a cut-off of th1 or a ratio of expansion of #d4 ,set 7,l4,h47 index of ,c to #d on ,d1 and on the reverse side of the slide read #jfjb on the logarithmic scale4 ,the characteristic #j2 hence to #jfjb on ,d set 7,r4,h47 index of ,c1 and under #bcjb on ,c read #achd on ,d4 ,add #a1 and to #bchd thus obtained on ,d set r 7 #d7 on ,c1 and under #a on ,c read #jeif1 the mean pressure constant required4 ,mean pressure constants for the most usual degrees of cut-off are given below3-- ,cut-off in fractions of stroke ,mean pressure constant #jifh #jieb #jicd #jiai #jiac #jhdf #jgff #jgej #jfii #jffd #jeif #jebb #jdfe #jdba #jche #jcee #jccj #jcji #jbij #jbgd #jbfj #jbdg #jbcf ,to find mean pressure3--,set #a on ,c to constant on ,d1 and under initial pressure on ,c read mean pressure on ,d4 ,given the absolute initial pressure1 length of stroke1 and admission period1 to find the absolute pressure at any point in the expansion period1 it being assumed that the steam expands adiabatically4 7,p#b 7,p#a7_/7,r7 in which ,p#a initial pressure and ,p#b the pressure corresponding to a ratio of expansion ,r47 ,set ,l4,h4 index of ,c to ratio of expansion on ,d1 and read on the back of the slide the decimal of the logarithm4 ,add the characteristic1 and to the number thus obtained on ,d set #i on ,c1 and read off the value found on ,d under the index of ,c4 ,set this number on the logarithmic scale to the index mark1 in the opening on the back of the rule1 and under ,l4,h4 index of ,c read the value of ,r on ,d4 ,the initial pressure divided by this value gives the corresponding pressure due to the expansion4 ,,ex4--,absolute initial pressure #abj lb4 per square inch2 stroke1 #d ft42 cut-off 4 ,find the respective pressures when and ths of the stroke have been completed4 ,in the first case ,r #b4 ,therefore setting the ,l4,h4 index of ,c to #b on ,d1 we find the decimal of the logarithm on the back of the slide to be #jcja4 ,the characteristic is #j1 so placing #i on ,c to #jcja on ,d1 we read #jccd as the value under the ,r4,h4 index of ,c4 7,n4,b4--,in locating the decimal point it is to be observed that the log4 of ,r has been multiplied by #aj1 in accordance with the terms of the above expression47 ,setting this number on the logarithmic scale to the back index1 the value of ,r is found on ,d1 under the ,l4,h4 index of ,c1 to be #baf4 ,setting #abj on ,c to this value1 it is found that the pressure at stroke1 read on ,c over the ,r4,h4 index of ,d1 is #eee lb4 per square inch4 ,in a similar manner1 the pressure when ths of the stroke is completed is found to be #ced lb4 per square inch4 ,for other conditions of expanding steam1 or for gas or air1 the method of procedure is similar to the above4 ,to find the horse-power of an engine1 having given the mean effective pressure1 the cylinder diameter1 stroke1 and number of revolutions per minute4 ,to cylinder diameter on ,d set #ade on ,c2 bring cursor to stroke in feet on ,b1 #a on ,b to cursor1 cursor to number of revolutions on ,b1 #a on ,b to cursor1 and over mean effective pressure on ,b find horse-power on ,a4 7,n4,b4--,if stroke is in inches1 use #ejb in place of #ade given above47 ,,ex4--,find the indicated horse-power1 given cylinder diameter #bg in41 mean effective pressure #ch lb4 per square inch1 stroke #cb in41 revolutions #eg per minute4 ,set #ejb on ,c to #bg on ,d1 bring cursor to #cb on ,b1 #a on ,b to cursor1 cursor to #eg on ,b1 #a on ,b to cursor1 and over #ch on ,b read #bjj ,i4,h4,p4 on ,a4 ,to determine the horse-power of a compound engine1 invert the slide and set the diameter of the high-pressure cylinder on to the cut-off in that cylinder on ,a4 ,use the number then found on ,a over the diameter of the low-pressure cylinder on as the cut-off in that cylinder1 working with the same pressure and piston speed1 and calculate the horse-power as for a single cylinder4 ,to find the cylinder ratio in compound engines1 invert the slide and set index of to diameter of the low-pressure cylinder on ,d4 ,then over the diameter of the high-pressure cylinder on ,c1 read cylinder ratio on ,a4 ,,ex4--,diameter of high-pressure cylinder #g;in41 low-pressure #ae in4 ,find cylinder ratio4 ,set index on to #ae on ,d1 and over #gge on read #cge1 the required ratio1 on ,a4 ,the cylinder ratios of triple or quadruple-expansion engines may be similarly determined4 ,,ex4--,in a quadruple-expansion engine1 the cylinders are #ah1 #bf1 #cg1 and #ed inches in diameter4 ,find the respective ratios of the high1 first intermediate1 and second intermediate cylinders to the low-pressure4 ,set 7,r4,h47 index of to #ed on ,d1 and over #ah1 #bf1 and #cg on read #i1 #dca1 and #bac1 the required ratios1 on ,a4 ,given the mean effective pressures in lb4 per square inch in each of the three cylinders of a triple-expansion engine1 the ,i4,h4,p4 to be developed in each cylinder1 and the piston speed1 to find the respective cylinder diameters4 ,set #db1jjj on ,b to piston speed on ,a2 bring cursor to mean effective pressure in low-pressure cylinder on ,b1 index of ,b to cursor1 and under ,i4,h4,p4 on ,a read low-pressure cylinder diameter on ,c4 ,to find the diameters of the high-pressure and intermediate-pressure cylinders1 invert the slide and place the mean pressure in the low-pressure cylinder on to the diameter of that cylinder on ,d4 ,then under the respective mean pressures on read corresponding cylinder diameters on ,d4 ,,ex4--,the mean effective pressures in the cylinders of a triple-expansion engine are3--,l4,p41 #ajcb2 ,i4,m4,p41 #bge2 and ,h4,p41 #gge lb4 per square inch4 ,the piston speed is #fej ft4 per minute1 and the ,i4,h4,p4 developed in each cylinder1 #gej4 ,find the cylinder diameters4 ,set #db1jjj on ,b to #fej on ,a1 and bring cursor to #ajcb on ,b4 ,bring index of ,b to cursor1 and under #gej on ,a read #fhe in4 on ,c1 the ,l4,p4 cylinder diameter4 ,invert the slide1 and placing #ajcb on to #fhe on ,d1 read1 under #bge on 1 the ,i4,m4,p4 cylinder diameter #db in41 on ,d2 also under #gge on read the ,h4,p4 cylinder diameter #be in41 on ,d4 ,to compute brake or dynamometrical horse-power4 ,set #ebe on ,c to the total weight in lb4 acting at the end of the lever 7or pull of spring balance in lb47 on ,d2 set cursor to length of lever in feet on ,c1 bring #a on ,c to cursor1 and under number of revolutions per minute on ,c find brake horse-power on ,d4 ,given cylinder diameter and piston speed in feet per minute1 to find diameter of steam pipe1 assuming the maximum velocity of the steam to be #fjjj ft4 per minute4 ,set #fjjj on ,b to cylinder diameter on ,d1 and under piston speed on ,b read steam pipe diameter on ,d4 ,given the number of revolutions per minute of a ,watt governor1 to find the vertical height in inches1 from the plane of revolution of the balls to the point of suspension4 ,set revolutions per minute on ,c to #ce1bjj on ,a1 and over index of ,b read height on ,a4 ,given the weight in lb4 of the rim of a cast-iron fly-wheel1 to find the sectional area of the rim in square inches4 ,set the mean diameter of the wheel in feet on ,c to #jajb on ,d1 and under weight of rim on ,c find area on ,d4 ,given the consumption of coal in tons per week of #ef hours1 and the ,i4,h4,p41 to find the coal consumed per ,i4,h4,p4 per hour4 ,set ,i4,h4,p4 on ,c to #dj on ,d1 and under weekly consumption on ,c read lb4 of coal per ,i4,h4,p41 per hour on ,d4 ,,ex4--,find coal used per ,i4,h4,p4 per hour1 when #bd tons is the weekly consumption for #cjj ,i4,h4,p4 ,set #cjj on ,c to #dj on ,d1 and under #bd on ,c read #cb lb4 per ,i4,h4,p4 per hour on ,d4 7,n4,b4--,for any other number of working hours per week divide #bbdj by the number of working hours1 and use the quotient in place of #dj as above47 ,to find the tractive force of a locomotive4 ,set diameter of driving wheel in inches on ,b to diameter of cylinder in inches on ,d1 and over the stroke in inches on ,b read on ,a1 tractive force in lb4 for each lb4 of effective pressure on the piston4 ,,steam ,,boilers4 ,to find the bursting pressure of a cylindrical boiler shell1 having given the diameter of shell and the thickness and ultimate strength of the material4 ,set the diameter of the shell in inches on ,c to twice the thickness of the plate on ,d1 and under strength of material per square inch on ,c read bursting pressure in lb4 per square inch on ,d4 ,,ex4--,find the bursting pressure of a cylindrical boiler shell #g ft4 #f in4 in diameter1 with plates in4 thick1 assuming an ultimate strength of #ej1jjj lb4 per square inch4 ,set #ij on ,c to #aj on ,d1 and under #ej1jjj on ,c find #eee lb4 on ,d4 ,to find working pressure for ,fox's corrugated furnaces by ,board of ,trade rule4 ,set the least outside diameter in inches on ,c to #ad1jjj on ,d1 and under thickness in inches on ,c read working pressure on ,d in lb4 per square inch4 ,to find diameter d in inches1 of round steel for safety valve springs by ,board of ,trade rule4 ,set #hjjj on ,c to load on spring in lb4 on ,d1 and under the mean diameter of the spring in inches on ,c read d#c on ,d4 ,then extract the cube root as per rule4 ,,speed ,,ratios ,,of ,,pulleys1 ,,etc4 ,given the diameter of a pulley and its number of revolutions per minute1 to find the circumferential velocity of the pulley or the speed of ropes1 belts1 etc41 driven thereby4 ,set diameter of pulley in inches on ,c to #chb on ,d1 and over revolutions per minute on ,d read speed in feet per minute on ,c4 ,,ex4--,find the speed of a belt driven by a pulley #ec in4 in diameter and running at #ahj revolutions per minute4 ,set #ec on ,c to #chb on ,d1 and over #ahj on ,d read #bejj ft4 per minute on ,c4 ,,ex4--,find the speed of the pitch line of a spur wheel #c ft4 #f in4 in diameter running at #fj revolutions per minute4 ,set #db in4 on ,c to #chb on ,d1 and over #fj on ,d read #ffj ft4 per minute on ,c4 ,given diameter and number of revolutions per minute of a driving pulley1 and the diameter of the driven pulley1 to find the number of revolutions of the latter4 ,invert the slide and set diameter of driving pulley on to given number of its revolutions on ,d2 then opposite diameter of any driven pulley on read its number of revolutions on ,d4 ,,ex4--,diameter of driving pulley #aj ft42 revolutions per minute #ee2 diameter of driven pulley #b ft4 #i in4 ,find number of revolutions per minute of latter4 ,set #aj on to #ee on ,d1 and opposite #bge on read #bjj revolutions on ,d4 ,,belts ,,and ,,ropes4 ,to find the ratio of tensions in the two sides of a belt1 given the coefficient of friction between belt and pulley and the number of degrees in the arc of contact 7log4 ,r 77_/7#acb774 ,set #acb on ,c to the coefficient of friction on ,d1 and read off the value found on ,d under the number of degrees in the arc of contact on ,c4 ,place this value on the scale of equal parts on the back of the slide1 to the index mark in the aperture1 and read the required ratio on ,d under the ,l4,h4 index of ,c4 ,,ex4--,find the tension ratio in a belt1 assuming a coefficient of friction of #jc and an arc of contact of #abj degrees4 ,set #acb on ,c to #jc on ,d1 and under #abj on ,c read #jbgc4 ,place this on the scale to the index on the back of the rule1 and under the ,l4,h4 index ,c read #ahge on ,d1 the required ratio4 ,given belt velocity and horse-power to be transmitted1 to find the requisite width of belt1 taking the effective tension at #ej lb4 per inch of width4 ,set #ffj on ,c to velocity in feet per minute on ,d1 and opposite horse-power on ,d find width of belt in inches on ,c4 ,given velocity and width of belt1 to find horse-power transmitted4 ,set #ffj on ,c to velocity on ,d1 and under width on ,c find horse-power transmitted on ,d4 7,n4,b4--,for any other effective tension1 instead of #ffj use as a gauge point3--#cc1jjj tension47 ,given speed and diameter of a cotton driving rope1 to find power transmitted1 disregarding centrifugal action1 and assuming an effective working tension of #bjj lb4 per square inch of rope4 ,set #baj on ,b to #age on ,d1 and over speed in feet per minute on ,b read horse-power on ,a4 ,,ex4--,find the power transmitted by a #a;in4 rope running at #djjj ft4 per minute4 ,set #baj on ,b to #age on ,d1 and over #djjj on ,b read #ehc horse-power on ,a4 ,find the 8centrifugal tension0 in the previous example1 taking the weight per foot of the rope as #jbg;d#b4 ,set #fee on ,c to the diameter1 #age in41 on ,d1 and over the speed1 #djjj ft4 on ,c1 read centrifugal tension #aad lb4 on ,a4 ,,spur ,,wheels4 ,given diameter and pitch of a spur wheel1 to find number of teeth4 ,set pitch on ,c to 7#cadaf7 on ,d1 and under any diameter on ,c read number of teeth on ,d4 ,given diameter and number of teeth in a spur wheel1 to find the pitch4 ,set diameter on ,c to number of teeth on ,d1 and read pitch on ,c opposite #cadaf on ,d4 ,given the distance between the centres of a pair of spur wheels and the number of revolutions of each1 to determine their diameters4 ,to twice the distance between the centres on ,d1 set the sum of the number of revolutions on ,c1 and under the revolutions of each wheel on ,c find the respective wheel diameters on ,d4 ,,ex4--,the distance between the centres of two spur wheels is #cge in41 and they are required to make #ba and #bd revolutions in the same time4 ,find their respective diameters4 ,set #ba #bd #de on ,c to #ge 7or #cge #b7 on ,d1 and under #ba and #bd on ,c find #ce and #dj in4 on ,d as the respective diameters4 ,to find the power transmitted by toothed wheels1 given the pitch diameter d in inches1 the number of revolutions per minute n1 and the pitch p in inches1 by the rule1 ,h4,p4 7n d p#b7_/7djj74 ,set #djj on ,b to pitch in inches on ,d2 set cursor to d on ,b1 #a on ,b to cursor1 and over any number of revolutions n on ,b read power transmitted on ,a4 ,,ex4--,find the horse-power capable of being transmitted by a spur wheel #g ft4 in diameter1 #c in4 pitch1 and running at #ij revolutions per minute4 ,set #djj on ,b to #c on ,d2 bring cursor to #hd in4 on ,b1 #a on ,b to cursor1 and over #ij revolutions on ,b read #agj1 the horse-power transmitted1 on ,a4 ,,screw-,,cutting4 ,given the number of threads per inch in the guide screw1 to find the wheels to cut a screw of given pitch4 ,set threads per inch in guide screw on ,c1 to the number of threads per inch to be cut on ,d4 ,then opposite any number of teeth in the wheel on the mandrel on ,c1 is the number of teeth in the wheel to be placed on the guide screw on ,d4 ,,strength ,,of ,,shafting4 ,given the diameter d of a steel shaft1 and the number of revolutions per minute n1 to find the horse-power from3-- ,h4,p4 d#c n #jjb4 ,set #a on ,c to d on ,d1 and bring cursor to d on ,b4 ,bring #ej on ,b to cursor1 and over number of revolutions on ,b read ,h4,p4 on ,a4 ,,ex4--,find horse-power transmitted by a #c in4 steel shaft at #aaj revolutions per minute4 ,set #a on ,c to #c on ,d1 and bring cursor to #c on ,b4 ,bring #ej on ,b to cursor1 and over #aaj on ,b read #eid horse-power on ,a4 ,given the horse-power to be transmitted and the number of revolutions of a steel shaft1 to find the diameter4 ,set revolutions on ,b to horse-power on ,a1 and bring cursor to #ej on ,b4 ,then move the slide until the same number is found on ,b under the cursor that is simultaneously found on ,d under the index of ,c4 ,this number is the diameter required4 ,to find the deflection k in inches1 of a round steel shaft of diameter d1 under a uniformly distributed load in lb4 w1 and supported by bearings1 the centres of which are l feet apart 7k 7w l#c7_/7gh1jjj;d#d774 ,modifying the form of this expression slightly1 we proceed as follows3--,set d on ,c to l on ,d1 and bring the cursor to the same number on ,b that is found on ,d under the index of ,c4 ,bring d on ,b to cursor1 cursor to w on ,b1 #gh1hjj on ,b to cursor1 and read deflection on ,a over index of ,b4 ,,ex4--,find the deflection in inches of a round steel shaft #c;in4 diameter1 carrying a uniformly distributed load of #cbjj lb41 the distance apart of the centres of support being #i ft4 ,set #ce on ,c to #i on ,d1 and read #beg on ,d1 under the ,l4,h4 index of ,c4 ,set cursor to #beg on ,b1 and bring #ce on ,b to cursor1 cursor to #cbjj on ,b1 #gh1jjj on ,b to cursor1 and over ,l4,h4 index of ,b read #jaii in41 the required deflection on ,a4 ,to find the diameter of a shaft subject to twisting only1 given the twisting moment in inch-lb4 and the allowable stress in lb4 per square inch4 ,set the stress in lb4 per square inch on ,b to the twisting moment in inch-lb4 on ,a1 and bring cursor to #ea on ,b4 ,then move the slide until the same number is found on ,b under the cursor that is simultaneously found on ,d under the index of ,c4 ,,ex4--,a steel shaft is subjected to a twisting moment of #b1gjj1jjj inch-lb4 ,determine the diameter if the allowable stress is taken at #ijjj lb4 per square inch4 ,set #ijjj on ,b to #b1gjj1jjj on ,a1 and bring the cursor to #ea on ,b4 ,moving the slide to the left1 it is found that when #aaea on the ,r4,h4 scale of ,b is under the cursor1 the ,l4,h4 index of ,c is opposite #aaea on ,d4 ,this1 then1 is the required diameter of the shaft4 7,n4,b4--,the rules for the scales to be used in finding the cube root 7page #db7 must be carefully observed in working these examples47 ,,moments ,,of ,,inertia4 ,to find the moment of inertia of a square section about an axis formed by one of its diagonals 7,i 7s#d7_/7ab774 ,set index of ,c to the length of the side of square s on ,d2 bring cursor to s on ,c1 #ab on ,b to cursor1 and over index of ,b read moment of inertia on ,a4 ,to find the moment of inertia of a rectangular section about an axis parallel to one side and perpendicular to the plane of bending4 ,set index of ,c to the height or depth h of the section1 and bring cursor to h on ,b4 ,set #ab on ,b to cursor1 and over breadth b of the section on ,b read moment of inertia on ,a4 ,,ex4--,find the moment of inertia of a rectangular section of which h #ad in4 and b #g in4 ,set index of ,c to #ad on ,d1 and cursor to #ad on ,b4 ,bring #ab on ,b to cursor1 and over #g on ,b read #afjj on ,a4 ,,discharge ,,from ,,pumps1 ,,pipes1 ,,etc4 ,to find the theoretical delivery of pumps1 in gallons per stroke4 ,set #bid on ,b to the diameter of the plunger in inches on ,d1 and over length of stroke in feet on ,b read theoretical delivery in gallons per stroke on ,a4 7,n4,b4--,a deduction of from #bj to #dj per cent4 should be made to allow for slip47 ,to find loss of head of water in feet due to friction in pipes 7,prony's rule74 ,set diameter of pipe in feet on ,b to velocity of water in feet per second on ,d and bring cursor to #bbe on ,b2 bring #a on ,b to cursor1 and over length of pipe in miles on ,b1 read loss of head of water in feet1 on ,a4 ,to find velocity in feet per second1 of water in pipes 7,blackwell's rule74 ,set #bc on ,b to diameter of pipe in feet on ,a1 and under inclination of pipe in feet per mile on ,b read velocity in feet per second on ,d4 ,to find the discharge over weirs in cubic feet per minute and per foot of width4 7,discharge #bad7;h#c77 ,set #jjjdfg on ,c to the head in feet h on ,d1 and under h on ,b read discharge on ,d4 ,to find the theoretical velocity of water flowing under a given head in feet4 ,set index of ,b to head in feet on ,a1 and under #fdd on ,b read theoretical velocity in feet per second on ,d4 ,,horse-,,power ,,of ,,water ,,wheels4 ,to find the effective horse-power of a ,poncelet water wheel4 ,set #hhj on ,c to cubic feet of flow of water per minute on ,d1 and under height of fall in feet on ,c1 read effective horse-power on ,d4 ,for breast water wheels use #ifj1 and for overshot wheels #gge1 in place of #hhj as above4 ,,electrical ,,engineering4 ,to find the resistance per mile1 in ohms1 of copper wire of high conductivity1 at #fj ,f4 the diameter being given in mils4 7#a mil4 #jjja in474 ,set diameter of wire in mils4 on ,c to #ed1ijj on ,a1 and over ,r4,h4 or ,l4,h4 index of ,b read resistance in ohms on ,a4 ,,ex4--,find the resistance per mile of a copper wire #fd mils4 in diameter4 ,set #fd on ,c to #ed1ijj on ,a1 and over ,r4,h4 index of ,b read #acd ohms on ,a4 ,to find the weight of copper wire in lb4 per mile4 ,set #gia on ,c to diameter of wire in mils4 on ,d1 and over index of ,b read weight per mile on ,a4 ,given electromotive force and current1 to find electrical horse-power4 ,set #gdf on ,c to electromotive force in volts on ,d1 and under current in ampres on ,c read electrical horse-power on ,d4 ,given the resistance of a circuit in ohms and current in ampres1 to find the energy absorbed in horse-power4 ,set #gdf on ,b to current on ,d1 and over resistance on ,b read energy absorbed in ,h4,p4 on ,a4 ,,ex4--,find the ,h4,p4 expended in sending a current of #ae ampres through a circuit of #bbj ohms resistance4 ,set #gdf on ,b to #ae on ,d1 and over #bbj on ,b read #ffc ,h4,p4 on ,a4 ,,commercial4 ,to add on percentages4 ,set #ajj on ,c to #ajj given percentage on ,d1 and under original number on ,c read result on ,d4 ,to deduct percentages4 ,set ,r4,h4 index of ,c to #ajj the given percentage on ,d1 and under original number on ,c read result on ,d4 ,,ex4--,from #af deduct #g per cent4 ,set #aj on ,c to #ibe on ,d and under #af on ,c1 read #adh #ad1 #afs4 on ,d4 ,to calculate simple interest4 ,set #a on ,c to rate per cent4 on ,d2 bring cursor to period on ,c and #a on ,c to cursor4 ,then opposite any sum on ,c find simple interest on ,d4 ,for interest per annum4 ,set ,r4,h4 index on ,c to rate on ,d1 and opposite principal on ,c read interest on ,d4 ,,ex4--,find the amount with simple interest of #bej at #h per cent41 and for a period of #a year and #i months4 ,set #a on ,c to #h on ,d2 bring cursor to #age on ,c1 and #a on ,c to cursor2 then opposite #bej on ,c read #ce1 the interest1 on ,d4 ,then #bej #ce #bhe the amount4 ,to calculate compound interest4 ,set the ,l4,h4 index of ,c to the amount of #a at the given rate of interest on ,d1 and find the logarithm of this by reading on the reverse side of the rule1 as explained on page #df4 ,multiply the logarithm1 so found1 by the period1 and set the result1 on the scale of equal parts1 to the index on the under-side of the rule2 then opposite any sum on ,c read the amount 7including compound interest7 on ,d4 ,,ex4--,find the amount of #ejj at #e per cent4 for #f years1 with compound interest4 ,set ,l4,h4 index of ,c to #aje on ,d1 and read at the index on the scale of equal parts on the under-side of rule1 #jjbab4 ,multiply by #f1 we obtain #jabgb1 which1 on the scale of equal parts1 is placed to the index in the notch at the end of the rule4 ,then opposite #ejj on ,c read #fgj on ,d1 the amount required1 including compound interest4 ,,miscellaneous ,,calculations4 ,to calculate percentages of compositions4 ,set weight 7or volume7 of sample on ,c1 to weight 7or volume7 of substance considered1 on ,d2 then under index of ,c read required percentage on ,d4 ,,ex4--,a sample of coal weighing #abe grms4 contains #jjddbe grm4 of ash4 ,find the percentage of ash4 ,set #abe on ,c to #jjddbe on ,d1 and under index on ,c read #ced1 the required percentage of ash on ,d4 ,given the steam pressure ,p and the diameter d in millimetres1 of the throat of an injector1 to find the weight ,w1 of water delivered in lb4 per hour from ,w 7d#b,p7_/7#jeje74 ,set #jeje on ,c to ,p on ,a2 bring cursor to d on ,c and index of ,c to cursor4 ,then under d on ,c read delivery of water on ,d4 ,to find the pressure of wind per square foot1 due to a given velocity in miles per hour4 ,set #a on ,b to #b on ,a1 and over the velocity in miles per hour on ,d read pressure in lb4 per square foot on ,b4 ,to find the kinetic energy of a moving body4 ,set #fdd on ,b to velocity in feet per second on ,d1 and over weight of body in lb4 on ,b read kinetic energy or accumulated work in foot-lb4 on ,a4 ,,trigonometrical ,,applications ,scales4--,not the least important feature of the modern slide rule is the provision of the special scales on the under-side of the slide1 and by the use of which1 in conjunction with the ordinary scales on the rule1 a large variety of trigonometrical computations may be readily performed4 ,three scales will be found on the reverse or under-side of the slide of the ordinary ,gravt or ,mannheim rule4 ,one of these is the evenly-divided scale or scale of equal parts referred to in previous sections1 and by which1 as explained1 the decimal parts or mantiss of logarithms of numbers may be obtained4 ,usually this scale is the centre one of the three1 but in some rules it will be found occupying the lowest position1 in which case some little modification of the following instructions will be necessary4 ,the requisite transpositions will1 however1 be evident when the purposes of the scales are understood4 ,the upper of the three scales1 usually distinguished by the letter ,s1 is a scale giving the logarithms of the sines of angles1 and is used to determine the natural sines of angles of from #ce minutes to #ij degrees4 ,the notation of this scale will be evident on inspection4 ,the main divisions #a1 #b1 #c1 etc41 represent the degrees of angles2 but the values of the subdivisions differ according to their position on the scale4 ,thus1 if any primary space is subdivided into #ab parts1 each of the latter will be read as #e minutes 7#e71 since #a #fj4 ,sines of ,angles4--,to find the sine of an angle the slide is placed in the groove1 with the under-side uppermost1 and the end division lines or indices on the slide1 coinciding with the right and left indices of the ,a scale4 ,then over the given angle on ,s is read the value of the sine of the angle on ,a4 ,if the result is found on the left scale of ,a 7#a to #aj71 the logarithmic characteristic is #b2 if it is found on the right-hand side 7#aj to #ajj71 it is #a4 ,in other words1 results on the right-hand scale are prefixed by the decimal point only1 while those on the left-hand scale are to be preceded by a cypher also4 ,thus3-- ,sine #b #dj #jjdfe2 sine #ae #dj #jbgj4 ,multiplication and division of the sines of angles are performed in the same manner as ordinary calculations1 excepting that the slide has its under-face placed uppermost1 as just explained4 ,thus to multiply sine #ae #dj by #ae1 the ,r4,h4 index of ,s is brought to #ae on ,a1 and opposite #ae #dj on ,s is found #dje on ,a4 ,again1 to divide #adb by sine #af #cj1 we place #af #cj on ,s to #adb on ,a1 and over ,r4,h4 index of ,s read #ejj on ,a4 ,the rules for the number of integers in the results are thus determined3 ,let ,n be the number of integers in the multiplier ,m or in the dividend ,d4 ,then the number of integers ,p1 in the product or ,q1 in the quotient are as follows3-- ,when the result is found to the right of ,m or ,d1 ,p ,n #b,q ,n and in the same scale ,when the result is found to the right of ,m or ,d1 ,p ,n #a,q ,n #a and in the other scale ,when the result is found to the left of ,m or ,d1 and,p ,n #a,q ,n #a in the other scale ,when the result is found to the left of ,m or ,d1 and,p ,n ,q ,n #b in the same scale ,if the division is of the form 7#bj #cj7_/7ej71 the result cannot be read off directly on the face of the rule4 ,thus1 if in the above example #bj #cj on ,s1 is placed to agree with #ej on the right-hand scale of ,a1 the result found on ,s under the ,r4,h4 index of ,a is #dd #cj4 ,the required numerical value can then be found3 7#a7 ,by placing the slide with all indices coincident when opposite #dd #cj on ,s will be found #jjjg on ,a2 or 7#b7 ,in the ordinary form of rule1 by reading off on the scale ,b opposite the index mark in the opening on the under-side of the rule4 ,the above rules for the number of integers in the quotient do not apply in this case4 ,if it is required to find the sine of an angle simply1 this may be done with the slide in its ordinary position1 with scale ,b under ,a4 ,the given angle on scale ,s is then set to the index on the under-side of the rule1 and the value of the sine is read off on ,b under the right index of ,a4 ,owing to the rapidly diminishing differences of the values of the sines as the upper end of the scale is approached1 the sines of angles between #fj and #ij cannot be accurately determined in the foregoing manner4 ,it is therefore advisable to calculate the value of the sine by means of the formula3 ,sine #a #b sin#b 7#ij 7_/7#b74 ,to determine the value of sin#b 7#ij 7_/7#b74 ,with the slide in the normal position1 set the value of 7#ij 7_/7#b74 on ,s to the index on the under-side of the rule1 and read off the value x on ,b under the ,r4,h4 index of ,a4 ,without moving the slide find x on ,a1 and read under it on ,b the value required4 ,,ex4--,find value of sine #gi #dj4 ,sine #gi #dj #a #bsin#b #e #aj4 ,but sine #e #aj #jjijj1 and under this value on ,a is #jjjha on ,b4 ,therefore sine #gi #dj #a #jjafb #jihch4 ,the sines of very small angles1 being very nearly proportional to the angles themselves1 are found by direct reading4 ,to facilitate this1 some rules are provided with two marks1 one of which1 a single accent 771 corresponds to the logarithm of 7#a7_/7sine #a7 and is found at the number #cdch4 ,the other mark--a double accent 77--corresponds to the logarithm of 7#a7_/7sine #a7 and is found at the number #bjf1bfe4 ,in some rules these marks are found on either the ,a or the ,b scales2 sometimes they are on both4 ,in either case the angle on the one scale is placed so as to coincide with the significant mark on the other1 and the result read off on the first-named scale opposite the index of the second4 ,in sines of angles under #c1 the number of integers in the result is #e2 while it is #d for angles from #c to #ba2 #c from #ba to #c #bg2 and #b from #c #bg to #cd #bc4 ,,ex4--,find sine #f4 ,placing the significant mark for minutes coincident with #f1 the value opposite the index is found to be #age1 and by the rule above this is to be read #jjjage4 ,for angles in seconds the other significant mark is used2 while angles expressed in minutes and seconds are to be first reduced to seconds4 ,thus1 #c #aj #aij4 ,tangents of ,angles4--,there remains to be considered the third scale found on the back of the slide1 and usually distinguished from the others by being lettered ,t4 ,in most of the more recent forms of rule this scale is placed near the lower edge of the slide1 but in some arrangements it is found to be the centre scale of the three4 ,again1 in some rules this scale is figured in the same direction as the scale of sines--viz41 from left to right1--while in others the ,t scale is reversed4 ,in both cases there is now usually an aperture formed in the back of the left extremity of the rule1 with an index mark similar to that already referred to in connection with the scale of sines4 ,considering what has been referred to as the more general arrangement1 the method of determining the tangents of angles may be thus explained3-- ,the tangent scale will be found to commence1 in some rules1 at about #cd1 or1 precisely1 at the angle whose tangent is #jja4 ,more usually1 however1 the scale will be found to commence at about #e #dc1 or at the angle whose tangent is #ja4 ,the other extremity of the scale corresponds in all cases to #de1 or the angle whose tangent is #a4 ,this explanation will suggest the method of using the scale1 however it may be arranged4 ,if the graduations commence with #cd1 the ,t scale is to be used in conjunction with the right and left scales of ,a2 while if they commence with #e #dc it is to be used in conjunction with the ,d scale4 ,in the former case the slide is to be placed in the rule so that the ,t scale is adjacent to the ,a scales1 and1 with the right and left indices coinciding1 when opposite any angle on ,t will be found its tangent on ,a4 ,from what has been said above1 it follows that the tangents read on the ,l4,h4 scale of ,a have values extending from #jja to #ja2 while those read on the ,r4,h4 scale of ,a have values from #ja to #aj4 ,otherwise expressed1 to the values of any tangent read on the ,l4,h4 scale of ,a a cypher is to be prefixed2 while if found on the ,r4,h4 scale1 it is read directly as a decimal4 ,,ex4--,find tan4 #c #ej4 ,placing the slide as directed1 the reading on ,a opposite #c #ej on ,t is found to be #fg4 ,as this is found on the ,l4,h4 scale of ,a1 it is to be read as #jjfg4 ,,ex4--,find tan4 #ag #de4 ,here the reading on ,a opposite #ag #de on ,t is #cb1 and as it is found on the ,r4,h4 scale of ,a it is read as #jcb4 ,as in the case of the scale of sines1 the tangents may be found without reversing the slide1 when a fixed index is provided in the back of the rule for the ,t scale4 ,we revert now to a consideration of those rules in which a single tangent scale is provided4 ,it will be understood that in this case the slide is placed so that the scale ,t is adjacent to the ,d scale1 and that when the indices of both are placed in agreement1 the value of the tangent of any angle on ,t 7from #e #dc to #de7 may be read off on ,d1 the result so found being read as wholly decimal4 ,thus tan4 #ac #bj is read #jbcg4 ,if a back index is provided1 the slide is used in its normal position1 when1 setting the angle on the tangent scale to this index1 the result can be read on ,c over the ,l4,h4 index of ,d4 ,the tangents of angles above #de are obtained by the formula3 ,tan4 7#a7_/7tan4 7#ij 774 ,for all angles from #de to 7#ij #e #dc7 we proceed as follows3--,place 7#ij 7 on ,t to the ,r4,h4 index of ,d1 and read tan4 on ,d under the ,l4,h4 index of ,t4 ,the first figure in the value thus obtained is to be read as an integer4 ,thus1 to find tan4 #ga #bj we place #ij #ga #bj #ah #dj on ,t1 to the ,r4,h4 index of ,d1 and under the ,l4,h4 index of ,t read #bif1 the required tangent4 ,the tangents of angles less than #dj are sensibly proportional to the angles themselves1 and as they may therefore be considered as sines1 their value is determined by the aid of the single and double accent marks on the sine scale1 as previously explained4 ,the rules for the number of integers are the same as for the sines4 ,multiplication and division of tangents may be quite readily effected4 ,,ex4--,tan4 #ba #ej #ae #f4 ,set ,l4,h4 index of ,t to #ae on ,d1 and under #ba #ej on ,t read #f on ,d4 ,,ex4--,tan4 #gb #dj #aag #cge4 ,set 7#ij #gb #dj7 #ag #bj on ,t to #aag on ,d1 and under ,r4,h4 index of ,t read #cge on ,d4 ,cosines of ,angles4--,the cosines of angles may be determined by placing the scale ,s with its indices coinciding with those of ,a1 and when opposite 7#ij 7 on ,s is read cos4 on ,a4 ,if the result is read on the ,l4,h4 scale of ,a1 a cypher is to be prefixed to the value read2 while if it is read on the ,r4,h4 scale of ,a1 the value is read directly as a decimal4 ,thus1 to determine cos4 #hf #cj we find opposite 7#ij #hf #cj7 #c #cj on ,s1 #fa on ,a1 and as this is on the ,l4,h4 scale the result is read #jjfa4 ,again1 to find cos4 #ei #bj we read opposite 7#ij #ei #bj7 or #cj #dj on ,s1 #ea on ,a1 and as this is found on the ,r4,h4 scale of ,a1 it is read #jea4 ,in finding the cosines of small angles it will be seen that direct reading on the rule becomes impossible for angles of less than #bj4 ,it is advisable in such cases to adopt the method described for determining the sines of the large angles of which the complements are sought4 ,cotangents of ,angles4--,from the methods of finding the tangents of angles previously described1 it will be apparent that the cotangents of angles may also be obtained with equal facility4 ,for angles between #e #de and #de1 the procedure is the same as that for finding tangents of angles greater than #de4 ,thus1 the angle on scale ,t is brought to the ,r4,h4 index of ,d1 and the cotangent read off on ,d under the ,l4,h4 index of ,t4 ,the first figure of the result so found is to be read as an integer4 ,if the angle 77 lies between #de and #hd #ae1 the slide is placed so that the indices of ,t coincide with those of ,d1 and the result is then read off on ,d opposite 7#ij 7 on ,t4 ,in this case the value is wholly decimal4 ,secants of ,angles4--,the secants of angles are readily found by bringing 7#ij 7 on ,s to the ,r4,h4 index of ,a and reading the result on ,a over the ,l4,h4 index of ,s4 ,if the value is found on the ,l4,h4 scale of ,a1 the first figure is to be read as an integer2 while if the result is read on the ,r4,h4 scale of ,a1 the first two figures are to be regarded as integers4 ,cosecants of ,angles4--,the cosecants of angles are found by placing the angle on ,s to the ,r4,h4 index of ,a1 and reading the value found on ,a over the ,l4,h4 index of ,s4 ,if the result is read on the ,l4,h4 scale of ,a1 the first figure is to be read as an integer2 while if the result is found on the ,r4,h4 scale of ,a1 the first two figures are to be read as integers4 ,it will be noted that some of the rules here given for determining the several trigonometrical functions of angles apply only to those forms of rules in which a single scale of tangents ,t is used1 reading from left to right4 ,for the other arrangements of the scale1 previously referred to1 some slight modification of the method of procedure in finding the tangents and cotangents of angles will be necessary2 but as in each case the nature and extent of this modification is evident1 no further directions are required4 ,,the ,,solution ,,of ,,right-,,angled ,,triangles4 ,from the foregoing explanation of the manner of determining the trigonometrical functions of angles1 the methods of solving right-angled triangles will be readily perceived1 and only a few examples need therefore be given4 ,let a and b represent the sides and c the hypothenuse of a right-angled triangle1 and a and b the angles opposite to the sides4 ,then of the possible cases we will take 7#a47 ,given c and a1 to find a1 b1 and b4 ,the angle b #ij a1 while a c sin a and b c sin b4 ,to find a1 therefore1 the index of ,s is set to c on ,a1 and the value of a read on ,a opposite a on ,s4 ,in the same manner the value of b is obtained4 ,,ex4--,given in a right-angled triangle c #i ft4 and a #cj4 ,find a1 b1 and b4 ,the angle b #ij #cj #fj4 ,to find a1 set ,r4,h4 index of ,s to #i on ,a1 and over #cj on ,s read a #de ft4 on ,a4 ,also1 with the slide in the same position1 read b #gh ft4 #ggid on ,a over #fj on ,s4 7#b47 ,given a and c1 to determine a1 b1 and b4 ,in this case advantage is taken of the fact that in every triangle the sides are proportional to the sines of the opposite angles4 ,therefore1 as in this case the hypothenuse c subtends a right angle1 of which the sine #a1 the ,r4,h4 index 7or #ij7 on ,s is set to the length of c on ,a1 when under a on ,a is found a on ,s4 ,hence b and b may be determined4 7#c47 ,given a and a1 to find b1 c1 and b4 ,here b 7#ij a71 and the solution is similar to the foregoing4 7#d47 ,given a and b1 to find a1 b1 and c4 ,to find a1 we have tan4 a a_/b1 which in the above example will be 7#de7_/7gh7 #jegg4 ,therefore1 placing the slide so that the indices of ,t coincide with those of ,d1 we read opposite #jegg on ,d the value of a #cj4 ,the hypothenuse c is readily obtained from c a_/7sin a74 ,,the ,,solution ,,of ,,oblique-,,angled ,,triangles4 ,using the same letters as before to designate the three sides and the subtending angles of oblique-angled triangles1 we have the following cases3-- 7#a47 ,given one side and two angles1 as a1 a1 and b1 to find b1 c1 and c4 ,in the first place1 c #ahj 7a b72 also we note that1 as the sides are proportional to the sines of the opposite angles1 b 7a sine b7_/7sine a7 and c 7a sine c7_/7sine a74 ,taking as an example1 a #de1 a #eg1 and b #fc1 we have c #ahj 7#eg #fc7 #fj4 ,to find b and c1 set a on ,s to a on ,a1 and read off on ,a above #fc and #fj the values of b 7 #dgh7 and c 7 #dfd7 respectively4 7#b47 ,given a1 b1 and a1 to find b1 c1 and c4 ,in this case the angle a on ,s is placed under the length of side a on ,a and under b on ,a is found the angle b on ,s4 ,the angle c #ahj 7a b71 whence the length c can be read off on ,a over c on ,s4 7#c47 ,given the sides and the included angle1 to find the other side and the remaining angles4 ,if1 for example1 there are given a #fe1 b #db1 and the included angle c #ee1 we have 7a b7 7a b7 tan4 7a b7_/7#b7 tan4 7a b7_/7#b74 ,then1 since a b #ahj #ee #abe1 it follows that 7a b7_/7#b7 7#abe7_/7b7 #fb #cj4 ,by the rule for tangents of angles greater than #de1 we find tan4 #fb #cj #aib4 ,inserting in the above proportion the values thus found1 we have #ajg #bc #aib tan4 7a b7_/7#b74 ,from this it is found that the value of the tangent is #jdab1 and placing the slide with all indices coinciding1 it is seen that this value on ,d corresponds to an angle of #bb #be4 ,therefore1 since 7a b7_/7#b7 #fb #cj1 and 7a b7_/7#b7 #bb #be1 it follows that a #hd #ee1 and b #dj #e4 ,finally1 to determine the side c1 we have c 7a sin c7_/7sin a7 as before4 ,,practical ,,trigonometrical ,,applications4 ,a few examples illustrative of the application of the methods of determining the functions of angles1 etc41 described in the preceding section1 will now be given4 ,to find the chord of an arc1 having given the included angle and the radius4 ,with the slide placed in the rule with the ,c and ,d scales outward1 bring one-half of the given angle on ,s to the index mark in the back of the rule1 and read the chord on ,b under twice the radius on ,a4 ,,ex4--,required the chord of an arc of #ae1 the radius being #bc in4 ,set #g #cj on ,s to the index mark in the back of the rule1 and under #df on ,a read #f in41 the required length of chord on ,b4 ,to find the area of a triangle1 given two sides and the included angle4 ,set the angle on ,s to the index mark on the back of the rule1 and bring cursor to #b on ,b4 ,then bring the length of one side on ,b to cursor1 cursor to #a on ,b1 the length of the other side on ,b to cursor1 and read area on ,b under index of ,a4 ,,ex4--,the sides of a triangle are #e and #f ft4 in length respectively1 and they include an angle of #bj4 ,find the area4 ,set #bj on ,s to index mark1 bring cursor to #b on ,b1 #e on ,b to cursor1 cursor to #a on ,b1 #f on ,b to cursor1 and under #a on ,a read the area #eac sq4 ft4 on ,b4 ,to find the number of degrees in a gradient1 given the rise per cent4 ,place the slide with the indices of ,t coincident with those of ,d1 and over the rate per cent4 on ,d read number of degrees in the slope on ,t4 ,as the arrangement of rule we have chiefly considered has only a single ,t scale1 it will be seen that only solutions of the above problem involving slopes between #aj and #ajj per cent4 can be directly read off4 ,for smaller angles1 one of the formul for the determination of the tangents of submultiple angles must be used4 ,in rules having a double ,t scale 7which is used with the ,a scale7 the value in degrees of any slope from #a to #ajj per cent4 can be directly read off on ,a4 ,to find the number of degrees1 when the gradient is expressed as #a in x4 ,place the index of ,t to x on ,d1 and over index of ,d read the required angle in degrees on ,t4 ,,ex4--,find the number of degrees in a gradient of #a in #ch4 ,set #a on ,t to #ch on ,d1 and over ,r4,h4 index of ,d read #ad #de on ,t4 ,given the lap1 the lead and the travel of an engine slide valve1 to find the angle of advance4 ,set 7lap lead7 on ,b to half the travel of the valve on ,a1 and read the angle of advance on ,s at the index mark on the back of the rule4 ,,ex4--,valve travel #d;in41 lap #a in41 lead in4 ,find angle of advance4 ,set #a #acab on ,b to #bbe on ,a1 and read #ce #dj on ,s opposite the index on the back of the rule4 ,given the angular advance 1 the lap and the travel of a slide valve1 to find the cut-off in percentage of the stroke4 ,place the lap on ,b to half the travel of valve on ,a1 and read on ,s the angle 7the supplement of the angle of the eccentric7 found opposite the index in the back of the rule4 ,to this angle1 add the angle of advance and deduct the sum from #ahj1 thus obtaining the angle of the crank at the point of cut-off4 ,to the cosine of the supplement of this angle1 add #a and multiply the result by #ej1 obtaining the percentage of stroke completed when cut-off occurs4 ,,ex4--,given the angular advance #ce #dj1 the valve travel #d;in41 and the lap #a in41 find the angle of the crank at cut-off and the admission period expressed as a percentage of the stroke4 ,set #a on ,b to #bbe on ,a1 and read off on ,s opposite the index1 the supplement of the angle of the eccentric #bf #bj4 ,then #ahj 7#ce #dj #bf #bj7 #aah the crank angle at the point of cut-off4 ,further1 cos4 #aah cos4 #fb sin 7#ij #fb7 sin #bh1 and placing #bh on ,s to the back index1 the cosine1 read on ,b under ,r4,h4 index of ,a1 is found to be #jdfi4 ,adding #a and placing the ,l4,h4 index of ,c to the result1 #adfi1 on ,d1 we read off under #ej on ,c1 the required period of admission #gcd per cent4 on ,d4 ,the trigonometrical scales are useful for evaluating certain formul4 ,thus in the following expressions1 if we find the angle a such that sin4 a k1 we can write3-- 7k7_/7#a k#b7 tan4 a2 7#a k#b7_/7k7 cot4 a2 7#a k#b7 cos4 a2 etc4 ,in the first expression1 take k #jbih4 ,place the slide with the sine scale outward and with its indices agreeing with the indices of the rule4 ,set the cursor to #jbih on the 7,r4,h47 scale of ,a1 and read #ag #bj on the sine scale as the angle required4 ,then under #ag #bj on the tangent scale1 read #jcab on ,d as the result4 ,,slide ,,rules ,,with ,,log4-,,log4 ,,scales4 ,for occasional requirements1 the method described on page #de of determining powers and roots other than the square and cube1 is quite satisfactory4 ,when1 however1 a number of such calculations are to be made1 the process may be simplified considerably by the use of what are known as log4-log41 logo-log41 or logometric scales1 in conjunction with the ordinary scales of the rule4 ,the principle involved will be understood from a consideration of those rules for logarithmic computation 7page #h7 which refer to powers and roots4 ,from these it is seen that while for the multiplication and division of numbers we add their logarithms1 for involution and evolution we require to multiply or divide the logarithms of the numbers by the exponent of the power or root as the case may be4 ,thus to find #cb4c1 we have 7log4 #c7 #bc log4 x1 and by the ordinary method described on page #de we should determine log4 #c by the aid of the scale ,l on the back of the slide1 multiply this by #bc by using the ,c and ,d scales in the usual manner1 transfer the result to scale ,l1 and read the value of x on ,d under #a on ,c4 ,by the simpler method1 first proposed by ,dr4 ,p4 ,m4 ,roget1#h the multiplication of log4 #c by #bc is effected in the same way as with any two ordinary factors--i4e41 by adding their logarithms and finding the number corresponding to the resulting logarithm4 ,in this case we have log4 7log4 #c7 log4 #bc log4 7log4 x74 ,the first of the three terms is obviously the logarithm of the logarithm of #c1 the second is the simple logarithm of #bc1 and the third the logarithm of the logarithm of the answer4 ,hence1 if we have a scale so graduated that the distances from the point of origin represent the logarithms of the logarithms 7the log4-logs47 of the numbers engraved upon it1 then by using this in conjunction with the ordinary scale of logarithms1 we can effect the required multiplication in a manner which is both expeditious and convenient4 ,slightly varying arrangements of the log4-log4 scale1 sometimes referred to as the 8,p line10 have been introduced from time to time1 but latterly the increasing use of exponential formul in thermodynamic1 electrical1 and physical calculations has led to a revival of interest in ,dr4 ,roget's invention1 and various arrangements of rules with log4-log4 scales are now available4 ,the ,davis ,log4-,log4 ,rule4--,in the rule introduced by ,messrs4 ,john ,davis ,son ,limited1 ,derby1 the log4-log4 scales are placed upon a separate slide--a plan which has the advantage of leaving the rule intact for all ordinary purposes1 while providing a length of #dj in4 for the log4-log4 scales4 ,in the #aj in4 ,davis rule one face of the slide1 marked ,e1 has two log4-log4 scales for numbers greater than unity1 the lower extending from #ajg to #b1 and the upper continuing the graduations from #b to #ajjj4 ,on the reverse face of the slide1 marked -,e1 are two log4-log4 scales for numbers less than unity1 the upper extending from #jjja to #je1 and the lower continuing the graduations from #je to #jicc4 ,both sets of scales are used in conjunction with the lower or ,d scale of the rule1 which is to be primarily regarded as running from #a to #aj1 and constitutes a scale of exponents4 ,in the #bj in4 rule the log4-log4 scales are more extensive1 and are used in conjunction with the upper or ,a scale of the rule 7#a to #ajj72 in what follows1 however1 the #aj in4 rule is more particularly referred to4 ,it has been explained that on the log4-log4 scale the distance of any numbered graduation from the point of origin represents the log4-log4 of the number4 ,the point of origin will obviously be that graduation whose log4-log4 #j4 ,this is seen to be #aj1 since log4 7log4 #aj7 log4 #a #j4 ,hence1 confining attention to the ,e scale1 to locate the graduation #bj1 we have log4 7log4 #bj7 log4 #acja #jaacig1 so that if the scale ,d is #be cm4 long1 the distance between #aj and #bj on the corresponding log4-log4 scale would be #aacig #d #bhdi mm4 ,for numbers less than #aj the resulting log4-logs4 will be negative1 and the distances will be spaced off from the point of origin in a negative direction--i4e41 from right to left4 ,thus1 to locate the graduation #e1 we have log4 7log4 #e7 log4 #jfii #ahdd2 i4e41 #a #jhdd or #jaef2 so that the graduation marked #e would be placed #aef #d #ci mm4 distant from #aj in a negative direction1 and proceeding in a similar manner1 the scale may be extended in either direction4 ,in the -,e scale1 the notation runs in the reverse direction to that of the ,e scale1 but in all other respects it is precisely analogous1 the distance from the point of origin 7#ja in this case7 to any graduation x representing log4 -log4 x44 ,it follows that of the similarly situated graduations on the two scales1 those on the -,e scale are the reciprocals of those on the ,e scale4 ,this may be readily verified by setting1 say1 #aj on ,e to 7,r4,h47 #a on ,d1 when turning to the back of the rule we find #ja on -,e agreeing with the index mark in the aperture at the right-hand extremity of the rule4 ,in using the log4-log4 scales it is important to observe 7#a7 that the values engraved on the scale are definite and unalterable 7e4g41 #ab can only be read as #ab and not as #abj1 #jjjab1 etc41 as with the ordinary scales72 7#b7 that the upper portion of each scale should be regarded as forming a prolongation to the right of the lower portion2 and 7#c7 that immediately above any value on the lower portion of the scale is found the #ajth power of that value on the upper portion of the scale4 ,keeping these points in view1 if we set #aa on ,e to #a on ,d we find over #b on ,d the value of #aab #aba on ,e4 ,similarly1 over #c we find #aac #acca1 and so on4 ,then1 reading across the slide1 we have1 over #b1 the value of #aab #aj #aabj #fgc1 and over #c we have #aac #aj #aacj #agd4 ,hence the rule3--,to find the value of xn1 set x on ,e to #a on ,d1 and over n on ,d read xn on ,e4 ,with the slide set as above1 the #hth1 #ith1 etc41 powers of #aa cannot be read off2 but it is seen that1 according to 7#b7 in the foregoing1 the missing portion of the ,e scale is that part of the upper scale 7#b to about #bf7 which is outside the rule to the left4 ,hence placing #aa to #aj on ,d1 the #hth1 #ith1 etc41 powers of #aa will be read off on the upper part of the ,e scale4 ,in general1 then1 ,if x on the lower line is set to #a on ,d1 then xn is read directly on that line and x#ajn on the upper line4 ,if x on the upper line is set to #a on ,d1 then xn is read directly on that line and x on the lower line4 ,if x on the lower line is set to #aj on ,d1 then x is read directly on that line and xn on the upper line4 ,if x on the upper line is set to #aj on ,d1 then x is read directly on that line and x on the lower line4 ,these rules are conveniently exhibited in the accompanying diagram 7,fig4 #ad74 ,they are equally applicable to both the ,e and -,e scales of the #aj in4 rule1 and include practically all the instruction required for determining the nth power or the nth root of a number4 ,they do not apply directly to the #bj in4 rule1 however1 for here the relation of the lower and upper scales will be xn and x#ajjn4 ,,ex4--,find #aafgbef4 ,set #aafg on ,e to #a on ,d1 and over #bef on ,d read #adhe on ,e4 ,,ex4--,find #dfafa4 ,set #df on upper ,e scale to #a on ,d1 and over #afa on ,d read #aag 7#aafg7 on ,e4 ,,ex4--,find #adjbg and #adbg4 ,set #ad on ,e to #aj on ,d1 and over #bg on ,d read #ajie #adjbg on lower ,e scale and #bdh #adbg on upper ,e scale4 ,illustration3 ,,fig4 #ad4 ,,ex4--,find #dfjjahd and #dfjahd4 ,set #df on upper ,e scale to #aj on ,d1 and over #ahd on ,d read #ajgc on lower ,e scale and #bjbb 7#bjbbh7 on upper ,e scale4 ,,ex4--,find #jjgdaae4 ,using the -,e scale1 set #jjgd to #a on ,d1 and over #aae on ,d read #jje on -,e4 ,the method of determining the root of a number will be obvious from the preceding examples4 ,,ex4--,find #a4d7ag7 and #ad7ag74 ,set #ag on ,e to #ad on ,d1 and over #a on ,d read #gef on upper ,e scale and #abbd on lower ,e scale4 ,,ex4--,find #jjca7jiad74 ,set #jiad on -,e to #ca on ,d1 and over #aj on ,d read #jjee on upper -,e scale4 ,when the exponent n is fractional1 it is often possible to obtain the result directly with one setting of the slide4 ,thus to determine #aace by the first method we find #ajfbe1 and placing #aace on ,e to #a on ,d1 read #aadd on ,e over #ajfbe on ,d4 ,by the direct method we place #aace on the ,e scale on #af on ,d1 and over #ag on ,d read #aadd on ,e4 ,it will be seen that since the scale ,d is assumed to run from #a to #aj we are unable to read #af and #ag on this scale2 but it is obvious that the ratios 7#ag7_/7af7 and 7#ag7_/7af7 are identical1 and it is with the ratio only that we are1 in effect1 concerned4 ,since an expression of the form x-n 7#a7_/7xn7 or 77#a7_/7x77n1 the required value may be obtained by first determining the reciprocal of x and proceeding as before4 ,by using both the direct and reciprocal log4-log4 scales 7,e and -,e7 in conjunction however1 the required value can be read directly from the rule1 and the preliminary calculation entirely avoided4 ,in the ,davis form of rule1 the result can be read on the -,e scale1 used in conjunction with the ,d scale of the rule1 x on ,e being set to the index mark in the aperture in the back of the rule4 ,,ex4--,find the value of #aaieafe4 ,set #aaie on ,e to the index in the left aperture in the back of the rule1 and over #afe on ,d read #jgde on the -,e scale4 ,it may be noted in passing that the log4-log4 scale affords a simple means for determining the logarithm or anti-logarithm of a number to any base4 ,for this purpose it is necessary to set the base of the given system on ,e to #a on ,d1 when under any number on ,e will be found its logarithm on ,d4 ,thus1 for common logs41 we set the base #aj on ,e to #a on ,d1 and under #ajj we find #b1 the required log4 ,similarly we read log4 #bj #acja2 log4 #ee #agd2 log4 #eej #bgd1 etc4 ,reading reversely1 over #ach on ,d we find its antilog4 #bd on ,e2 also antilog4 #aeh #ch2 antilog4 #aai #aee1 etc4 ,for logs4 of numbers under #aj we set the base #aj to #aj on ,d2 hence the readings on ,d will be read as one-tenth their apparent value4 ,thus log4 #c #jdgg2 log4 #ebe #jgb2 antilog4 #jdae #bf2 antilog4 #jebe #c4ce1 etc4 ,the logs4 of the numbers on the lower half of the ,e scale will also be found on the ,d scale2 but a consideration of ,fig4 #ad will show that this will be read as one-tenth its face value if the base is set to #a on ,d1 and as one-hundredth if the base is set to #aj4 ,for natural1 hyperbolic1 or ,napierian logarithms1 the base is #bgah4 ,a special line marked or e serves to locate the exact position of this value on the ,e scale1 and placing this to #a on ,d we read log4e #dce #adg2 log4e #gd #bj2 antilog4e #bhi #ah1 etc4 ,the other parts of the scale are read as already described for common logs4 ,calculations involving powers of e are frequently met with1 and these are facilitated by using the special graduation line referred to1 as will be readily understood4 ,if it is required to determine the power or root of a number which does not appear on either of the log4-log4 scales1 we may break up the number into factors4 ,usually it is convenient to make one of the factors a power of #aj4 ,,ex4--#ciejaig #cieaig #ajc #aig #cieaig #ajeia4 ,then #cieaig #ae1 and #ajeia 7or antilog47 #eia #hab1jjj4 ,hence1 #ae #hab1jjj #ab1ahj1jjj is the result sought4 ,numbers which are to be found in the higher part of the log4-log4 scale may often be factorised in this way1 and greater accuracy obtained than by direct reading4 ,the form of log4-log4 rule which has been mainly dealt with in the foregoing gives a scale of comparatively long range1 and the only objection to the arrangement adopted is the use of a separate slide4 ,the ,jackson-,davis ,double ,slide ,rule4--,in this instrument a pair of aluminium clips enable the log4-log4 slide to be temporarily attached to the lower edge of the ordinary rule1 and used1 by means of a special cursor1 in conjunction with the ,c scale of the ordinary slide4 ,in this way both the log4-log4 and ordinary scales are available without the trouble of replacing one slide by the other4 ,since the scale of exponents is now on the slide1 the value of xn will be obtained by setting #a on ,c to x on ,e and reading the result on ,e under n on ,c4 ,by using a pair of log4-log4 slides1 one in the rule and one clamped to the edge by the clips1 we have an arrangement which is very useful in deducing empirical formul of the type y xn4 ,the ,yokota ,slide ,rule4--,in this instrument the log4-log4 scales are placed on the face of the rule1 each set comprising three lines4 ,these1 for numbers greater than #a1 are found above the ,a scale while the three reciprocal log4-log4 lines are below the ,d scale4 ,both sets are used in conjunction with the ,c scale on the slide4 ,other features of this rule are3--,the ordinary scales are #aj in4 long instead of #be cm4 as hitherto usual2 hence the logarithms of numbers can be read on the ordinary scale of inches on the edge of the rule4 ,there is a scale of cubes in the centre of the slide and on the back of the slide there is a scale of secants in addition to the sine and tangent scales4 ,illustration3 ,,fig4 #ae4 ,the ,faber ,log4-log4 ,rule4--,in this instrument shown in ,fig4 #ae1 the two log4-log4 scales are placed on the face of the rule4 ,one section1 extending from #aa to #bi1 is placed above the ,a scale1 and the other section1 extending from #bi to #ajj1jjj1 is placed below the ,d scale4 ,these scales are used in conjunction with the ,c scale of the slide in the manner previously described4 ,the width of the rule is increased slightly1 but the arrangement is more convenient than that formerly employed1 wherein the log4-log4 scales were placed on the bevelled edge of the rule and read by a tongue projecting from the cursor4 ,illustration3 ,,fig4 #af4 ,another novel feature of this rule is the provision of two special scales at the bottom of the groove1 to which a bevelled metal index or marker on the left end of the slide can be set4 ,the upper of these scales is for determining the efficiency of dynamos and electric motors2 the lower for determining the loss of potential in an electric circuit4 ,the ,perry ,log4-log4 ,rule4--,in this rule1 introduced by ,messrs4 ,a4 ,g4 ,thornton1 ,limited1 ,manchester1 the log4-log4 scales are arranged as in ,fig4 #af1 the ,e scale1 running from #aa to #aj1jjj1 being placed above the ,a scale of the rule1 and the -,e or ,e#a scale running from #jic to #jjjja1 below the ,d scale of the rule4 ,these scales are read in conjunction with the ,b scales on the slide by the aid of the cursor4 ,the following tabular statement embodies all the instructions required for using this form of log4-log4 slide rule3-- ,when x is greater than #a4 xn ,set #a on ,b to x on ,e2 over n on ,b read xn on ,e x-n ,set #a on ,b to x on ,e2 under n on ,b read x-n on ,e#a x ,set n on ,b to x on ,e2 over #a on ,b read x on ,e x ,set n on ,b to x on ,e2 under #a on ,b read x on ,e#a ,when x is less than #a4 xn ,set #a on ,b to x on ,e#a2 under n on ,b read xn on ,e#a x-n ,set #a on ,b to x on ,e#a2 over n on ,b read x-n on ,e x ,set n on ,b to x on ,e#a2 under #a on ,b read x on ,e#a x ,set n on ,b to x on ,e#a2 over #a on ,b read x on ,e ,if #aj on ,b is used in place of #a on ,b1 read x in place of xn on ,e1 and x- in place of x-n on ,e#a4 ,if #ajj on ,b is used1 these readings are to be taken as x and x- respectively4 ,in rules with no -,e scale the value of x-n is obtained by the usual rules for reciprocals4 ,we may either determine xn and find its reciprocal or1 first find the reciprocal of x and raise it to the nth power4 ,the first method should be followed when the number x is found on the ,e scale4 ,,ex4--#cdeahb #jaje4 ,set #a on ,c to #cde on ,e1 and under #ahb on ,c read #iea on ,c4 ,then set #a on ,b to #ie on ,a1 and under index of ,a read #jaje on ,b4 ,when x is less than #a the second method is more suitable4 ,,ex4--#jbcagg 77#a7_/7jbc77agg #dceagg #ace ,set #a on ,b to #jbc on ,a1 and under index of ,a read 7#a7_/7jbc7 #dce on ,b4 ,set #a on ,c to #dce on ,e1 and under #agg on ,c read #ace on ,e4 ,as with the ,davis rule1 the exponent scale ,c will be read as th its face value if its ,r4,h4 index 7#aj7 is used in place of #a4 ,,special ,,types ,,of ,,slide ,,rules4 ,in addition1 to the new forms of log4-log4 slide rules previously described1 several other arrangements have been recently introduced1 notably a series by ,mr4 ,a4 ,nestler1 of ,lahr 7,london3 ,a4 ,fastlinger1 ,snow ,hill74 ,these comprise the 8,rietz10 the 8,precision10 the 8,universal10 and the 8,fix0 slide rules4 ,,the ,,rietz ,,rule4--,in this rule the usual scales ,a1 ,b1 ,c1 and ,d1 are provided1 while at the upper edge is a scale1 which1 being three times the range of the ,d scale1 enables cubes and cube roots to be directly evaluated and also n and n4 ,a scale at the lower edge of the rule gives the mantissa of the logarithms of the numbers on ,d4 ,,the ,,precision ,,slide ,,rule4--,in this rule the scales are so arranged that the accuracy of a #bj in4 rule is obtainable in a length of #aj in4 ,this is effected by dividing a #bj in4 7#ej cm47 scale length into two parts and placing these on the working edges of the rule and slide4 ,on the upper and lower margins of the face of the rule are the two parts of what corresponds to the ,a scale in the ordinary rule2 while in the centre of the slide is the scale of logarithms which1 used in conjunction with the #ej cm4 scales on the slide1 is virtually twice the length of that ordinarily obtainable in a #aj in4 rule4 ,the same remark applies to the trigonometrical scales on the under face of the slide4 ,both the sine and tangent scales are in two adjacent lengths1 while on the edge of the stock of the rule1 below the cursor groove1 is a scale of sines of small angles from #a #di to #e #dd4 ,this is referred to the #ej cm4 scales by an index projection on the cursor4 ,if ,c and ,c are the two parts of the scale on the slide and ,d and ,d the corresponding scales on the rule1 it is clear that in multiplying two factors #a on ,c can only be set directly to the upper scale ,d2 while #aj on ,c can only be set directly to the lower scale ,d4 ,hence if the first factor is greater than about #cb1 the cursor must be used to bring #a on ,c to the first factor on ,d4 ,similarly1 in division1 numerators and denominators which occur on ,c and ,d or on ,c and ,d cannot be placed in direct coincidence but must be set by the aid of the cursor4 ,any uncertainty in reading the result can be avoided by observing the following rule3 ,if in setting the index 7#a or #aj7 in multiplication1 or in setting the numerator to the denominator in division1 it is necessary to cross the slide1 then it will also be necessary to cross the slide to read the product or quotient4 ,,the ,,universal ,,slide ,,rule4--,in this instrument the stock carries two similar scales running from #a to #aj1 to which the slide can be set4 ,above the upper one is the logarithm scale and under the lower one the scale of squares #a to #ajj4 ,on the edge of the stock of the rule1 under the cursor groove1 is a scale running from #a to #ajjj4 ,an index projecting from the cursor enables this scale to be used with the scales on the face of the rule1 giving cubes1 cube roots1 etc4 ,on the slide1 the lower scale is an ordinary scale1 #a to #aj4 ,the centre scale is the first part of a scale giving the values of sin n cos n1 this scale being continued along the upper edge of the slide 7marked 8sin-cos07 up to the graduation #ej4 ,on the remainder of this line is a scale running from right to left 7#j to #ej7 and giving the value of cos#bn4 ,in surveying1 these scales greatly facilitate the calculations for the horizontal distance between the observer's station and any point1 and the difference in height of these two points4 ,on the back of the slide are scales for the sines and tangents of angles4 ,the values of the sines and tangents of angles from #cd to #e #dd differ little from one another1 and the one centre scale suffices for both functions of these small angles4 ,,the ,,fix ,,slide ,,rule4--,this is a standard rule in all respects1 except that the ,a scale is displaced by a distance 77_/7#d7 so that over #a on ,d is found #jghed on ,a4 ,this enables calculations relating to the area and cubic contents of cylinders to be determined very readily4 ,,the ,,beghin ,,slide ,,rule4--,we have seen that a disadvantage attending the use of the ordinary ,c and ,d scales1 is that it is occasionally necessary to traverse the slide through its own length in order to change the indices or to bring other parts of the slide into a readable position with regard to the stock4 ,to obviate this disadvantage1 ,tserepachinsky devised an ingenious arrangement which has since been used in various rules1 notably in the ,beghin slide rule made by ,messrs4 ,tavernier-,gravt of ,paris4 ,in this rule the ,c and ,d scales are used as in the standard rule1 but in place of the ,a and ,b scales1 we have another pair of ,c and ,d scales1 displaced by one-half the length of the rule4 ,the lower pair of scales may therefore be regarded as running from #ajn to #ajn #a1 and the upper pair as running from 7#aj7 #ajn to 7#aj7 #ajn #a4 ,with this arrangement1 without moving the slide more than half its length1 to the left or right1 it is always possible to compare all values between #a and #aj on the two scales4 ,this is a great advantage especially in continuous working4 ,another commendable feature of the ,beghin rule is the presence of a reversed ,c scale in the centre of the slide1 thus enabling such calculations as a b c to be made with one setting of the slide4 ,on the back of the slide are three scales1 the lowest of which1 used with the ,d scale1 is a scale of squares 7corresponding to the ordinary ,b scale71 while on the upper edge is a scale of sines from #e #dd to #ij1 and in the centre1 a scale of tangents from #e #dc to #de4 ,on the square edge of the stock1 under the cursor groove1 is the logarithm scale1 while on the same edge1 above the cursor groove1 are a series of gauge points4 ,all these values are referred to the face scales by index marks on the cursor4 ,,the ,,anderson ,,slide ,,rule4--,the principle of dividing a long scale into sections as in the ,precision rule1 has been extended in the ,anderson slide rule made by ,messrs4 ,casella ,co41 ,london1 and shown in ,fig4 #ag4 ,in this the slide carries a scale in four sections1 used in conjunction with an exactly similar set of scale-lines in the upper part of the stock4 ,on the lower part of the stock is a scale in eight sections giving the square roots of the upper values4 ,in order to set the index of the slide to values in the stock1 two indices of transparent celluloid are fixed to the slide extending over the face of the rule as shown in the illustration4 ,as each scale section is #cj cm4 in length1 the upper lines correspond to a single scale of nearly #d ft41 and the lower set to one of nearly #h ft4 in length1 giving a correspondingly large increase in the number of subdivisions of these scales1 and consequently much greater accuracy4 ,in order to decide upon which line a result is to be found1 sets of 8line numbers0 are marked at each end of the rule and slide and also on the metal frame of the cursor4 ,in multiplication1 the line number of the product is the sum of the line numbers of the factors if the left index is used1 or #a more than this sum if the right index is used4 ,the illustration shows the multiplication of #b by #d4 ,the left index is set to #b 7line number1 #a71 and the cursor set to #d on the slide 7line number1 #b72 hence1 as the left index is used1 the result is found on line ,no4 #c4 ,similar rules are readily established for division4 ,the column of line numbers headed #j is used for units1 that headed #d for tens1 and so on2 one column is given for tenths1 headed #d4 ,the square root scale bears similar line numbers1 so that the square root of any value on the upper scales is found on the correspondingly figured line below4 ,illustration3 ,,fig4 #ag4 ,,the ,,multiplex ,,slide ,,rule differs from the ordinary form of rule in the arrangement of the ,b scale4 ,the right-hand section of this scale runs from left to right as ordinarily arranged1 but the left-hand section runs in the reverse direction1 and so furnishes a reciprocal scale4 ,at the bottom of the groove1 under the slide1 there is a scale running from #a to #ajjj1 which is used in conjunction with the ,d scale1 readings being referred thereto by a metal index on the end of the slide4 ,by this means cubes1 cube roots1 etc41 can be read off directly4 ,messrs4 ,eugene ,dietzgen ,co41 ,new ,york1 are the makers4 ,,the 8,,long0 ,,slide ,,rule has one scale in two sections along the upper and lower parts of the stock1 as in the 8,precision0 rule4 ,the scale on the slide is similarly divided1 but the graduations run in the reverse direction1 corresponding to an inverted slide4 ,hence the rules for multiplication and division are the reverse of those usually followed 7page #cj74 ,on the back of the slide is a single scale #a-aj1 and a scale #a-ajjj1 giving cubes of this single scale4 ,by using the first in conjunction with the scales on the stock1 squares may be read1 while in conjunction with the cube scale1 various expressions involving squares1 cubes and their roots may be evaluated4 ,,hall',s ,,nautical ,,slide ,,rule consists of two slides fitting in grooves in the stock1 and provided with eight scales1 two on each slide1 and one on each edge of each groove4 ,while fulfilling the purposes of an ordinary slide rule1 it is of especial service to the practical navigator in connection with such problems as the 8reduction of an ex-meridian sight0 and the 8correction of chronometer sights for error in latitude40 ,the rule1 which has many other applications of a similar character1 is made by ,mr4 ,j4 ,h4 ,steward1 ,strand1 ,london4 ,,long-,,scale ,,slide ,,rules ,it has been shown that the degree of accuracy attainable in slide-rule calculations depends upon the length of scale employed4 ,considerations of general convenience1 however1 render simple straight-scale rules of more than #bj in4 in length inadmissible1 so that inventors of long-scale slide rules1 in order to obtain a high degree of precision1 combined with convenience in operation1 have been compelled to modify the arrangement of scales usually employed4 ,the principal methods adopted may be classed under three varieties3 7#a7 ,the use of a long scale in sectional lengths1 as in ,hannyngton's ,extended ,slide ,rule and ,thacher's ,calculating ,instrument2 7#b7 the employment of a long scale laid in spiral form upon a disc1 as in ,fearnley's ,universal ,calculator and ,schuerman's ,calculating ,instrument2 and 7#c7 the adoption of a long scale wound helically upon a cylinder1 of which ,fuller's and the 8,r4,h4,s40 ,calculating ,rules are examples4 ,,fuller',s ,,calculating ,,rule4--,this instrument1 which is shown in ,fig4 #ah1 consists of a cylinder d capable of being moved up and down and around the cylindrical stock f1 which is held by the handle4 ,the logarithmic scale-line is arranged in the form of a helix upon the surface of the cylinder d1 and as it is equivalent to a straight scale of #ejj inches1 or #da ft4 #h in41 it is possible to obtain four1 and frequently five1 figures in a result4 ,upon reference to the figure it will be seen that three indices are employed4 ,of these1 that lettered b is fixed to the handle2 while two others1 c and a 7whose distance apart is equal to the axial length of the complete helix71 are fixed to the innermost cylinder g4 ,this latter cylinder slides telescopically in the stock f1 enabling the indices to be placed in any required position relatively to d4 ,two other scales are provided1 one 7m7 at the upper end of the cylinder d1 and the other 7n7 on the movable index4 ,illustration3 ,,fig4 #ah4 ,in using the instrument a given number on d is set to the fixed index b1 and either a or c is brought to another number on the scale4 ,this establishes a ratio1 and if the cylinder is now moved so as to bring any number to b1 the fourth term of the proportion will be found under a or c4 ,of course1 in multiplication1 one factor is brought to b1 and a or c brought to #ajj4 ,the other factor is then brought to a or c1 and the result read off under b4 ,problems involving continuous multiplication1 or combined multiplication and division1 are very readily dealt with4 ,thus1 calling the fixed index ,f1 the upper movable index ,a1 and the lower movable index ,b1 we have for a b c3--,bring a to ,f2 ,a to #ajj2 b to ,a or ,b2 ,a to #ajj2 c to ,a or ,b and read the product at ,f4 ,the maximum number of figures in a product is the sum of the number of figures in the factors and this results when all the factors except the first have to be brought to ,b4 ,each time a factor is brought to ,a1 #a is to be deducted from that sum4 ,for division1 as a_/7m n71 bring a to ,f2 ,a or ,b to m2 #ajj to ,a2 ,a or ,b to a2 #ajj to ,a and read the quotient at ,f4 ,illustration3 ,,fig4 #ai4 ,the maximum number of figures in the quotient is the difference between the sum of the number of figures in the numerator factors and those of the denominator factors1 plus #a for each factor of the denominator and this results when ,a has to be set to all the factors of the denominator and all the factors of the numerator except the first brought to ,b4 ,each time ,b is set to a denominator factor or a numerator factor is brought to ,a1 #a is to be deducted4 ,logarithms of numbers are obtained by using the scales m and n and hence powers and roots of any magnitude may be obtained by the procedure already fully explained4 ,the instrument illustrated is made by ,messrs4 ,w4 ,f4 ,stanley ,co41 ,limited1 ,london4 ,,the 8,r4,h4,s40 ,,calculator4--,in this calculator1 designed by ,prof4 ,r4 ,h4 ,smith1 the scale-line1 which is #ej in4 long1 is also arranged in a spiral form 7,fig4 #ai71 but in this case it is wrapped around the central portion of a tube which is about in4 in diameter and #i;in4 long4 ,a slotted holder1 capable of sliding upon the plain portions of this tube1 is provided with four horns1 these being formed at the ends of the two wide openings through which the scale is read4 ,an outer ring carrying two horns completes the arrangement4 ,one of the horns of the holder being placed in agreement with the first factor1 and one of the horns of the ring with the second factor1 the holder is moved until the third factor falls under the same horn of the ring1 when the resulting fourth term will be found under the same 7right or left7 horn of the holder1 at either end of the slot4 ,in multiplication1 #ajj or #ajjj is taken for the second factor in the above proportion1 as already explained in connection with ,fuller's rule2 indeed1 generally1 the mode of operation is essentially similar to that followed with the former instrument4 ,the scale shown on one edge of the opening in the holder1 together with the circular scale at the top of the spiral1 enables the mantiss of logarithms of numbers to be obtained1 and thus problems involving powers and roots may be dealt with quite readily4 ,this instrument is supplied by ,mr4 ,j4 ,h4 ,steward1 ,london4 ,,thacher',s ,,calculating ,,instrument1 shown in ,fig4 #bj1 consists of a cylinder #d in4 in diameter and #ah in4 long1 which can be given both a rotary and a longitudinal movement within an open framework composed of twenty triangular bars4 ,these bars are connected to rings at their ends1 which can be rotated in standards fixed to the baseboard4 ,the scale on the cylinder consists of forty sectional lengths1 but of each scale-line that part which appears on the right-hand half of the cylinder is repeated on the left-hand half1 one line in advance4 ,hence each half of the cylinder virtually contains two complete scales following round in regular order4 ,on the lower lines of the triangular bars are scales exactly corresponding to those on the cylinder1 while upon the upper lines of the bars and not in contact with the slide is a scale of square roots4 ,illustration3 ,,fig4 #bj4 ,by rotating the slide any line on it may be brought opposite any line in frame and by a longitudinal movement any graduation on these lines may be brought into agreement4 ,the whole can be rotated in the supporting standards in order to bring any reading into view4 ,as shown in the illustration1 a magnifier is provided1 this being conveniently mounted on a bar1 along which it can be moved as required4 ,,sectional ,,length ,,or ,,gridiron ,,slide ,,rules4--,the idea of breaking up a long scale into sectional lengths is due to ,dr4 ,j4 ,d4 ,everett1 who described such a gridiron type of slide rule in #ahff4 ,hannyngton's ,extended ,slide ,rule is on the same principle4 ,both instruments have the lower scale repeated4 ,h4 ,cherry 7#ahhj7 appears to have been the first to show that such duplication could be avoided by providing two fixed index points in addition to the natural indices of the scale4 ,these additional indices are shown at #aj and #ajj in ,fig4 #ba1 which represents the lower sheet of ,cherry's ,calculator on a reduced scale4 ,the upper member of the calculator consists of a transparent sheet ruled with parallel lines1 which coincide with the lines of the lower scale when the indices of both are placed in agreement4 ,to multiply one number by another1 one of the indices on the upper sheet is placed to one of the factors1 and the position of whichever index falls under the transparent sheet is noted on the latter4 ,bringing the latter point to the other factor1 the result is found under whichever index lies on the card4 ,in other arrangements the inventor used transparent scales1 the graduations running in a reverse direction to those of the lower scale4 ,in this case1 a factor on the upper scale is set to the other factor on the lower1 and the result read at the available index4 ,illustration3 ,,fig4 #ba4 ,,proell',s ,,pocket ,,calculator is an application of the last-named principle4 ,it comprises a lower card arranged as ,fig4 #ba1 with an upper sheet of transparent celluloid on which is a similar scale running in the reverse direction4 ,for continued multiplication and division1 a needle 7supplied with the instrument7 is used as a substitute for a cursor1 to fix the position of the intermediate results4 ,a series of index points on the lower card enable square and cube roots to be extracted very readily4 ,this calculator is supplied by ,messrs4 ,john ,j4 ,griffin ,sons1 ,ltd41 ,london4 ,,circular ,,calculators4 ,although the #aj in4 slide rule is probably the most serviceable form of calculating instrument for general purposes1 many prefer the more portable circular calculator1 of which many varieties have been introduced during recent years4 ,the advantages of this type are3 ,it is more compact and conveniently carried in the waistcoat pocket4 ,the scales are continuous1 so that no traversing of the slide from #a to #aj is required4 ,the dial can be set quickly to any value2 there is no trouble with tight or ill-fitting slides4 ,the disadvantages of most forms are3 ,many problems involve more operations than a straight rule4 ,the results being read under fingers or pointers1 an error due to parallax is introduced1 so that the results generally are not so accurate as with a straight rule4 ,the inner scales are short1 and therefore are read with less accuracy4 ,special scale circles are needed for cubes and cube roots4 ,the slide cannot be reversed or inverted4 ,illustration3 ,,fig4 #bb4 ,illustration3 ,,fig4 #bc4 ,,the ,,boucher ,,calculator4--,this circular calculator resembles a stem-winding watch1 being about #b in4 in diameter and in4 in thickness4 ,the instrument has two dials1 the back one being fixed1 while the front one1 ,fig4 #bb 7showing the form made by ,messrs4 ,w4 ,f4 ,stanley1 ,london71 turns upon the large centre arbor shown4 ,this movement is effected by turning the milled head of the stem-winder4 ,the small centre axis1 which is turned by rotating the milled head at the side of the case1 carries two fine needle pointers1 one moving over each dial1 and so fixed on the axis that one pointer always lies evenly over the other4 ,a fine index or pointer fixed to the case in line with the axis of the winding stem1 extends over the four scales of the movable dial as shown4 ,of these scales1 the second from the outer is the ordinary logarithmic scale1 which in this instrument corresponds to a straight scale of about #d;in4 in length4 ,the two inner circles give the square roots of the numbers on the primary logarithmic scale1 the smaller circle containing the square roots of values between #a and #cafb 7 7#aj771 while the other section corresponds to values between #cafb and #aj4 ,the outer circle is a scale of logarithms of sines of angles1 the corresponding sines of which can be read off on the ordinary scale4 ,on the fixed or back dial there are also four scales1 these being arranged as in ,fig4 #bc4 ,the outer of these is a scale of equal parts1 while the three inner scales are separate sections of a scale giving the cube roots of the numbers taken on the ordinary logarithmic scale and referred thereto by means of the pointers4 ,in dividing this cube-root scale into sections1 the same method is adopted as in the case of the square-root scale4 ,thus1 the smallest circle contains the cube roots of numbers between #a and #aj1 and is therefore graduated from #a to #baed2 the second circle contains the cube roots of numbers between #aj and #ajj1 being graduated from #baed to #dfeg2 while the third section1 in which are found the cube roots of numbers between #ajj and #ajjj1 carries the graduations from #dfeg to #aj4 ,what has been said in an earlier section regarding the notation of the slide rule may in general be taken to apply to the scales of the ,boucher calculator4 ,the manner of using the instrument is1 however1 not quite so evident1 although from what follows it will be seen that the operative principle--that of variously combining lengths of a logarithmic scale--is essentially similar4 ,in this case1 however1 it is seen that in place of the straight scale-lengths shown in ,fig4 #d1 we require to add or subtract arc-lengths of the circular scales1 while1 further1 it is evident that in the absence of a fixed scale 7corresponding to the stock of the slide rule7 these operations cannot be directly performed as in the ordinary form of instrument4 ,however1 by the aid of the fixed index and the movable pointer1 we can effect the desired combination of the scale-lengths in the following manner4 ,assuming it is desired to multiply #b by #c1 the dial is turned in a backward direction until #b on the ordinary scale lies under the fixed index1 after which the movable pointer is set to #a on the scale4 ,as now set1 it is clear that the arc-length #a-b is spaced off between the fixed index and the movable pointer1 and it now only remains to add to this definite arc-length a further length of #a-c4 ,to do this we turn the dial still further backward until the arc #a-c has passed under the movable pointer1 when the result1 #f1 is read under the fixed index4 ,a little consideration will show that any other scale length may be added to that included between the fixed and movable pointers1 or1 in other words1 any number on the scale may be multiplied by #b by bringing the number to the movable pointer and reading the result under the fixed index4 ,the rule for multiplication is now evident4 ,rule for ,multiplication4--,set one factor to the fixed index and bring the pointer to #a on the scale2 set the other factor to the pointer and read the result under the fixed index4 ,with the explanation just given1 the process of division needs little explanation4 ,it is clear that to divide #f by #c1 an arc-length #a-c is to be taken from a length #a-f4 ,to this end we set #f to the index 7corresponding in effect to passing a length #a-f to the left of that reference point7 and set the pointer to the divisor #c4 ,as now set1 the arc #a-f is included between #a on the scale and the index1 while the arc #a-c is included between #a on the scale and the pointer4 ,obviously if the dial is now turned forward until #a on the scale agrees with the pointer1 an arc #a-c will have been deducted from the larger arc #a-f1 and the remainder1 representing the result of this operation1 will be read under the index as #b4 ,rule for ,division4--,set the dividend to the fixed index1 and the pointer to the divisor2 turn the dial until #a on the scale agrees with the pointer1 and read the result under the fixed index4 ,the foregoing method being an inversion of the rule for multiplication1 is easily remembered and is generally advised4 ,another plan is1 however1 preferable when a series of divisions are to be effected with a constant divisor--i4e41 when b in 7a7_/7b7 x is constant4 ,in this case #a on the scale is set to the index and the pointer set to b2 then if any value of a is brought to the pointer1 the quotient x will be found under the index4 ,combined ,multiplication and ,division1 as 7a b c7_/7m n7 x1 can be readily performed1 while cases of continued multiplication evidently come under the same category1 since a b c 7a b c7_/7#a #a7 x4 ,such cases as a_/7m n r7 x are regarded as 7a #a #a #a7_/7m n r7 x2 while 7a b c7_/7m7 x is similarly modified1 taking the form 7a b c7_/7m #a7 x4 ,in all cases the expression must be arranged so that there is one more factor in the numerator than in the denominator1 #a's being introduced as often as required4 ,the simple operations of multiplication and division involve a similar disposition of factors1 since from the rules given it is evident that m n is actually regarded as 7m n7_/7#a71 while 7m7_/7n7 becomes in effect 7m #a7_/7n74 ,it is important to note the general applicability of this arrangement-rule1 as it will be found of great assistance in solving more complicated expressions4 ,as with the ordinary form of slide rule1 the factors in such an expression as 7a b c7_/7m n7 x are taken in the order3--#ast factor of numerator2 #ast factor of denominator2 #bnd factor of numerator2 #bnd factor of denominator1 and so on2 the #ast factor as a being set to the index1 and the result x being finally read at the same point of reference4 ,,ex4--7#ci #adb #fc7_/7acg #ai7 #acd4 ,commence by setting #ci to the index1 and the pointer to #acg2 bring #adb to the pointer2 pointer to #ai2 #fc to the pointer1 and read the result #acd at the index4 ,it should be noted that after the first factor is set to the fixed index1 the pointer is set to each of the dividing factors as they enter into the calculation1 while the dial is moved for each of the multiplying factors4 ,thus the dial is first moved 7setting the first factor to the index71 then the pointer1 then the dial1 and so on4 ,number of ,digits in the ,result4--,if rules are preferred to the plan of roughly estimating the result1 the general rules given on pages #ba and #be should be employed for simple cases of multiplication and division4 ,for combined multiplication and division1 modify the expression1 if necessary1 by introducing #a's1 as already explained1 and subtract the sum of the denominator digits from the sum of numerator digits4 ,then proceed by the author's rule1 as follows3-- ,always turn dial to the ,,left2 i4e41 against the hands of a watch4 ,note dial movements only2 ignore those of the pointer4 ,each time #a on dial agrees with or passes fixed index1 ,,add #a to the above difference of digits4 ,each time #a on dial agrees with or passes pointer1 ,,deduct #a from the above difference of digits4 ,treat continued multiplication in the same way1 counting the #a's used as denominator digits as one less than the number of multiplied factors4 ,,ex4--7#hf #jgc #ajb7_/7ce #jbc7 #gie #giedgc4 ,set #hf to index and pointer to #ce4 ,bring #jgc to pointer 7noting that #a on the scale passes the index7 and set pointer to #jbc4 ,set #ajb to pointer 7noting that #a on the scale passes the pointer7 and read under index #gie4 ,there are #a #j #a #b numerator digits and #a #j #a denominator digit2 while #a is to be added and #a deducted as per rule4 ,but as the latter cancel1 the digits in the result will be #b #a #a4 ,when moving the dial to the left will cause #a on the dial to pass both index and pointer 7thus cancelling71 the dial may be turned back to make the setting4 ,it will be understood that when #a is the first numerator1 and #a on the dial is therefore set to the index1 no digit addition will be made for this1 as the actual operation of calculating has not been commenced4 ,in the ,stanley-,boucher calculator 7,fig4 #bc7 a small centre scale is added1 on which a finger indicates automatically the number of digits to be added or deducted2 the method of calculating1 however1 differs from the foregoing4 ,to avoid turning back to #j at the commencement of each calculation1 a circle is ground on the glass face1 so that a pencil mark can be made thereon to show the position of the finger when commencing a calculation4 ,to ,find the ,square of a ,number4-- ,set the number1 on one or other of the square root scales1 to the index1 and read the required square on the ordinary scale4 ,to ,find the ,square ,root of a ,number4--,set the number to the index1 and if there is an odd number of digits in the number1 read the root on the inner circle2 if an even number1 on the second circle4 ,to ,find the ,cube of a ,number4--,set #a on the ordinary scale to the index1 and the pointer 7on the back dial7 to the number on one of the three cube-root scales4 ,then under the pointer read the cube on the ordinary scale4 ,to ,find the ,cube ,root of a ,number4--,set #a to index1 and pointer to number4 ,then read the cube root under the pointer on one of the three inner circles on the back dial4 ,if the number has #a1 #d1 #g1 #aj or #b1 #e1 etc41 digits1 use the inner circle4 #b1 #e1 #h1 #aa or #a1 #d1 etc41 second circle4 #c1 #f1 #i1 #ab or #j1 #c1 etc41 third circle4 ,for ,powers or ,roots of ,higher ,denomination4--,set #a to index1 the pointer to the number on the ordinary scale1 and read on the outer circle on the back dial the mantissa of the logarithm4 ,add the characteristic 7see p4 #df71 multiply by the power or divide by the root1 and set the pointer to the mantissa of the result on this outer circle4 ,under the pointer on the ordinary scale read the number1 obtaining the number of figures from the characteristic4 ,to ,find the ,sines of ,angles4--,set #a to index1 pointer to the angle on the outer circle1 and read under the pointer the natural sine on the ordinary scale2 also under the pointer on the outer circle of the back dial read the logarithmic sine4 ,,the ,,halden ,,calculex4--,after the introduction of the ,boucher calculator in #ahgf1 circular instruments1 such as the ,charpentier calculator1 were introduced1 in which a disc turned within a fixed ring1 so that scales on the faces of both could be set together and ratios established as on the slide rule4 ,cultriss's ,calculating ,disc is another instrument on the same principle4 ,the ,halden ,calculex1 of which half-size illustrations are given in ,figs4 #bd and #be1 represents a considerable improvement upon these early instruments4 ,it consists of an outer metal ring carrying a fixed-scale ring1 within which is a dial4 ,on each side of this dial are flat milled heads1 so that by holding these between the thumb and forefinger the dial can be set quickly and conveniently4 ,the protecting glass discs1 which are not fixed in the metal ring but are arranged to turn therein1 carry fine cursor lines1 and as these are on the side next to the scales a very close setting can be made quite free from the effects of parallax4 ,this construction not only avoids the use of mechanism1 with its risk of derangement1 but reduces the bulk of the instrument very considerably1 the thickness being about in4 ,on the front face1 ,fig4 #bd1 the fixed ring carries an outer evenly-divided scale1 giving logarithms1 and an ordinary scale1 #a-aj1 which works in conjunction with a similar scale on the edge of the dial4 ,the two inner circles give the square roots of values on the main scales as in the ,boucher calculator4 ,on the back face1 ,fig4 #be1 the ring bears an outer scale1 giving sines of angles from #f to #ij and an ordinary scale1 #a-aj1 as on the front face4 ,the scales on the dial are all reversed in direction 7running from right to left71 the outer one consisting of an ordinary 7but inverse7 scale1 #a-aj1 while the three inner circles give the cube roots of values on this inverse scale4 ,as the fine cursor lines extend over all the scales1 a variety of calculations can be effected very readily and accurately4 ,illustration3 ,,fig4 #bd4 ,illustration3 ,,fig4 #be4 ,,sperry',s ,,pocket ,,calculator1 made by the ,keuffel and ,esser ,company1 ,new ,york 7,fig4 #bf71 has two rotating dials1 each with its own pointer and fixed index4 ,the ,s dial has an outer scale of equal parts1 an ordinary logarithmic scale1 and a square-root scale4 ,the ,l dial has a single logarithmic scale arranged spirally1 in three sections1 giving a scale length of #ab;in4 ,the pointers are turned by the small milled head1 which is concentric with the milled thumb-nut by which the two dials are rotated4 ,the gearing is such that both the ,l dial and its pointer rotate three times as fast as the ,s dial and pointer4 ,all the usual calculations can be made with the spiral scale1 as with the ,boucher calculator1 and the result read off on one or other of the three scale-sections4 ,frequently the point at which to read the result is obvious1 but otherwise a reference to the single scale on the ,s dial will show on which of the three spirals the result is to be found4 ,illustration3 ,,fig4 #bf4 ,the ,k and ,e ,calculator1 also made by the ,keuffel and ,esser ,company1 is shown in ,figs4 #bg and #bh4 ,it has two dials1 of which only one revolves4 ,this1 as shown in ,fig4 #bg1 has an ordinary logarithmic scale and a scale of squares4 ,there is an index line engraved on the glass of the instrument4 ,the fixed dial has a scale of tangents1 a scale of equal parts and a scale of sines1 the latter being on a two-turn spiral4 ,the pointers1 which move together1 are turned by a milled nut and the movable dial by a thumb-nut1 as in ,sperry's ,calculator1 ,fig4 #bf4 ,illustration3 ,,fig4 #bg4 ,illustration3 ,,fig4 #bh4 ,,slide ,,rules ,,for ,,special ,,calculations4 ,,engine ,,power ,,computer4--,a typical example of special slide rules is shown in ,fig4 #bi1 which represents1 on a scale of about half full size1 the author's ,power ,computer for ,steam1 ,gas1 and ,oil ,engines4 ,this1 as will be seen1 consists of a stock1 on the lower portion of which is a scale of cylinder diameters1 while the upper portion carries a scale of horse-powers4 ,in the groove between these scales are two slides1 also carrying scales1 and capable of sliding in edge contact with the stock and with each other4 ,this instrument gives directly the brake horse-power of any steam1 gas1 or oil engine2 the indicated horse-power1 the dimensions of an engine to develop a given power1 and the mechanical efficiency of an engine4 ,the calculation of piston speed1 velocity ratios of pulleys and gear wheels1 the circumferential speed of pulleys1 and the velocity of belts and ropes driven thereby1 are among the other principal purposes for which the computer may be employed4 ,illustration3 ,,fig4 #bi4 ,,the ,,smith-,,davis ,,piecework ,,balance ,,calculator has two scales1 #aa feet long1 having a range from #a;d4 to #bj1 and marked so that they can be used either for money or time calculations4 ,the scales are placed on the rims of two similar wheels and so arranged that the divided edges come together4 ,the wheels are mounted on a spindle carried at each end in the bearings of a supporting stand4 ,the wheels are pressed together by a spring1 and move as one4 ,to set the scales one to the other1 a treadle gear is arranged to take the pressure of the spring so that when the fixed wheel is held by the left hand the free wheel can be rotated by the right hand in either direction4 ,when the amount of the balance has been set to the combined weekly wage the treadle is released locking the two wheels together1 when the whole can be turned and the amounts respectively due to each man read off opposite his weekly wage4 ,the ,smith-,davis ,premium ,calculator is on the same principle but the scales are about #d feet #f inches long and the wheels spring-controlled4 ,both instruments are supplied by ,messrs4 ,john ,davis ,son1 ,ltd41 ,derby4 ,,the ,,baines ,,slide ,,rule4--,in this rule1 invented by ,mr4 ,h4 ,m4 ,baines1 ,lahore1 four slides carrying scales are arranged to move1 each in edge contact with the next4 ,the slides are kept in contact and given the desired relative movement one to the other1 by being attached 7at the back71 to a jointed parallelogram4 ,on this principle which is of general application1 the inventor has made a rule for the solution of problems covered by ,flamant's formula for the flow of water in cast-iron pipes3--,v #gfbh;ds1 in which s is the sine of the inclination or loss of head2 d the diameter of the pipe in inches and ,v the velocity in feet per second4 ,the formula ,q ,,av is also included in the scope of the rule1 ,q being the discharge in cubic feet per second and ,a the cross sectional area of the pipe in square inches4 ,,farmar',s ,,profit-,,calculating ,,rule4--,the application of the slide rule to commercial calculations has been often attempted1 but the degree of accuracy required necessitates the use of a long scale1 and generally this results in a cumbersome instrument4 ,in ,farmar's ,profit-calculating ,rule the money scale is arranged in ten sections1 these being mounted in parallel form on a roller which takes the place of the upper scale of an ordinary rule4 ,the roller1 which is in4 in diameter1 is carried in brackets secured to each end of the stock1 so that by rotating the roller any section of the money scale can be brought into reading with the scale on the upper edge of the slide and with which the roller is in contact4 ,this scale gives percentages1 and enables calculations to be made showing profit on turnover1 profit on cost1 and discount4 ,the lower scale on the slide1 and that on the stock adjacent to it1 are similar to the ,a and ,b scales of an ordinary rule4 ,the instrument is supplied by ,messrs4 ,j4 ,casartelli ,son1 ,manchester4 ,,constructional ,,improvements ,,in ,,slide ,,rules4 ,the attention of instrument makers is now being given to the devising of means for ensuring the smooth and even working of the slide in the stock of the rule4 ,in some cases very good results are obtained by slitting the back of the stock to give more elasticity4 ,in the rules made by ,messrs4 ,john ,davis ,son1 a metal strip1 slightly curved in cross section as shown at ,a 7,fig4 #cj71 runs for the full length of the stock to which it is fastened at intervals4 ,near each end of the rule1 openings about #a in4 long are made in the metal backing through which the scales on the back of the slide can be read4 ,to prevent warping under varying climatic conditions both the stock of the rule and the slide are of composite construction4 ,the base of the stock is of mahogany1 while the grooved sides1 firmly secured to the base1 are of boxwood4 ,similarly the centre portion of the slide is of mahogany and the tongued sides of boxwood4 ,celluloid also enters into the construction1 a strip of this material being laid along the bottom of the groove in the stock4 ,a fine groove runs along the centre of this strip in order to give elasticity and to allow the sides of the stock to be pressed together slightly to adjust the fitting of the slide4 ,as a further means of adjustment the makers fit metal clips at each end of the rule1 so that by tightening two small screws the stock can be closed on the slide when necessary4 ,illustration3 ,,fig4 #cj4 ,illustration3 ,,fig4 #cb4 ,illustration3 ,,fig4 #ca4 ,in the rule made by the ,keuffel and ,esser ,company of ,new ,york1 one strip is made adjustable 7,fig4 #cb74 ,,the ,,accuracy ,,of ,,slide ,,rule ,,results4 ,the degree of accuracy obtainable with the slide rule depends primarily upon the length of the scale employed1 but the accuracy of the graduations1 the eyesight of the operator1 and1 in particular1 his ability to estimate interpolated values1 are all factors which affect the result4 ,using the lower scales and working carefully the error should not greatly exceed #jae per cent4 with short calculations4 ,with successive settings1 the discrepancy need not necessarily be greater1 as the errors may be neutralised2 but with rapid working the percentage error may be doubled4 ,however1 much depends upon the graduation of the scales4 ,rules in which one or more of the indices have been thickened to conceal some slight inaccuracy should be avoided4 ,the line on the cursor should be sharp and fine and both slide and cursor should move smoothly or good work cannot be done4 ,occasionally a little vaseline or clean tallow should be applied to the edges of the slide and cursor4 ,that the percentage error is constant throughout the scale is seen by setting #a on ,c to #aja on ,d1 when under #b is #bjb2 under #c1 #cjc2 under #e1 #eje1 etc41 the several readings showing a uniform error of #a per cent4 ,a method of obtaining a closer reading of a first setting or of a result on ,d has been suggested to the author by ,mr4 ,m4 ,ainslie1 ,b4,sc4 ,if any graduation1 as #d on ,c1 is set to #c on ,d1 it is seen that #d main divisions on ,c 7#dj-dd7 are equal in scale length to #c main divisions on ,d 7#cj-cc74 ,hence1 very approximately1 #a division on ,c is equal to #jge of a division on ,d1 this ratio being shown1 of course1 on ,d under #aj on ,c4 ,suppose 7#dc7 to be required4 ,setting the cursor to #dc on ,a1 it is seen that the root is something more than #bjf4 ,move the slide until a main division is found on ,c1 which exactly corresponds to the interval between #b and the cursor line1 on ,d4 ,the division #bg-bh just fits1 giving a reading under #aj on ,c1 of #gd4 ,hence the root is read as #bjgd4 ,for the higher parts of the scale1 the subdivisions1 #a-aa1 etc41 are used in place of main divisions4 ,the method is probably more interesting than useful1 since in most operations the inaccuracies introduced in making settings will impose a limit on the reliable figures of the result4 ,for the majority of engineering calculations1 the slide rule will give an accuracy consistent with the accuracy of the data usually available4 ,for some purposes1 however1 logarithmic section paper 7the use of which the author has advocated for the last twenty years7 will be found especially useful1 more particularly in calculations involving exponential formul4 ,,appendix4 ,,new ,,slide ,,rules--,,fifth ,,roots1 ,,etc4--,,the ,,solution ,,of ,,algebraic ,,equations--,,gauge ,,points ,,and ,,signs ,,on ,,slide ,,rules--,,tables ,,and ,,data--,,slide ,,rule ,,data ,,slips4 ,,the ,,pickworth ,,slide ,,rule4--,in this rule1 made by ,mr4 ,a4 ,w4 ,faber1 the novel feature is the provision of a scale of cubes 7,f7 in the stock or body of the rule4 ,from ,fig4 #cc it will be seen that the scale is fixed on the bevelled side of a slotted recess in the back of the rule4 ,the slide carries an index mark1 which is seen through the slot and can be set to any graduation of the scale2 in its normal position it agrees with #a on the scale4 ,the ,c scale on the face of the rule is divided into three equal parts by two special division lines1 marked ,,ii4 and ,,iii41 which1 together with the initial graduation #a of the scale1 serve for setting or reading off values on the ,d scale4 ,similar division lines are marked on the ,d scale4 ,illustration3 ,,fig4 #cc4 ,in using the rule for cubes or cube roots the slide is drawn to the right1 this movement never exceeding one-third of the length of the ,d scale4 ,with this limited movement1 and with a single setting of the slide1 the values of a1 7a #aj71 and 7a #ajj77 7a being less than #aj and not less than #a7 are given simultaneously and without any uncertainty as to the scales to use or the values to be read off4 ,to ,find the ,cube of a ,number4--,the marks ,,ii4 and ,,iii4 on ,d divide that scale into three equal sections4 ,if the number to be cubed is in the first section1 ,i4 on ,c is set to it2 if in the second section1 ,,ii4 on ,c is set to it2 if in the third section1 ,,iii4 on ,c is set to it4 ,then1 under the index mark on the back of the slide will be found the significant figures of the cube on the scale ,f4 ,if ,i4 on ,c was used for the setting1 the cube contains #a digit2 if ,,ii4 was used1 #b digits2 if ,,iii4 was used1 #c digits4 ,if the first figure of the number to be cubed is not in the units place1 the decimal point is moved through n places so as to bring the first significant figure into the units place1 the cube found as above1 and the decimal point moved in the reverse direction through #cn places4 ,to ,find the ,cube ,root of a ,number4--,the index mark is set to the significant figures of the number on scale ,f1 and the cube root is read on ,d under ,i41 ,,ii4 or ,,iii4 on ,c1 according as the number has #a1 #b or #c digits preceding the decimal point4 ,numbers which have #a1 #b or #c figures preceding the decimal point are dealt with directly4 ,numbers of any other form are brought to one of the above forms by moving the decimal point #c places 7or such multiple of #c places as may be required71 the root found and its decimal point moved #a place for each #c-place movement1 but in the reverse direction4 ,,the 8,,electro0 ,,slide ,,rule4--,in this special rule for electrical calculations1 made by ,mr4 ,a4 ,nestler1 the upper scales run from #ja to #ajjj1 and are marked 8,amp40 and 8sq4 mm40 respectively4 ,the lower scale on the slide running from #a to #aj1jjj is marked ,m 7metres71 while the lower scale on the rule 7#ja to #ajj7 is marked 8,volt40 ,the latter scale is so displaced that #aj on ,m agrees with #jagc on the ,volt scale4 ,the four factors involved are the current strength 7in ,amp472 the area of a conductor 7in sq4 mm472 the length of the conductor 7in metres72 and the permissible loss of potential 7in volts74 ,having given any three of these1 the fourth can be found very readily4 ,on the back of the slide are a scale of squares1 a scale of cubes and a single scale corresponding to the ,d scale of an ordinary rule4 ,hence1 by reversing the slide1 it is possible to obtain the #bnd1 #crd and #dth powers and roots of numbers4 ,in another form of the rule1 the scale of metres is replaced by one of yards1 while instead of the area of the conductor in sq4 mm41 the corresponding 8gauge0 sizes of wires are given4 ,,the 8,,polyphase0 ,,slide ,,rule4--,this instrument1 made by the ,keuffel ,esser ,company1 ,new ,york1 has1 in addition to the usual scales1 a scale of cubes on the vertical edge of the stock of the rule1 while in the centre of the slide there is a reversed ,c scale2 i4e41 a scale exactly similar to an ordinary ,c scale but with the graduations running from right to left4 ,the rule is specially useful for the solution of problems containing combinations of three factors and problems involving squares1 square roots1 cubes1 cube roots and many of the higher powers and roots4 ,it is specially adapted for electrical and hydraulic work4 ,,the ,,log-,,log ,,duplex ,,slide ,,rule4--,the same makers have introduced a log-log duplex slide rule1 in which the log-log scale is in three sections1 placed one above the other1 these occupying the position usually taken up by the ,a scale4 ,these scales are used in the manner already described 7page #hf71 but some advantage is obtained by the manner in which the complete log-log scale is divided1 the limits being e to e 7on ,scale ,l4,l4 #a72 e to e 7on ,scale ,l4,l4 #b72 and e to e#aj 7on ,scale ,l4,l4 #c71 e being the base of natural or hyperbolic logarithms 7#bgahbh74 ,in this way a total log-log range of from #aja to #bb1jjj is provided1 meeting all practical requirements4 ,these log-log scales are read in conjunction with a ,c scale placed at the upper edge of the slide4 ,a similar ,c scale1 but reversed in direction1 is placed at the lower edge of the slide1 this having red figures to distinguish it readily4 ,the adjacent scale on the body of the rule is an ordinary ,d scale1 and under this is an equally-divided scale giving the common logarithms of values on ,d4 ,in the centre of the slide is a scale of tangents4 ,it will be understood that a 8duplex0 rule consists of two side strips securely clamped together at the two ends1 forming the body of the rule1 the slide moving between them2 hence both front and back faces of the rule and slide are available1 graduations on the one side being referred to those on the other by the cursor which extends around the whole4 ,in this instrument1 the scales on the back face are the ordinary scales of the standard rule with the addition of a scale of sines which is placed in the centre of the slide4 ,it will be evident that this instrument is capable of dealing with a very wide range of problems involving exponential and trigonometrical formul4 ,,small ,,slide ,,rules ,,with ,,magnifying ,,cursors4--,several makers now supply #e in4 rules having the full graduations of a #aj in4 rule1 and fitted with a magnifying cursor 7,fig4 #cd74 ,this forms a compact instrument for the pocket1 but owing to the closeness of the graduations it is not usually possible to make a setting of the slide without using the cursor4 ,this1 of course1 involves more movements than with the ordinary instrument4 ,it is also very necessary to use the magnifying cursor in a direct light1 if accurate readings are to be obtained4 ,if these slight inconveniences are to be tolerated1 the principle could be extended1 a #aj in4 rule being marked as fully as a #bj in41 and fitted with a magnifying cursor4 ,the author has endeavoured1 but without success1 to induce makers to introduce such a rule4 ,the magnifying cursor1 supplied by ,messrs4 ,a4 ,g4 ,thornton1 ,limited1 has a lens which fills the entire cursor4 ,it has a powerful magnifying effect1 and the change from the natural to the magnified reading is less abrupt than with the semicircular lens4 ,illustration3 ,,fig4 #cd4 ,,the ,,chemist',s ,,slide ,,rule4--,a slide rule1 specially adapted for chemical calculations1 has been introduced recently by ,mr4 ,a4 ,nestler4 ,in this instrument the ,c and ,d scales are as usually arranged2 but1 in place of the ,a and ,b scales1 there are a number of gauge points or marks denoting the atomic and molecular weights of the most important elements and combinations4 ,the scales on the back of the slide are similarly arranged1 so that by reversing the slide the operations can be extended very considerably4 ,the rule finds its chief use in the calculation of analyses4 ,thus1 to find the percentage of chlorine if s grammes of a substance have been used and the precipitate of ,ag4,cl4 weighs a grammes1 we have the equation1 x 7,cl47_/7,ag4,cl47 7a7_/7s74 ,hence1 the mark ,ag4,cl4 on the upper scale of the slide is set to the mark ,cl4 on the upper scale of the rule1 when under a on the ,c scale is found the quantity of chlorine on ,d4 ,by setting the cursor to this value and bringing s on ,c to the cursor1 the percentage required can be read on ,c over #aj on ,d4 ,the rule is also adapted to the solution of various other chemical and electro-chemical calculations4 ,,the ,,stelfox ,,slide ,,rule4--,this rule1 shown in ,fig4 #ce1 has a stock #e in4 long1 fitted with a #aj in4 slide jointed in the middle of its length by means of long dowels4 ,by separating the parts the compactness of a #e in4 rule is obtained4 ,the upper scales on the rule and slide resemble the usual ,a and ,b scales4 ,the ,d scale on the lower part of the stock is in two sections1 the second portion being placed below the first1 as shown in the illustration4 ,the centre scale on the slide corresponds to the usual ,c scale1 while on the lower edge of the slide is a similar scale1 but with the index 7#a7 in the middle of its length4 ,the arrangement avoids the necessity of resetting the slide1 as is sometimes necessary with the ordinary rule1 and in general it combines the accuracy of a #aj in4 rule with the compactness of a #e in4 rule2 but a more frequent use of the cursor is necessary4 ,this rule is made by ,messrs4 ,john ,davis ,son1 ,limited1 ,derby4 ,illustration3 ,,fig4 #ce4 ,,electrical ,,slide ,,rule4--,another rule by the same makers1 specially useful for electrical engineers1 has the usual scales on the working edges of the rule and slide1 while in the middle of the slide is placed a scale of cubes4 ,a log-log scale in two sections is provided2 the power portion1 running from #ajg to #b1 is found on the lower part of the stock1 and the upper portion1 running from #b to #ajc1 on the upper part of the stock4 ,the uppermost scale on the stock is in two parts1 of which that to the left1 running from #bj to #ajj and marked 8,dynamo10 gives the efficiencies of dynamos2 that on the right1 running from #bj to #ajj and marked 8,motor10 gives the efficiencies of electric motors4 ,the lowest scale on the stock1 marked 8,volt10 gives the loss of potential in copper conductors4 ,the ordinary upper scale on the stock is marked ,l 7length of lead7 at the left1 and ,,kw 7kilowatts7 at the right2 the ordinary upper scale on the slide is marked ,a 7ampres7 and mm#b 7sectional area7 at the left1 and ,,hp 7horse-power7 at the right4 ,additional lines on the cursor enable the electrical calculations to be made either in ,british or metric units4 ,,the ,,picolet ,,circular ,,slide ,,rule4--,a simple form of circular calculator1 made by ,mr4 ,l4 ,e4 ,picolet of ,philadelphia1 is shown in ,fig4 #cf4 ,it consists of a base disc of stout celluloid on which turns a smaller disc of thin celluloid4 ,a cursor formed of transparent celluloid is folded over the discs1 and is attached so that the friction between the cursor and the inner disc enables the latter to be turned by moving the former4 ,by holding both discs the cursor can be adjusted as required4 ,the adjacent scales run in opposite directions1 so that multiplication and division are performed as with the inverted slide in an ordinary rule4 ,the outer scale1 which is two-thirds the length of the main scale1 enables cube roots to be found4 ,square roots are readily determined and continuous multiplication and division conveniently effected4 ,modified forms of this neatly made little instrument are also available4 ,illustration3 ,,fig4 #cf4 ,,other ,,recent ,,slide ,,rules4--,among other special types of slide rule1 mention should be made of the ,jakin #aj in4 rule for surveyors1 made by ,messrs4 ,john ,davis ,son1 ,limited1 ,derby4 ,by the provision of a series of short subsidiary scales1 the multiplication of a sine or tangent of an angle by a number can be obtained to an accuracy of #a in #aj1jjj4 ,the ,davis-,lee-,bottomley slide rule1 by the same makers1 has special scales provided for circle spacing4 ,the division of a circle into a number of equal parts1 often required in spacing rivets1 bolts1 etc41 and in setting out the teeth of gearwheels1 is readily effected by the aid of this instrument4 ,the ,cuntz slide rule is a very comprehensive instrument1 having a stock about #b in4 wide1 with the slide near the lower edge4 ,above the slide are eleven scales1 referable to the main scales by the cursor4 ,these scales enable squares and square roots1 cubes and cube roots1 and areas and circumferences of circles to be obtained by direct reading4 ,a much more compact instrument could be obtained by removing one-half the scales to the back of the rule and using a double cursor4 ,illustration3 ,,fig4 #cg4 ,in one form of #aj in4 rule1 supplied by ,mr4 ,w4 ,h4 ,harling1 ,london1 the body of the rule is made of well-seasoned cane1 with the usual celluloid facings4 ,the rule has a metal back1 enabling the fit of the slide to be regulated4 ,this backing extends the full length of the rule1 openings about #a in4 long being provided at each end1 enabling the scales on the back of the slide to be set with greater facility than is possible with the notched recesses usually adopted4 ,the author has long endeavoured1 but without success1 to induce makers to fit windows of glass or celluloid in place of the notched recesses4 ,this would allow the graduation of the ,s and ,t scales to be set more accurately1 and enable both to be used at each end of the rule--an advantage in certain trigonometrical calculations4 ,it would have the further advantage of permitting each alternate graduation of the evenly-divided or logarithm scale to be placed at opposite sides of one central line1 enabling the reading to be made more accurately and conveniently4 ,many special slide rules have lately been devised for determining the time necessary to perform various machine-tool operations and for analogous purposes1 while attention has again been given to rules for calculating the weights of iron and steel bars1 plates1 etc4 ,,the ,,davis-,,stokes ,,field ,,gunnery ,,slide ,,rule4--,this rule1 which is adapted for calculations involved in 8encounter0 and 8entrenched0 field gunnery1 is designed for the #ah pr4 quick-firing gun4 ,the upper and lower portions of the boxwood stock are united by a flexible centre of celluloid1 thus providing grooves front and rear to receive boxwood slides4 ,each of the nineteen scales is marked with its name1 and corresponding scales are coloured red or black4 ,the front edge is bevelled and carries a scale of #a in #bj1jjj4 ,the rule solves displacement problems1 map angles of sight1 changes of corrector and range corrections for changes in temperature1 wind and barometer1 etc4 ,a special feature for displacement calculations is the provision of a #ej yd4 sub-base angle scale1 by which the apex angle is read at one setting4 ,,the ,,davis-,,martin ,,wireless ,,slide ,,rule4--,in wireless telegraphy it is frequently necessary to determine wave-length1 capacity or self-induction when one or other of the factors of the equation1 #eif7,,lc7 is unknown4 ,the ,davis-,martin wireless rule is designed to simplify such calculations4 ,the upper scale in the stock 7inductance7 runs from #aj1jjj to #a1jjj1jjj2 the adjacent scale on the slide 7capacity7 runs from #jjjja to #jja but in the reverse direction4 ,the lower scale on the stock 7wave-length7 runs from #ajj to #ajjj1 giving square roots of the upper scale2 while on the lower edge of the scale are several arrows to suit the various denominations in which the wave-length and capacity may be expressed4 ,,improved ,,cursors4--,in some slide-rule operations1 notably in those involved in solving quadratic and cubic equations1 it not infrequently happens that readings are obscured by the frame of the cursor4 ,frameless cursors have been introduced to obviate this defect4 ,a piece of thick transparent celluloid is sometimes employed1 but this is liable to become scratched in use4 ,fig4 #cg shows a recent form of frameless glass cursor made by the ,keuffel ,esser ,company1 ,hoboken1 ,n4,j41 which is satisfactory in every way4 ,cursors having three hair lines are now fitted to some rules1 the distance apart of the lines being equal to the interval #jghed-a on the ,a scale4 ,,the ,,davis-,,pletts ,,slide ,,rule4--,in this rule a single log4-log4 scale and its reciprocal scale are arranged opposite the ordinary upper log4 scale4 ,thus1 common logarithms can be read directly1 while by taking advantage of the properties of characteristics and mantissas of common logarithms1 the scale can be extended indefinitely4 ,as #aj is the highest number on the log4-log4 scale1 it is carried down to within #jjbe of unity4 ,the reading of log4-log4 values above #aj is effected in a very simple manner4 ,there is also a scale in the centre of the slide which1 used in conjunction with the upper log4 scale enables the natural logarithm of any number between #jjjja and #aj1jjj to be read direct1 while any number on the upper log4 scale can be multiplied or divided by ex if the latter is between these limits4 ,on the back of the slide are scales for all circular and hyperbolic functions1 these being used in conjunction with the upper log4 scales4 ,,the ,,crompton-,,gallagher ,,boiler ,,efficiency ,,calculator has a stock in the thickness of which is a slot admitting a chart which can be moved at right angles to the two separate slides4 ,on the bevelled edge of one slide1 the graduations are continued so as to read against curves on the chart1 through an opening in the stock4 ,,the ,,davis-,,grinsted ,,complex ,,calculator4--,this slide rule is of considerable service in connection with calculations involving the conversion of complex quantities from the form a j b to the form ,r1 and vice versa4 ,the usual process of conversion necessitates repeated reference to trigonometrical tables1 and is both tedious and time-taking4 ,the ,complex ,calculator enables the conversions to be effected without reference to tables and with the minimum expenditure of time and labour4 ,the rule1 which is about #af in4 long1 has five scales4 ,the upper one 7,a7 is an ordinary logarithmic scale thrice-repeated4 ,the adjacent scales on the slide comprise 7#a7 a logarithmic scale of tangents 7,b7 ranging from #ja to #de1 and 7#b7 a logarithmic scale of secants 7,c7 from #j to #de4 ,the lower scales ,d and ,e are identical with the ,a scale1 and are provided to enable multiplication1 etc41 to be performed without the need for a separate slide rule4 ,readings can be transferred from ,a to the lower scales by means of the cursor4 ,in using the rule to convert a j b to ,r1 the index 7#de7 of the ,b scale is set to the larger component and the cursor to the smaller component1 on scale ,a4 ,then 7or its complement if b is greater than a7 is read on ,b under the cursor4 ,the cursor is then set to on the ,c scale1 and ,r is read on ,a under the cursor4 ,the rule is made by ,messrs4 ,john ,davis ,son1 ,limited1 ,derby4 ,,the ,,solution ,,of ,,algebraic ,,equations4 ,the slide rule finds an interesting application in the solution of equations of the second and third degree2 and although the process is essentially one of trial and error1 it may often serve as an efficient substitute for the more laborious algebraic methods1 particularly when the conditions of the problem or the operator's knowledge of the theory of equations enables some idea to be obtained as to the character of the result sought4 ,the principle may be thus briefly explained3--,if #a on ,c is set to x on ,d 7,fig4 #ch71 we find x7x7 x#b on ,d under x on ,c4 ,if1 however1 with the slide set as before1 instead of reading under x1 we read under x m on ,c1 the result on ,d will now be x7x m7 x#b mx q4 ,hence to solve the equation x#b mx q #j1 we reverse the above process1 and setting the cursor to q on ,d1 we move the slide until the number on ,c under the cursor1 and that on ,d under #a on ,c1 differ by m4 ,it is obvious from the setting that the product of these numbers q1 and as their difference m1 they are seen to be the roots of the equation as required4 ,for the equation x#b mx q #j1 we require m to equal the sum of the roots4 ,hence1 setting the cursor as before to q on ,d1 we move the slide until the number on ,c under the cursor1 and that on ,d under #a on ,c1 are together equal to m1 these numbers being the roots sought4 ,the alternative equations x#b mx q #j1 and x#b mx q #j are deducible from the others by changing the signs of the roots1 and need not be further considered4 ,illustration3 ,,fig4 #ch4 ,,ex4--,find the roots of x#b #hx #i #j4 ,set the cursor to #i on ,d1 and move the slide to the right until when #ffd is found under the cursor1 #acee on ,d is under #a on ,c4 ,these numbers are the roots required4 ,the upper scales can of course be used2 indeed1 in general they are to be preferred4 ,,ex4--,find the roots of x#b #abhx #cid #j4 ,set the cursor to #cid on ,a1 and move the slide to the right until we read #gfe on ,b under the cursor1 and #eae on ,a over #a on ,b4 ,the roots are therefore #gfe and #e4ae4 ,with a little consideration of the relative value of the upper and lower scales1 the student interested will readily perceive how equations of the third degree may be similarly resolved4 ,the subject is not of sufficient general importance to warrant a detailed examination being made of the several expressions which can be dealt with in the manner suggested2 but the author gives the following example as affording some indication of the adaptability of the method to practical calculations4 ,,ex4--,a hollow copper ball1 #ge in4 in diameter and #b lb4 in weight1 floats in water4 ,to what depth will it sink8 ,the water displaced #bgg #b #eed cub4 in4 ,the cubic contents of the immersed segment will be 77_/7#c77cr x#b x#c71 r being the radius and x the depth of immersion4 ,hence 77_/7#c77cr x#b x#c7 #eed1 and #aabex#b x#c #ebi4 ,to solve this equation we place the cursor to #ebi on ,a1 and move the slide until the reading on ,d under #a and that on ,b under the cursor together amount to #aabe4 ,in this way find #bde on ,d under #a1 with #hh on ,b under the cursor c1 c1 as a pair of values of which the sum is #aabe4 ,hence we conclude that x #bde in4 is the result sought4 ,with the rule thus set 7,fig4 #ci7 the student will note that the slide is displaced to the right by an amount which represents x on ,d1 and therefore x#b on ,a2 while the length on ,b from #a to the cursor line represents #aabe x4 ,hence the upper scale setting gives x#b7aabe x7 #aabex#b x#c #ebi as required4 ,illustration3 ,,fig4 #ci4 ,when in doubt as to the method to be pursued in any given case1 the student should work synthetically1 building up a simple example of an analogous character to that under consideration1 and so deducing the plan to be followed in the reverse process4 ,,screw-,,cutting ,,gear ,,calculations4 ,the slide rule has long found a useful application in connection with the gear calculations necessary in screw-cutting1 helical gear-cutting1 and spiral gear work4 ,,single ,,gears4--,for simple cases of screw-cutting in the lathe it is only necessary to set the threads per inch to be cut to the threads per inch in the guide screw 7or the pitch in inches in each case1 if more convenient74 ,then any pair of coinciding values on the two scales will give possible pairs of wheels4 ,,ex4--,find wheels to cut a screw of #a threads per inch with a guide screw of #b threads per inch4 ,setting #afbe on ,c to #b on ,d1 it is seen that #hj 7driver7 and #fe 7driven7 are possible wheels4 ,,compound ,,gears4--,when wheels so found are of inconvenient size1 a compound train is used1 consisting 7usually7 of two drivers and two driven wheels1 the product of the two former and the product of the two latter being in the same ratio as the simple wheels4 ,thus with #fj and #dj as drivers1 and #fe and #cj as driven1 we have1 7#fj #dj7_/7fe #cj7 7#bdjj7_/7aiej7 7#b7_/7afbe7 as before4 ,with the slide set as above1 values convenient for splitting up into suitable wheels are readily obtainable4 ,thus1 7#afjj7_/7acjj72 7#bdjj7_/7aiej72 7#djjj7_/7cbej72 7#dhjj7_/7cijj7 are a few suggestive values which may be readily factorised4 ,,slide ,,rules ,,for ,,screw-,,cutting ,,calculations4--,special circular and straight slide rules for screw-cutting gear calculations have long been employed4 ,for compound gears these usually entail the use of six scales1 two on each of the two slides and two on the stock4 ,the upper scale on the stock may be a scale of threads per inch to be cut1 the adjacent scale 7on the upper slide7 a scale of threads per inch in the guide screw4 ,setting the guide screw-graduation to the threads to be cut1 the lower slide is adjusted until a convenient pair of drivers is found in coincidence on the central pair of scales1 while a pair of driven wheels are in coincidence on the two lower scales4 ,some years ago1 a slide rule was introduced by which compound gears could be obtained with a single slide4 ,assuming the set of wheels usually provided--#bj to #abj teeth advancing by #e teeth--the products of #bj #be1 #bj #cj1 etc41 up to #aae #abj were calculated4 ,these products were laid out along each of the two lower scales4 ,the upper scales were a scale of threads per inch to be cut and a scale of the threads per inch of various guide screws4 ,setting the guide screw-graduation to the threads to be cut1 any coinciding graduations on the lower scales gave the required pairs of drivers and driven wheels4 ,,fractional ,,pitch ,,calculations4--,the author has long advocated the use of the slide rule for determining the wheels necessary for cutting fractional pitch threads1 and it is gratifying to find its value in this connection is now being appreciated4 ,for the best results a good #bj in4 rule is desirable1 but with care very close approximations can be found with an accurate #aj in4 rule4 ,in any case a magnifying cursor or a hand reading-glass is of great assistance4 ,,ex4--,find wheels to cut a thread of #jgjiji in4 pitch2 guide screw1 #b threads per inch4 ,to #jgjiji on ,d1 set #je 7guide screw pitch in inches7 on ,c4 ,to make this setting as accurately as possible1 the method described on page #aab may be used4 ,set #aj on ,c to about #ia on ,d1 and note that the interval #gg-gh on ,c represents #jia of the interval #gj-ga on ,d4 ,set the cursor to #gh on ,c and bring #e to the cursor4 ,the slide is then set so that #e on ,c agrees with #gjia on ,d4 ,inspection of the two scales shows various coinciding factors in the ratio required4 ,the most accurate is seen to be 7#ee on ,c7_/7#gh on ,d74 ,these values may be split up into 7#ee #ej7_/7fe #fj7 to form a suitable compound train of gears4 ,,gauge ,,points ,,and ,,signs ,,on ,,slide ,,rules4 ,many slide rules have the sign 7,prod47_/7#a7 at the right-hand end of the ,d scale1 while on the left is 7,quot47_/7#a47 ,it is somewhat unfortunate that these signs refer to rules for determining the number of digits in products and quotients1 which are used to a considerable extent on the ,continent1 and conflict with those used in this country4 ,by the ,continental method the number of digits in a product is equal to the sum of the digits in the two factors1 if the result is obtained on the ,,left of the first factor2 but if the result is found on the ,,right of the first factor1 it is equal to this sum #a4 ,the sign 7,prod47_/7#a7 the right-hand end of the ,d scale provides a visible reminder of this rule4 ,similarly for division3--,the number of digits in a quotient is equal to the number of the digits in the dividend1 minus those in the divisor1 if the quotient appears on the ,,right of the dividend1 and to this difference #a1 if the quotient appears on the ,,left of the dividend4 ,the sign 7,quot47_/7#a7 at the left-hand end of the ,d scale provides a visible reminder of this rule4 ,the sign - - found at both ends of the ,a scale is of general application but of questionable utility4 ,it is assumed to represent a fraction1 the vertical line indicating the position of the decimal point4 ,if the number #dee is to be dealt with in a multiplication on the lower scales1 we may suppose the decimal point moved two places to the left1 giving #dee1 a value which can be actually found on the scale4 ,if we use this value1 then to the number of digits in this result1 as many must be added as the number of places 7two in this case7 by which the decimal point was moved4 ,if the point is moved to the right1 the number of places must be subtracted4 ,similarly1 in division1 if the decimal point in the divisor is moved n places to the left1 then n places must be subtracted at the end of the operation2 while if the point is moved through n places to the right1 then n places must be added4 ,the sign referred to1 which1 of course1 applies to all scales1 completely indicates these processes and is submitted as a reminder of the procedure to be followed by those using the method described4 ,the signs 1 c1 c1 and ,m are explained in the ,section on 8,gauge ,points10 p4 #ec4 ,on some rules additional signs are found on the ,d scale4 ,one1 locating the value 7#ahj #fj7_/77 #cdcggd and hence giving the number of minutes in a radian1 is marked 4 ,another1 representing the value 7#ahj #fj #fj7_/77 #bjfbfe1 and hence giving the number of seconds in a radian is marked 4 ,a third point1 marked 1 placed at the value 7#bjj #ajj #ajj7_/77 #fcffbj1 is used when the newer graduation of the circle is employed4 ,these gauge points are useful when converting angles into circular measure1 or vice versa1 and also for determining the functions of small angles4 ,a gauge point is sometimes marked at #aadf on the ,a and ,b scales4 ,this is known as the 8,gunner's ,mark10 and is used in artillery calculations involving angles of less than #bj1 when1 for the purpose in view1 the tangent and circular measure of the angle may be regarded as equal4 ,for this constant1 the angle is taken in minutes1 the auxiliary base in feet1 and the base in yards4 ,the auxiliary base in feet on ,b is set to the angle in minutes on ,a when over #aadf on ,b is the base in yards on ,a4 ,the value 7#a7_/7aadf7 7 #c7_/7ahj #fj74 ,,tables ,,and ,,data4 ,,mensuration ,,formulae4 ,area of a parallelogram base height4 ,area of rhombus product of the diagonals4 ,area of a triangle base perpendicular height4 ,area of equilateral triangle square of side #jdcc4 ,area of trapezium sum of two parallel sides height4 ,area of any right-lined figure of four or more unequal sides is found by dividing it into triangles1 finding area of each and adding together4 ,area of regular polygon 7#a7 length of one side number of sides radius of inscribed circle2 or 7#b7 the sum of the triangular areas into which the figures may be divided4 ,circumference of a circle diameter #cadaf4 ,circumference of circle circumscribing a square side #dddc4 ,circumference of circle side of equal square #cede4 ,length of arc of circle radius degrees in arc #jjagde4 ,area of a circle square of diameter #jghed4 ,area of sector of a circle length of arc radius4 ,area of segment of a circle area of sector area of triangle4 ,side of square of area equal to a circle diameter #jhhfb4 ,diameter of circle equal in area to square side of square #aabhd4 ,side of square inscribed in circle diameter of circle #jgjg4 ,diameter of circle circumscribing a square side of square #adad4 ,area of square area of inscribed circle #abgcb4 ,area of circle circumscribing square square of side #aegjh4 ,area of square area of circumscribing circle #jfcff4 ,area of a parabola base x height4 ,area of an ellipse major axis minor axis #jghed4 ,surface of prism or cylinder 7area of two ends7 7length perimeter74 ,volume of prism or cylinder area of base height4 ,surface of pyramid or cone 7slant height perimeter of base7 area of base4 ,volume of pyramid or cone 777area of base perpendicular height74 ,surface of sphere square of diameter #cadaf4 ,volume of sphere cube of diameter #jebcf4 ,volume of hexagonal prism square of side #beih height4 ,volume of paraboloid volume of circumscribing cylinder4 ,volume of ring 7circular section7 mean diameter of ring #bdg square of diameter of section4 ,,specific ,,gravity ,,and ,,weight ,,of ,,materials4 ,,metals4 ,,metal4 ,specific ,weight of #a ,weight of #a ,gravity4 ,cub4 ,ft4 7,lb474,cub4 ,in4 7,lb474 ,aluminium1 ,cast #bef #afj #jjibg ,aluminium1 ,bronze #gfh #dge #jbge ,antimony #fga #dah #jbdb ,bismuth #iij #fag #jceg ,brass1 ,cast #haj #eje #jbic ,wire #hedh #ecc #jcji ,copper1 ,sheet #hhje #edi #jcah ,wire #hhhj #eed #jcba ,gold #aibde #abjj #jfie ,gun metal #hef #ecd #jcaj ,iron1 ,wrought 7mean7 #gfih #dhj #jbgh ,cast 7mean7 #gbag #dej #jbfa ,lead1 ,milled ,sheet #aadah #gab #jdab ,manganese #hjab #dii #jbhi ,mercury #aceif #hdi #jdia ,nickel1 ,cast #hbh #eaf #jcjj ,phosphor ,bronze1 ,cast #hfj #ecfh #jcaj ,platinum #baebb #acdb #jggh ,silver #ajeje #fee #jchj ,steel 7mean7 #gheb #dhif #jbhc ,tin #gdji #dfb #jbfh ,zinc1 ,sheet #gbj #ddi #jbfj ,cast #fhf #dbh #jbdh ,,miscellaneous ,,substances4 ,,substance4 ,specific ,weight of ,gravity4 #a ,cub4 ,in4 7,lb474 ,asbestos #ba-bhj #jgf-aja ,brick #aij #jfi ,cement #bgb-cje #jihd-aji ,clay #bj #jgb ,coal #acg #jdie ,coke #je #jaha ,concrete #bj #jgb ,fire-brick #bcj #jhc ,granite #be-bge #jea-ajj ,graphite #ah-bce #jfe-jhe ,sand-stone #bc #jhc ,slate #bh #ajb ,wood-- ,beech #jge #jbga ,cork #jbd #jjhg ,elm #jeh #jba ,fir #jef #jbjc ,oak #fb-he #jbe-jca ,pine #jdg #jag ,teak #jhj #jbi ,,ultimate ,,strength ,,oe ,,materials4 ,,material4 ,tension in ,compression ,shearing in ,modulus of lb4 per sq4 in lb4 per lb4 per sq4 ,elasticity in4 sq4 in4 in4 in lb4 per sq4 in4 ,cast ,iron #aa1jjj to #ej1jjj to #ad1jjj1jjj #cj1jjj #acj1jjj to #bc1jjj1jjj aver4 #af1jjj #ie1jjj #aa1jjj ,wrought ,iron #dj1jjj to #bf1jjj1jjj #gj1jjj to #ca1jjj1jjj aver4 #ej1jjj #ej1jjj #dj1jjj ,soft ,steel #fj1jjj to #cj1jjj1jjj #ajj1jjj to #cf1jjj1jjj ,soft ,steel aver4 #hj1jjj #gj1jjj #ee1jjj ,cast ,steel aver4 #abj1jjj #ae1jjj1jjj to #ag1jjj1jjj ,copper1 ,cast #ai1jjj #eh1jjj ,wrought #cd1jjj #af1jjj1jjj ,brass1 ,cast #ah1jjj #aj1ejj #i1agj1jjj ,gun ,metal #cd1jjj #aa1ejj1jjj ,phosphor ,bronze #eh1jjj #dc1jjj #ac1ejj1jjj ,wood1 ,ash #ag1jjj #i1cjj #a1djj ,beech #af1jjj #h1ejj ,pine #aa1jjj #f1jjj #fej #a1djj1jjj ,oak #ae1jjj #aj1jjj #b1cjj #a1ejj1jjj ,leather #d1bjj #be1jjj ,,powers1 ,,roots1 ,,etc41 ,,of ,,useful ,,factors4 n 7#a7_/7n7 n#b n#c n 7#a7_/7n7 n 7#a7_/7n7 #cadb #jcah #ihgj #cajjf #aggb #jefd #adfe #jfhc #b #fbhc #jaei #cidgh #bdhjej #bejg #jcii #ahde #jedb 77_/7#b7 #aega #jfcg #bdfg #chgh #abec #jgih #aafb #jhfj 77_/7#c7 #ajdg #jiee #ajig #aadh #ajbc #jigg #ajaf #jihe 7#d7_/7c7 #dahi #jbci #agedf #gcdif #bjdg #jdhi #afab #jfbb 77_/7#d7 #jghe #abgd #jfag #jdhd #jhhf #aabh #jibc #ajhd 77_/7#f7 #jebd #aiaj #jbgd #jadd #jgbd #achb #jhjf #abda #b #ihgj #jaja #igdji #ifacij #cadb #jcah #bade #jdff #c #cajjf #jjcbifacijbi1hjiiaj #eefh #agif #cadb #jcah 77_/7#cb7 #jjih #ajahf #jjjie #jjja #jcac #caib #jdfa #bafh g #cbb #jjcaajcfhd #cc1chfbd #efgd #jagf #caha #jcad #b;g #fdd #jjaedadgcf #bfg1jij #hjbe #jabe #djjg #jbdi ,,hydraulic ,,equivalents4 #a foot head #jdcd lb4 per square inch4 #a lb4 per square inch #bca ft4 head4 #a imperial gallon #bggbgd cubic inches4 #a imperial gallon #jafjde cubic foot4 #a imperial gallon #aj lb4 #a cubic foot of water #fbcb lb4 #fbcb imperial gallons4 #a cubic foot of sea water #fdjj lb4 #a cubic inch of water #jjcfaf lb4 #a cubic inch of sea water #jjcgjcg lb4 #a cylindrical foot of water #dhif lb4 #a cylindrical inch of water #jjbhd lb4 ,a column of water #ab in4 long #a in4 square #jdcd lb4 ,a column of water #ab in4 long #a in4 diameter #jcdj lb4 ,capacity of a #ab in4 cube #fbcb gallons4 ,capacity of a #a in4 square #a ft4 long #jjdcd gallon4 ,capacity of a #a ft4 diameter #a ft4 long #dhif gallons4 ,capacity of a cylinder #a in4 diameter #a ft4 long #jjcd gallon4 ,capacity of a cylindrical inch #jjjbhcb gallon4 ,capacity of a cubic inch #jjjcfjf gallon4 ,capacity of a sphere #ab in4 diameter #cbfc gallons4 ,capacity of a sphere #a in4 diameter #jjjahh gallon4 #a imperial gallon #ab ,united ,states gallon4 #a imperial gallon #dedc litres of water4 #a ,united ,states gallon #bcaj cubic inches4 #a ,united ,states gallon #jhc imperial gallon4 #a ,united ,states gallon #ch litres of water4 #a cubic foot of water #gdgf ,united ,states gallons4 #a cubic foot of water #bhcge litres of water4 #a litre of water #jbb imperial gallon4 #a litre of water #jbfd ,united ,states gallon4 #a litre of water #faj cubic inches4 #a litre of water #jjcec cubic foot4 ,,equivalents ,,of ,,pounds ,,avoirdupois4 #aj #ajj #ajjj #aj1jjj #ajj1jjj qr4 lb4cwt4 qr4 lb4ton cwt4 qr4 lb4ton cwt4 qr4 lb4ton cwt4 qr4 lb4 #a #j #aj #j #c #af #j #h #c #bj #d #i #a #d #dd #ab #c #ab #b #j #bj #a #c #d #j #ag #c #ab #h #ah #b #h #hi #e #b #bd #c #a #b #b #b #bj #a #f #c #d #ac #g #c #abacc #ah #b #h #d #a #ab #c #b #h #a #ae #b #bd #ag #ag #j #afagh #aa #a #bj #e #a #bb #d #a #bd #b #d #b #af #bb #f #a #bjbbc #d #a #d #f #b #d #e #a #ab #b #ac #b #h #bf #ae #b #bdbfg #ag #j #af #g #b #ad #f #a #j #c #b #b #j #ca #e #j #jcab #aj #j #j #h #b #bd #g #j #af #c #aa #a #bj #ce #ad #a #dceg #b #c #ab #i #c #f #h #j #d #d #j #a #ab #dj #c #b #hdja #ae #b #bd ,,trigonometrical ,,functions4 ,,right-,,angled ,,triangles4 ,illustration3 ,right-angled ,triangle ,sin4 ,a 7a7_/7b7 ,sec4 ,a 7b7_/7c7 ,tan4 ,a 7a7_/7c7 ,cos4 ,a 7c7_/7b7 ,cosec4 ,a 7b7_/7a7 ,cotan4 ,a 7c7_/7a7 ,versin4 ,a 7b c7_/7b74 ,coversin4 ,a 7b a7_/7b74 ,given4 ,required4 ,formul4 a1b ,a1,c1c ,sin4 ,a 7a7_/7b7 ,cos4 ,c 7a7_/7b7 c 77b a77b a77 a1c ,a1,c1b ,tan4 ,a 7a7_/7c7 ,cotan4 ,b 7a7_/7c7 b 7a#b c#b7 ,a1a ,c1c1b ,c #ij ,a c a ,cotan4 ,a b 7a7_/7,sin4 ,a7 ,a1b ,c1a1c,c #ij ,a a b ,sin4 ,a c b ,cos4 ,a ,a1c ,c1a1b ,c #ij ,a a c ,tan4 ,a b 7c7_/7,cos4 ,a7 ,,oblique-,,angled ,,triangles4 s 7a b c7 ,illustration3 ,oblique-angled ,triangle ,given4 ,formul4 ,a1,b1,c1a ,area 7a#b ,sin4 ,b ,sin4 ,c7 #b ,sin4 ,a ,a1b1c 7c b ,sin4 ,a7 a1b1c 7s7s a77s b77s c77 ,given4 ,required4 ,formul4 ,a1,c1a c c a7,sin4 ,c7_/7,sin4 ,a7 ,a1a1c ,c ,sin4 ,c 7c ,sin4 ,a7_/7a7 a1c1,b ,a ,tan4 ,a 7a ,sin4 ,b7_/7c a ,cos4 ,b7 a1b1c ,a ,sin4 ,a 777s b77s c77_/7b c77 ,cos4 ,a 77s7s a77_/7b c772 ,tan4 ,a 777s b77s c77_/7s7s a777 ,,compound ,,angles4 ,sin4 7,a ,b7 ,sin4 ,a ,cos4 ,b ,cos4 ,a ,sin4 ,b4 ,sin4 7,a ,b7 ,sin4 ,a ,cos4 ,b ,cos4 ,a ,sin4 ,b4 ,cos4 7,a ,b7 ,cos4 ,a ,cos4 ,b ,sin4 ,a ,sin4 ,b4 ,cos4 7,a ,b7 ,cos4 ,a ,cos4 ,b ,sin4 ,a ,sin4 ,b4 ,tan4 7,a ,b7 7,tan4 ,a ,tan4 ,b7_/7#a ,tan4 ,a ,tan4 ,b74 ,tan4 7,a ,b7 7,tan4 ,a ,tan4 ,b7_/7#a ,tan4 ,a ,tan4 ,b74 ,,slide ,,rule ,,data ,,slips1 ,,compiled ,,by ,c4 ,n4 ,,pickworth1 ,,wh4,,sc4 7,it is suggested that this page be removed by cutting through the above line1 and selected portions of the ,sectional ,data ,slips attached to the back of the ,slide ,rule47 #jjcabe #jjfbe #jjicge #jabe #jaefbe #jahge #jbahge #jbe #jbhabe #jcabe #jcdcge #jcge #jdjfbe #jdcge #jdfhge #jecabe #jefbe #jeicge #jfbe #jfefbe #jfhge #jgahge #jge #jghabe #jhabe #jhdcge #jhge #jijfbe #jicge #jifhge ,circ4 of circle #cadaf d4 ,area #jghed d#b4 ,sq4 eq4 area to cir41 s #jhhf d4 ,circle eq4 to sq41 d #aabh s4 ,sq4 inscbd4 in circ41 s #jgjg d4 ,circsb4 circ4 of sq41 d #adad s4 ,area of ellipse #j4ghed a b4 ,surface of sphere #cadaf d#b4 ,volume #jebcf d#c4 cone #jbfah d#b h4 ,radian 7#ahj7_/77 #egbi deg4 ,base of nat4 or hyp4 log4 e #bgahc4 ,nat4 or hyp4 log4 com4 log4 #bcjbf4 g 7at ,london7 #cbah ft4 per sec41 per sec4 ,abs4 temp4 deg4 ,f4 #dfa deg4 ,c4 #bgd4 ,c4 7#e7_/7i77;,f4 #cb72 ,f4 7#i7_/7e7;,c4 #cb4 ,cal4 pr4--,ther4 units per lb43 ,coal1 #ad1cjj2 petrol'm1 #bj1jjj2 coal gas per cu4 ft41 #gjj4 ,sp4 heat3--,wt4 iron1 #jaach2 ,c4,i41 #jabih2 copper1 brass1 #jjie2 lead1 #jjcad4 ,inch #bed mil'metres2 mil'metre #jjcicg in4 ,foot #jcjdh metres2 metre #cbhji feet4 ,yard #jiadch metre2 metre #ajicf yards4 ,mile #afjic kilomtrs42 kilomtr4 #jfbac mile4 ,sq4 in4 #fdeac sq4 cm42 sq4 cm4 #jaee sq4 in4 ,sq4 ft4 #ibi sq4 decmtr42 sq4 decmtr4 #jajgf sq4 ft4 ,sq4 yd4 #jhcf sq4 metre2 sq4 metre #aaif sq4 yds4 ,sq4 ml4 #behi hectares2 hectare #jjjchf sq4 ml4 ,cu4 in4 #afchf c4 cm42 c4 cm4 #jjfajb cu4 in4 ,cu4 ft4 #jjbhc c4 metre2 c4 metre #cecaf cu4 ft4 ,grain #jjfdh gramme2 gram4 #aedc grs4 ,ounce #bhce grams42 #jjcebg oz4 ,pound #jdecf kilogm42 kilogm4 #bbjd lb4 ,ton #ajaf tonnes2 tonne #jihdb ton4 ,mile per hr4 #adff ft41 or #ddg cm41 per sec4 ,lb4 per cu4 in4 #jjbgf kilogram per cu4 cm4 ,kilogram per cu4 cm4 #cfabe lb4 per cu4 in4 ,lb4 per cu4 ft4 #afjai kilogm4 per cu4 mtre4 ,grain per gall4 #jjadbf gramme per litre4 ,gramme per litre #gjaaf grains per gall4 ,ultimate ,strength,lb4 per ,sq4 in4 ,tens'n4,comp'n4 ,wt4 iron #ej1jjj #ej1jjj ,cast #af1jjj #ie1jjj ,steel #hj1jjj #gj1jjj ,copper #ba1jjj #ej1jjj ,brass #ah1jjj #aj1ejj ,lead #b1ejj #g1jjj ,pine #aa1jjj #f1jjj ,oak #ae1jjj #aj1jjj ,weight of ,metals4 ,cub4 ,in4 ,cub4 ,ft4 #ab ,cu4 ,in4 ,wt4 iron #jbgg #dhj #ccc ,cast #jbfj #dej #cab ,steel #jbhc #dij #cdj ,copper #jcah #eej #chb ,brass #jcjj #ebj #cfa ,zinc #jbdh #dcj #bih ,alumin'm #j4jif #afh #aaf ,lead #j4daa #gaj #dic ,lb4 per sq4 in4 #bca ft4 water #bjd in4 mercury #jjgjc kilo4 per sq4 cm4 ,atmosphere #adg lb4 per sq4 in4 #ccid ft4 water #ajcce ,ft4 hd4 water #jdcc lb4 per sq4 in4 #fbce lb4 per sq4 ft4 #jjcjd ,cub4 ft4 of water #fbce lb4 #jjbgh ton #bhcae litres #gdh ,u4,s4 galls4 ,gall4 7,imp47 #bggbg cu4 in4 #jafjd cu4 ft4 #aj lb4 water #dedd litres4 ,litre #agf pints #jbb gall4 #fa cu4 in4 #jjcec cu4 ft4 #jbfd ,u4,s4 gall4 ,horse-power #cc1jjj ft4-lb4 per min4 #jgdf kilowatt #dbd heat units per min4 ,heat unit #ggh ft4-lb4 #ajee watt-sec4 #ajge kilogrammetres #jbeb calorie4 ,foot-pound #jjjabi heat unit #acf joules #jachc kilogrammetres4 ,kilowatt #acd ,h4,p4 #dd1bdj ft4-lb4 per min4 #cdab heat units per hour4 ----- ,footnote #a3 ,it will be recognised that n is the characteristic of the logarithm of the original number4 ,footnote #b3 ,the special case in which the numerator is #a1 #aj1 or any power of #aj must be treated by the rule for reciprocals 7page #bg74 ,footnote #c3 ,the possible need for traversing the slide1 to change the indices1 when using the ,c and ,d scales1 is not considered as a setting4 ,footnote #d3 ,the reader may be reminded that cross-multiplication of the factors in any such slide rule setting will give a constant product1 e4g41 #bj #ide #bg #gj4 ,footnote #e3 ,in this case cross dividing gives a constant quotient1 e4g41 #h #c #d #ae4 ,since the upper scale is now a scale of reciprocals1 the ratio is really ,o ,d #ae #c ,footnote #f3 ,these lines should not be brought to the working edge of the scale but should terminate in the horizontal line which forms the border of the finer graduations1 their value being read into the calculation by means of the cursor 7see page #ee74 ,footnote #g3 ,the same principle may be applied to the cursor4 ,footnote #h3 ,philosophical ,transactions of the ,royal ,society1 #ahae4 ---------------------------------------- -------------------------------- ,,by ,,the ,,same ,,author4 ,,logarithms ,,for ,,beginners4 8,an extremely useful and much-needed little work1 giving a complete explanation of the theory and use of logarithms1 by a teacher of great clearness and good style40--,the ,mining ,journal4 #as4 #h;d4 ,post ,free4 ,,the ,,indicator ,,handbook4 ,comprising 8,the ,indicator3 ,its ,construction and ,application0 and 8,the ,indicator ,diagram3 ,its ,analysis and ,calculation40 ,complete in ,one ,volume4 #gs4 #aj;d4 ,post ,free4 8,mr4 ,pickworth's judgment is always sound1 and is evidently derived from a personal acquaintance with indicator work40--,the ,engineer4 ,,power ,,computer ,,for ,,steam1 ,,gas ,,and ,,oil ,,engines1 ,etc4 8,accurate1 expeditious and thoroughly practical4444 ,we can confidently recommend it1 and engineers will find it a great boon in undertaking tests1 etc40--,the ,electrician4 #gs4 #f;d4 ,post ,free4 ---------------------------------------- -------------------------------- ,,advertisements4 ,,logarithms ,,for ,,beginners ,for a full and intelligent appreciation of the ,slide ,rule and its various applications an elementary knowledge of logarithms is necessary4 ,all that is required will be found in this little work1 which gives a simple1 detailed and practical explanation of logarithms and their uses1 particular care having been taken to elucidate all difficult points by the aid of a number of worked examples4 ,seventh ,edition1 #as4 #h;d4 ,post ,free4 ,,power ,,computer for ,,steam1 ,,gas1 ,,and ,,oil ,,engines1 ,etc4 ,gives ,the ,indicated ,horse-,power of ,steam1 ,gas1 and ,oil ,engines--,the ,brake ,horse-,power of ,steam1 ,gas1 and ,oil ,engines--,the ,size of ,engine ,necessary to ,develop any ,given ,power--,the ,mechanical ,efficiency of an ,engine--,the ,ratio of ,compound ,engine ,cylinders--,the ,piston ,speed of an ,engine--,the ,delivery of ,pumps with any ,efficiency--,the ,horse-,power of ,belting--,the ,rim ,speeds of ,wheels1 ,speeds of ,ropes1 ,belts1 etc4--,speed ,ratios of ,pulleys1 ,gearing1 etc4 ,pocket size1 in neat case1 with instructions and examples4 ,post ,free1 #gs4 #f;d4 net4 ,c4 ,n4 ,,pickworth1 ,withington1 ,manchester ,w4 ,p4 ,,thompson1 ,g4 ,c4 ,,dymond1 ,f4,c4,s41 ,m4,i4,mech4,e41 ,f4,i4,c4,p4,a4 ,m4,i4,mech4,e41 ,f4,i4,c4,p4,a4 ,w4 ,p4 ,thompson ,co41 #ab ,,church ,,street1 ,,liverpool1 ,,chartered ,,patent ,,agents4 ,h4 ,e4 ,,potts1 ,j4 ,v4 ,,armstrong1 ,m4,sc41 ,hon4 ,chem41 ,f4,i4,c4,p4,a4 ,m4,text4,i41 ,f4,i4,c4,p4,a4 ,w4 ,h4 ,,beeston1 ,r4,p4,a4 ---------------------------------------- -------------------------------- ,,british ,,slide ,,rules for all ,,arts and ,,industries including ,illustration3 ,slide ,rule ,,log-,o-,,log ,,dr4 ,,yokota',s ,,surveyors' ,,wireless ,,gunnery ,,electrical ,,rules1 ,etc4 ,,send ,,for ,,list #ee ,,made ,,by-- ,,john ,,davis ,,son 7,derby71 ,ltd4 ,,all ,,saints' ,,works1 ,,derby ---------------------------------------- -------------------------------- ,k ,e ,slide ,rules are constantly growing in popularity1 and they can now be obtained from the leading houses in our line throughout the ,united ,kingdom4 ,illustration3 ,slide ,rule ,we manufacture a complete line of ,,engine-,,divided ,,slide ,,rules1 and call special attention to our ,patent ,adjustment1 ensuring smooth working of the ,slide2 also to our new 8,frameless0 ,indicator1 which hides no figures on the ,rule4 ,illustration3 ,thacher's ,calculating ,instrument ,,thacher',s ,,calculating ,,instrument1 for solving problems in multiplication1 division1 or combinations of the two2 has upwards of #cc1jjj divisions4 ,results can be obtained to the fourth and usually to the fifth place of figures with a surprising degree of accuracy4 ,we also make ,,all ,,metal1 ,,circular1 ,,stadia1 ,,chemists'1 ,,electrical1 and ,,other ,,special ,,slide ,,rules ,,descriptive ,,circulars ,,on ,,request ,,keuffel ,,esser ,,co4 #abg ,fulton ,st41 ,,new ,,york ,general ,office and ,factories1 ,,hoboken1 ,n4,j4 ,,chicago ,,st4 ,,louis ,,san ,,francisco ,,montreal ,,drawing ,,materials ,,mathematical and ,,surveying ,,instruments ,,measuring ,,tapes ---------------------------------------- -------------------------------- ,illustration3 #f in4 ,standard with magnifying ,cursor complete in pocket case1 #e_/- ,,norton ,,gregory ,,ltd4 ,head ,office ,,castle ,,lane1 ,,westminster1 ,,london1 ,s4,w4 #a ,branches #ga ,,queen ,,street1 ,,glasgow4 ,,phoenix ,,house1 ,,queen ,,street and ,,sandhill1 ,,newcastle-,,on-,,tyne4 ,,slide ,,rules in ,stock1 from #ag_/f to #bg_/f ,special ,quotations to the ,trade for ,quantities ,for particulars of ,surveying1 ,measuring and ,mathematical ,instruments1 ,appliances and ,material of all kinds for the ,drawing ,office1 write to the ,head ,office4 ,illustration3 ,,norton ,,gregory ,,ltd ---------------------------------------- -------------------------------- ,,norton ,,gregory1 ,,ltd41 ,london4 8,,diamond0 ,,drawing ,,instruments ,manufactured at our ,london ,works4 ,,centre ,,screw ,,spring ,,bow ,,half ,,set4 ,illustration3 ,centre ,screw ,spring ,bow ,half ,set #d inch ,spring ,bow ,half ,set centre screw adjustment1 with interchangeable needle1 pen1 and pencil points ,price #ag_/f ,the ,centre ,screw ,spring ,bow ,half ,set of ,compasses1 as illustrated1 possesses the advantage of ,,combining ,,in ,,one ,,instrument ,,the ,,set ,,of ,,three ,,separate ,,spring ,,bows hitherto in use1 while the centre screw makes for ease and accuracy of manipulation1 at the same time providing a radius of over #b inches1 or double that of the old pattern4 ,this instrument is less expensive than the set of #c bows1 while considerably stronger in construction4 ,the fixed needle point is shouldered4 ,this illustration is given as an indication of the various ,drawing ,instruments manufactured by us4 ,illustrated ,booklet giving full particulars and prices of other ,instruments and ,cases of ,instruments sent on application4 ,specially arranged ,sets of ,instruments made for ,colleges1 ,schools1 ,technical ,institutes ,estimates submitted on ,application4 ,write to our ,head ,office3 ,,castle ,,lane1 ,,westminster1 ,,london1 ,s4,w4 #a4 ---------------------------------------- -------------------------------- ,,drawing ,,and ,,surveying ,,instruments ,a4 ,g4 ,,thornton ,ltd4 ,,slide ,,rules ,,for ,paragon ,works ,,accurate ,,sectional ,,engineers #b ,king ,st4 ,west ,,papers ,,and ,,cloths ,,manchester ,d #aiaf ,illustrated ,catalogue1 just published1 in two editions2 ,drawing ,office 7#ddh pages72 ,draughtsman's 7#afj pages73 the most complete ,catalogues in the trade4 ,,contractors ,,to ,h4,m4 ,,war ,,office ,,and ,,admiralty ,manufacturers also of ,drawing ,materials and ,drawing ,office ,stationary4 7,,also ,,at ,,minerva ,,works ,,and ,,albert ,,mills1 ,,manchester47 ,,mathematical ,,instruments ,,surveying ,,instruments ,,slide ,,rules ,for ,students and ,engineers ,,mannheim1 ,,polyphase1 ,,duplex1 ,,electrical1 ,,log-,,log1 ,,and ,,calculex ,j4 ,h4 ,,steward ,,ltd4 ,scientific ,instrument ,makers #djf ,,strand1 and #deg ,,west ,,strand ,,london1 ,w4,c4 #b ---------------------------------------- -------------------------------- ,a ,true ,friend and ,trusty ,guide ,,the #8,,halden ,,calculex' ,illustration3 ,halden ,calculex ,,actual ,,size ,,british ,,made ,the handiest and most perfect form of ,slide ,rule4 ,does all that can be done with a straight rule4 ,complete in ,case1 with book of instructions1 #bg_/f post free4 ,j4 ,,halden ,,co41 ,,ltd41 #h ,,albert ,,square ,,manchester ,depots--,london1 ,newcastle-on-,tyne1 ,birmingham1 ,glasgow1 and ,leeds ---------------------------------------- -------------------------------- ,illustration3 ,rope ,,engineering1 ,,surveying ,,and ,,mathematical ,,instruments1 ,,etc4 ,,slide ,,rules4 ,,joseph ,,casartelli ,,son1 #dc ,,market ,,street1 ,,manchester4 ,tel4 ,no4 #bieh ,city4 ,established #agij4 ,,rope ,,driving ,is the most ,,efficient and most ,,economical ,,method of ,power ,transmission4 ,the ,,lambeth ,cotton ,driving ,rope ,is the most ,,efficient and most ,,economical ,,rope for ,power ,transmission4 ,illustration3 ,made #d ,strand or #c ,strand4 ,,special ,,features3 ,,less ,,stretch ,,than ,,any ,,other ,,rope4 ,,more ,,pliable ,,than ,,any ,,other ,,rope4 ,,greater ,,driving ,,power ,,than ,,any ,,other ,,rope4 ,,thomas ,,hart ,,ltd41 ,lambeth ,works1 ,,blackburn4 ---------------------------------------- -------------------------------- ,,transcriber',s ,,notes ,page ,changed from ,changed to #bd the right1 so the number of the right1 so the number of digits in the answer #c #b #a digits in the answer #c #b #a #b #b #aaf grammes1 we have the equation1 grammes1 we have the equation1 x 7,cl47_/7,ag4,cl47 x 7,cl47_/7,ag4,cl47 7a7_/7s74 ,hence1 the mark 7a7_/7s74 ,hence1 the mark ,typos fixed2 non-standard spelling and dialect retained4 ,used numbers for footnotes1 placing them all at the end of the last chapter4 ,enclosed italics font in underscores4 ,enclosed bold font in equals4 ,the caret 77 serves as a superscript indicator1 applicable to individual characters 7like #b;d7 and even entire phrases 7like #ast74 ,subscripts are shown using an underscore 77 with curly braces 1 as in ,h#b,o4