Sm m:S ^EBm University of California • Berkeley The Theodore P. Hill Collection of Early American Mathematics Books ^ ' ■'■^^E ' ^^ift %B9i ^a^ ysbtM S^aS^ USIN ESS CALC ULATOR >AND( : ACCOUNTANTS ASSISTANT' A CYCLOPEDIA OF THB Most Concise AND Practical Metlioiis OF Business Calcinations. INCLUDINO MANY VALUABLE LABOS-SAVINQ TABLES, TOGETHER WITH Improved Interest Tables, DECIMAL SYSTEM : SHOWING THK INTEREST ON FROM $io to $10,000 — Rate, Ten per Cent, per Annum. BY HOY D. ORTON, and W. H. SADLER, President and Founder of Sadler"- " Bryant & Stratton" Business College, Baltimore, Md. Lightning Calculator, formerly teach- er of Rapid Calculations at the U. S. Naval Academy. Designed /or the practical use of the Banker, Merchant^ Accounta »■?, Mechanic^ Farmer^ Business Man and Student. Contaic^tng the shortest^ simplest ana most rapid methods 0/ Computing- Numbers, adapted to all kinds 0/ business and every-day life. Written and arranged so ?is to be within the comprehension of every one having the slightest knowledge of figures. PRICE ei.OO. Sent to any part of the World on receipt of same» BALTIMORE. MD,: W. H. Sadler, Publisher, Nos. 6 and 8 North Charles Street Entered according to Act of Congress, in the year ^877, by In the Office of the Librarian of Congrees, at Washington. The pagoB of reprint from "Orton's Liohtntno Calculator" are protected by copyrights of Hoy D. Orton, issued in 1866 and 1871. N. B.—AU rights reserved. Any infringement roill be prosecuted to the fullest txtent of the law. OENERAL AGENTS. p. 0. Address, Nos. 6 & 8 N. Charles St, Baltimore. JAMES G. MOULTON, WILLIAM CALLEN, Jr., JOHN G. SCOUTEN, WILLIAM DAVID. AGENTS A^TANTED.-For particu- lars and territory, apply in person to either of the above General Agents', or address the Publisher. ORDERS. Parties ordering the " Calculator " should be par- ticular to write plainly their Name, Residence, County and Suae. Upon receipt of One Dollar, a copy of the book will be forwarded, jx)st-paid, to any address. W. H. SADLER, PflMer, Nos. 6 & 8 N. Charles St., BALTIMORE, MD. ^-= The principal features embodied in this work are sim« plicity and brevity. There can be nothing new in principle, but so far as the authors' knowledge extends, their peculiar methods and abbreviations in the practical applications of the rules of Addition, Multiplication, Fractions, Percentage, Interest, Averaging Accounts, and Mensuration, have not hereto- fore been published, except such as are contained in -the work of the senior author, known as " Orton's Lightning Calculator." As an endorsement of Professor Orton's original work on the subject of rapid or lightning calcula- tions, it may be here stated that over 400,000 copies of that book have been sold. It is not the design of the authors of this work to make a text-book for the use of beginners in arithmetic, but to offer to those who have mastered the principles of addition, subtraction, multiplication and division, a guide to the practical application of that knowl- edge of arithmetic and calculations which is required daily in business and the affairs of life. There is no quali- fication more essential to success than facility in the rapid and accurate use of figures. In view of the increasing demand for the original work, which has been several times revised, the present authors have decided to enlarge upon the subject. They present in this new volume— the result of their joint labors— the most extensive and comprehensive work of the kind ever offered to the public, in the full assurance that who- ever will carefully study its pages will glean therefrom aa abundant reward. Baltimore. Jidy, 1877. 3 -d^!^ CONTENTS Prefacr 3 Frontispikce — Illustration ... 4 Introduction 9 Addition 12 *' Lightuing Method 13 —Table U " Illustration 15 An Easy Way to Add 17 to 26 Multiplication — lUuatrated 26 Short Methods 26" 31 ** Coutractions 31 " 33 ** " • — Curious and Useful 33 " T.^l)le of Squares 34 Fractions 35 " !M<'ntal Operations 36 " Wli.n tlu' Sinn of the Fractions is One 38 " 41 " M lien till' Fractions have a like Denominator. ... 41 *' lxai)iil Procf'ss of Multiplying Mixed Numbers 41" 48 " ^Vhea the Multiplier is an Aliqiiot Part of 100. . . 48 ♦♦ Tableof Aliquot Parts of 100 to 1,000 48 " Counting-Room Exorcises 49 " 53 " Illustration 53 " Division, with Analysis 53 '* 55 « ' by Boxing 55, 66 " Multiplication aud Division 56" 59 Percentagi;— As apjjlled to Business 69 " Illustration 60 ♦* Given Cost and Selling Price to find the Rate 62 " Given Profit and Rate to find the Cost 63 •* Given Amount and Rate to find the Cost .... 63 •' Given Proceeds, showing Loss and Rate, to find the Cost 64 Profit and Loss— Illustration 65 " Short Business Methods 66 *• Table of Aliquot Parts 66 IhTERESi— Showing Application of Percentage 67 " 72 5 6 CONTENTS. Discount— Commercial 73 True 74 Bank 76 Commission — Illustration 77 ' Siiowing Application of Percentage 77 to 80 Insurance — Showinu; Application of Percentage SO Investments — Illustration 81 " Capital and Stocks, showing Application of Percentage 81 " 85 " Table for Investors 86 Intfrf-st Discount and Average — Illustration 87 " Simplifietl by Cancellation 93 " Short Practical Rules 100 Banks and Banking— Illustration 104 Interest — Bankers' Method 105 '* 114 Lightning Method 114 " 117 Merchants' Method 117 " 124 Partial Pa vments — Notes, Bonds and Mortgages 124, 129 Equation of Pavmenis 129, 135 Averaging Accounts — Illustration 135 •' Lightning Method 135 " 139 Partnerships 13d ' Settlements by Three Diifferent Methods 140 " 144 Gold to Currency— Gold at a Premium 144 CuHRENCY TO Gold — " " " 144, 145 Maturing Notks, etc 145, 146 Sterling Exchange— Illustration 147 How Calculated 148 ' 151 " OldTable 151 " New Method and Tables 152, 154 Harking Goods— Illustration 154 " Asking Price and Discounts 155, 159 *♦ Rapid Process 159, 162 " Table for Marking all Goods Purchased by the Dozen 162 Basis of Success in Business 163 Ledger Accounts— Illustration, or the Science of Book-keep- ing Comprised in a Few Pages. . . .164 " 175 How TO Close the Ledger 175, 177 Balancing Bcok.s — Illustration 176 Errors in Trial Balances — Illustration 178 How to Detect Them 178 " l^'6 Lumber Measuring — Illustration 1^3 " Short, Practical Rules 183 ' li'ti Measuring Coed Wood— Illustration IbG Short, Practical Rules 186 " 189 RoundTimber— Measuring— Illustration ISW •' Short, Practical Rules 189 " 193 Flooring — ** ' " 193 Square Timber — Measuiing — Illustration 194 Short, Practical Rules 194 " 196 Cisterns and Reservoirs— Table < f Capacities l'.*6 *' Illustration 196 •* Iluw to Measure their Contents. 197 " 2U0 CONTENTS. 7 Cask Gauoino— Illustration 200 fSl'orr. I»ractical Rules £(»3 Mkasurino Geain— Jlliistratiou 203 " Si/o of Bins, IIow Ascertained 204 * Weights and Measures, U S. Standard . 204 "Wkiohts and Measures > Tablo of Avoirdupois Weij^hts, and BrsHEi.s TO Pounds j No. of Pounds to the Busliel 205 Ir*)N Weights — Used in Railroading — Table of Estimates.. . 206 Cons JN CuiBS — Measuring — Illustration 207 '• J'raetical Rules for Estimates 207 " 210 Measuuino Hai— Ilhistratiou. *' Estimating Quantity in Stacks, Mows and Meadows 210 " 213 Weight of Live Catile— Illustration. *' "Weights Estimated by Measure- ment 214 Builders' MEASUREMENXS-Illustration 215 " Bricklaying 216 Tiling or Slating 217 Walling 218 Ghi/ing 220 " Plumbing 221 « Masonry 222 " Plastering 223 Short Rules foh the Mechanic — Illustration — 225 Square and Cube Roots 226 " 237 Mensuration, or I'ractical Geometry 2U7 " 246 Tabus or Multiples 246 AvoiRDUPOis Weight— Illustration, with Tables 247 '' Ljngor Iron Ton 247 *' Iron and Lead 247 « Miscellaneous Table 248 Apothecaries' Weight— Illustration, with Tables 248 ' Ki.uiD Measure 249 Dry Measure — Illustration and Tables 250 Cubic or Solid Measure — Illustration, with Tallies 251 Measurements — Valuable Information Concerning 252 Nails — Sizes and Number to the Pound 252 Liquid Measure — Illustration, with Tables 253 3Ieasurements — Linear ob Long — Illustration, with Tables 254 Surveyors' Measure 255 Geographical and Astronomical Calculations — Tnblo. . . 255 Surface or Square Measure— Illustration, with Tubks. . . 256 Surveyors' Square Measure, with Tables 257 Contents of Fielps and Lots — Table 258 Fencing — Tablo showing the No. of Stakes, Rails and Posts Required in Fencing 258 Troy Weight — Illustration, with Tables 269 Diamond Weight , 259 Paper, Books AiND Stationery— Illustration, with viuiuus Tables 260 O CONTENTS. PfiiNTiNo — Typfi-eetting 261 '* Press-work 262 " CosttfPaper 268 " Table showing Cost cf Paper by the Quire 264 Books— Sizes and Styles 262 Shoemakers' Measure 263 Measurement OF Time— Illustration 265 Table 266 " " Circular Measure 266 LoNOiTCDE AND Time — Table 267 Time — How to Ascertain the Difference between Cities 267 Table — For Ascertaining the No. c f Days between Two Dates 268 •' Showing the Number of Days from any Day in one Month to the same Day in Another 269 ASTUONOMICAL CALCULATIONS 270 tO 274 MoN e V of tJie U nited States— Illustration 274 " Fiance 275 " the German Empire 276 Arbitration op Exchange 277 Value of Foreign Coins in U. S. Gold 278, 279 Gold and Currency Values 280 Bank Accounis — Illustration — How to Transact Business with Banks 281 " 284 Interfst — Commercial Rules , 284, 285 Interest Tables— Decimal System , . ,286 " 290 " Compound Interest .. 295 Time Required for Monet at Interest lo Doublk 296 U. S. Interest RATts and Penalties , 297, 298 How to OnrAiN Wealth 299 Wages— Value of Time 300 Ready Reckoning 301 '• •' Tables 302, 303 Taule of Illustuations 304 i fs/^^J G^^l^il^n^rr-*^ w^Sai^lli^ ,^r^ Quantity is that which can be increased oi diminished by augments or abatements of homo geneous parts. Quantities are of two essential kinds, Geometrical and Physical. 1. Geometrical quantities are those which occupy space ; as line$^ surfaces^ solidsj liquids, gases, etc. 2. Physical quantities are those which exist in the time, but occupy no space ; they are known by their character and action upon geometrical quan- tities, as attraction^ light, heat, electricity and mag- netism, colors, force, power, etc. To obtain the magnitude of a quantity we com- pare it with a part of the same ; this part is im- printed m our mind as a unit, by which the whole is measured and conceived. No quantity can be measured by a quantity of another kind, but any quantity can be compared with any other quantity, and by such comparison arises what we call calcit' lation or Mathematics^ 10 ORTON & Sadler's calculator. Introduction^. Arithmetic means reckoning by numbers, calculating. Notation means writing numbers. Numeration means reading numbers. Number is one or more things or units, as one, two, &c. Unit or one is a single thing. Numbers are represented by figures. Figures are characters used in Arithmetic to represent numbers. All numbers are represented by the ten following figures: (WrUten) <^ /. J. J. J. J". ^ /. cf. f. Cipher, one. two. three, four. five. six. scYen. eight, nine. (Printed) 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. These figures, except the cipher, are often called Digits. Digit means the measure of a finger's breadth. Figures were called digits from counting the fingers in reckoning. The character is called a cipher, from the Arabic word tsphara, which signifies a blank or void. The uses of this character in numeration are ko important, that its name, cipher, has been extended to the whole art of Arithmetic, which has been culled to cipJier, meaning to work with figures. INTRODUCTION. 11 Standard Measures, to prevent error are generalhf derived /rom nature. For example, measures of time. from the time of the revolution of the earth about itf axis : of space, from the length of a barley-corn, taken from the middle of a full-grown ear ; also, from th« circumference of the earth ; of weight, from the weigM of a grain of wheat, taken as above ; also, /rom th0 weight of a definite quantity of distilled water ; of heat, /rom the temperature of boiling water, &c. The four principal operations of Arithmetic are represented by the following signs : -f Plus or more, the sign of Addition — Minus or less, " Subtraction. X Fnto (multiplied by) " Multiplication. -J- By (divided by) " Division. When, in solving a question, only one operation if need, the answer has a distinctive name. In addition, the answer is called the sum. Subtraction, •• "I Difference or ( Remainder Multiplication, '* " Product. Division, " " Quotient. A sign made thus =, called Equal to or Equals, is placed between two quantities to show their equality ; Thus, 1 + 1 = 2 is read, one plus one, equal to two ; or, more commonly and perhaps better, one plus oti^, eq*uiU two. To BE able to add two, three or four columns of figures at once, is deemed by many to be a Herculean task, and only to be accomplished by the gifted few, or, in other words, by mathemati- cal prodigies. If we can succeed in dispelling this illusion, it will more than repay us ; and we feel very confident that we can, if the student will lay aside all prejudice, bearing steadily in mind that to become proficient in any new branch or principle a little wholesome application is necessary. On the contrary, we can not teach a student who takes no interest in the matter, one who will always be a drone in society. Such men have no need of this principle. If two, three, or more, columns can be carried up at a time, there must be some law or rule by which it is done. We have two principles of Addi- tion; one for adding short columns, and one for adding very long columns. They are much alike, differing only in detail. When one is thoroughly learned, it is very easy to learn the second. 12 ADDITION. 13 ADDITION TABLE. The design of the table on the following page is to familiarize the student with the combination or grouping of figui^es so as to enable him in- stantly to see or read the result without stopping to add each figure separately. In learning this table avoid spelling the figures, as 4 and 5 are 9, but take in the result 9 as soon as the eye catches the combination — do not con- sider the figures 4+5, but see them as 9. To il- lustrate: add 4+ 5 -f 6+2, instead of saying 4 and 5 are 9 and 6 are 15 and 2 are 17, consider the combinations 4+ 5 as 9, 6+ 2 as 8 ; thus you really have but 9+8 to add instead of 4+5+6 + 2, pro- ducing a saving in time and mental work. The science of RAPID OR "LIGHTNING" ADDITION Lies in the ability of the calculator to instantly see or take in the result of two or more figures regardless of their combination, without stopping to add each figure separately, i. e., To read the result of figures as in reading a book, the pro- nunciation of a word is known, or the meaning of a sentence without the necessity of spelling or pronunciation of syllables. After mastering this table the learner will be surprised at the rapidity he can add a column of figures, and he will soon find himself grouping or combining with ease and accuracy four and five figures at a time, instead of two as illus- trated by the table. 2 14 ORTON & Sadler's calculator. TABLE OF ADDITION, Showing the combination of the 9 significant figures^ in groups of two only^ and producing, tohen added together J results from \ to IS. PrcKluced by coin- Pro- bination or addi- Pro- ducts, tion of the 9 ducts, significant figures. 1 = 1 ~ I 2 = 1 1 == 2 3 == 2 1 =r 3 4 = 3 2 1 2 = 4 5 = 43 1 2 = 5 6 = 5 4 3 1 2 3 = 6 r> 5 4 1 2 3 7 6 6 4 1 2 3J^ rr'G 6 12 3 4 7 = 8 = 9 = in « 9 8 7 G 5 ,^ *" 1 2 3 4 fS '" 11 9 8 7 6 ,, " = 2 3 4 5 = 11 io _ 9 8 7 6 ,o *^ 3 4 5 6 ~ *^ 13 =- ??I =13 14 « 9 8 7 5 67 = 14 15 =- 9 8 6 7 = 15 16 =- 9 8 7 8 « 16 17 = 9 8 =3 17 18 = 9 = 18 ADDTTTON. N B. Tlie above process of addition is only re* commended for beginners. Process. — For adding the above example, cora- mence at the bottom of the right-hand column. Add thus : 12, 16, 22 ; then carry the 2 tens to the second column, then add thus, 8, 10, 18, 22, carry the two hundreds to the third column, and add the same way, 9, 13, 16, 23. Never permit yourself for once to add up a column in this man- ner, 3 and 9 are 12 and 4 are 16, and 6 are 22 ; it is just as easy to name the sura at once^ without naming the figures you add, and three times as rapid. 15 16 ORTON & Sadler's calculator. ADDITION OP SHORT COLUMNS OF FIGURES. Addition is the basis of all numerical opera* tions, and is used in all departments of business, To aid the business man in acquiring facility and accuracy in adding short columns of figures, the following method is presented as the best : Process. — Commence at the bottom of 274 040 the right-hand column, add thus: 16^ 22, 134 32 ; then carry the 3 tens to the second 342 column; then add thus: 7, 14, 25; carry "^27 the 2 hundreds to the third column, and ^^^ add the same way: 12, 16, 21. In this 2152 '^ay you name the sum of two figures at once, which is quite as easy as it is to add one figure at a time. Never permit yourself for once to add up a column in this manner : 9 and 7 are 16, and 2 are 18 and 4 are 22, and 6 are 28, and 4 are 32. It is just as easy to name the result of two figures at once and four times as rapid. The following method is recommended for the addition of long columns op figures. In the addition of long columns of figures which frequently occur in books of accounts, in order to add them with certainty, and, at the name time, with ease and expedition, study well the following method, which practice will rendei familiar, easy, rapid, and certain. ADDITION. 17 THS lAST WAT TO ADD. EXAMPLE 2— EXPLANATION. Commence at 9 to add, and add as near 20 as pos aible, thus: 9+2+4+3:^18, place the 8 to the right of the 3, as in example ; commence at 6 to 7' add 6+4+8=18 ; place the 8 to the right of 4 the 8, as in example ; commence at 6 to add 6 6+4+7=17 ; place the 7 to the right of the 3* 7. as in example ; commence at 4 to add 4+ 9 9+3—16 ; place the 6 to the right of the 3, 4 as in example ; commence at 6 to add 6+4 7^ +7=17 ; place the 7 to the right of the 7, 4 as in example; now, having arrived at the 6 top of the column, we add the figures in the 8* new column, thus: 7+6+7+8+8=36 ; place 4 the right hand figure of 36, which is a 6, 6 under the original column, as in example, and 3^ add the left hand figure, which is a 3, to the 4 number of figures in the new column; there 2 are 5 figures in the new column, therefore 9 3+5=8 ; prefix the 8 with the 6, under the — original column, as in example ; this makes 86 86, which is the sum of the column. Remark 1. — If, upon arriving at the top of the column, there should be one, two or three figure? whose sum will not equal 10, add them on to the •urn of the figures of the new column, never placing 18 OKTON & Sadler's calculator. an extra figure in the new column, unless it be aL excess of units over ten. Remark 2. — By this system of addition you can stop any place in the column, where the sum of the figures will equal 10 or the excess of 10 ; but the addition will be more rapid by your adding as near 20 as possible, because you will save the form- ing of extra figures in your new column. EXAMPLE— EXPLANATION. 2+6+7=15, drop 10, place the 5 to the right of the 7; 6+5+4=15, drop 10, place the 5 to the right of the 4, as in example ; 8+3+7=18, drop 10, place the 8 to the right of the 7, 4 as in example; now we have an extra figure, 7* which is 4 ; add this 4 to the top figure of the 3 new column, and this sum on the balance of 8 the figures in the new column, thus: 4+8+ 4* 6+5=1:22 ; place the right hand figure of 22 5 under the original column, as in example, and 6 add the left hand figure of 22 to the num- 7' ber of figures in the new column, which are 6 three, thus : 2+3=5 ; prefix this 5 to the 2 fip:ure 2, under the original colum:i ; this — uxakes 52, which is the sum of the column. 52 ADDITION. 19 Rule.'— Far adding two or more columris, com- rn^mce at the right handj or units^ column; proceed in the same manner as in adding one column ; after the sum of the first column is obtained^ add all except the right hand figure of this sum to the second column, adding the second column the same way you, added the first; proceed in like manner with all the columns, always adding to each successive column the sum of the column in the next lower order, minus the right hand figure, N. B. The small figures which '^e place to the right of the column when adding are called integers. The addition by integers or by forming a new column, as explained in the preceding examplea should be used only in adding very long columns of figuies, say a long ledger column, where the foot- ings of each column would be two or three hundred, In which case it is superior and much more easy than any other mode of addition ; but in adding short columns it would be useless to form an extra column, where there is only, say, six or eight fig- ures to be added. In making short additions, the following suggestions will, we trust, be of use to the accountant who seeks for information on this subject. In the addition of several columns of figures, where they are only four or five deep, or when their respective sums will range from twenty ^ve 20 ORTON & Sadler's calculator. to forty, the accountant should commence with the unit column, adding the sum of the first two figures to the sum of the next two, and so on, naming onlj the results, that is, the sum of every two figures. In the present example in adding the unit 346 column instead of saying 8 and 4 are 12 and 235 5 are 11 and '6 are 23, it is better to let the T24 eye glide up the column reading only, 8, 12, 598 lY, 23; and still better, instead of making a separate addition for each figure, group the figures thus: 12 and 11 are 23, and proceed in like man- ner with each column. For short columns this is a very expeditious way, and indeed to be preferred ; but for long columns, the addition by integers is the most useful, as the mind is relieved at intervals and the mental labor of retaining the whole amount, as you add, is avoided, which is very important to any person whose mind is constantly employed in various commercial calculations. In adding a long column, where the figures are of a medium size, that is, as many 8s and 9s as there are 2s and 3s, it is better to add about three figures at a time, because the eye will distinctly see that many at once, and the ingenious student will in a short time, if he adds by integers, be able to read the amount of three figures at a glance, or as quick, we might say, as he would read a single figure. ADDITION. 21 Here we begin to add at the bottom of the *26* anit column and add successively three fig- ^^ ores at a time, and place their respective ^o^ sums, minus 10, to the right of the last fig- 954 are added ; if the three figures do not make 62 10, add on more figures; if the three figures 87* make 20 or more, only add two of the fig- ^f ures. The little figures that are placed to j^^ 4 the right and left of the column are called 877 integers. The integers in the present ex- 33 ample, belonging to the units column, are 84'* 4, 4, 5, 4, 6, which we add together, making ^^ 23; place down 3 and add 2 to the number of integers, which gives 7, which we add to 803 the tens and proceed as before. Eeason. — In the above example, every time we placed down an integer w^e discarded a ten, and when we set down the 3 in the answer we dis- carded two tens; hence, we add 2 on to the num- ber of integers to ascertain how many tens were discarded; there being 5 integers it made 7 tens, which we now add to the column of tens; on the same principle we might add between 20 and 30, always setting down a figure before we got to 30; then every integer set down would count for 2 tens, being discarded in the same way, it does in the present instance for one ten. When we add be- tween 10 and 20, and in very long columns^ U 22 ORTON & Sadler's calculator. would be much better to go as near 30 as possible, and count 2 tens for every integer set down, in which case we would set down about one -half as many integers as when we write an integer foi ev^ery ten we discard. When adding long columns in a ledger or day- book, and where the accountant wishes to avoid the writing of extra figures in the book, he can place a strip of paper alongside of the column he wishes to add, and write the integers on the paper, and in this way the column can be added as convenient almost as if the integers were written in the book. Perhaps, too, this would be as proper a time as any other to urge the importance of another good habit; I mean that of making 'plain figures. Some persons accustom themselves to making mere scrawls, and important blunders are often the result. If letters be badly made you may judge from Buch as are known; but if one figure be illegible, its value can not be inferred from the others. The vexation of the man who wrote for 2 or 3 monkeys, and had 203 sent him, was of far less importance than errors and disappointments sometimes result- ing from this inexcusable practice. We will now proceed to give some methods of proof. Many persons are fond of proving the cor- rectness of work, and pupils are often instructed to do so, for the double purpose cf giving them ADDITION. 23 exercise in nalculation and saving their teacher the trouble of reviewing their work. There are special modes of proof of elementary operations, as by casting out threes or nines, or by changing the order of the operation, as in add- ing upward and then downward. In Addition, some prefer reviewing the work by performing the Addition downward, rather than repeating the ordinary operation. This is better, for if a mis- take be inadvertently made in any calculation, and the same routine be again followed, we are very liable to fall again into the same error. If, for instance, in running up a column of Addition you should say 84 and 8 are 93, you would be liable, rn going over the same again, in the same way to slide insensibly into a similar error ; but by begin- ning at a different point this is avoided. This fact is one of the strongest objections to the plan of cutting off the upper line and adding it to the sum of the rest, and hence some cut off the lower line by which the spell is broken. The most thoughtless can not fail to see that adding a line to the sum of the rest, is the same as adding it in with the rest. The mode off proof by casting out the nines and threes will be fully explained in a following ahapter. A very excellent mode of avoiding error in ad4- 24 ORTON & Sadler's calculator. ing long columDS is to set down the result of each column on some waste spot, observing to place the numbers successively a place further to the left each time, as in putting down the product figures m multiplication; and afterward add up the amount. In this way if the operator lose his count he is not compelled to go back to units, but ofily to the foot of the column on which he is op- erating. It is also true that the brisk accountant, who thinks on what he is doing, is less liable to err than the dilatory one who allows his mind to wander. Practice too will enable a person to read amounts without naming each figure, thus instead of saying 8 and 6 are 14, and 7 are 21 and 5 are 26. it is better to let the eye glide up the column, read- ing only 8, 14, 21, 26, etc.; and, still further, it is quite practicable to accustom one's self to group 87 the figures in adding, and thus proceed very rap- 23 idly. Thus in adding the units' column, instead 45 cf adding a figure at a time, we see at a glance 62 that 4 and 2 are 6, and that 5 and 3 are 8, then 21 6 and 8 arc 14 ; we may then, if expert, add — constantly the sum of two or three figures at a time, and with practice this will be found highly advan- tageous in long columns of figures; or two or three columns may be added at a time, as the practiced eye will see that 24 and 62 are 86 almost as readily AS that 4 and 2 are 6. ADDITION. 26 Teachers will find the following mode of match ' ng lines for beginners very convenient, as they can inspeoc them at a glance : Add 7654384 8786286 3408698 2345615 1213713 23408696 In placing the above the lines are matched in pairs, the digits constantly making 9. In the above, the first and fourth, second and fifth are matched; and the middle is the hey line^ the result being just like it, except the units' place, which ia as many less than the units in the key line as there are pairs of lines; and a similar number will oc- cupy the extreme left. Though sometimes used as a puzzle, it is chiefly useful in teaching learners ; and as the location of the key line may be changed in each successive example, if necessary, the arti- fice could not be detected. The number of lines U necessarily odd. a SHORT METHODS OF MULTIPLICATION. Rule. — Set down the smaller factor under the larger, units under units, tens under tens. Begin with the unit figure of the multiplier, multiply by it, first the units of the multiplicand, setting the units of the product, and reserviny the tens to be added to the next product ; now multiply the tens of the multiplicand by the unit figure of the multiplier^ and the unite of the multiplicand by tens figure of 26 MULTIPLICATION. 27 the multiplier; add these two products together ^ set- ting down the units of their sum, and reserving thi tens to he added to the next product ; now multijoly the tens of the multiplicand hy the tens figure of tJu multiplier, and set down the whole amount. This will be the complete product. Remark. — Always add in the tens that are re- served as soon as you form the first product. EXAMPLE 1.— EXPLANATION. 1. Multiply the units of the multiplicand 24 by the unit figure of the multiplier, thus : 31 1X4 is 4 ; set the 4 down as in example. 2. Multiply the tens in the multiplicand by 744 the unit figure in the multiplier, and the units in the multiplicand by the tens figure in the multi- plier, thus : 1x2 is 2; 3x4 are 12, add these two products together, 2-f-12 are 14, set the 4 down as in example, and reserve the 1 to be added to the next product. 3. Multiply the tens in the multi- plicand by the tens figure in the multiplier, and add in the tens that were reserved, thus: 3x2 are 6, and 6+1=7 ; now set down the whole amount, which is 7. EXAMPLE 2.— EXPLANATION. 1. Multiply units by units, thus: 4x3 53 are 12, set down the 2 and reserve the 1 to 84 carry. 2. Multiply tens by units, and units by tens, and add in the one to carry on the 4452 28 ORTON & Sadler's calculator. arst product, then add these two products together, thus: 4X5 are 20+1 are 21, and 8x3 are 24, and 21-f-24 are 45, set down the 5 and reserve the 4 to carry to the next product. 3. Multiply tens by tens, and add in what was reserved to carry, thus: 8x5 are 40-[-4 are 44, now set down the whole amount, which is 44. EXAMPLE 3.— EXPLANATION. 5x3 are 15, set down the 5 and carry the 43 1 to the next product; 5X4 are 20=1 25 are 21; 2x3 are 6, 21+6 are 27, set down the 7 and carry the 2; 2x4 are 8+2 are 1075 10 ; now set down the whole amount. When the multiplicand is composed of three fig- ures, and there are only two figures in the multi- plier, we obtain the product by the following Rule, — Set down the smaller factor under the larger J units under units, tens under tens ; now muU tiply the first upper figure hij the unit figure of the multiplier^ setting down the units of the product, and reserving the tens to he added to the next product; now multiply the second upper hy units, and the fij-si upper hy tens, add these two products together, set- ting down the units figure of their sum, and reserv- ing the tens to carry, as hefore ; now multiply the third upper hy units, and the second upper hy tens^ add these two products together, setting down the units figure of their sum, and reserving the tens to MULTIPLICATION. 20 carry, as usual ; now multiply the third upper hy tens, add in the reserved figure, if there is one, and set down the whole amount. This will he (he com - pleie product. Remark. — One of the principal errors with the beginner, in this system of multiplication, is neglecting to add in the reserved figure. The stu- dent must bear in mind that the reserved figure l« product. 30 ORTON & Sadler's calculator. Multiply 32 by 45 in a single line. Hero we multiply 5x2 and set down and 32 carry as usual ; then to what you carry add 45 5X3 and 4X 2, which gives 24; set down 4 and carry 2 to 4x3, which gives 14 and 1440 completes the product. Multiply 123 by 456 in a single line. Here the first and second places are 123 found as before; for the third, add 6X1, 456 5X2, 4X3, with the 2 you had to carry, making 30 ; set down and carry 3 ; then 56088 drop the units' place and multiply the hundreds and tens crosswise, as you did the tens and units, and you find the thousand figure ; then, dropping both units and tens, multiply the 4X1, adding the 1 you carried, and you have 5, which completes the product. The same principle may DC extended to any number of places ; but let each Btep be made perfectly familiar before advancing to another. Begin with two places, then take three, then four, but always practicing some time on each number, for any hesitation as you progress will, confuse you. N. B. The following mode of multiplying num- bers will only apply where the sum of the two la&l or unit figures equal ten, and tho other figures in both factors are the same. MULTIPLICATION. 3^1 CONTKACTIONS IN MULTIPLICATION. To multiply when the unit figures added equal *'I0) and the tens are alike as 72 ^^18, dc, 1st. Multiply the units and set down the result. 2d. Add 1 to either number in tens place and multiply by the other, and you have the complete product. EXAMPLE PROCESS. Here because the sum of the units 4 and 6 86 are ten and the tens are alike ; we simply say 84 4 times 6 are 24, and set down both figures of the product ; then because 4 and 6 make ten we '^224 add 1 to 8, making 9, and 9 times 8 are 72, which completes the product. Note. — If the product of units do not contain ten tbe plac« of tens must be filled with a cipher The above rule is useful in examples like the fd- lowing : 2. What will 93 acres of land cost at 97 dollars per acre? Ans. $9021. 3. What will 89 pounds of tea cost at 81 cents per pound? Ans. $72.09. In the above the product of 9 hi/ I did not amourU to ten^ therefore is placed in tens place. 4. Multiply 998 by 902. Ans. 990016. In the above, because 2 and 8 are 10, ive add 1 to 99, making 100; then 100 times 99 are 9900 3 32 ORTON & SADLER'S CALCULATOR^ EXAMPLE EIGHTEENTH. Multiply 79 by 71 in a single line. Here we multiply IX^* and set down the 79 result, then we multiply the 7 in the mul- 71 tiplicand, increased by 1 by the 7 in the multiplier, 7X8, which gives 56 and t>om- 6609 pletes the product. EXAMPLE. Multiply 197 by 193 in a single line. Here we multiply 3x7 and set down the 197 result, then we multiply the 19 in the 193 multiplicand, increased by 1 by the 19 in the multiplier, 19x20, which gives 380 38021 4nd completes the product. EXAMPLE. Multiply 996 by 994 in a single line. Here we multiply 4x6 and set down 996 the result, then we multiply the 99 in 994 the multiplicand, increased by 1 by the 99 in the multiplier, 99x100, which 990024 gives 9900 and completes the product. EXAMPLE. Multiply 1208 by 1202 in a single line. Here we multiply 2x8 and set down 1208 the result, then we multiply the 120 in 1202 the multiplicand, increased by 1 by tbc 120 in the multiplier, 120x121, which 1452016 gives 14520 and completes the product. ^ MULTIPLICATION. 33 CDEIOUS AND USEFUL CONTRACTIONS. To multiply any number, of two figures, by 11, Hole.-- Write the sum of the figures between them 1. xVlultipiy 45 by 11 Ans. 495. Here 4 and 5 are 9, which write between 4 & 5 2. Multiply 34 by 11. Ans. 61A N. B. When the sum of the two figures is over 9, incretise the left-hand figure by the 1 to carry. 3. Multiply 87 by 11. Ans. 957 To square any number of 9s instantaneously. and without multiplying. Rule. — Write down as many 98 less one as there are 9s in the given number ^ an 8, as many 0« as 9«, and a 1. 4. What is the square of 9999 ? Ans. 99980001. Explanation. — We have four 9s in the given number, so we write down three 9s, then an 8, then three Os, and a 1. 5. Square 999999. Ans. 999998000001. To square any number ending in 5, Rule. — Omit the 5 and multiply the number, as it will then stand by the next higher number^ and annex 25 to the product, 6. What is the square of 75 ? Ans. 5625. Explanation. — We simply say, 7 times 8 are 56, to which we annex 25. 7. What is the square of 95? Ans. 9026 TABLE OF SQUARES, PBOM 1 TO 104. l«-r 1 272= 729 532=2809 'i92= 6241 22= 4 282= 7g4 542=2916 802= 6400 32= 9 292= 841 552=3025 812= 6561 42= 16 302= 900 562=3136 822= 6724 52= 25 312= 9gi 572=3249 832= 6889 62= 36 322=1024 582=3364 842= 7056 72= 49 332=1089 592=3481 852= 7225 82= 64 342=1156 602=3600 862= 7396 92= 81 352=1225 612=3721 872= 7569 102=100 362=1296 622=3844 882= 7744 112=121 372=1369 632=3969 892= 7921 122=144 382=1444 642=4096 902= 8100 132=169 392=1521 652^4225 912= 828L 142=196 402=1600 662=4356 922= 8464 152=225 412=1681 672=4489 932= 8649 162=256 422=1764 682=4624 942= 8836 172=289 432=1869 692=4761 952= 9025 182=324 442=1936 702--4900 962= 9216 192=361 452=2025 712=5041 972= 9409 202=400 462=2116 722=5184 982= 9604 212=441 472=2209 732=5329 992= 9801 222=484 482=2304 742=5476 1002=10000 232=529 492=2401 752=5625 1012=10201 242=576 502=2500 762=5776 1022=10404 252=625 512=2601 772=5929 1032=10609 262=676 522=2704 782=6084 1042=10816 Note. — To become familiar with the numbers shown in the above table, from 1 to 25, requires but little study and application upon the part of the pupil, and will prove 0/ great benefit in mathematical calculations. 34 FRACTIONS Are one or more of the equal parts into which a unit or whole thing is divided. All fractions express the division of units or things. The fractional terms are, numerator and denominator. The Numerator expresses the number of parts or units taken (it is therefore the dividend), and is written above the line. The Denominator expresses the division of the equal parts or units (it is therefore the divisor), and is written below the line. Fractions are written and read as follows : i or one-half, J or one-third, | or three- fourths. The Quotient produced from dividing the numerator by the denominator of a fraction is its value. Thus, — I =- 3 — L2 =:^ 3 — ^^ -= 9 The vahie of a fraction is 'less than 1 when the numera- tor is less than the denominator, and equals or exceeds 1 when the numerator equals or is greater than the denomi- nator. GENERAL PRINCIPLES GOYERM\G FRACTIONS. To increase or multiply a fraction, Multiply the numerator or divide the denoviinater. To decrease or divide a fraction, jylvlde the munerator or multiply the denoviinator. Multiplying or dividing both terms of a fraction by the same number does not change its value. Fractions may be reduced, added, subtracted, multi- plied, and divided. 35 S6 ORTON <& Sadler's calculator. Mental Operations in Fractions. To square any number containing |, as 6^, 9-^, Rule. — Multiply the whole number hy the next hiyher whole number, and annex ^ to the product. Ex. 1. What is the square of 7^? Ans. m^. We simply say, 7 times 8 are 56, to which we addf 2. What will 9 J lbs. beef cost at 9^ cts. a lb.? 3. What will 12^ yds. tape cost at 12^ cts. a yd.? 4. What will 5^ ]bs. nails cost at 5| cts. a lb. ? 5. What will 11^ yds. tape cost at 11^ cts. a yd.? 6. What will 19^ bu. bran cost at 19^ cts. a bu.7 Reason. — We multiply the whole number by the next higher whole number, because half of any number taken twice and added to its square is the same as to multiply the given number by one moie than itself. The same principle will multiply any two like numbers together, when the sum of the fractions is one, as 8^ by 8|, or 11 J by 11|, etc It is obvious that to multiply any number by any two fractions whose sum is ONE, that the sum of the products must be the originr'' number y and adding the number to its square is simply to multiply it by ONE more than itself; for instance, to multiply 7^ by 7|, we simply say, 7 times 8 are 5f>^ and then, to complete the multiplication, we add, of course, the product of the fractions (J times J arc 3^), making 5 6^5 the answer. MULTIPLICATION OP FRACTIONS. 37 WTiere the sum of the Fractions is One. To multiply any two like numbers together when the sum of the fractions is one. Rule. — Multiple/ the whole number hy the next higher whole number; after whichy add the product of tlie fractions, N. B. In the following examples, the product of the fractions are obtained yrrs^ for convenience. PRACTICAL EXAMPLES FOR BUSINESS MEN. Multiply 3| by 3^ in a single line. Here we multiply |X|, which gives y^^, 3^ and set down the result; then we multiply 3| the 3 in the multiplicand, increased by unity, by the 3 in the multiplier, 3x4, 12 ^^ which gives 12 and completes the product. Multiply 7f by 7f in a single line. Here we multiply |Xf) which gives /^, 7| and set down the result; then we multiply 7| the 7 in the multiplicand, increased by unity, by the 7 in the multiplier, 7x8, which gives 56^ 56, and completes the product. Multiply 11^ by 11| in a single line. Here we multiply f X^, which gives J, and 11 J aet down the result; then we multiply the 11 11 J in the multiplicand, increased by unity, by the 11 in the multiplier, 11x12. which gives 132 J 132. and completes the product. 38 ORTON & Sadler's calculator. EXAMPLE. Multiply 16f by 16J in a single line. Here we multiply JXf which gives J, and 16| set down the result, then we multiply the 16J 16 m the multiplicand, increased by > unity by the 16 in the multiplier, 16x17, 2721 which gives 272 and completes the product. EXAMPLE. Multiply 29J by 29^ in a single line. Here we multiply JXi which gives J, 29J and set down the result, then we multiply 29^ the 29 in the multiplicand, increased by unity by the 29 in the multiplier, 29 x 870| 30, which gives 870 and completes the pro- duct. EXAMPLE. Multiply 999§ by 999f in a single line. Here we multiply fXf? which gives 999| J|, and set down the result, then we 999f multiply the 999 in the multiplicand, increased by unity by the 999 in the 999000JJ multiplier, 999x1000, which gives 999000 and completes the product. Note. — The system of multiplication introduced in the preceding examples, applies to all numbers. Where the sum of the fractions is owe, and the whole numbers are alike, or differ by one, the learner if requested to study well these useful properties of numboTP MULTIPLICATION OF FRACTIONS. 39 Where the sum of the Fractioiis is One. To multiply any two numbers whose diflerenoe is onSy and the sum of the fractions is one. Rule. — Multiply the larger number, increased ht^ ONE, bi/ the smaller number; then square the frac- tion of the larger number, and subtrac* its square from ONE. PRACTICAL EXAMPLES FOR BUSINESS MEN. 1. What will 9^ lbs. sugar cost at 8| cts. a lb. ? Here we multiply 9, increased by 1, by 8, 91 thus, 8x10 are 80, and set down the result; gl then from 1 we subtract the square of -J, thus, \ squared is J^, and 1 less ^ is f|. ^^H 2. What will 8| bu. coal cost at 7^ cts. a bu.? Here we multiply 8, increased by 1, by 8| 7j thus, 7 times 9 are 63, and set down the 7 J result ; then from 1 we subtract the square ~"^ of I, thus, I squared is |, and 1, less |, is |. » 3. What will 11-j^ bu. seed cost at $10} J a bu.? Here we multiply 11, increased by 1, by 10, thus, 10 times 12 are 120, and set ^^A down the result; then from 1 we subtract ^^T? the square of j^g, thus, ^ squared is j^^, lOQlta and 1 less ^-J^ is f«-f. 4. How many square inches in a floor 99| in wiie and 98| in. long? Ans. 9800^4 40 ORTON & Sadler's calculator. METHOD OF OPERATION. EXAMPLE. jdultiply 6J by 6J in a single line. Here we add 6J+ J, which gives 6 J ; this 6} multiplied by the 6 in the multiplier, 6J 6X6^, gives 39, to which we add the pro- — duct of the fractions, thus JX J gives j^^, added 89j'g to 39 completes the product. EXAMPLE. Multiply 11 J by llf in a single line. Here we would add llJ-[-f) which gives 11J 12; this multiplied by the 11 in the multi- Jl| plier gives 132, to which we add the product of the fractions, thus f Xi gives j\y which 132 j^^ added to 132 completes the product. EXAMPLE. Multiply 12J by 12f in a single line. Here we add 12^-|-f, which gives 13J; 12J this multiplied by the 12 in the multiplier, 12| 12X13J, gives 159, to which add the pro- duct of the fractions, thus fX^ gives §, 159j which added to 159 completes the product. MULTIPLICATION OF FRACTIONS. 41 Where the Fractions have a Like Denominator, To multiply any two like numbers together, each 3f which has a fraction with a like denominator, as l| by 4J, or 11^ by llf, or 10| by lOJ, eic. Rule. — Add to the multiplicand the fraction of the multiplier, and multiple/ this sum by the whole number; afterwhich, add the product of the fractions. PRACTICAL EXAMPLES FOR BUSINESS MEN. N. B. In the following example, the sum of the frao- tlons is ONE. 1. What will 9f lbs. beef cost at 9^ cts. a lb.? The sum of 9| and \ is ten, so we simply ^1 say, 9 times 10 are 90; then we add the f_ product of the fractions, \ times | are j^^r. ^O^g N. B. In the following example, the sum of the frac tlons is less than one. 2 What will 8^1^ yds. tape cost at 8f cts. a yd. ? The sum of 8J and | is 8|, so we simply ^ say, 8 times 8J are 70; then we add the _4 product of the fractions, \ times \ are ^ or ^. 70^ N. B. In the following example, the sum of the frac- tions is greater than one. 3. What will 4f yds. cloth cost at $4| a yd.? The sum of 4| and | is 5|^, so we simply 4| say, 4 times 5J are 21 ; then we add the » product of the fractions, ^ times |^ are |^. 21 J^ N. B. "Where the fractions have different ienominators. reduce them to a common denominator. 42 ORTON & Sadler's calculator. Rapid Process of Multiplying Mixed Kumhers, A valuable and useful rule for the accountant in the practical calculations of the counting-room. To multiply any two numbers together, each of which involves the fraction ^, as 7^ by 9^, etc., Rtjle. — To the product of the whole numbers add half their sum plus ^, EXAMPLES FOR MENTAL OPERATIONS. 1. What will 3^doz. eggs cost at 7^ cts. a doz.? Here the sum of 7 and 3 is 10, and half this 31 sum is 5, so we simply say, 7 times 3 are 21 7| and 5 are 26, to which we add 4-. ^ 264 N. B. If the sum be an odd number, call it one less to make it even, and in such cases the fraction must be |. 2. What will 11^ lbs. cheese cost at 9^ ets. a lb.? 3. What will 8^ yds. tape cost at 15^ cts. a yd.? 4. What will 7^ lbs. rice cost at 13| cts. a lb.? 6. What will 10^ bu. coal cost at 12| cts. a bu.? Reason. — In explaining the above rule, we add half their sum because half of either number added to half the other would be half their sum, and we add ^ because ^ by ^ is ^. The same principle will multiply any two numbers together, each of which has the same fraction; for instance, if the fraction was ^, we would add one-third their sum ; if |, we would add three-fourths their sum, etc.; and then, to complete the multiplication, we would add. of course, the product of the fractions. MULTIPLICATION OF FRACTIONS. 43 GENERAL RULE For multiplying any two numbers together, each of <7bich involves the same fraction. To the product of the whole numhers^ add the product of their sum by either fraction ; after which^ add thf' product of their fractions. EXAMPLES FOR MENTAL OPERATIONS. 1. What will llf lbs. rice cost at 9f cts. a lb.? Here the sum of 9 and 11 is 20, and three- i\s fourths of this sum is 15, so we simply say, 9} 9 times 11 are 99 and 15 are 114, to which we add the product of the fractions (3^). ■'^■^^^rfl 2. What will 7| doz. eggs cost at 8| cts. a doz. ? 3. What will 6| bu. coal cost at 6| cts. a bu. ? 4. What will 45| bu. seed cost at 3| dol. a bu.? 5 Af hat will 3| yds. cloth cost at 5f dol. a yd. ? 6. What will 17| ft. boards cost at 13| cts a ft.? 7. What will 18| lbs. butter cost at 18| cts. a lb. ? N. B. If the produt*t of the sum by either frac- tion in a whole number with a fraction, it is better to reserve the fraction until we are through with the whole numlers, and then add it to the product of the fractions; for instance, to multiply 3:J- by 7|, we find the sum of 7 and 3, which is 10, and one- fourth of this sum is 2^; setting the | down in gome waste spot, we simply say, 7 times 3 are 21 and 2 are 23 ; then, adding the i to the product of the fractions (^^), gives -j^g^, making 23^^, Ans. 44 ORTON & Sadler's calculator. t Rapid Process of Multiplying all Mixed Numbers. N. B. Let the student remember that this is a general and universal rule. GENELAL RULE. To multiply any two mixed numbers together, 1st. Multiply the whole numbers together. 2d. Multiply the upper digit by the lower fractio^i. 3d. Multiply the lower digit by the upper fraction. 4th. Multiply the fractions together. 5th. Add these FOUR products together, N. B. This rule is so simple, so useful, and so true that svery banker, broker, merchant, and clerk should post i^ up for reference and use. PRACTICAL EXAMPLES FOR BUSINESS MEN N. B. The following method is recommended to begin- ners: Example.— -Multiply 12| by 9|. 12f 1st We multiply the whole numbers. ^f 2d. Multiply 12 by f and write it down. ;[08 3d. Multiply 9 by | and write it down. 9 4th. Multiply | by | and write it down. 6 5th. Add these four products together, __A and we have the complete result. 123j^ N. B. When the student has become familiar T^ith the above process, it is better to do the intei- nediate work in the head, and, instead of setting iown the partial products, add them in the mind as you pass along, and thus proceed very rapidly. MULTIPLICATION OF MIXED NUMBERS. 45 Multiply 8| by lOf Here we simply say 10 times 8 arc 80 8J and I of 8 is 2, making 82, and | of 10 is 10} 2, which makes 84 ; then ^ times I is 2^^^, making 84^^^ the answer. ^'hh PRACTICAL BUSINESS METHOD For Multiplying all Mixed Numbers, Merchants, grocers, and business men generally, in multiplying the mixed numbers that arise Id the practical calculations of their business, only care about having the answer correct to the near- est cent ; that is, they disregard the fraction. When it is a half cent or more, they call it an- other cent ; if less than half a cent, they drop it. And the object of the following rule is to show the business man the easiest and most rapid process of finding the product to the nearest unit of any two numbers, one or both of which involves a fraction. GENERAL RULE. To multiply any two numbers to the nearest unit, 1st. Multiply the whole number in the multiplicand by the fraction in the multiplier to the nearest unit, 2d. Multiply the whole number in the multiplier hy the fraction in the multiplicand to the nearest unit 3d. Multiply the whole numbers together and add the three products in your mind as you proceed. N. B. In actual business the work can generally be done mentally for only easy fractions occiw in business 46 ORTON & Sadler's calculator. N B. This rule is so simple and so true, according tA all business usage, that every accountant should make himsel'f perfectly familiar with its application. There being no such thing as a fractian to add in, there is scarcely any liability to error or mistake. By no other arithmetical process can the result be obtained by so fer figures. EXAMi'LES FOR MENTAL OPERATION. EXAMPLE FIRST. Multiply 11^ by 8| by business method. 11^ Here :| of 1 1 to the nearest unit is 3, and ^ of 8J 8 to the nearest unit is 3, making 6, so we sim- • ply say, 8 times 11 are 88 and 6 are 94, Ans. 94 Reason. — \ of 11 is nearer 3 than 2, and J of 8 is nearer 3 than 2. Make the nearest whole number the quotient, EXAMPLE SECOND. Multiply 7| by 9| by business method. Here | of 7 to the nearest unit is 3, and J 7| of 9 to the nearest unit is 7 ; then 3 plus 7 9| is 10, so we simply say, 9 times 7 are 63 and 10 are 73, Ans. 73 EXAMPLE THIRD. Multiply 23-^ by 19^ by business method. Here ^ of 23 to the nearest unit is 6, and 23| ^ of 19 to the nearest unit is 6 ; then 6 plus 19| 6 is 12, so we simply say, 19 times 23 are 437 and 12 are 449, Ans. "^^^ N. B. In multiplying the whole numbers together, sA ways use the single-line method. MULTIPLICATION OF MIXED NUMBERS. 47 EXAMPLE FOUETH. Multiply 128| by 25 by business method. Here | of 25 to the oearest unit is 17, so 128 1 we simply say, 25 times 128 are 3200 and ^ ^ 17 are 3217, the answer. 3217 PRACTICAL EXAMPLES FOR BUSINESS MEN. 1. What is the cost of 17^ lbs. sugar at 18| cts. per lb. ? Here | of 17 to the nearest unit is 13, 17| and ^ of 18; is 9 13 plus 9 is 22, so we 18| simply say, 18 times 17 are 306 and 22 are 328, the answer. ^^'^^ 2. What is the cost of 11 lbs. 5 oz. of butter a*. 831 cts. per lb.? Here ^ of 11 to the nearest unit is 4, H^ and ^ of 33 to the nearest unit is 10 ; 331 then 4 plus 10 is 14, so we simply say, 33 times 11 are 363, and 14 are 377, Ans. ^^•'^'^ 3. What is the cost of 17 doz. and 9 eggs at 12^ cts. per doz.? Here ^ of 17 to the nearest unit is 9, 17/^ and ^ of 12 is 9 ;. then nine plus 9 is 18, 12^ BO we simply say, 12 times 17 are 204 and 18 are 222, the answer. ^^'^^ 4. What will be the cost of 15| yds. calico at 12^ cts. per yd.? Ans. $1.97. •48 ORTON & Sadler's calculator. Where the Multiplier is an Aliquot Part of 100. Merchants in selling goods generally make the price of an article some aliquot part of 100, as in selling sugar at 12A cents a pound or 8 pounds for 1 dollar, or in selling calico for 16J cents a yard or 6 yards for 1 dollar, etc. And to be- come familiar with all the aliquot parts of 100, so that you can apply them readily when occasion requires, is perhaps the most useful, and, at the same time, one of the easiest arrived at of all the computations the accountant must perform in the practical calculations of the counting-room. TABLE OF THE ALIQUOT PARTS OF 100 AND 1000 N. B. Most of these are used in business. m is J part of lOO H is A part of 100. 25 is f or i of 100. 16f is ^2^ or J of 100. 37i is f part of lOa 331 is i4 or I of 100. 50 is f or J of 100. 66| is T^ or 1 of 100. 62i is I part of 100. 83i is {^ or t of 100. 75 is f or } of 100. 125 is J part of lOOO 87J is I part of 100. 250 is f or J of 1000. 6i is ^ part of 100. 375 is J part of 1000. 18} is ^ part of 100. 625 is f part of 1000. 31J is A part of lOO 875 is I part of 1000. To multiply by an aliquot part of 100, Rule — Add two ciphers to the multiplicand, then take scich part of it as the multiplier is part of 100. N B. If the multiplicand is a mixed number reduc< the fraction to a decimal of two places bofore diyiding. COUNTmG-EOOM EXERCISES. Examples. — 1. Multiply 424 by 25. As 25 = ^ of 100, divide 42400 by 4 = 10600. N. B. If the multiplicand is a mixed number, reduce the reaction to a decimal of two places before dividing. 2. Give the cost of 12^ yds. cloth @ 18|c. per yd. Process. — 12^ = ^] changing 18| to a decimal, we have 18.75 -r- 8 = $2.34f. Note. — Aliquot parts may be conveniently used when the mul tiplier is but little more or less than an aliquot part. 3. Multiply 24 by 11%. 1st. Multiply 24 by 16f (the one-sixth of 100) Thus 24 X 16f = 2400 ^ 6 = 400 As nf = 16f + 1 multiply 24 by 1 =_24 Hence 24 x 17f = the two products, 424 ^ 49 60 ORTON & Sadler's calculator. 3. To multiply any number by 125 add three ciphers, and divide by 8. Multiply 3467 by 125. Product, 433375. 8 )3467000 433375 Note.— By annexing three ciphers the number is in- creased one thousand times; and by dividing by 8, the quotient will be only one-eighth of 1000, that is 125 times. 4. To multiply any number by 161 add two ciphers, and divide by 6. Multiply 3768 by 16?. Product, 62800. 6 )376800 62800 5. To multiply any number by 1661 add three ciphers, and divide by 6. Multiply 7875 by 166 1. Product, 1312500. 6 )7875000 1312500 6. To multiply any number by 83 J add two ciphers, and divide by 3. Multiply 9879 by 33J. Product, 32930C. 8)987900 829300 . COUNTING-ROOM EXERCISES. 51 Rationale. — As in the last case, by annexing two ciphers, we increase the multiplicand one hun- dred times ; and by dividing the number by 3, we only increase the multiplicand thirty-three and one-third times, because 33J is one-third of 100. 4. To multiply any number by 333J add three ciphers, and divide by 3. Multiply 4797 by 333J. Product, 1599000. 3)4797000 1599000 5. To multiply any number by 6§ add two ci- phers, and divide by 15 j or add one cipher and multiply by f . Multiply 1566 by 6f . 15)156600 ^'^«*— ^ 10440 First method. 15660 2 3)31320 10440 Second method. 6. To multiply any nunber by 66f add three ciphers, and divide by 15 ; or add two ciphers and multiply by f . 52 ORTON & Sadler's calculator. Multiply 3663 by 66^. 15)3663000 244200 First method. 366300 2 3)732600 244200 Second method. 7. To multiply any number by 8J add two ci- phers, and divide by 12. Multiply 2889 by 8J. Product, 24075. 12)288900 24075 8. To multiply any number by 83 J add thre€ ciphers, and divide by 12. Multiply 7695 by 83J. Product, 641250. 12)7695000 641250 9. To multiply any number by 6 J add t\i» phers, and divide by 16 or its f£.ctors — 4X4. Multiply 7696 by 6^. Product, 48100. 4)769600 4)192400 48100 DIVISION OF FRACTIONS, WITH ANALYSIS. As the base of all numbers, whether whole or fractional, are of the same value, inverting any number simply demonstrates the number of times it is contained in a single unit. As the unit takes the place of the number, so must the number take the 'place of the^iinit. Example 1. — Divide 6 by 7. Ans. f. By inverting the divisor we find j, j; now if 7 is contained in one unit ^ of one time, it is con- tained in 6, six times ;^, or ^ Ans, Example 2. — Divide 5 by ^. Ans, 61. I is contained in a unit | times, therefore it is contained in f ; JX4 = V» ^^ ^^ ^'^^• 53 54 orton & Sadler's calculator. Example 3. — Divide 8 J by I. A7is. Hi. By inverting f we have .j, the number of times it is contained in a unit ; therefore it is contained 8i, 8J times i or y X| = %® or lli=llj Ans. (Note. 8iX| = V^oi'lli-) Example 4. — Divide 6} by 3. A7is. 2i. As 3 is contained in a unit J of one time, 61 is contained 62 times J, or thus, y -^-3= y X J = V or 2i ^m. Rule. — To ascertain the number of times the divisor is co7itained in a unit, invert the divisor and multiply by the units in the dividend. Example 5. — Divide 3 by 4. Ans. 5i. Solution. — 4 ^^ contained in a unit | times, there- fore it is contained in 3, 3 times | or ^5' =5i Ans. Note. — Inverting the divisor shows how often it is contiiined in a unit, and multii)lying this number by the dividend gives the quotient of all examples in DWISION OF FRACTIONS. Example 6. — Divide \ by i. Ans. Solution. — } is contained in a unit 4 times, therefore it is contained in |, 1 of 4 times, or I times Ans. Example 7. — Divide f of f by | of J. Statement, J of |_ 3 Note. — Simply reject or cancel factors in the dividend and divisor, tJiaa you Juive i as the quotient. DIVISION OF FRACTIONS. 55 Example 8.— Divide J of f of 3 by ^^ of | of §. 1.^ Statement-^ of ? of f | Rejecting common fac- 304 n^ > toi-s and we have lor ^^ ^^ ^ "^^ the result f for 23^2. 2d Staieme-it. — $ fi a By boxiug the deuominators. 43^ % ^ i. = f|or2Jj^ns. i0 5 5 Rule. — Beject or cancel factors common to the divisor and dividend. Multiply the remaining terms between lines (or inside the box) together for the desired denominator; and the terms above and below the lines (or outside the box) for the desired numerator. Note.— The above rule is commended as the most sim- ple one devised for the division of fractions. Example 9.— Divide i of f by % of i. The application of the above rule to Example 9 produces the following statement, so simple that it can readily be understood by the most ordinary pupil. Draw a figure of four sides, representing a box, then write the example as follows ; 3 2 y_Tj==§8ort 56 ORTON & Sadler's calculator. The numerator of the fraction will be found outside the box, which, multiplied together, will give the desired numerator, thus, 3X2X5X2 = 60 The denominator will be found upon the inside of the box, which, multiplied together, will give the desired denomi- nator, thus, 4X5X4X1 = 80 Note. When factors are common to each other always take advantage of cancellation. An investigation of our new method of treating the division of fractions will prove, no doubt, to be the most simple and practical ever yet devised. MULITIPLICATION AND DIVISION. To multiply i, is to take the multiplicand J of one time; that is, take i of it, or divide it by 2. To multiply by J, take a third of the multipli- cand, that is, divide it by 3. To multiply by f, take i first, and multiply that by 2; or, multiply by 2 first, and divide the product by 3. Example. — If 1 cord of wood cost $6.00, what will be the cost of i of a cord ? 6^otoio/i.— $6.00 X 1= $4.50 3 1800-r-4= 4.50 What will j of a cord cost ? jXi=iH-f=$2.00 MULTIPLICATION AND DIVISION. 57 EXAMPLES. I. What will 360 barrels of flour come to ai 5| aoUars a barrel. At 1 dollar a barrel it would be 360 dollars ; at 5 J dollars, it would be 5 J times as much. 360 5 times, 1800 J of a time, 90 Ans. $1890 Before we attempt to divide by a mixed number, such as 2^, 3J, 5f , etc., we must explain, or rather observe the principle of division, namely: That the quotient will he the same if we multiply the divi- dend and divisor hy the same number. Thus 24 divided by 8, gives three for a quotient. Now, if we double 24 and 8, or multiply them by any num- ber wnatever, ana then divide, we shall still have 3 for a quotient. 16)48(3; 32)96(3, etc. Now, suppose we have 22 to be divided by 5^ ; we may double both these numbers, and thus be clear of the fraction, and have the same quotient. 5^)22(4 is the same as 11)44(4. How many times is \\ contained in 12? Ans, J usi as many times as 5 is contained in 48. The 5 is 4 times 1 J, and 48 is 4 times 12. From these observations, we draw the following rule for divid- ing by a mixed number. 58 ORTON & Sadler's calculator. EuLE. — Multiply the whole number by the lower term of the fraction ; add the upper term to the pro- duct for a divisor ; then multiply the dividend by the lower term of the fraction^ and then divide. How many times is 1^ contained in 36? An», 30 times. N. B. If we multiply both these numbers by 5, they will have the same relation as before, and a quotient is nothing but a relation between two numbers. After multiplication, the numbers may be considered as having the denomination of fifths. How many times is J contained in 12 ? Ans. 48 times. One-fourth multiplied by 4, gives 1 ; 12, multi- plied by 4, gives 48. Now, 1 in 48 is contained 48 times. Divide 132 by 2|. Ans. 48. Divide 121 by 15J. Ans. 8 How many times is | contained in 3 ? Ans. 4 times. By a little attention to the relation of numbers, we may often contract operations in multiplication. A dead uniformity of operation in all cases indi- cates a mechanical and not a scientific knowledge of numbers. As a uniform principle, it is much easier to multiply by the small numbers, 2, 3, 4, 5, tlian by 7, 8, 9. PERGENTA^GE. The greater portion of all arithmetical calcu- lations, as applied to every-day business transac- tions, being based upon percentage, it is important that the foregoing principles and illustrations be thoroughly mastered. Percentage is the process of computing by the hundred. Per cent, or rate per cent, means by the hun- dred, and is represented by the character % in- stead of being written thus, 5%, 20%, 100%. Any % less than 1%, can be written in the form of a fraction, thus, }%, f %, or expressed deci- mally, thus, .002%, .0025%. THE FIVE FACTS To be considered in percentage are : Is^. The Base, 2d, The Rate. M, The Per- centage, 4th. The Amount. 5th. The Difference. The Base — Is the number upon ^vhich the Percentage is calculated. The Rate — Is the number denoting the per cent, (or hundredths) of the Base taken, and is always used as the multiplier. 59 » o » Pi o H 3 P4 Ai PERCENTAGE. 61 The Percentage — Is the sum (in hundredths) obtained from multiplying the Base by the Bate, The Amount — Is the Base increased by add- ing the Percentage. The Difference — Is the Base diminished by subtracting the Percentage, APPLICATION OF PERCENTAGE. Given — The Base and Bate, To Find — T'he Percentage. Rule I. — Multiply the Base by the Rate, ex- pressed decimally^ and point off two places from the right PROBLEM. What is 9% (or the percentage) of $800?— jy Base, $800. Multiply by the rate. rrocess. Rate, .09 BXR= Percentage, $72.00— Point off two figures from the right, and the result is the per cent. $72. Given — The Base and Bate. To Find — The Amount. Rule II. — To the Base add the Percentage. PROBLEM. What is the amount of $800, increased 9% ? p. Base, $800. Plus the Percentage. Percentage, 72. Obtained under Rule I. B+P= Amount, $872. 6 62 ORTON & Sadler's calculator Given — The Base and Rate, To Find — The Difference, Rule III. — From the Base subtract the Per- centage. PROBLEM. What is the Difference (or proceeds) of $800, less 9%? p Base, $800. Minus the Percentage. Percentage, 72. Obtained under Rule I. B—P= Difference, $728.— or Proceeds. Given — The Base and Percentage. To Find— TAe Rate, Rule IV. — Divide the Percentage by the Base, expressed decimally. Note. — When cent* ^ire not shown in the percentage add two ciphers. PROBLEM. Bought of W. H. Sadler, invoice of Orton's Lightning Calculators, for $850, and sold them at a profit of $297.50— what % did I make ? Ans. 35%. Process, — Percentage divided by the Base. Base, $850. Percentage, $297.50. P-J-B==Rate. Base. Percentage. Rate. -850)297.50(35% 2550 4250 4250 PERCENTAGE. 63 Given — Rate and Percentage, To Find— TAe Base, Rule V. — Divide the Percentage by the Rate. Note. — When cents are not shown in the percentage annex two ciphers. PROBLEM. Sold William Callen, Jr., invoice of Orton'a LightDing Calculators, upon which I gained $297.50. Ascertaining my profit to be 35% — what was the amount or cost? Ans. $850. Process. — Percentage divided by the rate. Percentage, $297.50. Rate, 35%. P-T-R=Base. Bate. Percentage. Base. 35)297.50(850. 280 Given — Amount and Rate, To Find— T/ie Base, Rule VI. — Divide the Amount by 100, added to the Rate. Note. — When cents are not shown in the amount annex two ciphers. PROBLEM. Sold John G. Scouten, invoice of Orton^s Lightning Calculators, amounting to $1147.50, and made a profit of 35%. What did the books cost? Ans, $850. 64 ORTON & Sadler's calculator. Process. — Amount divided by 100 plus the rate. Amount, $1147.50. Rate, 35%. 100 in- creased by the rate 35=1.35. Rate. Amount. Base or Cost. 1.35)1147.50(850 1080 675 675 A-T-100+R=Base. Given — The Difference and Bate, To Find— T/ie Base. EuLE VII. — Divide the Amount by 100 less the Rate. Note. — When cents are not shown in the difference annex two ciphers. PROBLEM. Invoice of Orton's Lightning Calculators sold William David, were damaged by water, and he was compelled to sell them for $552.50, thereby losing 35%. What did they cost? Ans. $850. Process. — Difference divided by 100 less the rate. Difference, $552.50. Rate, 35%. 100 less the r.ite 35=65. Bate. Difference. Base or Cost. 65)552.50(850. 520 325 325 Dh-1.00— R=Base. q PROFIT AND LOSS Are terms denoting the gain or loss arising from business transactions. The preceding Rules under percentage are specially adapted to the majority of business transactions ; we therefore call attention to their application : Capital or Cod is treated as the Base. Per cent (%) of profit or loss is treated as the Rats. Sum Gained or Lost is treated as the Per- centage. Selling Price is treated as the Amount. Cost, less the Loss, is treated as the Difference. 65 ee ORTON & SADLER S CALCULATOR. SHORT METHODS in MERCHANDISING. When the Bate is an Aliquot part of $1.00 or 100, instead of following Hule I. of Percentage the labor will be greatly abridged by applying the short method, as explained on page 48. Note. — Aliquot parts of a number are such whole or mixed numbers as will divide it without a remainder. Thus 2, 2^, 3^, and 5 are aliquot parts of 10, being con- tained in it 5, 4, 3, and 2 times. TABLE OF ALIQUOT PARTS. 100 ITon 1 ft. Aliquot parts of 1 10 or 1000 of or 1 A. ~14 5 $1.00 2000 lb. 1 doz. One half is.... 60 500 1000 6 80 sq. rd. One third is. . v. 'i% 33>g 2 One eighth is. % iVi 12K 125 250 20 sq. rd. One tenth is . . A 1 10 100 200 16 " One twelfth is .\ Wz 83}^ I etc. Example.— Multiply 843 X 83}. Proces.?.— Since 83i is ,V of 1000, 83} times any number is j^^ ^^ 1000 times that number. Therefore, to multiply 843 by 83} we simply multiply 843 by 1000, and divide the product by 12, the quotient will be the required product thus: 843 X 1000 = 843G00 -i- 12 = 70250. Is the sura paid for the use of money. Simple Interest is interest on the principal only. Annual Interest is simple interest on the prin- cipal, and on each year's interest from the date of its accruing until paid. Compound Interest is interest allowed on inter- est and principal combined. ^^^ Calculations in compound interest may be abridged by use of tables on page 209. Note. — Interest may be compounded and added to the principal annually, semi-annually, or quarterly, as per agreement between lender and borrower. Accurate Interest is interest calculated on the basis of 365 days to the year. It is reckoned by the usual methods, and ^^^ of the sum de- ducted, except in case of leap year when g^j- of the interest is subtracted. Legal Interest is the rate fixed according to law. Usury is when a higher rate of interest is paid than is sanctioned by law. Note.— See pages 297-298. The Principal is the sum in use and upon which interest is paid. * For a better understanding of the practical calcula- tions of interest, the author refers to other portions of this work. 67 ^S ORTON & Sadler's calculator. The Bate of Interest is the price paid for the use of one dollar. The Amoxint is the principal with the accrued interest added. As in Percentage there are five facts to be considered, viz. : Principal^ Rate per annum, Literest, Time and Amount. APPLICATION OF PERCENTAGE. The Principal is treated as the Base, The Pate, or price paid per annum, is treated as the Pate. The Interest is treated as the Percentage. The Principal and Interest is treated as the Amount. The Time is an additional element in Interest. Given — Principal, RatCy and Time (in days). To Find — The Interest at any rate per cent. EuLE I. — Time in days. Multiply the Prin- cipal hy the Rate, and the product by the time {expressed in days) ; then divide the result hy 36 and the quotient will he the Interest in millsy or Tij^oo oj $1.00. Given — Principal, Rate, and Time (in months). To Find — The Interest at any rate per cent. INTEREST. 69 KuLE II. — Time in months. Multiply the Principal by the Rate, and the product by the Time {number of months); divide the result by 12, and the quotient will he the Interest in cents, or yj^ of $1.00. GiYB^— Principal, Mate, and Time (in yeare). To Find — The Interest at any rate per cent. Rule III. — Time in years. Multiply the Principal by the Rate, aiid the product by the Time {number of years), the result will be the Interest in cents, or yjo of $1.00. Note. — The above rules are not specially recoraraended for general business use, but are here presented to call attention to the principle (percentage) upon which all in- terest calculations are based. For Short Methods and Practical every-day rules the author refers to the portion of this work devoted exclusively to interest calculations. Given — The Principal, Rate, and Time, To Find — The Interest and Amount. Rule. — Calculate the Interest for the Time at the stated Rate, and add to the Principal, the product will be the Amount. Example. — What will $1000 amount to, in- vested for 8 months at 7% interest? PXRXT-r-12 = Interest. Principal, $1000 + 46.67 = $1046.67 Amount. 70 ORTON & Sadler's calculator. Process.— Principal, 1000 X Rate, 7%. 07 7000 X Time in months. 8 12)56000 Product-f-12=Interest $46,666 [S46.67. Given — The Principal, Interest, and Time, To Fi:sT>— The Bate, Rule. — Divide the stated Interest by the in- terest on the Principal, for the Time calctdated at 1% per annum, the quotient will be the Rate. If $5000, invested for 1 year and 6 months, gains $525 — what is the rate? Ans,7%. Proem.— Principal, $5000 X Rate, 1J%, for 1 year and 6 months =$75. Interest at 1% — Stated Interest divided by $75= the Rate. stated Int. Interest for time @ 1%=$75)525(7% Rate. 525 Given — The Bate, Time, and Interest To Find — The Principal Rule. — Divide the stated Interest by the in- terest on one dollar for the stated Time, at the dated Rate. Note. — When cents are shown in the interest annex two, and for mills three ciphers. INTEREST. 71 Example. — What principal will gain $525 interest in 1 year and 6 months, at 7%. Process. — Interest on $1.00 for 1 year and 6 months, is .105, or lOJ cents. stated Int. Interest on $1.00=.105)525000(5000 Principal. 525 000 Given — The Bate, Time, and Amount To Find — The Principal, Rule. — Dioide the Amount by 100, plus the interest on one dollar for the stated Time, at the stated Rate. Note. — When cents are shown in the interest annex two, and for mills three ciphers. Example. — What principal will amount to $5525, in 1 year and 6 months, at 7% ? Process. — Interest on $1.00 for 1 year and 6 months, at 7% =.105, or 10} cents. 100+. 105= 1.105)$525.000(5000 Principal. 525 f^' 000 Given — Principal, Bate, and Interest. To Find— T/ie Tiine. Rule. — Divide the stated Interest by the interest on the Pkincipal, for one year, at the stated Rate. 72 ORTON Sc Sadler's calculator. Note. — The integer or whole number in the quotient will be ihe time in years. For months, multiply the deci- mal or remainder by 12, and divide as before, the quotient will be the time in months. For days, multiply the deci- mal or remainder in months by 30, and divide again, the quotient will be the time in days. Example. — In what time will $5000 amount to $5455? Process. — Principal, $5000. Interest 1 year at 6% =$300. Stated Interest, $455-r-300=Time, 1 year, 6 mouths, 6 days. 300)455(1 year. 300 '155 12 300)1860(6 months. 1800 60 30 300)1800,6 days. 1800 TEST EXAMPLE. In what time will $3000, at 7%, amount to $3570.50? Ans. 2 years, 8 months, 18 days. Note. — When cents are shown in the stated interest annex two ciphers to the interest on the principal for one year, providing cents are not shown, and vice versa. >ii--i>i*ni.'Lj; Is the percentage off or allowance made for the payment of money before maturity. COMMERCIAL DISCOUNT. In this connection the term discount is used without reference to time. It is the sura or per- centage deducted from the Lid or asking price of goods. It is the allowance or deductions made from Invoices or Bills purchased, in consideration for prompt or ca^h payment. Note. — Certain goods usually sold on credit may be bought for less price, providing cash settlements are made. The sum or abatement from the credit price or terms, such as, 2i, 5, or 6% off, is termed discount. Again, on various classes of articles the retail price is fixed by the publisher or manufacturer, and certain de- ductions are allowed to importers or wholesale buyers, which is given in the form of a per cent, off, such as, 25, 33i, and 40%, with further allowances for Net Cash payment. Net Price of an article is the selling or asking price, less the discount. Net Proceeds, or cash value of a bill, is its face with the discount deducted. APPLICATION OF THE RULE of PERCENTAGE Base — The selling price or face of bill. Pate — The rate per cent, of deduction. Percentage — The discount or amount of de- duction. - rjQ 74 ORTON & Sadler's calculator. Rule I. — Multiply the selling price or face of the bill by the rate per cent, of dedxiction, and the product will be the Commercial discount Rule II. — From the selling price or face of the bill deduct the commercial discount, and the differ- ence will be the Net Price, Cash Value, or Net Proceeds. Note. — The practical application of the above rules has, previously, been so fully illustrated that examples here are not deemed necessary. TKUE DISCOUNT Is the difference between the face of the debt and its present worth or value, therefore it is evidently the interest on the present worth from date to the time of maturity. Note. — Every debt or note due at some future time, without interest, has some existing value now, and that value is termed Present Worth ; therefore, Present Worth is such a sum as being placed at interest to the date of maturity as will amount to the stated debt. APPLICATION OF PERCENTAGE. The Present Worth is treated as the Base. The Debt or Face of Bill is treated as the Amount The True Discount is treated as the Difference. PERCENTAGE. 75 TO ASCERTAIN THE PRESENT WORTH. Rule. — Divide the amount of the debt or face of bill by 100 plus the interest on $1.00 for the given time, at the stated rate. Note. — When cents are shown in the interest annex two, for mills three, ciphers, to the debt or bill. TO ASCERTAIN THE TRUE DISCOUNT. Rule. — From the debt or face of bill subtract the present worth. Example. — What is the present worth of S618— Note due in 6 mouths, at 6% ? Process.— Int. on $1.00 6 mos. @ 6% is .03 .03+100=1.03)618.00(600.— Present worth. 618_ 00 618-600=18.— True discount. Proo/.— 1.00X600=$ 618—600= 18. Face of note, $618. 76 ORTON & Sadler's calculator. BANK DISCOUNT Is the Interest paid in advance, or deducted from the face of a note or time draft. Note. — Should the paper offered for discount bear inter- est, bank discount is the interest on the amount due at maturity instead of on the face. In discounting, the time is reckoned by days, and the basis of calculation 360 days to the year. In discounting paper, banks include the day on which the note is discounted and the day on which it matures. Discount is the sum deducted from the face of a note or acceptance, which is the interest for the number of days from date of discount to maturity. Proceeds is the sum given or amount of the note or acceptance, discounted, less the interest. Maturity of a note is the time or date it be- comes due,. including days of grace. Days of Grace are the three days allowed by law for payment, after the expiration of the time specified in the note. Protest is the formal legal notice made by a Notary Public, notifying the maker and endorsers of the non-acceptance or payment of paper for which they are held liable. Note. — A protest for non-payment must be made on the last of three days of grace, unless that day should occur on Sunday, or legally authorized holiday, in which case protest must be made on the day previous. Non-Protest. — In case of non-protest wherein there are endorsers to commercial paper, they are legally released, and the holder can only look to the maker for payment. Protest Waived. — Consent of drawei*s or en- dorsers to hold themselves responsible for pay- ment without the necessity of protest. COMMISSION. The rate of commission or brokerage in gene- rality of cases is established by custom, ranging from i% to 1%. A commission merchant generally gets 2}% for selling, and an additional 2J% if he guarantees the payment. Commission — Is the sum paid by the principal to an agent for selling goodc or property, or col- lecting money. Consignment — Goods sent to a commission merchant to be sold. Consignor — The party sending the goods, or shipper. 77 78 ORTON & Sadler's calculator. Consignee — The party to whom the goods are sent. Proceeds — The sum remaining after all ex- penses are paid. Account Sales — Consignees' written statement to the consignor, showing at what price the goods were sold, the expenses, and the net proceeds. Guarantee — Pledge or security given by the commission merchant for all goods sold on credit. Broker — One who sells or purchases goods, stock, etc., by direction of another, without hav- ing them in his possession. Brokerage — The sum paid a broker for his services. The principles and workings of percentage involved in commission and brokerage are the same as those heretofore treated. CORKESPONDING TERMS. The Base is the amount of sales, investments, or collecting. The Rate is the per cent, allowed for services. The Percentage is the Commission or Brokerage. The Amount or Difference is the Net Proceeds. KuLE I. — To find the Commission or Broker- age, multiply the Base by the Rate. 9 COMMISSION. 79 Rule II. — To find the Rate^ divide the Com- mission by the Base. Rule III. — To find the Base, divide the Com- mission by the Bate. Note. — Wherein a certain sum is supplied a broker for investment or purchases, from which the pay for his com- mission is to be taken ; commission on his own commission not allowed, we have the following Rule. — Divide the sum supplied by 100 plus the rate % of commission^ the quotient will be the Net Proceeds; this sum subtracted from the Amount will give the Commission. Example. — James G. Moulton remits a broker $10,000 with instructions to invest in cotton, his commission, 2^%, which is to be de- ducted — what is the amount of cotton pur- chased? What is his commission? Working amount to be invested, = 100 % Commission on sum, = 2} Total on purchase, = 102i% Sum furnished, $10,000-r-1.025=$9756.10 Inv. $10,000- 9756.10=243.90 Commission. $9756.10 X. 025= 243.90 Commission. TEST example. How many bushels of corn can be purchased for $3485 — the market price being 68 cents per bushel, and your agent's commission for pur- chasing 2i % ? Ans. 5000 bushels. Is a contract issued by companies, wherein they agree for a certain consideration to indemnify the owner or holder of certain property against loss or damage by fire or shipwreck, etc. The Underwriter is the Insurance Agent who acts for the company. The Insured is the party asking for protection, and in whose favor the policy is issued. The Policy is the written contract issued by the company, describing the property, amount of risk, and conditions of indemnity. The Rate or per cent, of Premium is the cost of $100 of insurance. The Premium is the amount paid the com- pany for insurance, and is generally calculated at a certain per cent, on the amount of insurance. PERCENTAGE AS APPLIED TO INSURANCE. The Amount is treated as the Base, The per cent, of Premium is treated as the Rate, The Premium is treated as the Percentage. Given — Amount of Insurance and Rate. To Find — The Premium. Rule. — Amount X by the Rate = Premium, Given — Premium and Rate of Insurance, To Find — The Amount. Rule. — Premium expressed in cents -j- by the Rate = Amount. Given — Amount and Premium. To FmD— The Rate. Rule. — Premium expressed in cents -v- by the Amount = Rate. 80 INVESTMENTS. 81 INVESTMENTS. CAPITAL AND STOCKS. Capital is money invested in business or private enterprises, conducted under individual or co- partnership management. Capital Stock is money or property invested by sundry persons in manufacturing, railroading, banking, etc., and is generally divided into cer- tificates or shares of $100 each. The management of such enterprises is con- trolled by a Board of Directors, from among whose number executive oflSicers are elected or appointed. 82 ORTON & Sadler's calculator. Certificates of Stock are official documents issued by the corporation or company, represent- ing a certain number of shares of the joint capi tal to which the holder is entitled. The Par Value of stocks is the sura or nomi- nal value for which they were issued, as expressed on their face. The Market Value is the sum for which they can be sold. Stocks are at Par when their market value is the same as their face. Stocks are below Par when their market value is less than their face. Stocks are above Par when their market value is in excess of their face value. Note. — Shares representing $100 each, when quoted at $100, are worth par ; when at $110 or over $100 are above par, and when at $90 or less than $100 are below par. Market quotations of stocks are generally quoted at a certain per cent, above or below the par value. The value of stocks depend upon the success and pros- Serity of the business they represent, and per cent of ividend declared. Assessment is the sum called for from the stockholders to make up any deficiency or losses that may arise in conducting the business. Dividend is the sum paid the stockholders, and is a division of the profits of the company. Assessments and Dividends are calculated upon a certain per cent, of the par value of the stock. INVESTMENTS. Qo Brokerage. — The party buying and selling stocks is called a Broker or Stock Jobber, and the compensation received for his services is termed Brokerage. The usual rate of brokerage is i to i per cent, of the par value of stock pur- chased or sold. PERCENTAGE. The majority of business transactions that arise in connection with stocks may be readily calculated by applying the principles of percent- age heretofore shown, as an examination of the following illustrations will show : To ascertain the Cost, including Brokerage. Rule. — To the market value of one share add the brokerage^ and multiply by the number of shares. Example. — What will 100 shares of Balti- more & Ohio Railroad stock cost, market value 127S, brokerage i%? PROCESS. 127^ (Cost of 1 share) + }/^ (Brokerage) = 12-7)^ Cost of 1 sbar©. 1273^X 100 (Number of ahare8,) = $l.:J,750 Cost. To ascertain the number of Shares. Rule. — To the market value of one share ada the brokerage (if any), and divide the sum to be invested by the amount thus obtained. 84 ORTON & Sadler's calculator. Example. — How raany shares of Baltimore & Ohio Railroad stock can be purchased for $12,750? Market value 1271, brokerage i%. Ans. 100 shares. p:^ocess. 127% (Market value) -fV^ (Brokerage) = V2.Vyi Ccst of 1 share. ei2,750 ^ 127}^ = 100 Shares. To ascertain amount of Investment. Rule. — Divide the stated income by the income on one share (which will give the number of shares required). Multiply the number of shares by the cost per share, and the product will be the required investment. Example. — How much capital must Ije invest- ed in New York Central Railroad stocks @ 110, which pay semi-annual dividends of 6%, to realize ^n income of $900 per annum ? Ans. $8250. PROCESS. SOOO (Desired income) -T-$r2 (Income on 1 share) =75 No. of shares. $110 (Coat o! 1 share) X 75 (Number of shares) — ■$>S2o{) luvestmeut. To ascertain the Rate % of iiihome realized from investments. Rule. — Divide the annual dividend or income on one share by the cost per share. Example. — If Railroad shares paying annual dividends of 10% command a premium of 25% — what per cent, of income will be realized from investing in said shares? Ans, 8%. INVESTMENTS. 85 PROCESS. ^0 (Income from I share) ~- $125 (Ccst of 1 share^sr?^. To ascertain at what price stocks must be bought to produce a certain Income. Rule. — Divide the dividend or income on one share by the desired rate of interest or income. Example I. — What amount of premium must stocks bring, paying annual dividends of 12%, to net 9% income to the investor? Arts, 33 J % Premiunf. PROCESS. $12 (Income from 1 8haro)-^9^ (Keq. \nt)=$l23% Value of 1 share. 1333^—100 Par value=33>^ Premium. Example II. — At what price must stock pro- ducing annual dividends of 6% be bought so as to net the investor 9%. Ans. 33i% discount. PROCESS. S6 (Income from 1 8harc)-H9«< CRcq. int.)=?66^.< Value of 1 share. jlOO (Par value of 1 share) — CG% (Market value)=33)/;^ DifecouuL TEST EXAMPLE. At what price must stock of the par value of $50 per share, which pays annual dividends of $3 per share, be bought to produce an income of7J%? Ans.$iO. 8 S6 ORTON &, Sadler's calculator. TABLE FOR INVESTORS. The following Table shows the rate per cent, of Annual Income from Bonds bearing 5, 6, or 7 per cent. interest J and costing from 50 to 125. Purchase Price. 5% 6% n Purchase Price. 5% 6% 7% 50 10.00 12.00 14.00 88 6.68 6,81 7.94 61 9.80 11.76 13.72 89 6.61 6.74 7.86 52 9.61 11.53 13.46 90 5.65 6.66 7.77 53 9.43 11.32 13.20 91 5.49 6.69 7.69 54 9.25 11.11 12.96 92 6.43 6.52 7.60 55 9.00 10.90 12.72 93 6.37 6.4.5 7.52 56 8.92 10.70 12 60 94 5.31 6.38 T.44 57 8.77 10.52 12.27 95 6.26 6.31 7.36 68 8.62 10.34 12.06 96 5.20 6.25 7.29 69 8.47 10.16 11.86 97 S.15 6.18 7.21 60 8.33 10.00 11.66 98 6.10 6.12 7.14 61 8.19 9.83 11.47 99 5.05 6.06 7.07 62 8.06 9.67 11.29 ICO 6.C0 6.00 7.C0 63 7.93 9.52 11.11 101 4.95 5.94 6.93 64 7.81 9.37 10.93 102 4.90 5.88 6.86 65 7.69 9.23 10.76 103 4.85 5.82 6.79 66 7.57 9.09 10.60 104 4.80 6.76 6.72 67 7.46 8.95 10.44 105 4.76 5.71 6.66 6« 7.35 8.82 10.29 106 4.71 6C6 6.60 69 7.24 8.69 10.14 107 4.67 6.60 0.64 70 7.14 8.57 10.00 108 4.02 6.55 6.48 71 7.04 8.45 9.85 109 4.68 5.50 6.42 72 6.94 8.33 9.72 110 4.54 5.45 6.36 73 6.84 8.21 9.58 111 4.60 6.40 6.30 74 6.75 8.10 9.45 112 4.46 5.35 6.26 75 6.60 8.00 9.33 113 4.42 6.30 6.19 76 6.67 7.89 9.21 114 4.38 6.26 6.14 77 6.49 7.79 9.(10 115 4.35 5.21 6.08 78 6.41 7.69 8.97 116 4.31 5.17 6.03 79 6.32 7.59 8.86 117 4.27 6.12 5.98 80 6.25 7.cO 8.75 118 4.23 6.08 5.93 n 0.17 7.40 8.64 119 4.20 6.04 6.88 82 6.09 7.31 8.53 120 4.16 5.C0 5.^3 83 6.02 7.22 8.43 121 4.13 4.95 5.78 84 5.95 7.14 8.33 122 4.09 4.91 5 73 85 6.88 7.05 8.23 123 4.06 4.87 5.69 86 5.81 6.97 813 124 4.03 4.83 5.65 87 6.74 6.89 8.04 125 4.00 4.80 6.60 Definition of Terms Interest is premium paid for the use of money, goods, or property. It is computed by percentage — a certain per cent, on the money being paid for its use for a stated time. The money on which interest is paid is called the principal. The per cent, paid is called the rate ; the prin- cipal and interest added together is called the AMOUNT. When a rate per cent, is stated, without the mention of any term of time, the time is under- stood to be I year. The first important step in the calculation of simple interest is the arranging of the time for which it is computed. The student must study the 87 88 ORTON & Sadler's calculator. following Propositions carefully, if he would be expert in this important and useful branch of buB- iness calculations : PROPOSITION 1. £f the time consists of years^ multiply the principal by the rate per cent,, and that product by the number of years. Example 1. — Find the interest of $75 for 4 years at 6 per cent. Operation. $75 The decimal for 6 per cent, is .06 06. There being two places of decimals in the multiplier, we 4.50 point off two in the product. 4 $18.00 Ans. PROPOSITION 2. If the time consists of years and mx)nt}iSy reduce the time to months^ and multiply the principal hy the rate per cent, and number of months together^ and divide the result by \2. Note. — The work can always be abbreviated at 4, 6, 8, 9, 12, and 15 per cent., by canceling the per cent., or time, or principal, with the common diTiuor 12. INTEREST. 89 Example :i..-— Find the interest of J240 for 2 /ears and 7 months at 8 per cent. First method. Principal, Per cent., In. for lyr., 2yrs.+7mo3., $240 .08 19.20 31mos. 12)595.20 Second method : by cancellation. ^^0—20 8 rate. 31 time. It 49.60 Am, $49.60 Am. The operation by canceling is much more brief. We simply place the principal, rate, and time, an the right of the line, and 12 on the left; then we cancel 12 in 240, and the quotient 20 multiplied with 8 and 31 gives the interest at once. Note. — After 12 is canceled the product of the remaining numbers is always the interest. PROPOSITION 8. If the time consists of years^ months^ and days, re- duce the years to months, add in the given months^ and place one-third of the days to the right of this number, which we multiply by the principal and rate per cent., and divide by 12, as before, or cancel and divide by 12 before multiplying. Example 3. — Find the interest of $231 for 1 ^eur, 1 month, and 6 days, at 5 per cent. 90 OETON & SADLER'S CALCULATOR. First method. Second method : Principal, «231 by cancellation. Per cent., .05 231 pria i$ 5 rate. In, for lyr., 11.55 m-n lyr.+ lmo.+6da. , 13.2mo, - ei2.705 An$. 12)152.460 $12,705 Am. By the second method we cancel 12 in 132, and multiply the quotient 11 by 5 and 231. Note. — When the principal is $, and the time is in years or months, the interest is in cents ; if the time is in years, months, and days, the interest ife m mills, unless the days are less than 3, in which case it would be in cents, as before. Note. — The reason we divide the days by 3 is because we calculate 30 days for a month, and di- viding by 3 reduces the days to the tenth of months. Note. — The three preceding propositions will work any note in interest for any time and at any given rate per cent. How to Avoid Fractions in Interest, PROPOSITION 4. jy, when the time coiisists of years^ months^ and days, are not divisible by 3, you can divide the days by 3, and annex the mixed number as in F'^'tsyosition INTEREST. 91 3, on/ you wish to avoid fractions^ you can reduce the time to interest days, and multiply the principal^ rate and days together , and divide the result by 36 or its factors, 4X 9. Note. — la this case as in the preceding, the work can almost always be contracted by dividing the rate or time or principal with the divisor 36. Note. — We use the divisor 36, because we cal- culate 360 interest days to the year. We discard the 0, because it avails nothing to multiply or di- vide by. Example 4.— -Find the interest of $210 for 1 year, 4 months, and 8 days, at 9 per cent. Year. Months. Days. 1 4 8=16.2J months or 48? days Operation Operation By Prop. 3. By Prop. 4. $210 $210 .9 9 18.90 16.2S 12)307440 $25,620 Ans, 18.90 488 36)922320 $25,620 Am We will now work the example by cancellation to show its brevity. 92 ORTON & Sadler's calculator. Operation hy Gancellatum. Time 488 days. 210 4-J0 m$ 122 122 210 $25,620 Now cancel 9 in 36 goes 4 times, then 4 into 488 goes 122. Now multiply remaining numbers to- gether, thus, 210x122 and we have the interest at once. When the days are not divisible by 3 we reduce the whole time to days ; then we placo the princi- pal rate and time on the right of the line. Now, because the time is in days, we place 36, on the left of the line for a divisor. {1/ the time wcu months we would place 12 on the left.) Note. — A very short method of reducing time to interest days is to multiply the years by 36 ; add in 3 times the number of months and the tons* ftgure of the days, and annex the unit figure ; but If the days are less than 10 simply annex them. EXAMPLE 1. — Reduce 1 year, 2 months, and 6 days, to days. Tears. Months. Days. 36X1+3X2=42 annex 6=426 Am. SIMPLE INTEREST BT CANCELLATION. 93 Example 2. — Beduce 2 years, 3 months and 17 days, to interest days. Tears. M'ths. Days. Days. 36x2+3x3+1=82. annex 7=827 days Aua. Note. — The student should commit to meiiiory the multiplication of the number 36 up as far as 9 times 36, and then he can reduce almost in- stantly years, months, and days, to days. SIMPLE INTEREST BY CANCELLATION. Rule. — Place the principal^ time^ and rate per cent on the right hand side of the line. If the time consists of years and months^ reduce them to months j and place 12 (the number of months in a i/ear) on the left hand side cf the line. Should the tims con- mt of months and days^ reduce them to days or deci- mal parts of a m^onth. If reduced to days, place 36 on the left. If to decimals parts of a month, place 12 only as before. Point off two decimal plaices when the time is in months, and tnree decimal places when the time is in days. Note. If the principal contains cents, point oflF four decimal places when the time is in monthn and five decimal places when the time is in day^. 94 ORTON & Sadler's calculator. Note. — Wt 'place 36 on the left hecause there ^4 300 interest days in a year, (^Ctistom has made tht& lawful.') Example 1. — What is the interest on $60 foi 117 days at 6 per cent? Operation. Here 117X0 00 Both sixes on the must be the $0 right cancels 36 on answer. 117 the left, and we have nothinir left $1,170 Am. to divide by. In this case we point oflf three decimal places be- cause the time is in days. If the time had been 111 months, we would have pointed oflf but two deci mal places. Example 2. — What is the interest of S96.50 for 90 days at 6 per cent? Operation. 96.50 9650 0— $0 00 — 15 15 1.44.750 An$, Now cancel 6 in 36 and the quotient 6 into 90, and we have no divisor left. Hence 15x96.50 must be the answer. Note — As there are cents in the principal, we point off five decimals ; three for days and two foi oents Pay no attention to the decimal point trntil ihe close of the operation. SIMPLE INTEREST BT OANOELLAnON. 95 Example 3. — What is the interest of $480 for 361 days at 6 per cent? ^g0— 80 361 0— J0 361 80 828.880 Am. Now cancel 6 in 36 and the quotient 6 into 480, and we have no divisor left. Hence 80x361 must be the answer. Example 4.— What is the interest of $720 for 9 months at 7 per cent? tn—^^ 60 n 9 9 7 540 7 $37.80 Arts, Now cancel 12 in 720 there is nothing left to divide by. Hence 60x9x7 must be the answer. N. B. When interest is required on any sum for days only, it is a universal custom to consider 30 days a month, and 12 months a year ; and, as the unit of time is a year, the interest of any sum for one day is j^^, what it would be for a year. For 2 days, 3§o, etc.; hence if we multiply by the days, we must divide by 360, or divide by 36 and save labor. The old form of this method was to place 360, or 12 and 30, on the left of the line, but using 36 is much shorter. 96 OBTOK dc Sadler's calculator. WHEN THE DAYS ARE NOT DIVISIBLE BT THEEBi Note. — When the time consists of months and days, and the days are not divisible by three, re- duce the time to dayi. Example 5 —What is the interest of *960 fof 1 1 months and 20 days at 6 per cent? Months. DajB. Operation. 11 20=350 days. 000—160 350 ^ —36 350 160 6 $56,000 Now cancel 6 in 36 and the quotient 6 into 960, and we have no divisor left. Hence 160X 350 must be the answer. Example 6. — What is the interest of $173 for 8 months and 16 days at 9 per cent? Months. DayB. Operation. 8 16=256 dayg, 173 173 4r-H 64 ??0— 64 $11,072 Aru, Now cancel 9 in 36 and the quotient 4 into 256, and we have no divisor left. Hence 64X173 must be the answer. N. B. Let the puj il remember that this is a gen- eral and universal method, equally applicable to any per cent, or any required time, and all other rules must be reconcilable to it ; and, in &ct, all other rules are but modifications of this. SIMPLE INTEREST BY CANCELLATION. 97 Example 7. — What is the interest on $1080 Tor 7 months and 11 days at 7 per cent? UoQtks. Days. 7 11=221 days. Operation. 10$0— 30 221 10 221 30 7 6630 7 $46,410 Ans. Now cancel 36 in 1080 and we have no divisor left, hence 30X221X7 must be the answer. WITH more difficult TIME AND RATE PER CENT. Example 8. — What is the interest of $160 for 19 months and 23 days at 4^ per cent? Months. Dajb. 19 23 =-593 days. Operation. 160—20 593 t^H 593 20 ^ $11,860 Ans. Now cancel 4| in 36 and the quotient 8 into 160 we have no divisor left, hence 20x593 must be the interest. when the DATS ARE DIVISIBLE BY THREE. EuLE. — Place taken decimally and annexed to the number of months, and this number, divided by 12, carried out decimally. But this makes the multiplier very large ; hence, to avoid this large number id 120 ORTON & Sadler's calculator. the multiplier, where the days are divisible by 3, and this number, annexed to the months, is not divisible by 12, we use the following rule, called our base at 12 per cent. : KuLE. — Reduce the years to months^ add in the months^ take one- third of the number of days and annex to this number^ multiply the principal by the number thus formed ; if there are cents in the prin- cipal^ point off five decimal places ; if there are no cents in the principal^ point off three decimal places ; this gives the interest at 12 per cent. For any other rate per cent.^ take such part of the base before mul- tiplying as the required rate is a part of 12, EXAMPLE. Required the interest on $123, at 12 per cent for 2 years, 2 months and six days. SOLUTION. Reduce the 2 years to months gives us 24 months, add on the 2 months gives us 26 aionths, take one-third of the days, J of $123 6=2., annexed to the 26 months gives 262 262, which constitutes the base ; multiply — the principal by this base, and you have $32 .226 the interest at 12 per cent. EXAMPLE. Required the interest on $144, at 6 per cent., for 4 years, 5 months and 12 days. MERCHANTS* METHOD OP COMPUTING INT. 121 SOLUTION. Reduce the 4 years to months gives 48 months, add in the 5 months gives 53 months, take one- third of the days and annex to the number of months, J of 12=4. annex to the 53 months, 534 ; this number multiplied into the principal would give the interest at 12 per cent. But we want it at 6 per cent. We will now take such part of either principal or base as 6 is a part of 12 ; 6 is J of 12, therefore we will take J of 144=72 otie-half of the principal, and mul- 534 tiply it by the base, which will give the interest at 6 per cent, $38,448 EXAMPLE. Required the interest on $347 25, at 8 per cent., for 2 years, 3 months and 9 days. SOLUTION. Reduce the 2 years to months, 24 months, add the 3 months, 27 months, take one-third of the days, J of 9=3, annex to th^ months, 273, the base; this, multiplied into the principal, would give the interest at 12 per cent. But we want the interest at 8 per cent ; we will take two-thirds of the base before multiply- $347 25 ing! f of 273=182; the principal 182 multiplied by this number gives the interest at 8 per cent. $63.19950 Remark. — This base is used where the days are divisible by 3, and the number formed by annex- 11 122 ORTON & Sadler's calculator. ing one-third of the days to the months not divisi • ble by 12. We now come to time in which neithei days nor months are divisible. Where such time as this occurs, we use a base at 36 per cent. Rule. — Reduce the time to dai/s, hy multiplying the years hy 12, adding in the months^ if any^ and multiplying this number 6y 30, adding in the days, if any; multiply the principal hy this number^ pointing off 5 decimal places^ where cents are given in the principal, and 3 places where no cents are given. This will give the interest at 36 per cent EXAMPLE. Required the interest on $144, at 36 per cent., for 3 years, 2 months and 2 days. SOLUTION. Reduce the time to days gives 1142 $144 days ; multiply the principal by this base, 1142 and you have the interest at 36 per - cent $164,448 EXAMPLE. Required the interest on $144, at 9 per cent., lor 5 years, 7 months and 5 days. SOLUTION. Reduce the time to days gives 2,015 days; if we multiply the principal by this base, we would get the interest at 36 per cent.; but we want it at .9 pei cent. We can ^ake such part of either MERCHANTS' METHOD OF COMPUTING INT. 123 principal oi base as 9 is a part of 36 before multi- plying ; 9 is J of 36 ; wo will take J of the prin- cipal, it being divisible by 4 ; J of 144=36, 2915 which, multiplied into the base, will give 36 the interest at 9 per cent., by pointing ofi 3 decimal places. S72.540 EXAMPLE. Bequired the interest on $875 15, at 6 per cent.» for 5 years, 7 months and 12 days. SOLUTION. Reduce the time to days gives 2022 days ; 6 is J of 36 ; take one sixth of the base, i of 2022=337; multiply the prin- $875 15 cipal by this number, point off 5 dec- 337 imal places, and you have the interest at 6 per cent., the required rate. $294.92555 Remark. — We have now fully explained our method of computing interest at the three different bases. Any and every problem in interest can be solved by one of these three bases. Some prob- lems can be solved easier by one base than another. Where the days are divisible by 3, and their num- ber, annexed to the months, divisible by .12, it is the shortest and best method to use the base at 1 per cent. By using one or the other of these three bases, the student can avoid the use of vulgar fractions. The student must study these three principles carefully, and learn to adopt readily the base best suited to the problem to be solved. To compute interest on notes, bonds, and mon gages, on which partial payments have been made, two or three rules are given. The following is called the common rule, and applies to cases where the time is short, and payments made within a yeai cf each other. This rule is sanctioned by custom and common law; it is true to the principles of simple interest, and requires no special enactment. The other rules are rules of law, made to suit such cases as require (either expressed or implied) an- nual interest to be paid, and of course apply to no business transactions closed within a year. Rule. — Compute the interest of the principal sum for the whole time to the day of settlement, and find the amount. Compute the interest on the several pay- ments, from the time each was paid to the day oj settlement ; add the several payments and the vater- est on each together, and call the sum the amount oJ the payments. Subtract the amount of the payments from the a/mount of the principal, will lextvc the sitm due, 124 PAETIAL PAYMENTS. 125 EXAMPLES. I. A gave his Dote to B for $10,000 ; at the end 0^4 months, A paid $6,000; and at the expiration of another 4 months, he paid an additional sum of $3,000 ; how much did he owe B at the close of the year? By the Oommon Rule, Principal $10,000 Interest for the whole time 600 Amount $10,600 1st payment $6,000 Interest, 8 months 240 2d payment 3,000 Interest, 4 months 60 Amount $9,300 9,300 Due $1300 PROBLEMS IN INTEREST. There are four parts or quantities connected with each operation in interest : these are, the Principal, Mate per cent, Time, Interest or Amount If any three of them are given the other may be found. Principal, interest, and time given, to find the rate per cent. 1. At what rate per cent, must $500 be put on interest to gain $120 in 4 years ? 126 ORTON & Sadler's calculator. Operation. Bj analysis 8500 The interest of .01 $1 for the given — - time at 1 per cent. 6.00 is 4 cents. 8500 4 will be 500 times asmuch=:500X.04 20.00)120.00(6 per cent., Ans. =$20.00. Then if 120.00 $20 give 1 per cent., $120 will give ^^jj^ =z6 per cent. Rule. — Divide the given interest hy the interest of the given sum at I per cent, for the given time, and the quotient will be the rate per cent, required Principal, interest, and rate per cent, given, to find the time. 2. How long must $500 be on interest at 6 per cent, to gain $120 ? Operation By analysis. $500 We find the in- .06 terest of $1.00 at the given rate for 30.00)120.00(4 years, Ans. 1 year is 6 cents 120.00 $500, will therefore be 500 times as much=500X .06=:$30.00. Now, if it take 1 year to gain $30, it will require ^V* to gain $120=4 years, Ans, PARTIAL PAYMENTS. 127 Rule. — Divide the given interest hy th^ interest of the principal for 1 ^ear, and the quotient is the time. Given the amount, time, and rate per cent., to find the 'principal. Rule. — Divide the given amount hy the amount o/"?!, at the given rate per cent., for the given time. Remark. — This rule is deduced from the fact that the amount of different principals for the same time and at the same rate per cent., are to each other as those principals. BANK DISCOUNT. Bank Discount is the sum paid to a bank for the payment of a note before it becomes due. The amount named in a note is called the face of the note. The discount is the interest on the face of the note for 3 days more than the time specified, and is paid in advance. These 3 days are called days of grace, as the borrower is not obliged to make payment until their expiration. Hence, to compute bank discount, we have the fo^ lowing Rule. — Find the interest on the face of the note for 3 days more than the time specif ^d ; this will he the discount. From the face of the note deduct the discount, and thi remainder will be the PRESENT VALUE of (he note. 128 ORTON & Sadler's calculator. DISCO riNT, OR COUNTING BACK. The object of discount is to show us what al« lowance should be made when any sum of money Is paid before it becomes due. The present worth of any sum is the principal that must be put at interest to amount to that sum in the given time. That is, $100 is the 'present worth of $106 due one year hence; because SlOO at 6 per cent, will amount to $106 j and $6 is the discount, 1. What is the present worth of $12.72 due one year hence ? First method. Second method. $12.72 $ 100 1.06)12.72($12 Am. 10.6 106)1272.00($12 Am, 106 2.12 2.12 212 212 As $100 will amount to $106 in one year at 6 per cent., it is evident that if |gg of any sum be taken, it will be its present worth for one year, and that ygg will be the discount. And as $1 is the present worth of $1.06 due one year hence, it r evident that the present worth of $12.72 must be equal to the number of times $12.72 will contain n.06. EQUATION OP PAYMENTS. 129 Rule. — Divide the given sum hy the amount of A for the given rate and timCy and the quotient wilt >e the present worth. If the present worth be sub- tracted from the given sum, the remainder will be the discount. ijEDUATlDNOFPAYMENTSi Equation op Payments is the process of find- ing the equalized or average time for the payment of several sums due at different times, without losa to either party. To find the average or mean time of payment, when the several sums have the same date. Rule. — Multiply each payment by the time thai must elapse before it becomes due; then divide the sum of these products by the sum of the payments, and the quotient will be the averaged time required. Note. — When a payment is to be made down, it has no product, but it must be added with the other payments in finding the average time. Example 1. — I purchased goods to the amount of $1200; $300 of which I am to pay in 4 months, $i00 in 5 months, and $500 in 8 montls. IIow long a credit ought 1 to receive, if 1 pay th* wbolo sum at once? -4ns. 6 uoaths. 130 ORTON & Sadler's calculator. Mo. Mo. r A credit on $300 for* xflDnthsIl ^vyOAA 1 OAA •< the same as the credit on $1 foi 4X'3UU=1ZUU h200 months. ' i" A credit on 8400 for 6 months ii 5<400r=:2000 -{the same as the credit oa 81 for < 2000 months. Qv^ r AA Af\(\f\ \ A credit on 8500 for 8 months is O/^UUU ^Kjyjyj ^ the same as the credit on $1 for — (4000 months. 1 0AA\ TOAA /a Therefore, I should have th© IZUUj os., 9 00 •'"'^ ^ ^''"i " Iday, 05 JunelO...S500| » iq days, 1 67 " 30.... 300 j " 6 mos., 18 00 Total Cr. J^ "800 Tot Cr.Int. 41 CT Total Dr.") /Total Dr.") of * j 1357) of Int. i66 26 Total Cr. ) / Total Cr. ) of ^ / 800 \ of Int. J 44 67 Bal. of * ~lui Bal. of Int. 21 59(3 months. 16 7 1 4 88 30 14640(26 + days. 1114 3500 334 3 158 Ans. — Balance of 5^ due in 3 mos., 26 days, from Dec. 31, or April 26. We regret that our limited space will not permit a more detailed exhibit of the method, either as to its philosophy or its facts. The above working must carry its own sug- gestions. The theory is, that if a settlement were made as on the preceding 31st of December (the assumed day of settlement), the debit side of the account, as shown, would be entitled to $66.26 discount, and the credit side to $44.67, making a balauce of $21.59 in favor of the debit 138 ORTON «& Sadler's calculator. side. As the balance of account is also in favor of the debit side, it is only necessary to know how long it would take the balance of account to produce the balance of in- terest (at the rate of 1% a month, or 12% per annum*) ; to know the time — reckoned forward — when the balance of ac- count falls due. Now, as the rate named (1% a month), the interest for one month can be had by merely cutting off two figures from the right of dollars, we have the bal- ance of account thus divided ($5.57), a ready divisor of the balance of interest ($21.59), the quotient being the number of months and parts of a month it will take the balance of account to produce the balance of interest. This time reckoned forward from the assumed focal date, will get the average date of payment. Thus, 3 months, 26 days from December 31 will bring the average date as stated — April 26th. *FORMULA. For calculating Interest at\2^o per annum. Time — Months and days. Multiplied by the 1 __Interest expressed The Principal J number of months. J in cents. Dollars only. ] Multiplied by i ) __Interest expressed the number of days. J in mills. J the Principal f Multiplied by any | __Interest expressed Dollars only. ( number of days. J in mills. Note. — In the above application, when the j)rincipal contains cents, point off two additional decimal places. ^^=- cow7i\ Classification, Real ; Closed To or By Balance, Debit oj Account shows our charges against them or indebtedness in our favor. Credit of Account shows their charges against us, or our indebtedness in favor of others. The Difference, if in favor of the debit side, shows a Resource, or the sum due us from the parties represented by the account. If in favor of the credit side, a Liability, or amount we owe them. Note. — When the balance is in our favor the account is termed a Personal Account Receivable; when against us a Personal Account Payable. u. ond'tanryien^ Accounts representing consignments of mer- chandise or property received from others, to be sold for their account and risk, are treated the same as Personal Accounts. The debit side of LEDGER ACCOUNTS. 169 the account showing amount of expenses in- curred, and the credit side the returns. Q/iiett>fhwriwu^. Classification, Kepresentative ; Closed To or By Loss and Gain, Debit of Account shows Cost of goods pur- chased. Credit of Account shows Sales or Proceeds. Note. — If the goods are not all sold or disposed of, it will be necessary to credit the account with the inventory or market value of the unsold quantity. The Difference shows a gain or loss according to the excess of the sides. If in favor of the credit, Si Gain; of the debit, ^ Loss. Note. — This account may be made to comprise all properties purchased for traffic, such as groceries, dry- goods, flour, produce, hardware, crockery, etc. ; or, if de- sired, each kind may be represented under its own separate heading. ^^m^7 fun/menu ¥(> A special account under the above heading is frequently used to represent merchandise or prop- erty which we have consigned away to be sold for our account and risk. The account is treated similar to Merchandise. It is 15 170 ORTON & Sadler's calculator. Debited with the cost or outlay. Credited with the returns. Note. — At closing the Ledger, if account sales or full returns have not been received, it will be necessary to credit the account with the value represented, or inven- tory, same as in Merchandise. The Difference shows a Gain or Loss, (See Merchandise account.) Classification, Representative; Closed To or By Loss and Gain. Debit of Account shows Cost or Outlay. Credit of Account shows Proceeds from Sales, or Income. Note. — Add inventory, if any, same as in Merchandise. The Difference shows Gain or Loss according to the excess of sides. (See Merchandise ac't.) ^^foU Q^io^'^wted Classification, Representative ; Closed By Loss mid Gain. Debit of Account shows Cost or Outlay. Credit of Account shows Returns, if any. If LEDGER ACCOUNTS, 171 value is estimated, add as Inventory in Mer- chandise. The Difference will show a Loss, Classification, Eepresentative ; Closed By Loss and Gain. Debit of Account shows Expenditures and out- lay for conducting the business. Note. — In case there should be returns from expense expenditures, the account would be credited. The Difference shows a Loss. n¥etm Classification, Representative; Closed To or By Loss and Gain. Debit of Account shows sums paid for use of others' money. Credit of Account shows sums received for use of our money. The Difference, if in favor of the credit side, shows a gain ; when in favor of the debit side, a loss. 172 ORTON & Sadler's calculator. t^'t>ovo7^¥, Classification, Representative ; Closed To or By Loss and Gain. Debit of Account shows sums paid or allowed by us. Credit of Account shows sums received or al- lowed us. The Difference, if in favor of the credit side, shows a gain ; when in favor of the debit side, a nydw^wnt>&. Classification, Representative; Closed To or By Loss and Gain, Debit of Account shows amount paid for in- surance. Credit of Account shows amount received from others for insurance. The Difference, if in favor of the credit side, shows a gain ; when in favor of the debit side, a LEDGER ACCOUNTS. 173' Classification, Representative; Closed To or By Loss and Gain. Debit of Account shows suras paid or allowed for services in buying or selling. xjredit of Account shows suras received or al- lowed for services in buying or selling. Tlie Difference, if in favor of the credit side, shows a gain ; when in favor of the debit side, a loss, (or, Frofii and Loss) exhibits Net Gains or Losses. Closed To or By Stock or Capital^ wherein single proprietorship is represented. Li Partnership business the account is closed To or By Partners^ Account, showing each partner's respective share in the division of gains or losses. Debit of Account shows the total Losses arising from the business. Credit' of Account shows the total Gains pro- duced from the business. The Difference shows a Net Gain or Net Loss, according to the excess of sides ; a Net Gain if 174 ORTON &, Sadler's calculator. the difference is in favor of the credit side, and a Net Loss when in favor of the debit side. Note. — In closing the account, the Net Gain is carried to the credit of Stock or Partners^ account, showing an increase of capital. The Net Loss is carried to the debit of Stock or Partners^ account, showing a diminution of capital. Gains or Losses are divided between partners in accordance with the articles of copartnership or agreement made between them. The methods of adjustment will be found fully illustrated un- der the head of Partnership ^ page 139 of this work. t>ct>wrVv- Is a special account kept with the proprietor to show his transactions with the business as an in- dividual. The account is DEBITED With his withdrawals of money or the appro- priation of any value belonging to the business to his private use, such as personal, household, or living expenses. The account is closed By Stock or Capital^ and total amount carried to the debit of the Stock or Capital account. CLOSING THE LEDGER. 175 Are special accounts kept with each partner, showing their transactions with the firm as indi- viduals. The accounts are DEBITED For precisely the same reasons as explained in the individual proprietor's Private Account. They are closed By Partner Sy and each total debit, as shown, carried to the debit of the respective partner's general or Capital Account. HOW TO CLOSE THE LEDGER. Inventories. — Take an inventory of all unsold property and credit the account, showing the cost of same when purchased. This entry is made with red ink, thus : By Bal- ance Inventory, or By Inventory. After ruling, the amount is brought down to the debit of the new account in black ink. Representative Accounts. — Those showing Gains or Losses are closed after crediting the inventory, if any, by writing on the smaller side, with red ink, To Loss and Gain, or By Loss and CLOSING LEDGER ACCOUNTS. 176 CLOSING THE LEDGER. 177 Gain, The excess as shown is carried to the debit or credit of the Loss and Gain account, the entry being made in black iuk. Real Accounts. — Those showing Resources or Liabilities are closed by writing on the smaller side, in red ink, To Balance or Bfj Balance, After ruling, the balance of excess as shown is brought to the debit or credit of the new account. Loss and Gain. — After closing all Bepresenta- live accounts, and the excess of gains or losses having all been brought forward, write on the smaller side, in red ink, To Stock or Partners' or By Stock or Partners'. The difference will show the Net Gain or Net Loss, which carry to the debit or credit of Stock or Partners* account. Stock or Partners' Account. — Write on the smaller side, in red ink. To Balance or By Bal- ance. Rule up the account and bring down the balance as shown to the debit or credit of the new account in black ink. Proof of Work. — Take a Trial Balance after closing, and if your work is correct, the Ledger will be in equilibrium. ERKOES IN TRIAL BALANCES. VALUABLE RULES FOR THEIR DETECTION. trom Bryant & Stratton's Business Arithmetic, with kind permission of the author, 11, B. Bryant, President Jiryant & Stratloa Buainetjs College, Chicago, Illinois. In keeping accounts by Double Entry, each item appears in at least two different accounts, on the Dr. side in one and on the Cr. side in the other ; hence the sum of all the debit entries in all the accounts will equal the sum of all the credit entries in all the accounts, and the sum of the Dr. balances will equal the sum of the Cr, balances, if all the entries are properly made. 178 ERROES IN TRIAL BALANCKS. 179 ILLUSTRATION OF DOUBLE ENTRY. Dr. merchandise. Cr. May $ 75 380 1200 725 1120 $3500 Apr. I 1 111 ay 10 By Ciisfi $3000 500 Dr. Al>r. i 1 ToMdse. May 10 CASH. $3000 500 Cr. 380 liOO 725 1120 A TRIAL BALANCE Is a summary of the entire amounts entered on the Dr. and Cr. side of each account, or simply of the balances of all the accounts in detail, and if the sums of the debit and credit entries in the Trial Balance are not equal, then there is some error in the accounts or in making up the Trial Balance, which should be discovered and cor- rected. The first rule of the book-keeper should be to make no error; but for such as are fallible the following suggestions may be of some practical utility. 180 ORTON & Sadler's calculator. 1. If the error be found in one figure only, it is probably an error of adding or copying. 2. If it involve several figures, it may have arisen from the omission of an entire entry or from making the same entry twice. 3. If it be divisible by 2, without a remainder, it may have arisen from posting an item to the wrong side of the account, in which case the item would be half of the apparent error. 4. If the error be divisible by 9, without a re- mainder, it may have arisen from transposition, three cases of which may be easily detected by rules founded on the peculiar property of the number 9. These cases are — 1st. When two figures are made to exchange places with each other, the orders in notation re- maining the same : e. g., 372 made to read 327, or 732, or 273. 2d. When two or more figures are made to change their places in notation, their arrangement in re- spect to each other remaining the same : e. gf., $4275 made to read $42750, or $42.75, or $427.50. 3d. When two significant figures are made to change position both with respect to each other and also the orders of notation : e. g., $14 made to read $0.41. 5. To detect the first and second cases of trans- position divide the amount of the error hi the trial balance successively hy 9, 99, 999, 9099, e^a, as far as ERBORS IN TRIAL BALANCES. 181 pofi'iible without a remainder, rejecting all ciphers at the right of the last significant Jigure in the error. The quotients that contain hut one digit figure will express the difiference between the two digit figures transposed, which will be adjacent to each other it* the divisor consist of but one 9, separated by one other figure if it consist of two 9's, by two other figures if it consist of three 9*s, and so an. 'I'hose quotients wliich contain two or more figures will express the member itself, which is transposed in notation simply, the arrangement of the significant figures remaining the same. In either case the order of the last significant figure in the error will be the lowest order of the figures transposed. The orders of the other figures can be easily determined by referring to the error and applying the principles of notation. 6. To detect the third case, divide the error in the balance by as many 9's as is passible so as to give only a single figure in the quotient, and then the remainder in the same way, rejecting all ciphers at the right of the last significant figure in both dividends, after which there should be no remainder. The first quotient will be the figure filling both the highest and lowest order in the transposition ; the second quotient will be the other figure. NoiF, —If tlie error of the tria^ baianco ba not divisible by 9 it caa- tiot be the n'diilt of tVAnsposition alono. Hut whenever the error becomes n;> re'hiced m to be divisible by 9 without a remainder, a tva'ii;)06Ui(ju being then possible, the above todta boduIU bo a])i)lied. 16 182 ORTON & Sadler's calculator. WE ADD THE FOLLOWING IMPORTANT SUGGESTIONS. First. — Examine the Cash and Bills Receivable accounts ; the balance can never appear on the credit side, and should equal the amount of cash and notes on hand, as shown by the 0. B. and B. B. Second. — Refer to the Bills Payable account ; the balance shown must appear in favor of the credit side and equal the amount of our outstanding paper. Third. — If the cash book is kept as an original book of entry, see that the balance on hand from previous month has been deducted. Fourth. — If the error appears only in the cent or dollar columns, it is not necessary to add the columns to the left. Fifth. — If, after applying the above tests, the error still exists, it will be necessary to go over the entire work. Re-add carefully the debit and credit sides of all the accounts, as the error undoubtedly lies in the addition. If not found, examine each posting separately, check from the journal or book posted from to the ledger, and vice versa, as you proceed with your work. Note. — In the use of these rules in practice, not cnJy the balances of the ledger accounts as they appear on the balance-sheet should be examined, but also all the separate postings, as a transpQsitiou there will equally affect the final balance. MEASUREMENT OF LUMBER. The unit of board measure is a square foot 1 inch thick. To measure inch hoards. Rule. — Multiply the length of the board in feet by its breadth in inches, and divide the product by 12 ; the quotient is the contents in square feet. Note. — When the board is wider at one end than the other, add the width of the two ends together, and take half the sum for a mean width. Example. — How many square feet in a hoard 10 feet long, 13 inches wide at one end, a^^d 9 inches wide at the other ? Process. — (13 + 9) -j- 2 = 11 (mean width} hep 10 length X 11 = 110 -^ 12 = 9^ feet. Arts. 183 184 ORTON & Sadler's calculator. Sawed lumber, as joists, plank, and scantlings, are now generally bought and sold by boai^d meaS' ure. The dimensions of a foot of board measure are 1 foot long, 1 foot wide, and 1 inch thick. To ascertain the contents (board measure) of boardSy scantling, and plank. Rule. — Multiply the width in inches by the thickness in inches, and that produc<^ by the length Infeety which last product divide by 12. Example. — How many feet of lumber in 14 planks 16 feet long, 18 inches wide and 4 inches thick? Process.— 16 feet X 18 inches X 4 inches =1152, then 1152 -T- 12 = 96 feet = contents of one plank. 96 X 14 = 1344 feet. Ans. To find hoiv many feet of lumber can be sawed fro'n a log. (Gauge of saw \ inch.) Rule. — From the diameter of the log in inches, subtract 4 (for slabs), one-fourth of this remainder squared and multiplied by the length in feet will give the correct amount of lumber that can be made from any log whatever. Example. — How many feet of lumber can be made from a log which is 36 inches in diameter, and 10 feet long ? Process. — From 36 (diameter) subtract 4 (f<»r slabs) = 32, then divide the 32 by 4, making 8. which squared = 64, then multiply the 64 by 10 (length) = 640 feet. Am. MEASUREMENT OF LUMBER. 185 To find how many fed of lumber there are left in a log after it is made perfectly square. Rule. — Multiply the diameter in inches at the small end by one-half the number of inches, and this product by the length of the log in feet, which last product divide by 12. Example. — If the diameter of a round stick of timber be 22 inches, and its length 20 feet, how much lumber will it contain when hewn square? 22 X 11 X 20 Half diameter = 11, and = 403 J ft., 12 the lumber when hewn square. To find how many feet of square edged boards, of a given thickness j can be sawed from a log of a given diameter. Rule. — Find the quantity of lumber in the log, when made square, by the last Rule ; then divide by the thickness of the board, including the saw calf, the quotient is the number of feet of boards. Example. — How many feet of square edged boards, li inch thick, including the saw calf, can be sawn from a log 20 feet long, and 24 inches diameter? 24 X 12 X 20 = 480 ft., the lumber when hewn sq. 12 Then 480 divided by It »= 384 feet. Ans. MEASUREMENT OF WOOD. Wood is measured by the cord, which contains 128 cubic feet. Wood is bought and sold by the cord and frac- tions of a cord. Pine and spruce spars from 10 to 4 inches in liameter inclusive, are measured by taking the dia- meter, clear of bark, at one-third of their length from the large end. Spars are usually purchased by the inch diame- ter ; all under 4 inches are considered poles. Spruce spars of T inches and less, should have 5 f-^et in length for every inch in diameter 186 MEASUREMENT OP WOOD. 187 Note. — A pile of wood that is 8 feet long, 4 fecJl high, and 4 feet wide, contains 128 cubic feet, or a cord, and every cord contains 8 cord-feet; and as 8 is j'g of 128, every cord-foot contains 16 cubic feet ; therefore, dividing the cubic feet in a pile of wood by 16, the quotient is the cord-feet; and if cord-feet be divided by 8, the quotient is cords. Note. — If we wish to find the circumference of a tree, which will hew any given number of inches square, we divide the given side of the square by .225, and the quotient is the circumference re- quired. What must be the circumference of a tree that will make a beam 10 inches square ? Note. — When wood is " corded" in a pile 4 feet wide, by multiplying its length by its hight, and dividing the product by 4, the quotient is the cord- feet ; and if a load of wood be 8 feet long, and its hight be multiplied by its width, and the product divided by 2, the quotient is the cord-feet. IIow many cords of wood in a pile 4 feet wide, 70 feet 6 inches long, and 5 feet 3 inches high? Note. — Small fractions rejected. To find how large a cube may be cut from any given sphere, or be inscribed in it. Rule. — Square the diameter of the sphere^ divicU that product hy 3, and extract the square root of tht quotient for the avswer. 188 ORTON & Sadler's caloulator. I have a piece of timber, 30 inches in diameter, how large a square stick can be hewn from it? Rule. — Multiply the diameter by .7071, and thi product is the side of a square inscribed, I have a circular field, 360 rods in circumference; what must be the side of a square field that shall contain the same quantity? Rule. — Multiply the circumference by .282, and the product is the side of an equal square. I have a round field, 50 rods in diameter; what is the side of a square field that shall contain the same area? Ans, 44.31 135-|- rods. Rule. — Midtiply the diameter by .886, and the product is the side of an equal square. There is a certain piece of round timber, 30 inches in diameter ; required the side of an equi- lateral triangular beam that may be hewn from it. Rule. — Multiply the diameter by .866, a:xd the product is the side of an inscribed equilateral tri- angle. To find the area of a globe or sphere. Definition. — A sphere or globe is a round solid body, in the middle or center of which is an imag- inary point, from which every part of the surface is equally distant. An apple, or a ball used by ohildren in some of their pastimes^ may be called a sphere or globe. ROUND TIMBER. Round timber, when squared, is estimated to lose one-fifth ; hence (50 cubic feet, or) a ton of round timber is said to contain only 40 cubic feet. Round, sawed, and hewn timber is bought and sold by the cubic foot. To measure round timber. Rule.* — Take the girth in feet, at both the large and small ends, add them, and divide their sum by two for the mean girth ; then multiply the length in feet by the square of one-fourth of the mean girth, and the quotient will be the contents in cubic ''eet, according to the common practice. * This rule gives ahont four-ffths of the true contents, oti0- yth being allowed to the buyer for waste in hewing. 189 190 ORTON <& Sadler's calculator. Example. — What are the cubic contents of a round log 20 feet long, 9 feet girth at the large end, and 7 feet at the small end ? Solution. — 9 + 7 = 16-5-2 = 8 mean girth. Then 20 length x 4 feet (the square of ^ mean girth) == 80 cubic feet. Ans. Note. — If the girth be taken in inches, and the length in feet, divide the last product by 144. Example. — What are the cubic contents of a round log 12 feet long, 50 inches girth at the large end, 38 inches at the small end? Work.— 50 -f 38 = 88 -^ 2 = 44 mean girth. Then 12 length x 121 inches (the square of ^ mean girth) = 1452 -:- 144 = 10 j^^ cubic feet. To measure round timber as the frustum of a cone : that is, to measure all the timber in the log. Rule. — Multiply the square of the circumference at the middle of the log in feet by 8 times the length, and the product divided by 100 will be the contents. Extremely near the truth. Note. — The above rule makes 1 foot more timber in every 190 cubic feet a log contains if ciphered out by the long and tedious rules of Geometry. It Is therefore suflBciently correct for all practical pur- poses, and this rule being so short and simple in comparison with all others, every lumberman, ship- builder, carpenter, inspector or surveyor of timber, should post it up for reference and use. TIMBER MEASURE. 191 A TABLE FOR MEASURING TIMBER. Qnart«»r Oirt. Area. Quarter Girt. Area. Quarter Girt. Area. Inches. 6 6i 6J Feet. .250 .272 .294 .317 Inches. 12 m 12| Feet. 1.000 1.042 1.085 1.129 Inches. 18 18} 19 19} Feet. 2.250 2.376 2.506 2.640 7 7i 71 .340 .364 .390 .417 13 13i 13} 13i 1.174 1.219 1.265 1.313 20 20} 21 21} 2.777 2.917 3.062 3.209 8 8} 8} 8i .444 .472 .501 .531 14 14^ 14} 14i 1.361 1.410 1.460 1.511 22 22} 23 23} 3.362 3.516 3.673 3.835 9 9i n .562 .594 .626 .659 15 15J 15} 15| 1.562 1.615 1.668 1.722 24 24} 25 25} 4.000 4.168 4.340 I 4.516 10 10} 10} lOJ .694 .730 .766 .803 16 m 16} 16i 1.777 1.833 1.890 1.948 26 26} 27 27} 4.694 4.876 5.062 5.252 11 11} Hi iij .840 .878 .918 .959 17 m 17} 171 2.006 2.066 2.126 2.187 28 28} 29 29} 30 5.444 5.640 5.840 6.044 6.250 To measure round timber by the table. Multiply the area corresponding to the quarte^ girt in inches by the length of the log in feet 192 ORTON & Sadler's calculator. Note. — If the quarter-girt exceed the table, take half of it, and four times the contents thus formed 5jrill be the answer. EXAMPLE 1. If a piece of round timber be 18 feet long, and the quarter girt 24 inches, how many feet of timber are contained therein ? 24 quarter 24 girt. 96 48 By the Table. 576 square. 18 Against 24 stands 4.00 Length, 18 4608 Product, 72.00 576 An<» 72 fpptL 144)10368(72 feet 1008 xXUS. 1 ii ICCli* 288 288 This table gives the oustomary, but only about foit>r -fifths ol the true contents, one-fifth being allowed the buyer for waste n hewing or sawing to make the timber square. The following rule gives the true contents ; — Multiply square of girth by .08 times length. In the above example the whole gi/th is 8 feet, squared is 64 x (.08 x 18 length) ^ 92.16 feet TIMBER MEASURE. 193 I. Of Flooring. Joists are measured by multiplying their breadth by their depth, and that product by their »ength They receive various names, according to the posi- tion in which they are laid to form a floor, such as trimming joists, common joists, girders, binding joists, bridging joists and ceiling joists. Grirders and joists of floors, designed to bear fi;reat weights, should be let into the walls at each end about two-thirds of the wall's thickness. In boarded flooring, the dimensions must be taken to the extreme parts, and the number of squares of 100 feet must be calculated from thesd dimensions. Deductions must be made for stair- cases, chimneys, etc. Example 1. If a floor be 57 feet 3 inches long and 28 feet 6 inches broad, how many squarsa oJ flooring are there in that room ? By Decimals, 57.25 28.5 28625 By Duodecimah F. I. 57 : 3 28 : 6 45800 11450 456 114 28 : 7 : 6 7:0:0 100)1631.625 feet. Squares 16.31625 16:31 : 7 : 6 Ana. 16 squares and 31 feet 17 SQUARE TIMBER To measure square timber. IluLE. — Multiply the breadth in feet by the iepth in feet, and that by the length in feet, and the quotient will be the contents in cubic feet. Example. — How many cubic feet in a square log 12 feet long by 2 feet broad and 1^ feet deep ? Explanation. — 2 feet breadth x 1^ feet depth X 12 feet length = 36 cubic feet. Ans. Note. — If the breadth and depth be taken in inches, divide the last product by 144. Example. — :How many cubic feet in a square log 24 feet long, 30 inches broad, and 20 inches deep ? Solution. — 30 inches breadth x 20 inches depth X 24 feet length = 14400 -h 144 - 100 cubic feet 194 TIMBER MEASURE. 195 PROBLEM To find iht solid contents of squared or four-sided Timber, By the Carpenters Rule, As 12 on D ; length on c : Quarter girt on D : dolidity on 0. Rule I. — Multiply the breadth in the middle by the depth in the middle^ and that product by the length for the solidity. Note. — If the tree taper regularly from one end to the other, half the sum of the breadths of the two ends will be the breadth in the middle, and half the sum of the depths of the two ends will bo the depth in the middle. Rule II. — Multiply the sum of the breadths of the two ends by the sum of the depths^ to which add the product of the breadth and depth of each end ; one-sixth of this sum multiplied by the length, will give the correct solidity of any piece of squared tim- ber tapering regularly, PROBLEM To find how much in length will make a solid foot, or any other asssigned quantity, of squared timber, of equal dimensions from end to end. Rule. — Divide 1728, the solid inches in a foot^ or the solidity to be cut off^ by the area of the end in inches, and the quotient will be the length in inches 196 ORTON & SADLER S CALCULATOR. Note. — To answer the purpose of the above rule, some carpenters' rules have a little table upoD them, in the following form, called a table of tim her measure. 1 9 1 11 1 3 9 1 inches. 144 "T" 36 16 9 1 6 4 1 2 1 2 1 1 feet. 2 3 4 1 5 6 1 7 1 8 9 1 side of the square. This table shows^ that if the side of the square be 1 inch, the length must be 144 feet ; if 2 inches be the side of the square, the length must be 3^6 feet, to make a solid foot. CAPACITY OF CISTERNS OR WELLS. Tabular view of the number of gallons contained in the clear between the brickwork for each ten inches of depth : Diameter. Gallons. 2 feet equal 19 2i " " 3 " " 3i " " 4 " " 4i " '' '^''ZZ 5 " " 5i " " 'Z'''Z 6 " " Qh " " n t( It 7i " " 'ZZ.Z 30 44 8i 9 60 9^ 78 10 97 11 122 12 148 13 176 14 207 15 240 20 275 25 Diameter. Gallons. 8 feet equal 313 "" " " 353 461 489 692 705 827 959 1101 1958 3059 CISTERNS AND RESEBVOIBS. How to measure their contents. 1st. Find the solid contents of the cistern in cubic inches. 2d. Divide the contents so found by 14553, and the quo- tient will be the number of hogsheads. If the height of the cistern be given, how do you find the diameter, so that the cistern shall contain a given number of hogsheads? 1st. Reduce the height of the cistern to inches, and the contents to cubic inches. 2d. Multiply the height by the decimal .7854. 3d. Divide the contents by the last result, and extract the square root of the quotient, which will be the diameter of the cistern in inches. Note.— In tHtiniating the capacity of cisterns, reservoir?, etc., tb« fblluwing table is ustd: 31i gallons 1 barrel. 63 '* 1 hogshead. The barrels used in commerce vary from 30 to 45 gallon^ and the hogshead from 40 to tJO gallons. 197 198 ORTON & SADLER'S CALCULATOR. CISTERNS AND RESERVOIRS. HOW TO MEASURE THEIR CONTENTS. Cisterns and Reservoirs are constructed for the purpose of holding large quantities of water or fluids, and are in the form of a tub cylinder, or solid square. They are generally built in the ground, and mortised on all sides, except the opening, with brick or stone, and are chiefly permanent in their construction. TO MEASURE ROUND OR CYLINDRICAL CISTERNS. Rule I. — Multiply the square of the diameter in feet by the depth in feet, which wilt give the number of cylindrical feet in the cistern. Rule II. — Multiply the cylindrical feet by 373 for Hogsheads. 4000 373 — for Barrels, or divide by 5. 2000 47 — for Gallons. 8 The result will be the contents either in hogsheads, barrels, or gallons, depending upon the fractions used. Example I. — What is the contents in hogsheads of a cistern 20 feet in diameter and 10 feet deep ? Solution. — 20 ft. X 20 ft. = 400 ft. square of diameter. 400 " X 10 " = 4000 cylindrical ft. 373 4000 cylindrical ft. X =z 373 hhds.— J«j». 4000 Example II. — How many gallons in a cistern 10 feet in diameter and 16 feet deep? TO MEASURE CISTERNS. 199 Solution.— 10 X 10 = 100 X 16 = 1600 cylindrical ft 47 1600 X — = 9400 gallons.— ^W5. Or, 373 1600 X — = 298f barrels.— ^715. 2000 Or 373 1600 X = 149 i hogsheads.— ^n«. 4000 TO MEASURE SQUARE CISTERNS. Rule I. — Multiply the width in feet hy the length in feet, and, that 2)roduct by the depth in feet; this last product will be the number of cubic feet in the cistern. Rule II. — Multiply the cubic feet thus obtained by 19 — for Hogsheads. 160 19 — for Barrels. 80 7 -jYjj for Gallons. This will give the contents either in hogsheads^ barrels, or geUlonSj as desired. Example. — How much water will a cistern contain which is 6 feet wide, 8 feet long, and 10 feet deep ? Solution.— 6 X 8 = 48 X 10 = 480 cubic ft. in cistern. 19 480 X — = 57 hogsheads. — Ans. 160 19 Or, 480 X — = 114 barrels.— ^rw. 80 40 Or, 480X 7 A8ff or 480X7.48 =rr 3590- -gaii.—^««. 100 OASK-GAUGING. ixauging is the art of measuring the capacity o{ casks and vessels of any form. In commerce, most of the gauging is done by the use of the diagonal rod, which gives only approximate results, but suf- ficiently accurate for ordinary purposes. Ullage is the difference between the actual con- tents of a vessel and its capacity, or that part which is empty. To measure small cylindrical vessels. Rule. — Multiply the square of the diameter, in inches, by 34, and that by the height, in inches, and point off four figures ; the result will be th< capacity, in wine gallons and decimals of a gallon For beer gallons multiply by 28 instead of 34. 200 CASK-GAUGING. 201 Example. — A can measures 15 inches in diame- ter, and is 2 feet 2 inches in heiglit. How many gal Ions will it contain ? 15x15 = 225 X 26 height = 5850 ; 5850 X 34 = 19.8900. Ans. 19yVu galls. Casks are usually regarded as the two equal frus- tums of a cone, and are very accurately gauged by three dimensions as follows : — To measure a cask by three dimensions. 1st. Add the bung and head diameters in inches, and divide by 2 for the mean diameter. 2d. Multiply the square of the mean diameter by the length of the cask in inches. 3d. Multiply the last product by .0034 for wine gallons, .0028 for beer gallons. Example. — How many wine gallons in a cask, the bung diameter of which is 22 inches, the head diameter 20 inches, and the length 32 inches ? Work.— 22 -f 20 = 42 -t- 2 = 21 (mean diame- ter) : then 21 x 21 = 441 (square of mean diame ter), x 32 length = 14112 x .0034 = 47.9808. Ans. Note. — If the cask is not full, stand it on the end, and multiply by the height of the liquid, in- stead of the length of the cask, for actual contents When the cask is much bilged or rounded front the bung to the head, a more accurate way is to gauge by four iimcnsions, as follows : — 202 ORTON & Sadler's calculator. To measure a cask by four dimensions, 1st. Add the bung and head diameters in inches, and the diameter in inches between bung and head. 2d. Divide their sum by 3 for the mean diameter. 3d. Multiply the square of the mean diameter by the length of the cask in inches. 4*th. Multiply the last product by .0034 for wine gallons, .0028 for beer gallons. Example. — What are the contents in gallons of a cask, the bung diameter of which is 24 inches, the middle diameter 20 inches, the head diameter 16 inches, and its length 40 inches? Work.— 24 + 20 -f 16 = 60 -r- 3 = 20 (mean diameter), then 20 X 20 = 400 (square of mean dia- meter) X 40 length = 16000 x .0034 = 54. 4 gallons 1. The ale gallon contains 282 cubic inches. 2. The wine gallon contains 231 cubic inches. 3 The bushel contains 2150.4 cubic inches. 4. A cubic foot of pure water weighs 1000 ounces = 62^ pounds avoirdupois. 5. To find what weight of water may be put into a given vessel. Multiply the cubic feet by 1000 for the ounces, or by 6 2^ /or the pounds avoirdupois. 6. What weight of water can be put irto a cis- icrn ti feet square ? Ans, 26,367 lbs. 3 o% MEASURING GRAIN. By the United States standard, 2150 cubic inchef make a bushel. Now, as a cubic foot contain? TT28 cubic inches, a bushel is to a cubic foot as 2150 to 1728; or, for practical purposes, as 4 to 5. Therefore, to convert cubic feet to bushels, it is necessary only to multiply by |.or 8. To measure the bushels of grain in a granary Rule. — Multiply the length in feet by tht breadth in feet, and that again by the depth in feet, and that again by J. The last product will be the number of bushels tJie granary contains. Example. — How many bushels in a bin 10 feet long, 4 feet wide, and 4 feet deep. Work. — 10 feet length X 4 feet breadth x 4 fee/ depth a. IGO <'ubic fe^t ; then IBO x J = 128. Ans. 203 204 ORTON & Sadler's calculator. SIZE OF BINS To contain a given number of bushels. Having any number of bushels, how then will jcu find the corresponding number of cubic feet? Increase the number of bushels one-fourth itself, and the result will be the number of cubic feet. How will you find the number of bushels which a bin of a given size will hold ? Find the content of the bin in cubic feet ; then diminish the content by one-fifth, and the result will be the content in bushels. How will you find the dimensions of a bin which shall contain a given number of bushels ? Increase the number of bushels one-fourth itself, and the result will show the number of cubic feet which the bin will contain. Then, when two dimensions of the bin are known, divide the last result by their product, and the quotient will be the other dimension. If you wish the contents of a pile of ears of corn, or roots, in heaped bushels, ascertain the cubic feet, and multiply by iVu. WEIGHTS AND MEASURES Recognized by the Laws of the United States, In some States dried apples and peaches are sold by the heaping bushel as are other of farm products. A bushel of corn in the ear is three heaped half-bushels, or four even-full. TABLE OF AVOIRDUPOIS POUNDS ITI A BUSHEL, As prescribed by statute in the several States named. 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Bnckwlieat Castor Beans Clover Seed Dried Apples Dried Peaches Flax Seed IlenipSeed Indian Corn Indian ( om in ear Indian Corn Meal.. Oafs il 5^^ 1 1 :i 111 1^^ In Pennsylvania 80 lbs coarse, 70 lbs. ground, or 62 lbs. fine salt make 1 bnshel ; and in Illinois, 50 lbs. common, or 55 lbs. fine salt make 1 bushel. In Tennessee 100 ears of coiu are a bushel. A heaping bushel contains 2S15 cubic inches. In Maine 64 lbs. of ruta baga turnips or beets make 1 bushel. A cask of lime is 240 lbs Lime in slacking absorbs 2)^ times its volume, and 2>^ times its weight in water. 18 205 RAILROAD FREIGHT. TABLE OF GROSS WEIGHTS. The Articles named are Billed at actual weights, if possible, but usually at the weights in the Table below when it is not convenient to weigh them.. • Ale and Beer 320 lb. per bbl. " 170 ** " 100 Apples, dried 24 " green 56 « " 150 Barley 48 Beans, wliilo 60 " castor 46 Beef 320 Bran 20 Brooms 40 Buckwheat 52 Cider 350 Charcoal 22 Clover Seed 60 Corn 56 " in car 70 " Meal 48 " « 220 Eggfi 200 FvA\ 300 Flax Seed 56 Flour 200 Hemp Seed 44 K' bbl. bu. bbl. bu. doz. bu. bbl. bu. bbl. bu. bbl. bu. High wines 350 lb. per bbl. Hungarian Grass Seed 45 " bu. Lime 200 " bbl. Malt 38 " bu. Millet 45 " Nails 108 " kog. Oats 32 " bu. Oil 400 " bbl. Onions 57 '* bu. Peaches, dried. . . 33 " *' Pork 320 " bbl. Potatoes, common. 150 " " . 60 " bu. " sweet. . . 55 " " Rye 56 « " Salt, fine 56 * « " " 300 " coarse " in sacks.. Timothy Seed. Turnips Vinegar Wheat Whiskey 350 203 4") 56 350 60 350 bbl. sack, bu. bbl. bu. bbl. One ton weight is 2000 lbs. ESTIMATED WEIGHTS OF lUMBER AND OTHER ARTICLES. Note.— From 18,000 to 20,000 lb. is considered a car-load in most places, each car itself also weighing about 20,000 lb. 206 rO MEASURE CORN ON THE COB IN CRIBS Corn is generally put up in cribs made of rails ; but the rule will apply to a crib of any size or kind, whether equilateral, or flared at the sides. When the crib is equilateral. Rule. — Multiply the length in feet by the breadth in feet, and that again by the height in feet, which last product multiply by .63 (the frac- tional part of a heaped bushel in a cubic foot), and the result will be the heaped bushels of ears. For the number of bushels of shelled corn multiply by 42 (two-thirds of .63), instead of .63, 207 208 ORTON & Sadler's calculator. Example. — Required the number of bushels of shelled corn contained in a crib of ears, 15 feet long, by 5 feet wide, and 10 feet high ? 15 length x 5 width, x 10 height = 150 cubic feet. Then Y50 X .63 = 412.50 heaped bushels A ears. Also 750 X .42 = 315 bushels of shelled corn. In measuring the height, of course, the height of the corn is intenaed. And there will be found to be a difference in measuring corn in this mode, be- tween fall and spring, because it shrinks very much in the winter and spring, and settles down. Wlien the crib zs flared at the sides. Rule. — Multiply half the sum of the top and bot- tom widths in feet by the perpendicul-ar height in feet, and that again by the length in feet, which last product multiply by .63 for heaped bushels of ears, and by .42 for the number of bushels of shelled corn. Note. — The above rule assumes that three heap- ing hclf bushels of ears make one struck bushel of shelled corn. This proportion has been adopted upon the authority of the major part of our best agricultural journals. Nevertheless, some journals claim that two heaping bushels of ears to one of shelled corn is a more correct proportion, and it is the custom in many parts of the country to buy MEASUREMENT OF CORN. 209 and sell at that rate. Of course much will de- pend upon the kind of corn, the shape of the ear, the size of the cob, &c. Some samples are to be found, three heaping half bushels of which will even overrun one bushel shelled ; while others again are to be found, two bushels of which will fall short of one bushel shelled. Every farmer must judge for himself, from the sample on hand, whether to allow one and a half or two bushels of ears to one of shelled corn. In either case, it is onl^ an approximate measurement, but sufficient for all ordi- nary purposes of estimation. The only true? vay of measuring all such products is by weight MEASUREMENT OF HAY- The only correct mode of measuring hay is to «veigh it. This, on account of its bulk and cha- racter, is very difficult, unless it is baled or other- wise compajted. This difficulty has led farmers to estimate the weight by the bulk or cubic con- tents, a mode which is only approximately correct. Some kinds of hay are light, while others are heavy, their equal bulks varying in weight. Bui for all ordinary farming purposes of estimating the amount of hay in meadows, mows, and stacks, the following rules will be found sufficient : — As nearly as can be ascertained, 25 cubic yards of average meadow hay, in windrows, make a ton 210 MEASUREMENT OF HAY. 211 When loaded on wag( ns, or stored in barns, 20 cubic yards make a ton. When well settled in mows, or stacks, 15 cubic yards make a ton. Note. — These estimates are for mecfium-eii t%i mows or stacks ; if the hay is piled to a great height, as it often is where horse hay-forks are used, the row will be much heavier per cubic yard. When hay is haled, or closely packed for ship- ping, 10 cubic yards will weigh a ton. To find the number of tons in long square stacks. Rule. — Multiply the length in yards by the width in yards, and that by half the altitude in yards, and divide the product by 15. Example. — How many tons of hay in a square stack 10 yards long, 5 wide, and 9 high ? Solution.— 10 X 5 X 4^ = 225-^ 15 = 15 tons. Ans. To find the number of tons in circular stacks. Rule. — Multiply the square of the circumference in yards by 4 times the altitude in yards, and di- vide by 100 ; the quotient v/ill be the number of cubic yards in the stack ; then divide by 15 for the number of tons. Example. — How many tons of hay in a circular Btack, whose circumference at the base is 25 yards, and height 9 vards ? 212 ORTON A Sadler's calculator. Solution. — 25 x 25 = 625, the square of the circumference ; then 625 x 36 (four times the length), = 225000 ^ 100 = 225 (the number oi cubic yards), then 225 -r- 15 = 15, the number of tons. An easy mode of asceiHaining the value of a given number of lbs, of hay, at a given price per ton of 2000 lbs. Rule. — Multiply the number of pounds of hay (coal, or anything else which is bought and sold by the ton) by one-half the price per ton, pointing off three figures from the right hand ; the remain- ing figures will be the price of the hay (or any article by the ton). Example. — What will 658 lbs. of hay cost, @ $T 50 per ton ? Solution. — $7 50 divided by 2 equals $3 75, by which multiply the number of ponnlg, thus : 658 x $3 75 = 246.750, or $2 46. An, Note. — The principle in this rule is the same as in interest — dividing the price by two gives us the price of half a ton, or 1000 lbs. ; and pointing jfl three figures to the right is dividing by 1000. A truss of hay, new, is 60 lbs. ; old, 50 lbs. ; straw, 40 lbs. A load of hay is 36 trusses. A bale of hay is 300 lbs. RULES FOR DETERMINING THE WEIGHT OF LIVE CATTLE. For cattle of a girth of from 5 to Y feet, allow 23 lbs. to the superficial foot. For cattle of a girth of from t to 9 feet, allow 31 lbs. to the superficial foot. For small cattle and calves of a girth of from 3 to 5 feet, allow 16 lbs. to the superficial foot. For pigs, sheep, and all cattle measuring less than 3 feet girth, allow 11 lbs. to the superficial foot Measure in inches the girth round the breast, just behind the shoulder-blade, and the length of the back from the tail to the forepart of the shoui- der-blade Multiply the girth by the length, and divide by 144 for the superficial feet, and then n ul- 213 214 ORTON & Sadler's calculator. tiplj the superficial feei by the number of lbs, allowed for cattle of different girths, and the pro- duct will be the number of lbs. of beef, veal, or pork, in the four quarters of the animal. To find the number of stone, divide the number of lbs. by 14. Example — What is the estimated weight oi beef in a steer, whose girth is 6 feet 4 inches, and length 5 feet 3 inches ? Solution. — 76 inches girth, x 63 inches length, = 4788 -r- 144 = 33J square feet, X 23 = 764| lbs., or 54| stone. Ans, Note. — When the animal is but half fattened, a deduction of one lb. in every 20 must be made; and if very fat, one lb. for every 20 must be added. Where great numbers of cattle are annually bought and sold under circumstances that forbid ascertaining their weight with positive accuracy, the estimated weight may be thus taken with ap- proximate exactness — at least with as much accu- racy as is necessary in tlie aggregate valuation of stock. No rules or tables can, however, be at all times implicitly relied on, as there are many cir* cumstances connected with the build of the animal, the mode of fattening, its condition, breed, &c.. that will influence the measurement, and conse* quently the weight. A person skilled in estimate irig the weiixht of stock soon learns, however, to make allowance for all these circumstances. BRICK BUILDING. A perch of stone is 24. Y5 cubic feet ; when bnilt In the wall, 22 cubic feet make 1 perch, 2J cubic feet being allowed for the mortar and filling. Threo pecks of lime and four bushels of sand to a perch of wall. To find the number of perches of stone in walls. Rule. — Multiply the length in feet by the height in feet, and that by the thickness in feet, and divide tlie product by 22. Example. — How many perches of stone con- tained in a wall 40 feet long, 20 feet high, and 18 inches thick ? Solution. — 40 feet length x 20 feet height x ^ feet thick = 1200 .4- 22 « 54.54 perches. Anf^. 215 216 ORTON & Sadler's calculator. Note. — To find the perches of masonry, divide the cubic feet by 24.75, instead of 22. Brick'ivorJc, The dimensions of common bricks are from 71 to 8 inches long, by 41 wide, and 2| thick. Front bricks are 8i inches long, by 4J wide, and 2} tliick. The usual size of fire-bricks is 9 J inches long, by 4f wide, by 2f thick. 22 to 23 common bricks to a cubic foot when laid ; 15 common bricks to a foot of 8-inch wall when laid. To find the nmnher of common bricks in a wall. Rule. — Multiply the length of the wall in feet by the height in feet, and that by its thickness in feet, and that again by 22. Example. — How many common bricks in a w^all 40 feet long by 20 feet high, and 12 inches thick? Solution.— 40 feet length X 20 feet height, X 1 foot thick, X 22 == 17,600. Ans. Note. — For walls 8 inches thick, multiply the length in feet by the height in feet, and that by 15. When the wall is perforated by doors ard win- iows, deduct the sum of their cubic feet from the ;ubic contents of the wall, including the openings, aefore multiplying by 22 or 15 as before. Laths. Laths are l\io\h inches wide, by 4 feet long, are usually set i inch apart, and a bundle contains 100. bricklayers' work. 217 BBICKLAYEBS' WORK. The principal is tiling, slating, walling and chim- oey work. Of Tiling or Slating, Tiling and slating are measured by the square 3f 100 feet, as flooring, partitioning and roofing were in the Carpenters' work ; so that there is not much difference between the roofing and tiling ; yet the tiling will be the most ; for the bricklayers sometimes will require to have double measure for hips and valleys. When gutters are allowed double measure, the way is to measure the length along the ridge- tile, and add it to the content of the roof: this makes an allowance of one foot in breadth, the whole length of the hips or valleys. It is usual also to allow double measure at the eaves, so much as the projector is over the plate, which is commonly about 18 or 20 inches. Sky-lights and chimney shafts are generally de- ducted, if they be large, otherwise not. Example 1. There is a roof covered with ti.es, whose depth on both sides (with the usual alt iw* ance at the eaves) is 37 feet 3 inches, and che length 45 feet ; how many squares of tiling ar^ oontained therein ? 19 218 ORTON & Sadler's calculator. BY DUODECIMALS FKET. INCHES. 37 3 45 185 148 11 3 BY DECIMALS. 37.25 45 18625 14900 16 76.25 16 76 3 2. Of Walling. Bricklayers commonly measure their work by the rod of 16 J feet, or 272 J square feet. In some places it is a custom to allow 18 feet to the rod ; that is, 324 square feet. Sometimes the work is measured by the rod of 21 feet long and 3 feer high, that is, 63 square feet ; and then no regard is paid to the thickness of the wall in measuring* but the price is regulated according to the thick- ness. When you measure a piece of brick-work, the first thing is to inquire by which of these ways it must be measured; then, having multiplied the length and breadth in feet together, divide the pro- duct by the proper divisor, viz.: 272.25, 324 or 63, according to the measure of the rod, and the quo- tient will be the answer in square rods of that measure. But, commonly, brick walls that are measured by the rod are to be rodnced to a standard thick- bricklayers' work. 219 Dess of a brick and a-half, which may be done by the following Rule. — Multiply the number of superficial feet that are contained in the wall by the number of half bricks which that wall is in thickness; one- third part of that product will be the content vi feet The dimensions of a building are generally taken by measuring half round the outside and half round the inside, for the whole length of the wall ; this length, being multiplied by the hight, gives the superficies. And to reduce it to the standard thickness, etc., proceed as above. All the vacuities, such as doors, windows, window backs, etc., must be deducted. To measure any arched way, arched window or door, etc., take the hight of the window or dooi trom the crown or middle of the arch to the bot- tom or sill, and likewise from the bottom or sill to the spring of the arch ; that is, where the arch begins to turn. Then to the latter hight add twice the former, and multiply the sum by the width of the window, door, etc., and one-third of the pro- duct will be the area, sufficiently near for practice. Example 1. If a wall be 72 feet 6 inches long, and 19 feet 3 inches high, and 5 J bricks thick, how many rods of brick work are contained therein, when reduced to the standard ? 220 ORTON & Sadler's calculator. glaziers' wobk. GlajBiera take their dimensions in feet, inches iod eights or tenths, or else in feet and hundredth parts of a foot, and estimate their work by the i*quare foot. Windows are sometimes measured by taking the dimensions of one pane, and multiplying its super- ficies by the number of panes. But, more gen- erally, they measure the length and breadth of the window over all the panes and their frames for the length and breadth of the glazing. Circular or oval windows, as fan lights, etc., are . measured as if they were square, taking for their dimensions the greatest length and breadth, as a compensation for the waste of glass and labor in cutting it to the necessary forms. Example 1. If a pane of glass be 4 feet 8| inches long, and 1 foot 4^ inches broad, how many feet of glass are in that pane ? ' DUODECIMALS . BY DECIMALS. FT. IN. P. 4.729 4 8 9 1.354 1 4 3 18916 23645 4 8 9 1 6 11 14187 1 2 2 3 4729 6 4 10 2 3 6.403066 PLUMBERS WORK. 221 PLUMBERS WORK. Plumbers* work is generally rated at so much per pound, or by the liundred weight of 112 pounds, and the price is regulated according to the value of lead at the time when the work is per- formed. Sheet lead, used in roofing, guttering, etc., weighs from 6 to 12 pounds per square foot, ac- cording to the thickness, and leaden pipe varies in weight per yard, according to the diameter of its bore in inches. The following table shows the weight of a square foot of sheet lead, according to its thickness, reck- oned in parts of an inch, and the common weight of a yard of leaden pipe corresponding to the diameter of its bore in inches: Thickness of Lead. Pounds to a Square foot. Bore of Leaden Pipe. Pounds per yard. 5't 5.899 i 10 6.554 1 12 1 7.373 H 16 1 8.427 li 18 1 9.831 If 21 1 11.797 2 - 222 ORTON & SADLER 8 CALCULATOR. Example 1. A piece of sheet lead measures 16 feet 9 inclief? in length, and 6 feet 6 inches in breadth ; what is its weight at 8 J pounds to a square foot ? BY DUODECIMA FEET. INCHES. 16 9 6 6 LS 6 BY DECIMALS FEET. 16.75 6.5 100 8 6 4 8375 10050 108 10 6 108.875 feet. Then 1 foot : 8J pounds : : 108.875 feet 898.21875 pounds=:8 cwt. 2 J pounds nearly. MASON 8 WORJL Masons measure their work sometimes by the foot solid, sometimes by the foot superficial, and sometimes by the foot in length. In taking dimensions they girt all their moldings as joiners do. The solids consist of blocks of marble, stone pillars, columns, etc. The superficies are pave- ments, slabs, chimney-pieces, etc. masons' work. 223 PLASTERERS WORK. Plasterers' work is principally of two kinds, namely, plastering upon laths, called ceiliny^ and plastering upon walls or partitions made of framed timber, called rendering. In plastering upon walls, no deductions are mado except for doors and windows, because cornices, festoons, enriched moldings, etc., are put on aft^r the room is plastered. In plastering tiriiber partitions, in large ware- houses, etc., where several of the braces and larger timbers project from the plastering, a fifth part is commonly deducted. Plastering between their timbers is generally called rendering between •juarters. Whitening and coloring are measured in tho same manner as plastering ; and in timbered par- titions, one-fourth, or one-fifth of the whole area is commonly added, for the trouble of coloring the sides of the quarters and braces. Plasterers' work is measured by the yard square, consisting of nine square feet. In arches, the girt round them, multiplied by the length, will give the superficies. Example I. — If a ceiling be 59 feet 6 inches long, and 24 feet 6 inches broad ; how many yards does that ceiling contain ? 224 ORTON se squares is 25, the square root of which is 5 ; consequently, when one leg of a right-angled triangle is 3, and the other 4, the hypotenuse must be 5. And if 3, 4, and 5, be multiplied by any other numbers, each by the same, the products will be sides of true right-angled tri- angles. Multiplying them by 2, gives 6, 8, and 10, by 3, gives 9, 12, and 15 ; by 4, gives 12, 16, and 20, etc.; all which are sides of right-angled tri- angles. Hence architects, in setting off the corners of buildings, commonly measure 6 feet on one side, and 8 feet on the other ; then, laying a 10-foot pole across from those two points, it makes the corner a true right-angle. N. B. — The solutions of the foregoing problems are all very brief by canceling. 244 ORTON & Sadler's calculator, 7b find the area of any triangle when the three ndes only are given. Rule. — From half the sum of the three sides sub- tract each side severally; multiply these three re- mainders and the said half sum continually together ; ^hen the square root of the last product will he the area of the triangle, EXAMPLE. Suppose I have a triangular fish-pond, whose three sides measure 400, 348, and 312yds; what quantity of ground does it cover ? Ans. 10 acres, 3 roods, S-frods. Note. — If a stick of timber be hewn three square, and be equal from end to end, you find the area of the base, as in the last question, in inches ; multiply that area by the whole length, and divide the product by 144, to obtain the bolid content* If a stick of timber be hewn three square, be 12 feet long, and each side of the base 10 inches, the perpendicular of the base being 8f inches, what ia its solidity? Ans. 3,6-[-feet. PROBLEM 1. The diameter given, to find the circumference. Rule. — As 7 are to 22, so is the given diameter t-o the circumference ; or^ more exactly^ o^ 113 are to 355, so is the diameter to the circumference, eta MENSURATION OR PRACTICAL GEOMETRY. 245 EXAMPLES. 1 . What is the circumference of a wheel, whose diameter is 4 feet? As 7 : 22 : : 4 : 12,57+ft., the cir-um., Am, 2. What is the circumference of a circle, whose liameter is 35 rods ? As 7 : 22 : : 35 : 110 rods, Am, Note. — To find the diameter when the circum- ference is given, reverse the foregoing rule, and say, a« 22 are to 7, so is the given circumference to the required diameter; or, as 355 are to 113, so is the circumference to the diameter. 3. What is the diameter of a circle, whose cir- cumference is 110 rods? As 22 : 7 : : 110 : 35 rods, the diam., Ans, Case 5. — To find how many solid fed a round stick of timber, equally thick from end to end^ will contain^ when hewn square. Rule. — Multiply twice the square of its serm-dv- ameler, in inches, by the length in the feet; then divide the product by 144, and the quotient will be the an- swer. N B. — When multiplication and division are combined, always cancel like factors. When the numbers are properly arranged, a few clips with the pencil, and, perhaps, a irijling multiplication will suffictj. 246 ORTON & Sadler's calculator. For the practical convenience of those who have occasion to refer to mensuration, we have arranged the following useful -able 3f multiples. It covers the whole ground of practical geometry,, and should be studied carefully by those who wish to be skilled in this beautiful branch of matliematics : TABLE OF MULTIPLES. Diameter of a circle X ^-1416 — Circumference. Radius of a circle X 6.283186 — Circumference. Square of the radius of a circle X 3.1416 — Area. Square of the diameter of a circle X 0.7854 =- Area. Square of the circumference of a circle X 0-07958 = Area. Half the circumference of a circle X by half its diameter -« Ai^ea. Circumference of a circle X 0.159165 =- Radius. Square root of the area of a circle X 0.66419 «- Radius. Circumference of a circle X 0.31831 — Diameter. Square root of the area of a circle X 112838 =« Diameter. Diameter of a circle X 0.86 -= Side of inscribed equilateral trianglo. Diameter of a circle X 0.7071 — Side of an inscribed square. Circumference of a circle X 0.225 — S\de of an inscribed square. Circumference of a circle X 0.282 — Side of an equal square. Diameter of a circle X 0.88C2 — Side of an equal square. Base of a triangle X t»y K *^® altitude — Area. Multiplying both diameters and .7854 together = Area of an elllpso. Surface of a sphere X by 3^ of its diameter =« Solidity. Circumference of a sphere X by its diameter — Surface. Square of the diameter of a sphere X 3.1416 — Surface. Square of the circumference of a sphere X 0.3183 « Surface. Cube of the diameter of a sphere X 0.5236 — Solidity. Cube of the radius of a sphere X 41888 — Solidity. Cube of the circumference of a sphere X 0.016887 — Solidity. Square root of the surface of a sphere X 0.56419 — Diameter, Square root of the surface of a sphere X 1.772454 — Circumference. Cube root of the solidity of a sphere X 1-2407 — Diameter. Cube root of tlie solidity of a sphere X 3.8978 — Circumference, Radius of a sphere X 1.1547— Side of inscribed cube. Square root of {% of the square of) the diameter of a sphere — Side of inscribed cube. Area of its base X by >«J of its altitude — Solidity of a cone or pyr- arrid, whether round, squaie, or triangular. irea if one of its sides X 6 >=- Surface of a cube. AHitude of Tapezoid X % the sum ct its parallel sides — Area. WEIGHTS AND MEASURES. 247 AVOIRDUPOIS WEIGHT Is the standard weight for weighing the greater por- tion of articles used in trade and commerce, such asgroceries,produce, iron, coal, hay, cotton, etc. TABLE. 437V^ grains ((;rr).l ounce — ox. IG oz 1 pound 76. 25 lb 1 quiirter. . .qr. 4 qr 1 \\\\.\\''hX,cwt. EQUIVALENTS. T. 1 = ciot. 20 = 1 = qr. 80 4 1 lb. 02. ~ 2000 = 32000 = 100 = 1600 = 25 = 400 1 = 16 1 = 14000000 = 700000 = 175000 = 7000 ^ 437i Scale of units : — 437J, 10, 25, 4, 20. The dram is now seldom used, except with silk manu- facturers. The ounce is divided into k and ^. The following denominations are also used; LONG OB IRON TON. 28 lbs 1 quarter. 4 qr., or 112 lbs 1 hundredweight. 20 cwt., or 2240 lbs 1 ton. This measurement is nearly obsolete. It is allowed at the Custom House in estimating duties, and in whole- Bale transactions of iron and coal. Note. — The grain avolnlniwis, though never used, is tlie same as the grain in Troy weight. 7(Kio gmina make the avuirUuiKjis pound, and 5700 grains the Truy pound. IRON, LEAD, Etc. 14 lbs 1 stone. 21i stone 1 pig. 8 pigs 1 fother. 248 ORTON & SADLER S CALCULATOR. AVOIKDUPOIS AT^-EIGHT. Miscellaneous Table. 14 pounds of Iron or Lead 1 stone. 100 " '' Grain or Flour 1 cental. loo " " Raisins 1 cask. 100 " " Dry Fish 1 quintal. 100 " "Nails 1 keg. 196 " "Flour 1 barrel. 200 " " Pork, Beef, or Fish 1 barreL 240 " "Lime 1 cask. 280 " " Salt 1 barrel of Salt at the N. Y. Salt Works. APOTHECAKIES' WEIGHT Is used by Apothecaries and Physicians in dis- pensing medicines, not liquid. The grains men- tioned in the following table are Troy. TABLE. 20 Grains [gr. xx.) 1 scruple, 9 3 Scruplos O iij) 1 dram, g 8 Drams (.5 viij) 1 ounce, ^ 12 Ounces (5 xij) 1 pound, lb EQUIVALENTS. ft> .? 5 9 gr. 1 z:== 12 = 96 = 288 = 57G0 1 = 8 == 24 = 480 1 == 3 = 60 1 = 20 ScALR OF Units :— 20, 3, 8, 12. The only difference between Troy and Apothecaries* weight is the division of the ounce. The pound, ounce, and grain are the same. Drugs and medicines are bought and sold in quantities by Avoirdupois weight; WEIGHTS AND MEASURES. 249 APOTHECARIES' FLUID MEASURE, Used ill mixing liquid medicines by measure. 60 Minims (m) 1 fluid drachm,/,^ 8 /.^ 1 fluid ounce, /g 16/5 1 pint, 0. ( Oct arms.) 8 1 gallon, ( Coiig. Congius.) EQUIVALENTS. 128 Cong. 1 = 0. 8 = 1 ^ 16 1 /5 1024 128 8 1 m 61440 7680 480 60 Scale of Units:— 60, 8, 16, 8. Note. — One fluid ounce ^455.0944 Troy grains. The minim is a droi) of pure water, and is equal to alx)ut i^^^j af < grain Troy. An ordinary teacupful is about 4 fluid ounces. Common tablespoonful ^ a fluid ounce. Teaspoon contains about 45 drops. A^OOD MEASURE, For measuring wood, rough stone, fences, etc. 16 Cu. Ft 1 cord ft. 8 Cord Feet or 128 ( u >ie Ft..l cord. 243 Cubic Feet 1 ]>erch of stone or masonry. A cord of wood is a pile 8 feet long, by 4 feet wide, and 4 feet high. A cord foot is one foot of the running pile, or h of a cord. A ])erch of stone or masonry is 16i feet long, U feet wide, and 1 foot high. Wood. Measuring wood in the load. If the rack is narrower nt the bottom than at the top, the width of the logd should be measured half-way from base to top; this will ijive the average width. 250 OETON & Sadler's calculator. DKY MEASURE Is used in measuring arti- cles not fluid, such as grains, seeds, vegetables, fruit, salt, etc. TABLE. 2 ^inis{pt.).l quart .... qt, 8 qt 1 peck.....pk, 4pk 1 bushel....6w. 36 bu lcli'ldron,cA, EQUIVALENTS. ch. 1 = bu. pk. 36 = 144 1 = 4 1 qt, 1152 32 8 1 pL 2304 64 16 2 Scale of units :— 2, 8, 4, 36. Note. — 1 gal. Wine Measure contains 231 cu. in., 1 gal. Ale or Beer Measure (nearly obsolete), 282 cu. in., and 1 bu. 2150-j^^j^y cu. in. The legal bushel of the United States is the old Win- chester measure, cylindrical in form, 18i inches in diame- ter and 8 inches deep, containing 2150^* ^^ cubic inches. The Imperial bushel of England is 2218^'jjYo cubic inches. 32 English bushels equal 33 of the United States. Heaped Pleasure is the contents of bushel heaped in the shape of a cone. Corn in the ear, large fruits, vegetables, and bulky articles are sold by this measure. Stricken Measure is the bushel even full, having been stricken off by a rule or striker. Grain, seeds, etc., are sold by this measure. It is customary to allow 5 stricken measures for 4 heaped ones. It is usual to quote the price of grain, etc., by the bushel, but more frequently to deter- njiue their value by weight. WEIGHTS AND MEASURES. 251 CUBIC OR SOLID MEASURE, Used iu measuring any- thing containing length, breadth, depth, and thick- ness, such as timber, wood, stone, boxes, stor- age capacity of rooms, bins, cisterns, etc. EQUIVALENTS. cu,ft. cu. in, 1 =: 1728 Scale of units ;— 1, 1728.ei"iv 3i " 34 '* " bjukes 4 " 16 " " Si>ikes 4i " 12 " " Spikes 5 " 10 " From this table au estimate of quantity aud suitable ti/ts Icr on^ Job of work cau be made. 252 WEIGHTS AND MEASURES. 253 LIQUID OR RATINE MEASURE, Used in measuring liquids, such as liquors, vinegar, molasses, oils, etc., and estimating the capacity of vessels de- signed to contain them. TABLE. 4 gills {gi.) 1 pint, pt. 2 pints 1 quart, qt. 4 quarts ....1 gallon, gal, 3H gallons.. .1 barrel, bbl. 2 barrels.. .1 hogs'd, hhd, ' hhd. 1 = bbL 2 1 EQUIVALENTS. gal. 63 3U 1 qt 252 126 4 1 pt 504 252 8 2 1 gi. ^ 2016 = 1008 = 32 =^ 8 = 4 Scale of Units:— 4, 2, 4, 31 J, 2. Note.— The gallon must contain exactly 10 pounds Avoirdupois, of pure water, at a teinperatnro of 62°, the barometer being at 30 inches. It is the standard unit of measure of capacity for liquids and dry goods of every description, and is y larger than the old wine measure, tj^j larger than the old dry measure, and -^^ less than the old ale measure. The wine gallon must contain 231 cubic inches. Barrels used in commerce are made of various sizes, from 30 to 45, and even 56 gallons. There is no definite measure called a hogshead, they are usually gauged, and have their capacities in gallons marked on them. The Standard gallon, United States, contains 231 cubic inches. The Imperial gallon, Great Britain, 277.274 cubic inches, and is equal to j more than the United States' measure. In measuring cisterns, reservoirs, vats, etc., the barrel is estimated at 31i gallons, and the hogshead 63 gallons. A gallon of water weighs nearly 8 J pounds, avoirdupois. A. pint is generally estimated as a pound. 22 254 ORTON & Sadler's calculator. LINEAK OK LONG MEASURE. For measuring lengths, distances and dimensions of objects. TABLE. 12 inches 1 foot. 3 feet lyard. 6^ yards 1 rod. m feet Irod. 1 320 rods 1 mile. ■ ^H ■ ■ ■ H| Note.— The inch is divided 1 ■ 1 I 1 HI lntoH,K,^^- EQUIVALENTS. mt. fur rd. yd. ft. m. 1 : = 8 = 320 = 1760 = 6280 = 63360 1 = 40 = 220 = 660 r= 7920 1 ^ 5i 1 = 16^ = 3 1 = 198 36 12 Scale of units :— 12, 3, 5i, 40, 8. Note. — Cloth Meamre is practically out of use. In measuring goods sold by the yard, the yard is divided into halves, fourth^^ eighths, and sixteenths. At United States Custom Houses, in estimating duties, the yard is divided into tenths and hundredths. ^ FOR MEASURING HEIGHTS AND DISTANCES. 3 inches 1 palm, l 9 inches 1 span. 4 " Ihand. | 3t'o feet 1 pace. MARINER'S MEASURE. Table used by mariners in calculating distances on water, and the speed of vessels. 9 in 1 span. 8 spans, or 6 ft....l fathom. 120 fath 1 cable's length. 7 J cables 1 mile or knot. 51 ft. nearly.... 1 " " " 3 miles 1 league. Note.— The number of knots of the log line run off in half a minute indicates the number of knots of distance a vessel goes per hour. WEIGHTS AND MEASURES. 255 SURVEYOR'S LONG MEASURE, For measuring boundaries of land, areas, railroads, canals. Tjort Inches I link. I 4 Rods 1 chain. 25 Links 1 rod. | SO Chains 1 mile. EQUIVALENTS. mi, ch. rd, I. hi, 1 = 80 = 320 = 8000 ^ 63360 1 ^ 4 =r 100 -= 792 1 r= 25 --= 198 1 = 7.92 Scale of Units :— 7.92, 25, 4, 80. 10 chains long by 1 broad, or 10 square chains, 1 acre. Gunter's Chain, which is the unit of measure used by surveyors, is QQ feet long, consisting of 100 links. Measurements are recorded in chains and hundredths. Latterly a steel measuring tape 100 feet long, with each foot divided into tenths, is used by engineers as a sub- stitute for the cumbersome chain. Note. — By scientific persons and revenue officers, the inch is divided into tenths^ hundredlhs, etc. Anionji; mechanics, the inch is divideJ into nffhths. The division of the inch into 12 parts, callel lines, is not now in use A standard Kiiulisb mile, wliich is the measure that we use, is 5280 feet in U-U!;ih, ITiJO yards, or 320 rods. A strip, one rod wide and ono mile long, is tN\o acr(-s. By this it is easy to calcnlate the quantity of laud taken up by roads, and also how much is wabteJ by fenced. TABLE For Geographical and Astronomical Calculations. 1 Geographic mile 1 .15 statute miles. 3 *' " 1 league. 60 " " or 69.16 statute miles.l degree. 360 Degrees Circumference of the earth. Note. — The earth's circumference is 24,8553/^ miles, nearly. The nautical mile ijG'J75| feet, or 71)5^ fiet longer than the common mile. 256 orton & Sadler's calculator. SUKFACE OB SQUABE MEASUKE Used in ascertaining the extent of surfaces, such as land, boards, plaster- ing, paving, etc. TABLE. 144 Square Inches (sq. in.). ..I square foot, sq. ft. 9 Square Feet 1 square ysird,sq.yd. 30i Square Yards 1 sq. rod or perch, sq. rd.; P. 160 Square Rods 1 acre, A. 640 Acres 1 square mile, sq. mi. »q.mi. A. sq. rd. sq. yd. sq.ft. sq. in. 1 — 640 =: 102400 = 3097600 = 27878400 ==4014489600 1 =^ 160 =:^ 4840 = 43560 = 6272640 1= 30i= 2721-== 39204 1 ~ 9 == 1296 1 ^ 144 Scale of Units :— 144, 9, 30i, 40, 4, 640. Measure 209 feet on each side, and you have a square acre within an inch. Note. — The following gives the comparative size, in square yards, of acres in different countries: English acre, 4840 square yards; Scotch, G150; Irish, 7840; Ham- bnrgh, 11,545; Amsterdam, 9722; Dautzic, G650; Frunce (hectare), 11,900; Prussia (morgen), 3053. This difference hhould be borne in mind in reading of the products per acre in different countries. Our laud measure is that of England. Artificers estimate their work as follows: By the square foot ; as in glazini,', stone-cutting, etc. By the square yard, or by the square of 100 square feet; as in plastering, flooring, roofing, paving, etc. One thousand shingles, averaging 4 inches wide, and laid 5 inches to the weather, are estimated to be a square. WEIGHTS AND MEASURED. 257 SURVEYORS' SQUARE MEASURE, Used for measuring the area or contents of fields, farms, and government lands. TABLE. 625 Square Links (s<7. ^.) 1 pole, P. 16 Poles I square chain, 6<7. ch, 10 Square Chains 1 acre, A. 640 Acres 1 square mile, sq^mi. 36 Square Miles (6 miles sq.)....l township, Tp, EQUIVALENTS. Tp, sq. mi. A. sq. ch. P. sq. I. 1 == 36 == 23040 = 230400 ^ 3686400 == 2304000000 1 = 640 = 6400 = 102400 ^ 64000000 1 = 10 ■= 160 ^ 10000 11= 1(3 z=: 1000 I = 625 Scale of Units :— 625, 16, 10, 640, 36. The acre is the unit of land measure. GOVERNMENT LAND MEASURE. A township — 36 sections, each a mile square. A section— 640 acres. A quarter section, half a mile square — 160 acres. An eighth section, half a mile long, north and south, and a quarter of a mile wide — 80 acres. X sixteenth section, a quarter of a mile square— 40 acres. The sections are all numbered 1 to 36, commencing at the northeast corner, thus : The sections are all di- vided into quarters, which are named by the cardinal points, as in section 1 . The quarters are divided in the same way. The descrip- tion of a forty-acre lot would read: The south half of the west half of the south-west (quarter of sec- tion 1 in townsliip 24, north of range 7 west, or as the case might be; and sometimes will fall short, and sometimes overrun the number of acres it is supposed to contain. 6 6 4 3 2 NW 1 NE SW 1 SE 7 8 9 10 11 12 18 17 16* 15. 14 13 19 30 20 29 32 21 28 33 22 27 34 23 35 24 25 131 36 258 ORTOIS^ « SADLER 8 CALCULATOR. CONTENTS OF FIELDS AND LOTS. For the convenience of farmers and others who desire lay off small lots of land for sale, or to ascertain the amount of land in fields, the following table is prepared, and will be found accurate : 52^ ft. square or 2722^ square ft. - = 1 A 735 - " " 5445 10890 120-.V *' " " 14520 U7^^ a " <^ 21780 2085 * " " 43560 10 rods X 16 rods z=: 1 8 " X 20 " ==:: 1 6 " X 32 " = 1 4 " X 40 " zr= 1 5 yds. X 968 yds. 10 ^' X 484 " 1 20 " X 242 " = 1 40 " X 121 *' = 1 80 " X 60.^ " = 1 70 " X 69;^ " = 1 220 feet X 198 feet = 1 440 " X 99 " = 1 ^ig of an acre. 1 ^l a n = i *' " u _ ^ u a a ^ 1 110 feet X 396 feet = 1 A 60 " X 726 " z= 1 120 " X 363 " = 1 240 " X 18U " z=^ 1 200 " X 108^9/' = i 100 " X145,V' z= J 100 " X 108^0-" — i 25 ft. X 100 ft. = .0574 25 " X 110 " := .0631 25 " X 120 " = .0688 25 " X 125 " = .0717 25 '* X 150 " = .109 TABLE Showing the number of Rails, Stakes and Posts required for 10 rods of Post-and-Rail Fence. Length of rail. ft. Length of pan el. ft. No. of panels. No. of posts. Number of rails. 6 rails high. I 7 rails high. 10 12 14 163^ 8 10 12 14^ 20% 21 17 14 12 124 99 83 68 145 116 96 79 Note. — In arranging the above table 12 inches lap have been al- lowed. The greater the lap, the stronger and more durable the fence. To ascertain the number of rails, etc., for any desired length offence, multiply the numbers given in the above table by the length in feet, and point off one figure from the left, and you have the desired result. WEIGHTS AND MEASURES 259 TKOY WEIGHT Is used for weighing gold, silver, platina, and pre- cious stones, except dia- monds ; also in philosoph- ical experiments. TABLE. 24 grains {gr.) make 1 pen- nyweight, pwt. 20 pwt. make 1 ounce, oz. 12 oz. " 1 i)Ound, Ih. ^'t-i iffiS ^gntii EQUIVALENTS. Ih. OZ. pivt. 1 = 12 = 240 1 = 20 1 gr. 6760 480 24 Scale of units:— 24, 20, 12. Note — Troy "Weight contains 5760 grains to the pound, or 1240 grains less tlian the avoirduiwis pound. In mixing medicines, not Uquid, apotliecaries use Troy grains. DIAMOND WEIGHT TABLE. 16 parts 1 grain. 4 gr 1 carat. 1 K 3i Troy grains, nearly Zh The word carat is used to express the fineness of gold, and means ^^ part. Pure gold is said to be 24 carats fine ; if there be 22 parts of pure gold and 2 parts of alloy, it is said to be 22 carats fine. The standard of American coin is nine-tenths pure gold, and is worth $20.67. What is called the neiv standard, used for watch cases, etc., is 18 carats fine. The term carat is also applied to a weight of 3i grains Troy, used in weighing diamonds; it is divided into 4 parts, called grains ; 4 grains Troy are thus equal to 5 grains diamond weight. 260 ORTON & SADLER S CALCULATOR. PAPER AND BOOKS. The followiDg de- nominations of meas- ure are used by the paper manufacturer, book and stationery trade. TABLE. 24 Sheets I quire. 20 Quires 1 ream. 2 Reams 1 bundle. 5 Bundles 1 bale. 1 Bale contains 200 quires or 4800 sheets. SIZES OF PAPER. Paper manufactured to order can be made any desired size. The following are regular or trade sizes, and can generally be found in stock at any of the wholesale paper houses. WRITING PAPERS— FLAT CAP. Name. Size, In. Law Blank 13x16 Flat Cap 14X17 Crown 15X19 Demy 16X21 Folio Post 17X22 Check Folio 17X24 Double Cap 17X28 Ex. Size Folio.... 19X23 Name. Size, In. Medium 18 X23 Royal 19 X24 Super Royal... 20 X28 Imperial 22 X30 Elephant 22^X271 Columbia 23 X33j Atlas 26 X33 Doub. Elephant. 26 X40 PRINTERS AND STATIONERS. 261 WRITING PAPERS— FOLDED, j^ame. Size, In. | Name. Size, In. Billet Note 6 X 8 Octavo Note 7 X 9 Commercial Note .8 X 10 Bath Note 8^x14 Letter 10 X16 Oommerc'l Let.ll x 11 Packet Post.... 11^X18 Packet Note 9 Xll Ex. Pack. Post. lUxlSJ Foolscap 12^X16 PRINTING PAPER. Used in Printing Newspapers and Books. Name. Size, In. ] Name. Size, In. Medium 19x 24 i Double Medium. . 24x 38 Royal 20x25 ! Double Royal. . . , 26x40 Super Royal 22x28! Doub. Sup. Royal . 28 X 42 Imperial 22x32 " " " .29x43 Medium-and-half.. 24x30 Broad Twelves. . .23x41 Small Doub. Med. 24x36 ■ Double Imperial. .32x46 COPYING. In the copying of papers, manuscripts, and documents for official record, clerks and copyists are usually paid by the folio. A folio varies in quantity in different States and sections of the world, but is generally esti- mated from 75 to 100 words. PRINTING— TYPE-SETTING. Printers generally charge for setting type, or the composition of matter, as it is technically termed, by the number of ems it contains, rated by the 1000 enis; an em is the square of the body of the type. 262 ORTON & SADLER^S CALCULATOR. PRINTING-PRESS-WORK. Press-work is generally charged by the token of 250 impressions, or 125 sheets, printed on both sides. The value or cost of press- work depends upon the style, quantity, and quality of ink used. MISCELLANEOUS TABLES. FOR COUNTING CERTAIN ARTICLES. 12 Units or pieces 1 dozen. 20 '* " 1 score. 12 Dozen 1 gross. 12 Gross .1 great-gross. BOOKS. Names and Sizes as classified hy Publishers, The number of folds and pages in a single sheet when manufactured. Name of book. Ji^illtol'S.Vi. Contain. Folio 2 leaves,... 4 pages. Quarto or 4to 4 " 8 '* Octavo or 8vo 8 '' 16 '' Duodecimo or 12 mo 12 ** .... 24 " 16 mo 32 " ....64 " 18 '* * 18 " ....36 *' 24 " 24 " ....48 " 32 " 32 " ....64 '' * Note. — This book is an 18mo., there being 36 pages to the sheet. The terms folio, quarto, octavo, etc., denote the number of leaves in which a sheet of paper is folded. The marks A, B, C ; 1, 2, 3 ; lA, 2A ; l-% 2*, etc., oo- casionally found at the bottom of pages, are what printers term signature marks, thus, 3*, being printed for the direc- tion of binders in folding the sheets. PRINTERS AND STATIONERS. 263 Printers and stationers generally procure their supplies of paper by the quantity, the cost per ream varying according to the quality and weight. The table on page 264 will be found invaluable in making up their estimate for small job work, as it shows at a glance the cost per quire of paper purchased at 15 to 30 cents per pound, and weighing from 10 to 60 pounds to the ream. Example. — What does one quire of paper cost purchased at 23 cents per pound, and weigh- ing 40 pounds to the ream ? Am, 46 cents. Explanation. — Refer to the weight column on the left and the purchasing price at the top, and you have the cost per quire shown in the purchasing column on the line with the weight. SHOEMAKER'S MEASUBE. 3 Barleycorns or sizes 1 inch. Number one, children's measure^ is 4| Inches, and that every additional number calls for an increase of i of an inch in length. Number one, adults* measure, is 8^ inches long, with a gradual increase of \ of an inch for additional numbers, so that, for example, number ten measures Hi inches. This measure corresponds to the number of the last^ and not to the length of the sole. TABLE To Ascertain the Cost of One Quire of Paper. CV3 1 OCCrH^t^CeOtOOl'MOaO^Ttil^OTCOOiCMtfliOOi-i-^t-O CD cq iq ■'i; M r^ I-; q CO t^ O ic rf CO (N r- q oo t-^ «0 o ^^ « iM^ ^, ■^' t- O ri '^ O •— ' -r t- C C5 CO ci 'm' in OC O* CO O Cl (M O OO — ' ^' I- S 00 O Tf; 0^ CO «q -t; -M^ OO O •*, ■>! 00 O "* C^ CO O -^, CM Tt O OS ?4 O » O CO « ci ^1 -f" »- C." CO O od r-I -T-' t^ O ci O CO r-< ■^ r-HrHr-»icocococo'r^iroo.oiO'x>0'£t-t-t-t-cx)oo g5 iC 0^1 q O CO l^ rt< r-i CO lO 'M O O CO t- ■<*i r-t CC iC 01 05 O CO CO* d QO r-J Tji' t^ 05 «m" lO t^ d CO lO CO i-i Tf> d d (m' -t !-■ d (N d 00 r- i-M-.rH(Mcq(N'McococoTrTtirri'roio«ouoo»oio»c>o»o oi lO t^ o m' lO t-^ o oi lO t- o oj ic t-^ o w .o t-^ o (m" lo 1- o c4 lO r-(,^r-i(MO^-MClCOCOCOOO'i d i^ o oi tt' t-^ d (M ^r-r-r-iCMCl«** t- r- d d CO* d CO d t-^ d ^i T^ d d 1— ' CO o CO d c^i d i-l d c4 -:t d oi ,-( ^ r- ^ (M d o-i '^^ d 00 d oi -*■ d oo c; I-* c? d t-" d r-i d d t- 1 Or-.i-H.-.r-tr-icMOICNC^G^COCOCOeOCOCOTj.rJirfTti'^tO.OOO s 1 oqcO'iiioixooO'^o^ ooco'+co cocorj-oi oqco-*, oi q d (M* -+■ d oc* d r-* d d h- 00* d oi -r CO 1-^ d .-^ d ic d d d oi -* 1 (-^i-ii— i-H.— r-r-'0^*i-<*-<^-i''*0»Oi.O ti 1 >.0>(MOCOrO t-'*r-OOiO<"•• " &P MEASUREMENT OF TIME. 267 The greatest distance across a circle is called its diameter. The dis- tance around it is called its circumference. Any part of the circum- ference is called an arc. LONGITUDE AND TIME. TABLE. For a difiference of | There is a difference of l^iu Long 4 m. in Time. 1' " 4 sec. " V « ^mc. " 1 hr. in Time 15° in Long. Im. « 15' Isec. " lb" NoTB. — Add difference of time for places east and subtract it for places weM of any given place. HOW TO ASCERTAIN The Difference of Time between Cities. BASIS OF CALCULATION. 360 degrees = 1 revolution of the earth, or 1 day. 1440 minutes = 1 " " " '' " 1 " 1440 -7- 360 = 4 minutes, or 1 degree. Refer to your map and notice the difference in degrees of longitude between places. Multiply the number of degrees by 4 ; the product will be the difference in time. Degrees of Longitude East, time increases. *' " " West, " decreases. PROBLEM. When it is 12 o'clock noon at Washington, what time Is it at Boston? A ns., 12.24. Per map, difference in degrees, 6 east, which increase the time 6 X 4 = 24 differences in time, or 24 minutes past 12. PROBLEM. When it is 12 o'clock noon at Washington, what is the time in San Frjincisco, Califonii i ? Ans., 8.58 a. m. Per map, difference 45^ degrees, West; Ab}/^ X 4 = 182 min., or 3h. 2m. less. Ans., 8 o'clock, 58 min., a. m. A telegram sent from Washington to San Francisco at 12 m. will be received at 9 o'clock a. m. (i. e., three hours lifore it is stnt)^ calculat- ing Son i" rancisco time. 268 ORTON & Sadler's calculator. TABLE For ascertaining the number of days between two dates. Jan. Feb. Mar. Apl. May June July Aug. Sept. Oct. Nov. Dec. 1 32 60 91 121 152 182 213 244 274 305 335 2 33 61 92 122 153 183 214 245 275 306 3:56 3 34 62 93 123 154 184 215 246 276 307 337 4 35 63 94 124 155 185 216 247 277 30S 338 5 30 64 95 125 156 186 217 248 278 309 339 6 37 65 96 126 157 187 218 249 279 310 340 7 38 66 97 127 158 188 219 250 280 311 341 8 39 67 98 128 159 189 220 251 281 312 342 9 40 08 99 129 160 190 221 252 282 313 343 10 41 09 100 130 101 191 222 253 283 314 344 U 42 70 101 131 162 192 223 254 284 315 345 12 43 71 102 132 163 193 224 255 285 316 346 Vi 44 72 103 133 104 194 225 256 286 317 347 14 45 73 104 134 165 195 226 257 287 318 348 15 46 74 105 135 166 196 227 258 288 319 349 16 47 75 lOo 136 167 197 228 259 289 320 350 17 48 76 107 137 168 198 229 260 290 321 351 18 49 77 108 138 169 199 230 261 291 322 352 VJ 50 78 109 139 170 200 231 262 292 323 353 20 51 79 110 140 171 201 232 263 293 324 354 21 62 80 111 141 172 202 233 264 294 325 355 22 53 81 112 142 173 203 234 265 295 326 356 23 54 82 113 143 174 204 235 266 296 327 357 24 55 83 114 144 175 205 236 267 297 328 358 25 56 84 115 145 176 206 237 268 298 329 359 26 57 85 116 146 177 207 238 269 299 330 300 27 58 86 117 147 178 208 239 270 300 331 361 28 69 87 118 148 179 209 240 271 301 332 362 29 88 119 149 180 210 241 272 302 333 363 80 89 120 150 181 211 242 273 303 334 364 31 90 151 212 243 304 365 Note. — To find from the above table the number of days between two dates, we give the following — RuLR I. — WJien the dates are in the same year, subtract the number of days of the parlier date from the number of days of the later date; the result will be the number of days required. II. When the dates are in consecutive years, subtract the number of days of the earlier date from 365, and add to the remainder the number of days of the later date; the residt will be the number of days required. When the year is a leap year, add one day to the result. MEASUREMENT OF TIME. 269 TABLE Showing the number of days from any day in one month to the same day in any other. 4 03 t < eg p s >, s be < ^ A ^ 1 > o 55 365 31 59 90 120 151 181 212 243 273 304 334 365 28 59 89 120 150 181 212 242 273 306 337 365 31 61 92 122 153 184 214 245 275 306 334 365 30 61 91 122 153 183 214 245 276 304 335 365 31 61 92 123 153 184 214 245 273 304 334 365 30 61 92 122 153 184 2151243 274 304 335 365 31 62 92 123 153 184 212 243 273 304 334 365 31 61 92 122 153|181 212 242 273 303 334 365 30 61 92 123 1 151 182 212 243 273 304 335 365 31 61 921120 151 181 212 242 273 304 334 365 31 62 90 121 151 182 212 243 274 304 335 Jan.. . Feb... March April. May . . June . July . Aug... Sept . . Oct . . . Nov.., Dec... 334 303 275 244 214 183 153 122 91 61 30 365 Note. — Find in the left-hand column the month from any day of which you wish to compute the number of days to the same day in any other month ; then follow the line along until under the desired month, and you have the required number of days. Example. — How many days from March 15 to July 15 ? Ans., 122 days. In leap-year, when the month of February occurs in the calculation, one day extra must be added. Example. — 1876. How many days from January 13 to May 13 ? Ans., Per table, 120 ; one day added for leap- year = 121 days. ASTRONOMICAL CALCULATiaNS A scientific method of telling immediately what dau of the week any date transpired or will transpire^ from the commencement of the Christian Era, for the term of three tnousand years. MONTHLY TABLE. The ratio to add for each month will be found in the following table: Ratio of October is 3 Ratio of May is 4 Ratio of August is ^.6 Ratio of March is 6 Ratio of February is 6 Ratio of November is 6 Ratio o€ June is Ratio of September is 1 Ratio of December is 1 Ratio of April is 2 Ratio of July is 2 Ratio of January is ..3 Note. — On Leap Year the ratio of January is 2, and the ratio of February is 6. The ratio of the other ten months do not change on Leap Years. CENTENNIAL TABLE. The ratio to add for each century will be found In the following table: o 200, 900, 1800, 2200, 2600, 3000, ratio is 800, 1000, ratio is 6 5- 400, 1100, 1900, 2300, 2700, ratio is 6 ^ 600 1200, 1600, 2000, 2400, 2800, ratio is 4 2 600 1300, ratio is 3 000, 700, 1400, 1700, 2100, 2500, 2900, ratio is 2 loo, 800, 150a ratio IB 1 270 ASTRONOMICAL CALCULATIONS. 271 Note. — The figure opposite each century is itg ratio; thus the ratio for 200, 900, etc., is 0. To find tho ratie of any century, first find the century in the above table, then run the eye along the line until you arrive at the end; the small figure at the end is its ratio. METHOT) OP OPERATION. Rule.* — To the given year add its fourth part. Tweeting the fractions ; to this sum add the day oj the month; then add the ratio of the month and the ratio of the century. Divide this sum hy 7 ; the remfiainder is the day of the week^ counting Sunday a» the first^ Monday as the second^ Tuesday as the third, Wednesday as the fourth, Thursday as the fifth, Friday as the sixth, Saturday as the seventh; the remainder for Saturday will he or zero. Example 1. — Required the day of the week for the 4th of July, 1810. To the given year, which is 10 Add its fourth part, rejecting fractions 2 Now add the day of the month, which is 4 Now add the ratio of July, which is 2 Now add the ratio of 1800, which is DiTide the whole sum by 7. 7 | 18 — 4 2 We have 4 for a remainder, which signifies the fourth day of the week, or Wednesday. • Wheti dividing the year by 4, always leave off ttie coatories* Wb #lYid« by i to find the number of Leap Years. 272 ORTON & Sadler's calculator. Note. — In finding the day of the week for the present century, no attention need be paid u) the centennial raUo^ as it is 0. Example 2. — Required the day of the week for the 2d of June, 1805. To the given year, which ia 6 Add its fourth part, rejecting fractions ..,.. 1 Now add the day of the month, which is 2 Now add the ratio of June, which is Divide the whole sum by 7. 7 | 8^i T We have 1 for a remainder, which signifies the first day of the week, or Sunday. The Declaration of American Independence was signed July 4, 1776. Required the day of the week. To the given year, which is 76 Add its fourth part, rejecting fractions 19 Now add the day of the month, which is 4 Now add the ratio of July, which is 2 Xow add the ratio of 1700, which is 2 Divide the whole sum by 7. 7 | 103 — 6 14 We have 5 for a remainder, which signifies the fifth day of the week, or Thursday. The Pilgrim Fathers landed on Plymouth Rock Doc 20, ltf20. Reqmired the day of the w^eeV ASTRONOMICAL CALCULATIONS. 273 i'o the given year, which is JiO Add its fourth part, rejectiug fractions 6 How add the day of the monih, which is 20 Now add the ratio of December, which is 1 Now add the ratio of 1600, which is 4 Divide the whole sum by 7. 7 | 50- -1 7 We have 1 for a remainder, which signifies th* first day of the week, or Sunday. On what day will happen the 8th of January, 1815? Ans. Sunday. On what day will happen the 4th of May, 1810? On what day will happen the 3d of December, 1423? Ans, Friday. On what day of the week were you born? The earth revolves round the sun once in 365 days, 5 hours, 48 minutes, 48 seconds ; this period is, therefore, a Solar year. In order to keep pace with the 6o\sLT. year, in our reckoning, we make ©very fourth to contain 366 days, and call it Leap Year. Still greater accuracy requires, howevei, that the leap day be dispensed with three times (D every 400 years. Hence, every year (except the centennial years) that is divisible by 4 is a Leap Year, and every centennial year that i? divisible by 400 is also a Leap Year. The next i^«Dtenn.'al year that will be a Leap Year h 2000 274 ORTON & SADLER S CALCULATOR. MONEY OP THE UNITED STATES Is the measure of value of all kinds, such as property, merchandise, services, etc. It is the medium of exchange in business. Coin or Specie is metal stamped and au- thorized by government to be used as money. Paper Money con- sists of notes issued by the United States Treas- ury and banks, and used as money. United States money is the legal currency of the United States. U. S. MONEY. 10 mills {M,) 1 cent ct. or ^. 10^ 1 dime 10 dimes ...1 dollar i>.or$. 10 dollars 1 eagle E. Note. — The mill is not coined. The Coin of the United States consists of gold, silver, nickel and bronze, and as fixed by the " New Coinage Act" of 1873 is as follows : Gold. The double-eagle, eagle, half-eagle, quarter- eagle, three-dollar, and one-dollar pieces. Silver. The jTmcZ^-doUar, half-dollar, quarter-dollar, the twenty-cent, and the ten-cent pieces. Nickel. The five-cent and three-cent pieces. Bronze. The one-cent piece. Note. — The term shilling is frequently used in the United States in stating the price of articles, and it indicates old divisions or equivalents of parts of the dollar. Its value vnries in dilferent States as follows: In the New England States, and in Indiana, Illinois, Missouri, Missis- sippi, Texas, Virginia, Kentucky, and Tennessee, ls.=:16% cts., and $1. =6s, ; in New York, Ohio, Michigan, and North Caiolina, l6.=123/^ct8. and $1 .=8s. ; in Pennsylvania, New Jersey, Delaware, and Maryland, l8.=13i^ cts., and $i.=7^8. ; in Georgia and South Carolina, l8.= 21 1 cts., and $1 . =i%B. These rates are liable to variations by custom ; as, in Illinois, tho shilling is rated frequently at 12}^. Canada Money consists, like United States money, of dollars and cents. The Canada coins are twenty-cent, ten-cent, five-cent, silver; and one-cent, bronze. MONEY OF FRANCE. 275 MONEY OF FRANCE. The money of account of France is the Franc of 100 Centimes or 1000 Sous, and is arranged on the decimal system. The principal gold coins in circulation are as follows : Louis d'or, Forty Franc piece, Twenty Franc piece, and Six Franc piece. The principal silver coins are, the Crown, } Crown, i Crown, Five Franc piece, Two Franc piece. Franc, i Franc of 50 Centimes, i Franc of 26 Centimes. The par value of the Franc is 19 cents and 3 mills, but its commercial value varies according to the fluctuations of the money market. When exchange is quoted at say 5.17i, it is understood to mean that 5 Francs and 17 i Centimes are equal to the gold dollar. Tlie gold cost of any given number of francs is, therefore, ascertained by dividing that number by the quotation or rate of exchange. For example, the cost of 9463 Francs at 5.21}=9463 fr.-T-5.215 fr.=$1814.57. The premium on gold is then added to find the cost in currency. 276 ORTON & Sadler's calculator. MONEY OF THE GERMAN EMPIRE. The German Empire has, within the past few years, issued a new coin called the Mark, which is' now adopted as the money of account of that nation. The gold and silver coins are quite numerous, embracing, as they do, those in use in about twenty-two States, and are as follows : Gold, — Ducat, Quintuple Ducat, Five Thaler piece, Ten Thaler piece. Double d'or, \ Caroline, i Caroline, Caroline, Five Gilder piece, Twenty Mark piece, Ten Mark piece, and Twelve Mark piece. Silver. — Mark, Thaler, Double Thaler, Crown Thaler, J Thaler, Double Gilder, Florin, i Florin, Twelve Grote piece, Grote, Rix Dollar, Crown, Kreutzer Groschen, 6 Pfen, 1 Schilling, 48 Schilling piece, 30 Kreutzer piece, 8 Schilling piece. In commercial transactions the Mark is not generally reckoned at its par value, but is gov- erned by the quotations which range at present between 91 cents and $1.00 for 4 Marks. The market value of the Mark, in gold, is found by dividing the quotation by 4. For example, 400 Marks at $.95 i = $.955 H- 4 m. X400 = $95.50. The premium on gold is then added to find the cost in currency. , ARBITRATION OF EXCHANGE. The method of finding the value, in gold and currency, of the moneys of the principal nations has already been briefly explained ; but the fluctuations of foreign exchange sometimes render it to the advantage of the merchant to remit indirectly. For example, suppose it is required to pay a debt due in France, and that the balance of trade between Great Britain and the United States is in favor of the latter nation. This, of course, would cause a decline in the value of the £ sterling in United States money. Now, if there is no corresponding difference in value be- tween the moneys of Great Britain and France, it would be cheaper first to purchase sterling and then remit through London. Such exchanges are termed direct and indirect, as the case will indicate, and are treated under the rule of ARBITRATION OF EXCHANGE. The limits of a work of this kind prevent an extended elucidation of this subject, and it is not thought best to pre- sent any set rules for memorizing. In the following examples, with solutions, indirect ex- change will be found much simplified. Indeed, when taken in the order of the different nations involved, there will appear but little distinction from the method of work- ing ordinary exchange. Example 1.— When sterling is quoted at $4.83^ U. S. money, and 26.73 fr., French, how many franca are equal to the gold dollar by in- direct exchange ? Solution.— U £1= $4,831^, and also 25.73 fr., $4.83>^ must equal 25.73 fr. Therefore 25.73 fr. -4- $4,835 = 5.32 fr., ana. Example 2.— When French exchange is quoted b.W/,fr,&nd ster- ling 25.73 fr., what is the value in gold of the £ sttrling by indirect exchange ? ^/M^ion.— Since $l=5.16Ufr., and £1=25.73 fr., therefore 25.73 fr. -5- 5.165 = $4.98. Example 3.— What would be the gain in sterling on $6000 by re- mitting through France, Germany, and Netherlands, with the follow- ing quotations: $1= 5.18 fr. ; 1.22 fr. = 1 mark ; 1.71 marks = 1 guil. ; 11.8^uil.=£l, whcMi the £ sterling by direct exchange is quoted $4.85? Solution.— As Franco is the first country, $6000 X 6.18 fr. = oldSO fr. -*-1.22fr. = 24575.475 m. -r- 1.71 m. =14847.94 guil. -=- 11.8 guil.= £1202.537 = £1262 78. 4%-|-d., indirect exchange. $G000-=-$4.85=£1237. 1134 = £1237 28. 3i<-f .1. direct exrlmngo. £1262 7«J. 4^d. less £1237 2i. Z]4d. = £25 68. 13^. gain. 24 277 278 ORTON & Sadler's calculator. J5 ?J .si s^ ;^ II ^1.00 .45,3 .19,3 .96,5 .54,5 1.00 .91,2 .91,8 .91,2 .92,5 .26,8 .91,8 4.97,4 .19,3 4.86,6] 3 c > Gold and silver Gold and silver Cin\(\ Gold Gold Silver Gold Gold Silver Gold and silver Gold ; ^ 2 o > E>H (^ Q B : < ; -c ^ i -^ -s 2 3 § S fc2 S 2i v c P ^ pi! ^ a VALUE OF FOREIGN COINS. 279 « 00, M i oT eo" lo rH CM. O 00^ O^ tH^ -^^ t-^ cm" oo" ro i Oi 05 O t~ ' eooofoooococco " -*" i-T ■^'" O 05 OJ .2 * ^ ^ is -s ^ s :s : ^ g SiJ -; » U U U O eS lU O O S S. ! S ^ «• « ° 3 GOLD AND CURRENCY. Gold is usually represented as rising and falling, but being the standard of value, it does not vary. The varia- tion is in the currency substituted for gold or specie ; hence, when gold is said to be at a premium, the currency or circulating medium is made the standard, while it is in fact below par. TABLE Showing the Comparative Value of Gold and Currency, The Amount in When $1 in Gold is sold for The Discount on Gold \vlii :;h can be Currency at Currency is boujrht forSlCO ^ in Cu 'lency. 1.01 or 1 per cent. Prem. 1.00 per cent. $99.99 or 99^.1^ 1.05 5 " " 4.77 « 95.23 " 1.10 « 10 «« " 9.10 " 90.90 ' 1.16 " 15 « «* 13.04 « 86.96 » 1.20 « 20 « " 16.67 u 83.33 " 1.25 « 25 u 20.00 tt 80.00 ' 80 1.30 « 30 u 23.08 «« 76.92 • 75^^ 1.331^ " 331^ " 25.00 '« 75.00 " 1.40 " 40 M 28.58 « 71.42 ' 71 if 1.50 " 50 " 33.33 «* 66.66 " 66% l.GO " 60 « 37.50 " 62.50 " G2y^ 1.66% " 66K « 40.C0 " 60.00 « 60 1.70 " 70 " 41.18 " 58.82 " 58}f 1.80 '' 80 « 44.45 " 55.55 " 55f 1.90 " 90 « 47.37 " 62.63 " %i^ 2.00 " 100 *' 50.00 « 50.00 " 2.50 " 150 " 60.00 « 40.00 * 40 6.00 " 400 " " 80.00 " 20.00 ' 20 7.50 " 650 « " 86.67 " 13.33 ' 135^ 10.00 " 9C0 '« K 90.00 «♦ 10.00 10 50.00 •' 4900 ♦^ " 98.00 (( 2.00 ' 2 100.00 " 990O « " 99.00 " 1.00 ' 1 TO ASCERTAIN HOW MUCH GOLD Can be Bought for a Staled Amount of Currency. Rule. — Add two ciphers to the amount of currency (in dollars), and divide by 100, increased by the premium, rate on gold; the quotient will be the gold sum. 280 BANK ACCOUNTS. 281 HOW TO TRANSACT BUSINESS WITH BANKS. BANK ACCOUNTS. HINTS TO MANY AND PRACTICAL ADVICE TO THOUSANDS. It is the belief of many observant philosophic persons who are well on their way tlirough life, that it* people generally knew more they would behave better, though few, if any of them, believe that knowledge and morality are synonymous terms. Acting on this conviction, the following "Hints to those who keep Bank Accounts'' have been suggested by a gentleman well qualified by general intelligence and long practical experience to ad- vise the young and untaught of the several matters. HINTS TO THOSE WHO KEEP BANK ACCOUNTS. 1. If you wish to open an account with a bank, pro- 282 ORTON & Sadler's calculator. vide yourself with a proper introduction. Well-managed banks do not open accounts with strangers. 2. Do not draw a check unless you have the money in bank or in your possession to deposit. Don't test the courage or generosity of your bank by presenting, or al- lowing to be presented, your check for a larger sum than your balance. 3. Do not draw a check and send it to a person out of the city, expecting to make it good before it can possibly get back. Sometimes telegraphic advice is asked about such checks. 4. Do not exchange checks with anybody. This is soon discovered by your bank ; it does your friend no good and discredits you. 5. Do not give your check to a friend with the condi- tion that he is not to use it until a certain time. He is sure to betray you, for obvious reasons. Do not take an out-of-town check from a neighbor, pass it through your bank without charge, and give him your check for it. You are sure to get caught. 6. Do not give your check to a stranger. This is an open door for fraud, and if your bank loses through you, it will not feel kindly to you. 7. When you send your checks out of the city to pay bills, write the name and residence of your pnyee, thus: Pay to Jno. Smith & Co., of Boston. This will put your bank on its guard, if presented at the counter. 8. Don't commit the folly of supposing that, because you trust the bank with your money, the bank ought to trust you by paying your overdrafts. 9. Don't suppose you can behave badly in one bank and stand well with the others. You forget there is a Clearing House. BANK ACCOUNTS. 283 10. Don't quarrel with your bank. If you are not treated well, go somewhere else, but don't go and leave your discount line unprotected. Don't think it unreason- able if your bank declines to discount an accommodation note. Have a clear definition of an accommodation note — in the meaning of a bank, it is a note for which no value has passed from the endorser to the drawer. 11. If you want an accommodation note discounted, tell your bank frankly that it is not, in their definition, a business note. If you take a note from a debtor with an agreement, verbal or written, that it is to be renewed in whole or in part, and if you get that note discounted, and then ask to have a new one discounted to take up the old one, tell your bank all about it. 12. Don't commit the folly of saying that you will guarantee the payment of a note which you have already endorsed, 13. Give your bank credit for being intelligent gener- ally and understanding its own business particularly. It is much better informed, probably, than you suppose. 14. Don't try to convince your bank that the paper or security which has already been declined is better than the bank supposes. This is only chaff. 15. Don't quarrel with a teller because he does not pay you in money exactly as you wish. As a rule, he does the best he can. 16. In all your intercourse with bank officers, treat them with the same courtesy and candor that you would expect and desire if the situations were reversed. 17. Don't send ignorant and stupid messengers to bank 10 transact your business. 284 ORTON & Sadler's calculator. \ INTEREST-COMMERCIAL RULE. Commercial Tear. — Calculations based on 360 days to the year, or 30 days to the month. SIX PER CENT. RuLK. — Multiply the given number of dollars hy the number of days of interest required ; divide the product by 6, and point off three figures from the right. Note — If cents appear in the principal, it will be necessary to point off five figures. The result is yf^ more than the true interest bashed on the calculation of 365 days per annum. To ascertain the true amount, it will only be neces- sary to deduct ^5 from the result obtained. Example. — What is the interest on $1000 for 219 days? Process.— 1000 X 219 = 219000 219000 -r- 6 = 36500 Point off three figures from the right, gives $36.50 interest. To ascertain the true interest (365 days), subtract Vy, thus ; 36.50 -j- 73 == 50. $36.50 — 50 = $36. Having ascertained the interest at 6 per cent., that for 7, 8, and 9 per cent, is readily foimd, by adding to it i, §, ^, etc., etc. To find the interest at any per cent., divide the INTEREST — COMMERCIAL RULE. 285 interest procured at 6 per cent, by 6, and multiply the amount by the required rate. VALUABLE INTEREST RULES. Basis Commercial Year of 360 days, or 30 days per month. 4 per cent. — Multiply the principal by the required number of days, divide by 9, and point off. 5 per cent. — Multiply by the number of days, and divide by 72. 6 per cent. — Multiply by the number of days, divide by 6, and point off three figures from the right. 8 per cent. — Multiply by the number of days, and divide by 45. 9 per cent. — Multiply by the number of days, divide by 4, and point off three figures from the right. 10 per cent. — Multiply by the number of days, and divide by 36. 12 per cent. — Multiply by the number of days, divide by 3, and point off three figures from the right. 15 per cent.-^ Multiply by the number of days, and divide by 24. 18 per cen^— Multiply by the number of daya, divide by 2, and point off three figures from the right. 20 per cew/.— Multiply by the number of days, and dividxi by 18. iS^The interest in each case will be in dollars and c^nlB. Rate, 10 per cent, per annum of 360 days. These Tables are arranged with a view to supply a want long existing among the great majority of busi- ness men and accountants, and are specially designed to supersede the numerous high-priced Interest Tables now before the public. CALCULATIONS. The basis is at the decimal rate of 10 per cent, per annum on the commercial year of 360 days, and the for which our calculations of interest are made is from 1 to 30 days, and from 1 to 12 months, on amounts ranging from $10 to $10,000, thereby meet- ing the wants of the capitalist as well as those of the more moderate tradesman. THE CALCUIJITIONS are of the most minute accuracy, being carried out in each instance to the fraction of one tenth of one mill ; 286 INTEREST TABLES. 287 hence, in gnmming up the araoiint of interest, it will always be necessary to point o^ four figures from the left. When amounts from $1 to $10 appear in the principal, the interest is obtained by taking the sum in the interest column opposite the required amounts from $10 to $90, and point off five figures from the left. CENTS IN THE PRINCIPAL. When there are cents in the principal in excess of 50, add $1.00, if less, reject them ; in the calcula- tion of interest this is in accordance with usual cus- tom. It is also customary when the fraction of interest is 5 mills, or in excess, to add one cent; when less, drop it. TO ASCERTAIN INTEREST on the basis of 365 days per annum. Subtract i^^ from the results obtained by tlie calcu- lations on the basis of 360 days per annum, which is equivalent to a reduction of IJ cents for every dollar of interest. TO ASCERTAIN INTEREST at any rate other than 10 per cent. Multiply the amount of interest obtained from the Tables by the required rate, and point off five figures from the left. INSTRUCTION. It will only be necessary to trace the work of the following examples to enable any one to become ex* pert in the 288 ORTON & Sadler's calculator. USE OF THE TABLES. In calculating interest, refer to the Table containing at its head the number of days or months for which interest is required, and opposite the principal in the column of dollars will be found the interest in dollars, cents, mills and tenths of mills. Example I. — Ascertain the interest on $3570 for 28 days at 10 per cent. See Table, page 204. Dollar Colnmn. Interest. $3000 $23.3333 500 3.8889 70 .5444 $3570 $27.7666 Interest at 10 per cent., 28 day*. To find the Interest on above amount at 6 per cent: Interest at 10 per cent., $27.7666 Multiply by required rate, 6 per cent. $16.65996 Point off five figures from the left and you will have the interest at 6 per cent., $16.66. OR, THUS : By removing the decimal point one place to the left we have the interest at 1 per cent. ; hence, by simply m^zltiplying by the required rate, the product will be the desired interest. INTEREST TABLES. 289 Examiile IL — Ascertain the interest on $2007 for 7 months, 17 days, at 10 per cent, ^ee Table, page 206, 7 months. Dollar Column. Interest. $2000 $116.6667 7 ($70,— $4.0833) .4083 See Table, page 202, 17 days. Dollar Column. $2000 9.4444 7 ($70.-- .3306) .0330 $2007 $126.5524 Interest 7 months, 17 days, 10 per cent. To find the interest on above amount at 7 per cent : Interest at 10 per cent., $126.5524 Multiply by required rate, 7 per cent. $88.58668 Point off five figures from the left, and we have the interest at 7 per cent., $88.59. 25 1 o or. O M -f r- t- C -M t 'S Q0 «5 O t' C4 r-» c\ I^ o So o S ^. C^. 2. ^;;=S'iq e-.^q 1 --COS OC l- -- d 00 I- f£ »o -r ro (M* i-i •«*r5Ci— • C- c: 00 r- '£> C oc I- -w .o -^2;]- S'£M^M 5.«.^^. ^qo^o c '^ q = CO ??Sls??S 'J^^IS?? 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CO oq tJj o io — q r; »^ co oo ^ - 'X> (M* I-' eo' ao ■^ -*»« CO CO* (N (N i-« r-i ^-gJL^J__g^8"o ^n^ s^ trg ? ?^ g? ?^-g^ igg.g?^ « o £: cc c CO -^ o CO -J 5-; lO CO — q Qc q uo CO f- zz ui 25 ^ ^i P ^' - * * "* o CO r- - a> I- rO C5 iC o 5S: CC C.-<(N ?0~ cc q F-< 00 rr o i^ co roi'ieo eoc4CM'c4i-i r-I^ * ' £i 5S JG r: » ^^^S ,5 hS'5J?S ^US^n SSIVB'S ??8SS5 CQ £?' 2 S d S o co' d oooo ooooo j^ I 5^55^^ OOOOO «_^<-Jtjt.ic:3 O OOOOO OOOOO OOOC^COiO (^ OC»00C^CO lO^COCSli-H 291 } n^^i^s Ort ooco gfe??8S SI 85?? S5?5§& SS!SS3 g§S82 CO O«0Q0 CO WrH ^gig^q §^Si2§ i. d Ci eg i??J'o::g 8:?S??fS g:is:=S g^^f??; ?]J::n^ I- o »d •>* "<* cl^ow co:^r-i ti3S^.^. "l^.-.o. 1 SS?i:5:-!o s:^??Si^ O CI -t 1- o C M -f- b >0 ::n^,^ 8;i:?5S r:??§?2 I- O C^l ^ o iJ c ir-t 1- C CI .O I-. ^£ O O ''J". CO ^. «-'. ^. = 1' c^gj^j5t5 ::::3ls:;?'. s??!2.r?- ^:\3ZT] Sr^'S. i? :: ssq:?? f^ «o lo o "^ S i- S r5 o O O C5 -* O «; -. -^. ^. «.^.'^.^.^. S.^i^.q :?:Sg=:5 Sl?2s:g5 S:^::rjo?:? J^ToS:? ?: o S' to t- t- X) X 0> CO c5c5fH O 't O ~ 1^ CI O CO r- -f » O <♦ 'i^ CO cicirH « S ^. 5 CO ^.s.^.i 1 ?§8^5g ?i\Bsn^ S??15S?2 58??£ 8 CO 5o 8 ?5 ?S8?I5 2§rA8S n -s) ■=> fi o n -^. o c:: o O -O O CO « 8??58f? o o -^a^ t:^ ro §8r?3 C^cir-; ^^^^^ sgqs 1 Si S§::?]?? ^SSi^S 8n:????^ g'c^?2S .*?:=?^f?5 «0 t-OOOi -f »0 O 1^ 00 cr. iC f- t- TO »0 «0 -i<; CO M cir-;^ iq 5 -(Jj M CO s.?2a 1 :::S?^Jo S^???3:2 gSSl^J? 3:???]:: s^n^^ Tssf — IC 2C -m' O »o T«5 Tji" co' co" drirt ^^.^m 1 5^ ??8?^??S 5??S&?? 85^?85 ??85S? ^^SISV: CO O O 00 ^' 'vi 2 2 ^ W5 "* "^ CO ci CO O «5 OO So ^^-V CO CI St:R3 1 ooooo •-• ooooo ooooo ooooo lOsHCOWr-l illii 88§8 ooooo OOOOC^tOlO oooo 292 1 l-H ooooo ooooo in^CONiH ooooo O500C^«OlO §§§§ SSSSS ssss a 1 a S CO « £8??58 S?^S??5 8??fe8?? &8??& 8??S8?3 58??5 $666.66 600 00 533.33 466.66 400.00 333.33 266.6ei 200 00 133.33 66.66 60.00 53.33 46.66 40.00 33.33 26.66 20.00 13.33 6.66 2.66 2.00 1.33 .66 7 months, 210°days. S?85?38 5??8?o?? 8?HS?SS ?S8I^S? 85S585 S585S5 $583.33 525.00 466.66 408.33 350.00 291.66 233.33 175.00 116.66 68.33 52.50 46.06 40.83 35.00 29.16 23.33 17.50 11.66 5.83 tartan tN* 2.33 1.75 1.16 .58 6 months, 180°days. 888S8 88888 88888 8888 88888 8888 S|8|8 250.00 200 00 150.00 100.00 50.00 45.00 4000 35.00 30 00 25.00 20.00 15 00 10.00 5.00 S888S "•t Ti^ CO CO ci 2.00 1.50 1.00 .50 gsf a g U3 rH 5S?558 ??5s8eS5 8S5?58S? 58??5 sn^su ?5853& $416.66 375.00 333.33 291.66 250.00 208.33 166.66 125.00 83.33 41.66 37 50 33.33 29.16 25.00 20.83 «o* (>i 00* Tji 3.75 3.33 2.91 2.50 2.08 1.66 1.25 .83 .41 4 months, or 120 days. ?38lo?2g 5S58':o'c? 8?o??S& ?28?o?? S^?^SIB ?585?3 ^s's^s 166.66 133.33 100.00 66.66 33.33 30-00 26.66 23.33 20.00 16.66 1333 10.00 6.66 3.33 3.00 2.66 2.33 2.00 1.66 1.33 1.00 .66 .33 3 months, 90 days. S88SS 88838 88888 8888 88888 8888 $250.00 225.00 200.00 175.00 150.00 125.00 100.00 7500 50.00 25.00 22.50 20.00 17.50 15.00 12.50 10.00 7.50 5.00 2.50 2.25 2.00 1.75 1.50 1.25 1.00 .75 .50 .26 2 months, or 60 days. •coS??SS S5'c5S??!o 8??5S?? ^SU^ S^^S^ ?o8?35 $166.66 150.00 133.33 116.66 100.00 83.33 66.66 60.00 33 33 16.66 o JO — * d 00 99*1 88 8 00*9 99*9 1.50 1.33 1.16 1.00 .83 SgS5S 1 month, or 30 days. ??85??8 5??85?? §l=??SJo ?58&S? g5??S5 g585?3 $83.33 75.00 66.66 68.33 60.00 41.66 33.33 25.00 16.66 8.33 3.33 2.50 1.66 .83 ^.8Sg5! ??.{nSS 1 0008 0006 OOOOI iiiii imm 8SSS oaooocoio SS22 293 294 ORTON & SADLER 8 CALCULATOR. Orton & Sadler's Interest Tables. Rate 10 per cent, 360 days per annum. 9 montlis, 10 montlis, 11 months, 12 months, Dolls. or or or or Dolls, 270 days. 300 days. 330 days. 360 days. 10000 STnO.OO 00 ^^■■iWM :53 Srll6 66 67 $1000.00:00 i 10000 9000 675.00 00 7511.00 00 825.00 00 900.00 00 9000 8000 600.00 0.) 6GG.GG G7 733.33 33 800.00 00 8000 7000 52.5.00 00 58:',.:53 33 641.66 67 700.7 18.33 33 20.00 00 200 100 7.50 00 8.33 33 0.16 67 10.10 00 100 90 6 75 00 7.50 00 8.25 00 O.f'O 00 90 80 0.0) 00 6. d 67 7.33 33 8.00 00 80 70 5.25 00 6.83 33 6.41 67 7.00 00 70 60 4.50 m 5.00 ; 00 6..50 00 6.00 00 60 50 S.75 0. 4.16; 67 4.58 33 6.00 00 50 40 8.00 00 8.33 33 3.66 67 400 00 40 30 i-'.25 •!0 2.50 (M) 2.75 00 3.00 00 30 20 1.50 00 1.66 67 1.83 33 2.00 00 20 ID .75 00 .83 33 .Ul 67 1.00 00 10 COMPOUND INTEREST. 295 COMPOUND INTEREST TABLE, SlwAoiiig the amount of $1.00 at Compound Interestf from 1 to 20 year*. Hate 5 to 10 -per cent. Trars. 5 Per Cent. 6 Percent. 7 Per Cent. 10 Percent. 1 1.050000 1.060000 1.070000 1.100000 2 1.102000 1.123600 1.144900 1.210000 3 1.157G25 1.191016 1225043 1.331000 4 1.215506 1.262477 1.310796 1.464100 l> 1.2762B2 1.338226 1.402552 1.610510 6 1.340096 1.418519 1.500730 1.771561 ••7 1.407100 1.503030 1.605781 1.948717 8 1.477455 1.593848 1.71H186 2.143589 9 1.551328 1.689479 1.838459 2.357948 10 1.628895 1.790848 1.967151 2.593742 11 1.710339 1.898299 2.104852 2.853117 12 1.795856 2.012196 2.25-il9J 3.138428 13 1.885619 2.132928 2.409845 3.4."227l 14 1.979932 2.200904 2.578534 3.79749S" . 1» 2.078928 2.396558 2.759031 4.177248 16 2.182S75 2.540352 2.952164 4.594973 17 2.292018 2.692773 3.158815 5.054470 18 2.400019 2.854339 3.379912 5.559917 19 2.520950 3.025599 3.616527 6.115909 20 2.653298 3.207135 3.869684 6.727500 N. B. In the calculations of Compound Interest, much labor will be saved by use of the above Table. RuLr:. — Refer to the Ttible, ascertain the amount of $1.00 for (he giv^n time at the specified rate, and multiply same hy the principal. 296 ORTON f our young inc-i whc desire success in life. WAGES-VALUE OF TIME. For Days and Hours, at Stated Rates Per Week Rat» $^1 3K $44>^ $5 5M $6|63^ ♦7 VA $8 $9 10 11 12 ^1 5 6 7 8 8 9 .10 .11 .12 .13 .13 .15 .17 .18 .20 S'-i .10 .12 .13 .15 .17 .18 .20 .22 .23 .25 .27 .30 .33 .37 .40 ri .15 .18 .20 .23 .25 .28 .30 .33 35 .38 .40 .45 .50 .55 .00 .'20 .23 .27 .30 .33 .37 .40 .43 .47 .50 .53 .60 .67 .73 .80 5 .'25 .29 .33 .38 .42 .46 .50 .54 .58 .63 .67 .75 .83 .92 1.00 6 .ao .35 .40 .45 .50 .55 .60 .60 .70 .75 .80 .90 l.{!0 1.10 1.20 7 .35 .41 .47 .53 .58 .64 .70 .76 .82 .88 .93 1.05 1.17 1.28 1.40 8 .40 .47 .53 .60 .67 .73 .80 .87 .93 1.00 1.07 1.20 1.33 1.47 1.60 V .45 .53 .GO .68 .75 .83 .90 .98 1.05 1.13 1.20 1.35 1.50 1.65 1.80 1^ .50 .58 .07 .75 .83 .92 1.00 1.08 1.17 1.25 1.33 1.50 1.67 1,83 2.00 1.00 1.17 1.33 1.50 1.67 1.83 2.00 2.17 •3.33 2.50 2.67 3.00 3.3.1 i 67 4.00 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.50 o.t!0 5.50 6.00 4 2.00 2.33 2.67 3.00 3.33 3.67 4.00 4.33 4.67 5.00 5.33 6.00 6.67 7. .33 8.00 5 2.50 2.92 3.33 3.75 4.17 4.58 5.00 5.42 1 5.83 6.25 0.67 7.50 8.33 .).ll $10. For Day Sy at Stated Bates Per Month. Rat^. $14 $15 $16 »n $18 m $20 $21 $22 $23 $24 $25 ^ 3 .54 .58 .62 .65 .69 .73 .77 .81 .85 .88 .92 .96 1.08 1.15 1.23 1.31 1.38 1.46 1.54 1.62 1.69 1.77 1.85 1.92 1.62 1.73 1.85 1.96 2.08 2.19 2.31 2.42 2.54 2.65 2.77 2.88 4 2.15 2.31 2.46 2.62 2.77 2.92 3.08 3.23 3.38 3.54 3.69 3.85 6 2.69 2.88 3.08 3.27 3.46 3.65 3.85 4.04 4.23 4.42 4.62 4.81 e 3.23 3.46 3.69 3.92 4.15 4.38 4.62 4.85 5.08 5.31 5.54 5.77 7 3.77 4.04 4.31 4.58 4.85 6.12 5.38 5.65 5.92 6.19 6.46 6.73 8 4.31 4.62 4.92 5.23 5.54 5.85 0.15 6.46 6.77 7.08 7.38 7.69 9 4.85 5.19 5.54 5.88 6.23 6.58 6.92 7.2T 7.62 7.96 8.31 8.65 10 5.38 5.77 6.15 6.54 6.92 7.31 7.69 8.08 8.46 8.85 9.23 9.62 11 6.92 6.35 6.77 7.19 7.62 8.04 8.46 8.8^ 9.31 9.73110.15 10.58 12 6.46 6.92 7.38 7.85 8.31 8.77 9.23 9.69 10.15 10.02 11.08 11.54 13 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 11.50 12.00 12.60 14 7.54 8.08 8.62 9.15 9.69 10.23 10.77 11.31 11.85 12.38 12.92 13.46 n •8.08 8.65 9.23 9.81 10.38 10.96 11.54 12.12 12.69 13.27 13.85 14.42 16 • 8.62 9.23 9.85 10.46 11.08 11.69 12.31 12.92 13.54 14.15 14.77 15.38 17 9.15 9.81 10.46 11.12 11.77 12.42 13.08 1 3.73114.38 16.04 15.69 16.35 18 9.69 10.38 11.08 11.77 12.46 13.15 13.85 14.54 15.23 16.92 16.62 17.31 19 10.23 10.96 11.69 12.42 13.15 13.88 14.62 15.35 16.08 16.81 17.54 18.27 20 10.77 111.54 12.31 13.08 13.85 14.62 15.38 16.15 16.92 17.69 18.46 19.23 21 11.31 12.12 12.92 13.73 14.54 16.35 16.15 16.96 17.77 18.58 19.38 120.19 22 11.85 12.69 13.54 14.38 15.23 16.08 16,92 17.77 18.62 i 19.46 20.31:21.15 23 12.38 13.27 14.15 15.04 15.92 16.81 17.69118.58 19.46 20.35 21.23122.12 24 12.92 13.85 14.77 15.69 16.62 17.54 18.46 119.38 '20.31 21.23 22.15 j 23.08 25 13.46 14.42 15.38 16.35 17.31 18.27^19.23 1 20.19! 21.15 22.12 23.08 1 24.04 26 14.00 15.00 16.00 17.00 I8.00! 19.00 20.00 121. 00 ; 22.UO 123.00 i 24.00 125.00 500 BEADY EECKONINQ. 301 VALUABLE TABLES For the Merchant, Farmer^ and Purchaser, showing at sight the Value of Articles Sold by the Pound, Dozen, Yard, or Piece, as Groceries, Produce, Dry Goods, etc. These Tables embody nearly all of the practical features comprised in publications devoted exclusively to the sub- ject of Ready Reckoning, for which prices are asked nearly equal to the cost of this entire work. They will be found invaluable in ascertaining the value of articles usually sold by the Business Trader and Farmer and con- sumed in families. APPLICATION OF THE TABLES. The outside perpendicular columns to the left and right show the price of the article, and the upper and lower lines the quavtUy. When advisable in securing results the working can be reversed. Example.— What will 14 lbs. of Coffee cost at 29 cts. per pound? See price in column to the left, 29, follow the finger along the line until the sum under the column 14 is reached, and you have the amount, $4.06. Note I. — When the price or quantity required is not shown in the tables, reduce the number of either or both to such amounts aa are shown in the extremes; ascertain the product from the table, and multiply by the number or numbers used as a divisor in reduction of the orlgiual amounts. Example. — ^What will 450 bushels of screenings cost at 23 cts. per bushel? As 450 is not contained in the table, we reduce it to 45 by dividing by 10. Referring to 45 in column ^t the left, and on tho same line under the head number 23, we have 10.35, the cost of 45 bushels. 10.35 X 10 -- $103.50, the cost of 450 bushels. Note TI.— "VThen the price or quantity is a fractionnl part of a whole number, ascertain the amount from U)e tnble by URinp: the nnmorator fts a whole number, aud then divide by the duaominutor, adding lbs result to the product already obtulued. 26 TABLE OP EBADY CALCULATIONS. See page 301 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 12 24 36 48 60 72 84 90 108 120 132 144 156 168 180 13 2Q 39 52 (^0 78 91 104 117 130 M3 156 169 182 195 14 28 42 66 70 84 98 112 126 140 154 168 182 196 210 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 17 34 51 68 So 102 119 136 153 170 187 204 221 238 255 18 36 64 72 90 108 126 144 162 180 198 216 234 252 270 19 38 67 76 95 114 133 152 171 190 209 228 247 266 285 20 40 00 80 100 120 140 160 180 200 220 240 260 280 300 21 42 63 84 105 126 147 168 189 210 231 252 273 294 315 22 44 66 88 110 132 154 176 198 220 242 264 286 308 330 23 46 69 92 115 138 161 184 207 230 253 276 299 322 345 24 48 72 96 120 H4 168 192 216 240 264 288 312 336 360 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 26 52 78 104 130 156 182 208 234 260 286 312 338 364 390 27 54 81 108 135 162 189 216 243 270 297 324 351 378 405 28 56 84 112 140 168 196 224 252 280 308 336 364 392 420 29 58 B>7 116 145 174 203 232 261 290 319 348 377 406 435 30 GO 90 120 150 180 210 240 270 300 330 360 390 420 450 31 02 93 124 \bo 186 217 248 279 310 341 372 403 434 465 32 64 96 128 160 192 224 256 288 320 352 384 416 448 480 33 m 99 132 165 198 231 264 297 330 363 396 429 462 495 34 Q^ 102 136 170 204 238 272 306 340 374 408 442 476 510 85 70 105 140 175 210 2-15 280 315 350 385 420 4bd 490 625 36 72 108 144 180 216 252 288 324 360 396 432 468 504 540 37 74 111 148 185 222 259 296 333 370 407 444 481 618 655 38 76 114 152 190 228 2(36 304 342 380 418 4b^ 494 532 670 39 78 117 156 195 234 273 312 351 390 429 468 507 546 585 40 80 120 160 200 240 280 320 360 400 440 480 520 560 600 41 82 123 164 205 246 287 328 369 410 451 492 533 574 615 42 84 126 168 210 252 204 336 378 420 462 5 (.'4 546 588 630 43 ^Q l-:9 172 215 258 301 344 387 430 473 516 559 602 645 44 S^ 132 176 220 264 308 352 396 440 484 528 572 Q\Q 660 45 00 135 180 225 270 315 360 405 450 495 540 b^5 630 675 46 lica(ion 26 Counting llooni 49 Division, or Boxing Fractio'is 63 Percentaj^e as Applied to liiisincsa 60 Profit and Loss 65 Commission 77 Stoclcs and Investments 81 Interest 87 Banking 104 Averaging Accounts 135 Sterling Excliango 147 Marking Goods , 154 Ledger Accounts 164 Closing Ledger 176 Trial Balances — Detecting Errors 178 Measuring Lumber 183 Measurement of Wood 186 Round Timber 189 Squai'e Timber 194 Cisterns and Reservoirs 197 Cask (Jauging 200 Measuring {J rain 203 Corn Cribs and Contents 207 Hay in tbe Stack 210 Weigbt of Cattle by Measurement 213 Building 215 Short Rules for ^lechanics 226 Avoirdupois Weight and Table 247 Apothecaries' Weiglit and Table 248 Cord Wood and Table 249 Dry Measure " " 250 Cubic " " " 251 Li