C Zi

J^iacttcal Creattöe

ON THE

SLIDING RULE:

IN TWO PARTS.

Part the First being an Introduction to the Use of the Rule generally as adapted for Calculations that usually occur tonbsp;Persons in Trade.

Part the Second containing Formulae for the Use of Surveyors, Architects, Civil Engineers, and Scientific Gentlemen.

BY B. BEVAN,

CIVIL ENGINEER AND ARCHITECT.

LONDON:

PRINTED FOR THE AUTHOR;

AND SOLI) BY

r.ONGMAN, HURST, REES, ORME, AND BROWN, PATERNOSTER ROW; ALSO BY W. CARY, OPTICIAN, 182, STRAND.

1822.

The two late Acts of Parliament for regulating Weights and Measures, make an alteration necessary in allnbsp;measures of capacity, in the United Kingdom, onnbsp;the first of January 1826; these alterations in thenbsp;size of the measures, require a corresponding changenbsp;in the formal^ of the Sliding Rule.

The following list of errata, will enable any person to correct the formulas, and fit it for the newnbsp;measures, with the same simplicity and accuracy asnbsp;to the former measures.

PREFACE.

r HE Sliding Rule is an instrument of general utility, for all purposes of expeditious calculations; and it may be said,nbsp;that few Instruments require less time and application fornbsp;attaining a sufficient knowledge of their principles, to enable any person of common education to become able tonbsp;resolve all questions in common arithmetic with great easenbsp;and despatch: a few hours’ attention is sufficient to instructnbsp;a common schoolboy in the use of the Rule, for the usualnbsp;questions that occur in common business ; after which thenbsp;progress is perfectly easy, to that of the more refined calculations required by the professional gentleman and man ofnbsp;science.

I presume, it has hitherto been considered a matter of difficulty to acquire a knowledge of the ready use of thenbsp;Rule; and this consideration has prevented persons engaged innbsp;the common pursuits of life from using so valuable an instrument; and I trust I may, without vanity, claim a smallnbsp;portion of public approbation, for the simple and readynbsp;method I have adopted, in explaining the mode of applyingnbsp;the Sliding Rule to purposes of calculations.

Eighteen years’ practice in the use of this instrument, in very extensive public undertakings, and in teaching its usenbsp;to a number of young persons, has proved the superioritynbsp;of my method over any former. It will also be found, thatnbsp;in addition to the greater clearness and facility of expressingnbsp;and rendering plain the use of the Rule, may be mentionednbsp;the conciseness of the space necessary to express the formulae,nbsp;whereby a person may copy on a small piece of paper, for

A 2

4 nbsp;nbsp;nbsp;PREFACE.

his pocket-book, the rules and examples suited to his own business or pursuits, without the incumbrance of a specialnbsp;book for that purpose. It frequently happens that calculations are required in the field, or in the market, or fair,nbsp;when persons even of considerable practice are liable to greatnbsp;mistakes, from the shortness of time allowed to performnbsp;the work in; but by a person in a very small degree acquainted with the use of the Rule, these mistakes may benbsp;avoided and much time saved.

It is not necessary to enumerate all the particular professions and trades that would be assisted by the use of the Slide Rule, but there are few that would not occasionally be muchnbsp;assisted by it; amongst others, the following may be namednbsp;alphabetically, viz. Accountants, Appraisers, Architects,nbsp;Astronomers, Auctioneers, Barge and Boat Builders, Brewers, Bricklayers, Brokers, Builders, Cabinet Makers, Carpenters, Chemists, Coach Makers, Coopers, Copper-smiths,nbsp;Corn Dealers, Drapers, Druggists, Dyers, Engineers,nbsp;Glaziers, Grocers, Ironmongers, Land Surveyors, Linennbsp;Drapers, Mechanists, Merchants, Military Officers, Millwrights, Sawyers, Schoolmasters, Ship Builders, Statuaries, Stonemasons, Surveyors, Timber Dealers, Watchmakers, Wheelwrights, and all persons having calculationsnbsp;to make either in money, weights, or measures. Personsnbsp;having logarithmic calculations to make will find the aidnbsp;of the Sliding Rule highly useful in proportioning the differences, or giving the inferior parts, either in sexagesimalsnbsp;or decimals. A few examples will be found in the followingnbsp;pages, to render more obvious the advantages of an acquaintance with the Rule. The object of the present publication is to comprise as much useful matter, in a smallnbsp;compass, as will be consistent with perspicuity; for thisnbsp;reason, a more extended preface would be a departure fromnbsp;the intention of the Author.

INTRODUCTION.

1 HE first requisite, in the practical use of the Rule, is to learn the notation, or the signification of the divisions andnbsp;figures on the several lines of the Rule. For this purposenbsp;it will be necessary for the person to have before him a Rule,nbsp;with its several divisions, to inspect and refer to. As it isnbsp;known that there are some varieties in the form of slidingnbsp;Rules now sold, although in general they are constructed onnbsp;similar principles, and will in most cases answer the samenbsp;purpose, yet it may give less trouble to the learner if Inbsp;adopt the common twelve inch Rule, having four lines of divisions upon its face, and which, for the purpose of reference, are generally marked with the letters A. B. C. D. atnbsp;one end ; the lines A and D being generally upon the stocknbsp;or Rule, and the lines B and C upon the slide. It will benbsp;seen that the lines marked A, B, and C, are similar, andnbsp;have their divisions and figures alike, consisting of twonbsp;series, viz.

1 nbsp;nbsp;nbsp;2nbsp;nbsp;nbsp;nbsp;3nbsp;nbsp;nbsp;nbsp;45 6 78910nbsp;nbsp;nbsp;nbsp;2nbsp;nbsp;nbsp;nbsp;3 4S 6' 78 9nbsp;nbsp;nbsp;nbsp;10

and that the line marked D has but one series or set of figures, viz.

1 nbsp;nbsp;nbsp;2nbsp;nbsp;nbsp;nbsp;34567’99nbsp;nbsp;nbsp;nbsp;10

the divisions or spaces on which are exactly double in extent of those on the preceding lines, and as they occupy the samenbsp;extent, are but half the number. This line, marked D, isnbsp;often called the square line, because it is used in calculationsnbsp;involving the square of one of the dimensions; by others, it isnbsp;called the girt line; but it will be niorc proper to call it the

o nbsp;nbsp;nbsp;INTRODUCTION.

line of single radius; the Jines marked A, B, and C, may then be called lines of double radii.

It may be right to observe, that the figure 1, at the beginning on the left hand end of the scale, may be considered as 1, or unit, and in this case the other figures will follownbsp;as they are stamped upon the scales; the divisions betweennbsp;the figures will be tenths, or decimals.

When any question requires more than two figures to express it, for instance, if a greater number than 100 is tonbsp;be used, the first figure on the Rule may be considered asnbsp;100 or 1000, amp;c. keeping in mind this simple rule, that thenbsp;other figures and divisions on the same line are to be proportional.

Thus, if the first figure be called 10, the second will represent SO, the third stand for 30, amp;c. and the divisionsnbsp;will be counted as units.

On the lines A, B, and C, which have doiihle radii, the second series of numbers and divisions will of course represent ten-fold the first series: thus, in the last example, wherenbsp;the figures on the first radius are made to stand for 10, SO,nbsp;.SO, amp;c. the figures on the second radius will stand for 100,nbsp;SOO, 300, amp;c. After this manner may figures large enoughnbsp;for any practical sum be considered on the Rule; and in tliisnbsp;case the second series must of necessity be considered as 100,nbsp;SOO, 300, and not have any arbitrary meaning attached tonbsp;them: thus, it would not be proper to call the first seriesnbsp;100, SOO, 300, amp;c. and in the same operation call thenbsp;second series 10,000, S0,000, 80,000, because you willnbsp;thereby disturb the due proportion between the said figures;nbsp;it being essential that the figures on the same line shouldnbsp;maintain a regular and uniform progression.

This uniformity of progression of the series of figures applies only to the same line, and not to different lines onnbsp;the same Rule; for example, if on the line marked A we call

INTRODUCTION. nbsp;nbsp;nbsp;/

the series 100, 200, 300, amp;c. it does not follow that the line marked B, although used in the same calculations,nbsp;should be of the same proportionate value with those on A,nbsp;PROVIDED all the figures on the line B are but esteemed in thenbsp;same uniform progression among themselves: thus, althoughnbsp;the figures on the line A may be called 10, 20, 30, amp;c.nbsp;those on the line B may be considered 1000, 2000, 3000,nbsp;amp;c. if the question should require such high numbers, whichnbsp;is seldom the case.

In regard to the numbers on the line C and D, when used together, the person using them will soon discover, thatnbsp;a certain proportion must be maintained in their application.

The above will be sufficient to explain generally the notation, or reading of the figures. The next thing to become acquainted with will be the divisions on the lines, which will be explained in one general way, as applicable tonbsp;all the lines upon the Rule, and is as follows:

Assuming that the figures are made to stand for 10, 20, 30, amp;c. it will follow, from the least reflection, that the primenbsp;subdivisions between the said numbers must stand for units;nbsp;and if the figures on the lines represent 100, 200, 300, amp;c.nbsp;the prime divisions must of course signify tens, always havingnbsp;a decimal proportion to those of the long or figured divisions.

A few examples will serve to make the above precepts understood.

Taking, in the first place, the lines lettered A and B, draw out the slide B to the right hand, until the figure 1 isnbsp;immediately under figure 3 on the line lettered A; the following proportion will be exhibited among the numbered, ornbsp;Jigured divisions:

A I nbsp;nbsp;nbsp;3_nbsp;nbsp;nbsp;nbsp;6nbsp;nbsp;nbsp;nbsp;9nbsp;nbsp;nbsp;nbsp;30nbsp;nbsp;nbsp;nbsp;60nbsp;nbsp;nbsp;nbsp;amp;c.

B I nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;2nbsp;nbsp;nbsp;nbsp;3nbsp;nbsp;nbsp;nbsp;10nbsp;nbsp;nbsp;nbsp;20

It will be observed, that there are several figured divisions on the two lines not coinciding with figured divisions on the alternate lines, as in the following expressions, which

INTRODUCTION.

will serve as examples to explain the signification of the primes and subdivisions ; thus,

B 1 nbsp;nbsp;nbsp;14nbsp;nbsp;nbsp;nbsp;5nbsp;nbsp;nbsp;nbsp;6nbsp;nbsp;nbsp;nbsp;7nbsp;nbsp;nbsp;nbsp;8 amp;c.

Those marked with the letter p will be found among the prime divisions. Also, by the same set of the slide, will benbsp;read,

A I nbsp;nbsp;nbsp;3nbsp;nbsp;nbsp;nbsp;4nbsp;nbsp;nbsp;nbsp;5nbsp;nbsp;nbsp;nbsp;7nbsp;nbsp;nbsp;nbsp;8nbsp;nbsp;nbsp;nbsp;amp;c.

B I nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;1.33nbsp;nbsp;nbsp;nbsp;1.66nbsp;nbsp;nbsp;nbsp;2.33nbsp;nbsp;nbsp;nbsp;2.66nbsp;nbsp;nbsp;nbsp;amp;c

Taking the same set of the slide, but assuming a different proportion between the first pair of divisions on the two lines,nbsp;they may be read as follows:

A few exercises of this kind will enable any person to become familiar with the use of the Rule, and with my method of description, as distinguished from that in all the publishednbsp;books on this subject. I consider the simple figurative representation of the different lines, but little incumbered withnbsp;literal explanation, as the characteristic merit of the presentnbsp;treatise; not only as the speedy means of teaching the use ofnbsp;the Rule, but also in the comparative suitability of expressionnbsp;for all practicable formulae, and for its universal applicationnbsp;to the business of common life. The simplicity and universality of the mode renders it almost essential to the man ofnbsp;business, and bears the same proportion to the practice ofnbsp;calculations by former methods, as that of algebraic formulaenbsp;does to the literal rules given in vulgar arithmetic.

As the whole requisite instruction for the complete use of the Rule depends upon one simple principle, it probably willnbsp;not be thought a waste of time in any person to bestow anbsp;few hours in learning this simple point, upon which thenbsp;whole is rendered plain. I will therefore take another example, and present it under a few modifications.

B I nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;2nbsp;nbsp;nbsp;nbsp;3nbsp;nbsp;nbsp;nbsp;4nbsp;nbsp;nbsp;nbsp;5 amp;c.

In this example the 12 and 16 on the line are not figured divisions, but among the prime subdivisions.

The same set of the slide being preserved, the reading of the figures and divisions may be,

A I nbsp;nbsp;nbsp;40nbsp;nbsp;nbsp;nbsp;60nbsp;nbsp;nbsp;nbsp;80__100 nbsp;nbsp;nbsp;120**nbsp;nbsp;nbsp;nbsp;14o'’ amp;c.

B I nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;1§nbsp;nbsp;nbsp;nbsp;2 2é 3

Still keeping the same set, the reading may be, A I 4nbsp;nbsp;nbsp;nbsp;5nbsp;nbsp;nbsp;nbsp;6nbsp;nbsp;nbsp;nbsp;7nbsp;nbsp;nbsp;nbsp;8nbsp;nbsp;nbsp;nbsp;10

B 1 nbsp;nbsp;nbsp;100 125 150nbsp;nbsp;nbsp;nbsp;175nbsp;nbsp;nbsp;nbsp;200nbsp;nbsp;nbsp;nbsp;250nbsp;nbsp;nbsp;nbsp;600 amp;c.

In all the above examples, it will be found that the relative value of the figures and divisions on each separatenbsp;line is the object to be attended to.

An example or two on the lines C, and D, may further illustrate the case.

When the first division on the line C coincides with the division numbered 1 on the line D, the figures may be readnbsp;as follows:

Cl nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;4nbsp;nbsp;nbsp;nbsp;9nbsp;nbsp;nbsp;nbsp;16nbsp;nbsp;nbsp;nbsp;25 36nbsp;nbsp;nbsp;nbsp;49 amp;c.

4 nbsp;nbsp;nbsp;5nbsp;nbsp;nbsp;nbsp;6nbsp;nbsp;nbsp;nbsp;7 amp;c.

On the common two feet carpenter’s Rule, the first figured division on the line D is marked 4, the others increasing upnbsp;to 40; this has been done to render calculations of the contentnbsp;of timber less difficult, to persons generally using the instrument for that purpose. The better sort of calculating Rulesnbsp;have the first figured division on the line D numbered 1;nbsp;but supposing the lines placed as above directed, the figuresnbsp;on the line C, will be found the common square numbers to

B

INTRODUCTION.

the figures on the line D, and ivill hereafter be expressed in the general formulae; thus,

C nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;power

D nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;square root

signifying, when 1 on C is upon 1 on D, that any power on C will have its square root on D.

The same thing will apply to the said lines, if the first divisions have a difiêrent value put upon them, provided thenbsp;number on C is equal to the square of the coinciding numbernbsp;on D; thus,

C I 100 nbsp;nbsp;nbsp;144nbsp;nbsp;nbsp;nbsp;3M 400 amp;c.

D I nbsp;nbsp;nbsp;10nbsp;nbsp;nbsp;nbsp;12nbsp;nbsp;nbsp;nbsp;15nbsp;nbsp;nbsp;nbsp;18nbsp;nbsp;nbsp;nbsp;20 amp;c.

The same, on the other hand, may apply decimally; thus,

C } nbsp;nbsp;nbsp;.01nbsp;nbsp;nbsp;nbsp;.04nbsp;nbsp;nbsp;nbsp;,09nbsp;nbsp;nbsp;nbsp;.16nbsp;nbsp;nbsp;nbsp;.49 amp;c.

.7 amp;c.

To persons acquainted with logarithms, it will not be necessary for me to offer any explanation, further than tonbsp;state, the divisions upon the Rule are simply the logarithmicnbsp;numbers, laid down from a scale of equal parts, as will benbsp;further illustrated in the second part of this Treatise.

GENERAL FORMULAE,

WITH

EXAMPLES.

MULTIPLICATION.

Bl nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;multiplier

and is tlius explained.

In this formula the line represents the divisions between tlie, double radius marked A upon the stock, and the double radius marked B on the upper edge of the slide,nbsp;and is to be read as follows; Unit, upon B, under the multiplicand upon A ; then over the multiplier upon B, will benbsp;found the product upon A.

Example I. Let it be required to multiply 17 by 5, referring to the general form for multiplication above, and substituting the figures for the particular case, it will standnbsp;thus:

85 answer

B j nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;5

or, 1 upon B under 17 upon A, over 5 upon B, is the product 85 upon A.

Example II, Multiply SJ by 75-A I nbsp;nbsp;nbsp;3.5nbsp;nbsp;nbsp;nbsp;26J answer

g_l_ ^ -

Place unit on B under tbe fifth division between 3 and 4 upon A, then over the fifth division between 7 and 8 uponnbsp;B, will be tlie answer 26| upon the line A; or according tonbsp;the decimal divisions of the Rule 26.25-

B 2

dividend.

divisor.

and it is to be understood as follows: under the dividend upon the line A, place the divisor, upon the line B; thennbsp;over unit, on the line B, will be found the quotient, or answer upon the line A.

Example I. Let it be required to divide 135 by 5. A Inbsp;nbsp;nbsp;nbsp;27 answernbsp;nbsp;nbsp;nbsp;135

B 1 nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;5

that is, draw out the slide until 5 on the line B corresponds with the middle of the space between 130 and 140, uponnbsp;the line A; then over unit, upon the line B, will be thenbsp;answer 27.

Example II. Divide 84 by 3§. A Inbsp;nbsp;nbsp;nbsp;24 answer

B I nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;3i

that is, place the nuddle division between 3 and 4, on the line B, under the fourth division between 8 and 9, upon thenbsp;line A; then over unit, on the line B, will be the answer 24,nbsp;on the line A.

DIRECT PROPORTION, OR RUDE OP THREE. 13

DIRECT PROPORTION,

OR

RULE OF THREE.

Rule.

second term

B [ nbsp;nbsp;nbsp;first termnbsp;nbsp;nbsp;nbsp;third term.

Example 1. If IjIS. cost lOd. what will 32Tb. cost.'*

A I nbsp;nbsp;nbsp;10nbsp;nbsp;nbsp;nbsp;80d. or 65. 8d. answer.

Place 4 on B, under 10 upon A; then over 32, on B, will be 80, the answer on A.

Example II. If a quarter of wheat cost 72^. how much per sack of 5 bushels ?

45s. answer

B [ nbsp;nbsp;nbsp;5 bushelsnbsp;nbsp;nbsp;nbsp;8 bushels.

That is, place 8 on B, under 72 on A; then over 5 on B, will be the answer, 45s. upon A. It may be right to observe, in this place, that occasionally the proper number innbsp;the order of progression on the line B, may be so far drawnnbsp;out, as not to have any part of the line A in contact; in suchnbsp;cases, the value of the numbers, both on the line B, andnbsp;also on the line A, may be increased tenfold, whereby thenbsp;left hand numbers may be adopted as substitutes for thosenbsp;required on the right hand.

Example III. If 2 will give 9, what will 30 yield.?

A I nbsp;nbsp;nbsp;9nbsp;nbsp;nbsp;nbsp;135 answer

B“| nbsp;nbsp;nbsp;2nbsp;nbsp;nbsp;nbsp;^0

In this case, the proper 30 is on that part of the slide drawn out of, and beyond the stock, and the 3 considered as 30 innbsp;lieu.

To square any number, and to find the root of any number, these will be performed on the lines marked C and D, the first being a line of double radii on the lower edge of thenbsp;slide, and the line D a single radius on the stock; the general form is,

square

D I nbsp;nbsp;nbsp;1 root

that is, place unit upon the line C, over unit on the line D; then over any number on D will be its square on the line C;nbsp;and, under any number on the line C, will be its root on thenbsp;line D.

Example I. Find the square of 2, 5, 7, and 8|.

Cl nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;4nbsp;nbsp;nbsp;nbsp;25nbsp;nbsp;nbsp;nbsp;49nbsp;nbsp;nbsp;nbsp;72| answer square.

8| roots.

As the common carpenter’s Rule begins the D line with the figure 4, and finishes at 40; it wil, therefore, be necessary to be particular in placing the units together, onnbsp;the lines C and D.

•r nbsp;nbsp;nbsp;¦

17

MEAT.

In the last example, there are several answers, serving to show the general use of the Rule, which needs but little farther explanation; but as it will save room, and lessen thenbsp;size of the book, this mode will be occasionally adopted. Thenbsp;meaning of the example is, that when 8 on B is placed undernbsp;12 on A, all the figures and divisions expressing the pricenbsp;per lb. on the line A, will have under them the shillings per

c

REDUCTION.

stone on the line B; which, in the example above, will be read,

9d. per ft. equals 6s. per nbsp;nbsp;nbsp;stone.

8.....5s. nbsp;nbsp;nbsp;éd.

7.....4s. nbsp;nbsp;nbsp;8d.

6.....'4s.

'-S.....3s. nbsp;nbsp;nbsp;4d.nbsp;nbsp;nbsp;nbsp;amp;c.

If the stone be 14fts. instead of 8 as above, the formulas and example will be,

A I 14 shillings per stone Ss.lOd. 7s. 8s. 2d. 9s. 4d. 10s. 6d.

5 nbsp;nbsp;nbsp;6““7nbsp;nbsp;nbsp;nbsp;8nbsp;nbsp;nbsp;nbsp;9

In these last examples, it will be observed that the answers are set down in shillings and pence; but upon the Rule they are shown in shillings and decimals, all the subdivisionsnbsp;on the Rule being decimals;

10 of which stand for 1 shilling,

5......6 pence,

21.......3 pence, amp;c.

BEER.

WINE.

REDUCTION.

MENSURATION.

MENSURATION.

RECTANGULAR FIGURES.

Example. Let the base be 7 and perpendicular 6; qu. area.?

27

D S

GAUGING.

GAUGING.

SQUARE PRISMS.

31

40

Rule.

hewn to dockyards.

B I nbsp;nbsp;nbsp;5nbsp;nbsp;nbsp;nbsp;customary measure by i girt.

Example. In 350 feet of timber, measured by the customary method of quarter girt, how much will there be if callipered at dockyard measure.^

A I nbsp;nbsp;nbsp;6nbsp;nbsp;nbsp;nbsp;420 cubic feet answer.

B I nbsp;nbsp;nbsp;5nbsp;nbsp;nbsp;nbsp;350

Scantlings, or unequal-sided square timber, may be measured readily by the area of its .section! which is obtainednbsp;by multiplying its breadth by its depth.

Rule.

Example. Required thé cubic feet in a plank 18 feet long, and 11 inches wide, and 3 inches thick. Here thenbsp;area of the section is 3x11=33.

18

4.125

41 cubic feet, answer.

Square timber is measured by the first of these formulae, substituting the .side of the square for the quarter girt.

What is called calliper measure, is the custom of considering timber partially hewn as die square, and measuring it as if it was perfectly square.

The side callipered, in timber sold to public dockyards, is said to be in proportion to the diameter as 19- to 22.

Rule.

cu. feet dockyard measure.

inches in diameter.

SIMPLE INTEREST.

Example. Required the interest of 73/. for 25 days. A I 365nbsp;nbsp;nbsp;nbsp;25

73/.

It will be seen that the whole of the formulae and examples in the present Part apply to all common Sliding Rules, and may be used with the common carpenter’s Rulenbsp;as well as with the more accurate and improved calculatingnbsp;Rule. It should be remembered, that these common Rulesnbsp;being sold at a low price, the divisions are not made with thenbsp;care and accuracy bestowed upon the better kind of Rules.

EKD OF THE FIRST PART.

PART II.

INTRODUCTION.

In the following Part, I shall explain the divisions and use of the improved calculating Rule, and give a number ofnbsp;useful formulae, to assist the more scientific calculator. Inbsp;shall not occupy the pages of this Treatise in describing thenbsp;construction of all the various Rules in use, because they arenbsp;mostly made upon the same principle; and when a personnbsp;understands the use of one Rule, it will be easy for him properly to apply the peculiar lines upon any other Rule. Fornbsp;the same reason I shall not particularly describe the commonnbsp;as used by navigators; or the double slidingnbsp;Rule, used by officers in the Excise; for the rules and examples in the present work will in most instances be equallynbsp;applicable to the Gunter’s Scale, or Excise Rule.

As several of the formulEe in this Part require the use of four lines in the same operation, it will be necessarynbsp;to adopt an instrument which has the divisions all commencing at the left hand end of die Rule. Having aboutnbsp;eight years since calculated all the divisions for a Rule tonbsp;three places of decimals, which were laid down by one ofnbsp;Ramsden’s dividing machines; and having used the Rulesnbsp;made from these calculations for many years, and foundnbsp;them sufficiently accurate for all common purposes, I shallnbsp;in the present work adopt this Rule as now made by Mr.nbsp;Cary.

Having already explained the lines marked A, B, C, D, which are common to most Rules, it remains only for me tonbsp;notice the additional lines laid down upon the said Rule, thenbsp;principal of which arc; Two on the extra slide, marked E,

H

INTRODUCTION TO PART II.

being lines of tripU radii, and adapted to calculations in which the cube of one of the dimensions, enters into the data ;nbsp;the several divisions on these lines, being of the same kind asnbsp;those on the line marked A, B, C, and D, but one third innbsp;point of space to those on the line D, and are used for finding the cubes of numbers, when applied to the line D; thüs,

E I nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;cubes.

roots.

signifying, that when unit on E is placed over unit on D, any number on D will have its cube on E; and the contrary, any number on E, will have its corresponding cubenbsp;root on D.

There is a line marked F, at the bottom of the groove usually occupied by the slide E. Another line, G, is laidnbsp;down on the stock under the said slide E. These lines,nbsp;F and G, are used in calculations of Annuities and Compound Interest; which will be more particularly explmnednbsp;under these respective titles.

There are various other uses to which these Rules are applicable; but I trust the reader will consider the numerousnbsp;formulae and examples, already given, as sufficient for a practical guide. The algebraic formulae will be interesting to thenbsp;mathematician, and point out to the abstruse calculator, thenbsp;means of forming rules for more special and difficult calculations.

Persons who are in the constant habit of pen calculations, will find the knowledge of the Rule highly useful as a checlc to their results, more expeditious and certain thannbsp;any repetition of the process by the pen.

The formulae for Annuities are given for 5 percent, only; but it is obvious that divisions may readily be calculated fornbsp;any other rate per cent., and laid down upon the Rule.nbsp;The rules for finding the comparative strength of Building Scantlings, will be found of great practical use to the

INTRODUCTION TO PART II.

carpenter, and may be modified to any particular circumstance with little trouble.

The divisors for weights and measures are, to a small extent, often put on one side of the Rule; a few of suchnbsp;divisors are placed on the back of the slide marked E, in thenbsp;Rule before mentioned; but as these divisors are very useful,nbsp;and would occupy too much room on a Rule, I shall give anbsp;more extended table at the end of the book, whence workmen in particular trades may have those divisors inserted onnbsp;the Rule which are most frequently called into use.

A table of gauge points also will be added for obtaining the same result, on the line C and D, by which persons innbsp;possession of the common carpenter’s Rule will be able tonbsp;solve all questions of this nature.

The succeeding chapters will be useful to the engineer and mechanic. The Rules referred to in the present Partnbsp;being usually made twelve inches in length, answer most ofnbsp;the purposes of measuring instruments, and may, if desired,nbsp;have the lines of varieties and areas of segments put on themnbsp;for Excise officers; or divisions to answer the purpose of pocketnbsp;balances for small weights, for persons who prefer such linesnbsp;to any of those at present on the Rule.

B. SEVAN.

im Nm. 1821.

ALGEBRAIC FORMüLjE.

By inverting the slide, the following formula will produce the same result.

D 1 nbsp;nbsp;nbsp;bnbsp;nbsp;nbsp;nbsp;d

3 _

therefore, when a=b and d=l a= y'C, thus,

B I_cu. root_power

cu. root

the meaning of which is this: place unit on D, under any power on B, and where the same number coincides on B andnbsp;D will be the cube root sought.

In operations of this nature, it will be observed, that three sets of numbers will be found to meet, one of whichnbsp;will be the proper root, and the other the root of 10 timesnbsp;the number sought, and the third of 100 times, amp;c.

Example. Find the cube root of 8. root

B I 8 nbsp;nbsp;nbsp;2nbsp;nbsp;nbsp;nbsp;4.3nbsp;nbsp;nbsp;nbsp;9.3

The improved calculating Rules have a line of treble radii, marked E, by the help of which, questions involvingnbsp;the cube of quantities are readily solved; some examples ofnbsp;which will be given in this book.

ARCHITECTURAL. ORDERS.

ARCHITECTURAL ORDERS.

These may be drawn to any proportion by the aid of the Slide Rule, by placing the proposed diameter, or height,nbsp;on the line A to the standard number on the line B; afternbsp;which, all the respective required numbers will be found onnbsp;the line A coinciding with the standard numbers on thenbsp;line B.

Exampk. Find, in inches and decimals, the principal dimensions of a Tuscan column, to a diameter of 12 inches.

answer.

My object being that of comprising as much useful information in a small compass as will be consistent with perspicuity, it will not be necessary to give examples to all the orders. The practical architect will be able to supply thenbsp;formulae for other orders.

ASTRONOMICAL CALCULATIONS.

ASTRONOMICAL CALCULATIONS.

In taking out logarithms, sines, and tangents, to the fraction of a minute, by those who are not possessed of Taylor’s Tables, much time is consumed in calculating the proportional parts. The Sliding Rule considerably shortensnbsp;this labour.

Example. Let it be required to find the logarithmic sine of 24° 41' 7quot; by Gardner’s Tables.

9.6207955 logarithm required.

9.6207955 Taylor’s Tables.

But it frequently happens, that fractions of seconds are necessary to the calculation; in which case the Rule isnbsp;equally serviceable and expeditious.

Let it be required to find the sun’s right ascension, declination, and the equation of time, for 4^ h. P. M. on the 13th November 1821, at Greenwich.

By the Nautical Almanack.

.^R

h. m. quot;

15 13 30.7 15 17 36.3

4 5.6 245.6

BUILDING SCANTLINGS.

BUILDING SCANTLINGS.

The strength of common building scantlings is made to vary according to the intended durability and special purpose of the building, and will generally be found within thenbsp;following limits.

A comparative formula for the strength of any scantling. Anbsp;nbsp;nbsp;nbsp;strength.

inches thickness, feet length.

inches depth.

Example I. Find the strength of a beam which is 24 feet long, 18 inches thick, and 20 inches deep.

A nbsp;nbsp;nbsp;30 answer, or a little above the average.

18

20

Same Example on carpenter’s Rule.

• A nbsp;nbsp;nbsp;187, answer.

18

20

24

BUILDING SCANTLINGS.

Example II. In a common joist of 15 feet bearing, and 3 inches thick, what ought to be the depth to give a strengthnbsp;of 10, and also of 20, on improved Rule?

10

15

7.1 inches, answer.

20

15 10 inches, answer.

Example III. To find the proper thickness of ceiling joist, the strength to be 7, having the length 8 feet, andnbsp;depth 4 inches, on improved Rule.

A| nbsp;nbsp;nbsp;7

3^ inches, answer.

Example IV. To find the width of a summer having a bearing of 18 feet, and being 16 inches depth, the strengthnbsp;to be 150, on improved Rule.

__

10| inches, the answer.

18 16

The last example on the common Rule will give the answer to strength 937.

DIVISORS FOR WEIGHTS AND MEASURES.

DIVISORS FOR WEIGHTS AND MEASURES.

On some Rules, there are divisors placed for facilitating calculations of certain measures , of capacity, and the weightsnbsp;of metallic and other substances, which I shall here explain.nbsp;They are usually divided into seven columns, and marked asnbsp;below.

The first column is under the word Square, marked FFF, signifying that in the mensuration of square or rectangularnbsp;bodies, all the dimensions are taken in feet.

Column 2, marked FII, signifies that the length only is to be taken in feet, and the width and thickness in inches.

Column 3, marked III, signifies that all the dimensions are to taken in inches.

Columns 4 and 5 apply in the same manner to cylinders, and columns 6 and 7 to globes: these last having but onenbsp;diameter, shows that when the divisors in column 6 are used,nbsp;the diameter is to be taken in feet, and when the diameter isnbsp;taken in inches the divisors in column 7 are to be used. Anbsp;few examples will show the convenience of these divisors.

DIVISORS FOR ^^EIGMTS AND MEASURES.

Example. A rectangular cistern, 6 feet long, 20 inches high, and 16 inches wide; query the ale gallons it will hold.?

First obtain the product of any two of the dimensions; for instance, 20x6=120, the divisors for ale-gallons innbsp;column 2 will be found 265.

If the object to be measured is square, the content or weight may be found by a single operation, by inverting thenbsp;slide; thus,

_ content, or weight.

side of the square.

Example. Let a piece of lead measure 6 inches long, 4 inches square ; query its weight in tbs.

By looking to the Rule, the divisor for lead will be found 243.

nbsp;nbsp;nbsp;391 lbs. answer.

2.43

For divisors in

inches diameter.

HYDRAULICS.

HYDRAULICS.

The quantity of water flowing through apertures depends upon the shape of the channel and of the aperture, and generally reduces the quantity, as determined from the theorynbsp;of falling bodies. The practical engineer will be able to applynbsp;the proper allowances, according to the circumstances of thenbsp;case. The following formulae will enable a person to makenbsp;the calculations in an expeditious manner; he will only havenbsp;to substitute the practical factor suited to the case, in thenbsp;place of the theoretic number here given.

Rulefyr apertures at the surface.

I 12 nbsp;nbsp;nbsp;inches depth to bottom of aperture.

5.35 nbsp;nbsp;nbsp;feet per second mean velocity.

4 nbsp;nbsp;nbsp;inches depth to bottom of aperture.

37 nbsp;nbsp;nbsp;inches per second mean velocity.

12 nbsp;nbsp;nbsp;inches depth of aperture.

32Ï nbsp;nbsp;nbsp;cubic feet per minute at l^lnchcs wide.

73

if the head is constant, or double that time if the head gradually diminishes to nothing.

HYDROSTATICS.

HYDROSTATICS.

INACCESSIBLE ÖISTANCES.

INACCESSIBLE DISTANCES.

It frequently happens, that calculations are required to determine inaccessible distances, from angles observed to annbsp;object of known subtense, and tables are given in some publications for that purpose: the following formulae will provenbsp;equal to a volume of such tables, when the angle does notnbsp;exceed 5 degrees.

LEVELLING.

LEVELLING.

As the line of sight, in tlie practice of levelling, is a tangent to the surface of the earth, a correction is necessary when the distance of the staff from the instrument is considerable. There is also a small correction necessary for refraction of the atmosphere.

The following formulae combine both these corrections.

The difference of level, as above found, to be subtracted from apparent level, to obtain the true level.

PERSPECTIVE.

PERSPECTIVE.

By ordinates at right angles to the principal ray, and distances let fall on the same.

To find the vanishing point to any vertical plane, let the angle formed by the plane with the principal ray = a. thenbsp;distance of eye from picture = A. the distance of the vanishing point from the principal ray will be, Tang, a x A.

For a plane making an angle with the horizon of b.

Tang. b.A _ v j. vertical above or helo cos. anbsp;nbsp;nbsp;nbsp;”

vanishing point of the ground line of the said plane.

face, bringing the line of sines, marked S, in contact with the line A.

A I nbsp;nbsp;nbsp;opposite side_its opposite side.

S I nbsp;nbsp;nbsp;sine of an anglenbsp;nbsp;nbsp;nbsp;sine of another angle.

Example I. Let the base of an oblique triangle be 40, and vertical angle 711°, the other side 32 and 36 respectively; required the angle at the base.

A I nbsp;nbsp;nbsp;40nbsp;nbsp;nbsp;nbsp;36nbsp;nbsp;nbsp;nbsp;32

58|°

TRIGONOMETRY.

Example II. Given the angles 32° and 110, and included side 60 ; to find the other side.

SPHERICAL TRIGONOMETRY.

Notwithstanding the improved Pocket Sliding Rule has but one set of lines of sines and tangents, it may, with the aidnbsp;of a small slip of paper, or pair of compasses, be used innbsp;solving all the common cases of sphOTcal trigonometry andnbsp;navigation.

For instance: Sine 8° : Sine 18° : : Sine 13 ; S. 30. The compasses or a slip of paper extending on a line of sines fromnbsp;8° to 18°, will also reach from 13 to 30. Again, Sine 6° :nbsp;Tang. 10 : : Sine 20 : Tang. 30; that is, the distance on thenbsp;line of sines between 6 and 20, will extend on the line ofnbsp;tangents from 10 to 30.

To find the latitude of the place from the sun’s declination and amplitude.

The extent on the line S. from the degrees of amplitude to 90°, will extend from the degrees of declination to the degrees of co-latitude.

WEIGHTS AND MEASURES.

FRENCH AND ENGLISH.

It will be seen that nearly the whole of the formulm and examples in the present volume apply to all common Slidingnbsp;Rules, and may be used with the common carpenter’s Rulenbsp;as well as with the more accurate and improved calculatingnbsp;Rule.

MISCELLANEOUS.

MISCELLANEOUS.

When many calculations are required, in cubeing up quantities from three dimensions, it will be convenient to havenbsp;an extra slide, to be placed inverted upon the line D, innbsp;such a manner that the divisor for the integer of the bulknbsp;coincides with the beginning of the divisions on the linesnbsp;A, B, and C, from which the cubical quantities will be foundnbsp;by a single operation.

MlSCELLi\NEOUS.

If the depth is taken in inches, and the length and width are taken in ƒeet, the divisor to be placed at the beginning ofnbsp;the Rule will then be 324 ; thus,

A. j nbsp;nbsp;nbsp;feet length.

B

Inv. C 324

If the depth and width are both taken in inches and length in yards, the divisor is 3888, on the inverted C line,nbsp;to be placed at the beginning of the Rule; thus,

A I nbsp;nbsp;nbsp;feet long.

cubic yards, inches wide.

If the quantity of malt in bushels were required, and the dimensions taken in yeet, as to length and width, and depthnbsp;in inches, the inverted line must be placed with the divisionnbsp;14.93 at the beginning; thus,

A I nbsp;nbsp;nbsp;feet long.

malt bushels, feet wide.

Iny. C 14.93

If cubic yeet are wanted, when the length is given in feet, and the width and thickness in inches, the divisor to benbsp;placed at the end of the Rule, on the inverted line will benbsp;144; thus,

A I feet length.

Inv. C 144 inches thickness.

MISCELI^VNKOUS.

Example. If a plank measure 20 feet long, lOi wide, and thick ; required the cubic feet.'

A I__20______

B • nbsp;nbsp;nbsp;cubic feet.

lOi

Inv. C 144

The general rule will be to place any of the divisors in columns I. TI. and III. from the table at the end of thisnbsp;book, on the line C inverted, upon the beginning of the Dnbsp;line, and using the other dimensions, at a single operation;nbsp;thus, for a general formula,

A I nbsp;nbsp;nbsp;anbsp;nbsp;nbsp;nbsp;length. '

The algebraic equation for this position of the lines will be,

Some persons may prefer a Rule capable of solving all the questions of double multiplication, and may not have occasion for trigonometrical calculations; in this case, the backnbsp;of the slide E, may have a line of double radii laid downnbsp;upon it, in lieu of the divisors at present, and the divisorsnbsp;may be placed on the back of the slide C, D. This arrangement will leave the improved Rule fully equal to all thenbsp;other purposes it is now adapted for; or the trigonometricalnbsp;lines may remain, as at present, and the additional line ofnbsp;double radii placed on the back of the slide E. A few divisors may then be placed on one of the edges of the Rule,nbsp;or at the bottom of the groove in lieu of the Annuity scales.

MISCELLANKOUS.

Some of the better kind of carpenters’ Rules have a double slide on the same face, the lower slide having thenbsp;inverted double radii ,• and for many purposes this modification of the Rule is preferable to the common one, which hasnbsp;the single radius, or D line, called by workmen the girt line,nbsp;it being adapted for the multiplication of three dimensions,nbsp;and dividing by a fourth. It becomes particularly fitted fornbsp;the cubical measure of all kinds of timber, either round,nbsp;square, or unequal-sided, and also for working up brickwork to the rod or cube yard.

A few examples will render the use of these Rules intelligible to the learner.

Suppose a piece of timber measures 33 feet long, and 14 inches by ; to find the cubic feet.

In this case the divisor is 144, which is placed at the beginning of the Rule, on the lower slide, as before described.

A I nbsp;nbsp;nbsp;33

B ?

Inv. C 144 nbsp;nbsp;nbsp;12i

As cubic feet will be the principal object of the carpenter and builder, the lower slide might be so divided as to be evennbsp;at the ends with the stock when the point of 144 ranges withnbsp;1 upon the line A.

An example in the measuring of brickwork may also be of service.

Let a wall be 58 feet long, 14 feet high, and 4 half bricks in thickness; required the rods contained.

A I nbsp;nbsp;nbsp;feet long 58

B i nbsp;nbsp;nbsp;rods reduced 4 answer, nearly.

C ; nbsp;nbsp;nbsp;feet highnbsp;nbsp;nbsp;nbsp;14

half bricks thick nbsp;nbsp;nbsp;4

Sec p. 37.

MISCELLANEOUS.

Again, let a wall be 90 feet long, 13 feet high, and 3 feet, or 8 half bricks in thickness; to find the number ofnbsp;rods.

MISCELLANEOUS,

DIVISION.

Perhaps one of the most general uses of the Rule to persons having many pen calculations to make, may be thatnbsp;of facilitating the operations of Long Division, serving tonbsp;point out without loss of time the successive figures of thenbsp;quotient by simple inspection. It is well known to the calculator, that he is frequently obliged to renew the process ofnbsp;multiplication when he arrives at the last figure, and to rubnbsp;out or erase the line, and repeat the work with a new figure :nbsp;a very slight acquaintance with the Rule will save all thisnbsp;trouble, and prevent much loss of time; nothing more isnbsp;necessary than to place any common Rule with the lines Anbsp;and B in the following form:

A [ nbsp;nbsp;nbsp;divisornbsp;nbsp;nbsp;nbsp;dividend.

B 1 nbsp;nbsp;nbsp;1nbsp;nbsp;nbsp;nbsp;quotient.

From which each separate part of the dividend will be fitted with its proper pirt of the quotient; and as the dmsor isnbsp;constant, the position of the slide will remain the same fornbsp;the whole operation, and towards the end of the process thenbsp;calculator will be furnished with the last two figures of thenbsp;quotient perfectly correct, without the trouble of actuallynbsp;performing the multiplication ; thus,

Divide 4965825 by 365.

365)4965825(13605

1315

2208

1825

The successive figures, as shown by the Rule, will be 13605.

MISCELLANEOUS.

A List of Gmige Points for Measures of Capacity, to he used on the Line D; the Dimensions all to be taken innbsp;Inches.

General Formula.

inches long nbsp;nbsp;nbsp;contents.

gauge point

The following Table of Divisors and Gauge Points for determining the weight in pounds avoirdupois is derived fromnbsp;the specific gravities of the respective substances; and asnbsp;there are small variations in the specific gravities, dependingnbsp;upon the quality of the substance, it will be easy to makenbsp;the correction to any of the numbers, as circumstances maynbsp;require, or to add other articles not comprised in the presentnbsp;Table.

For ascertaining die weight of substances by means of the Gauge Points in the following Table, the general formula is,

C I length or diameter_lbs, weight.

gauge point

TABLES OF DIVISORS AND GAUGE POINTS,

For ascertaining the Weight in Pounds Avoirdupois.

GAUGE POINTS.