BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF Menrg W, Sage 1891 ^^/^.^^^^ /^.^a.. 6896-1 Cornell University Library TA 159.A33 The assistant engineer, 3 1924 004 614 974 The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004614974 The Assistant Engineer BY Prof. JEAN P. GENTHON Assistant HtJ£ritieer, Aqueduct Commissioners Member of The Municipal Engineers of the City of New York BOOK I THE AXEMAN WRITTEN KOR THE CHIEF Journal of the Civil Service PtlBLISHED BY THE CHIEF PUBLISHING CO. 45 CENTRE ST., NEW YORK 1907 The Assistant Engineer BY Prof. JEAN P. GENTHON Asisstant Engineer, Aqueduct Commissioners Member of The Municipal Engineers of the City of New York BOOK I THE AXEMAN WRITTEN KOR THE CHIEF Journal, of the Civil Service PUBLISHED BY THE CHIEF PUBLISHING CO. 45 CENTRE ST., NEW YORK 1907 THE ASSIiSTAiNT ENGINEER. Book I. Copyrighted, 1M7 by THE CHIEffi' PUBLISHING CO. preface: This series is designied for the young man who, desirous of mak- ing engineering in the Public Service his career, wishes to talte a Civil Service examination. If successful, his name will be placed on the eligible list and he will in 'dme see open to him the doors of a Crov- ernment, iState or City Department. The plan of , this work is as follows: The Assistant Engineer must know not only his dtfties but those of the men under his charge. Therefore each position is taken or each rung of the engineering ladder is ascended in turn beginning with that most easily obtainable, and .In. that work, are- explained to the man occupying that position or striving for it. 1<^. The requirements for the Government, the State and Ooun'ty of New York, and the City of New York Civil Service, with the scope of the examinations, the ratings and questions given at prsvious ex- aminations. 2°. The scientific requirements or what the candidate should know. 3^. The technical requirements, or knowledge and use of the :n- siruDLents, and duties in the field and in the office. Although the requirements for the higher posiMans demand a knowledge of higher mathematics, the author has had the same ex- perience as Mr. Wm. F. Shunk. In his long practice there had never come before him a question, which could not be satisfactorily fOjved by elementary mathematics. It will be noticedi that certain technical examination .questions for a position may seem to belong properly to the grade next higher. This is dut to the necessity where the examiners are placed of raising oc- casionally the standard of an examinatin in order to draw out the very best material among the always' increasing number of applicants. V\> therefore recommend the prospecHve candidate to study a littla furilher than would "seem necetsary. J. P. GENTHON. July 1, 1907. Niew York. THE ASSIiSTAiNT BNGHNEEIR. 4a PRBLIMINARY CHAPTER. GENERAL QUALIFICATIONS REQUIRED. The principal qualifications required of a young man who wislhes to enter the Public Service in an, engineering department are: 1°. Aptitude for mathematics. 2°. Habit of observation. 3°. Good miemory i°. System. 5^. Readiness for work. APTITUDE 'POiR MATHEMATICS— He njay not have at the start more than a common 'Sic.hoo) education, but he must constantly in- crease his mathematical stock and keep on studying in order to fit' bimself for the next higher grade or position. HABIT OIF OBSERiVATIOiN — This habit may be in the man— I have seen it in children — If not, It has to be coaxed and cultivated. Keep your eyes open when a new problem or a new solution are pre- sented; when a new material or a comibinatlon of materials, or new appliances and processes are lused for old or new purposes. Notice the sieveral kinds of labor and of labor-saving devices employed. Re- mark the professional disousslonf! which arise before you and take part in them. When in doubt, ask questions. GOOD MEJMIORY. — Habit of observation strengthens memory wVich faculty may be Improved to a high degree. Remember names and faces of persons; ways and nreans you have seen made use of by others to attain certain results ; principal formulas employed in, your line of engineering; stations of remarkaible points, and the like. S'YlSTBM.^Thls quality means arrangement, classification, or- ganization and will show itself in the following instances: field-book clean and clear; calculations well arranged, ©ntertd in blank-books always checked and summarized when necessary; plans and drawings 071 regular sizes of paper aRccrding to the classes to which they be- long, with figures and letters of standard form and size, the proper titles, scale, assigned number, border 'and a uniform margin; regular steps taken to reach an end, as dividing a work Into suoh sections and employing on them such force as to complete it in the shortest time. In the most economical way and the most homogeneous manner; a pihoe for everything and everything in its place. RB.AlDINEJSS FOR WORK, — Be ready, when called upon, not only to perform your own work but to assist others. Help in checking fig- ures and calculatioms; in cleaning and packing iustrjments, in filing driiwings and papers. Give Infcrnration or advice, go for supplies. Fii ally be of even and genial tenrper and all around will feel better bnt none more than yourself- OFfGANIZATION OF AN ENGINEERING CORPS. All departments the duties of which are the erection of public 4'b THE ASSISTAiNT EINGINEEIR. works have a Chief Engineer -who prepares the vorlc and directs its construction- The Chief Engineer is aifelsted and ad^ited by a Deputy Ci ief EJngineer and one or more ConBulting Engineers. When the work is very ex'teESive. as a railroad, a system of high- ways, of sewers for a large city, a canal, an aqueduct, it is divided irrxi approximately equal portions called diviisiions, and to each of these a Division Engineer is assigned whose duty it is lo prepare the work w'thin that division and to direct its construction, subject to the or- ders and approval of the Chief Bmgineer. A division may be subdivided into sections with an Assistant En- gineer in charge having under him as aids, transitmen, levelers, topo- graphers and inspectors. The Transitman is aasiisted by chainn^en and an axeman. The Topographer is assisted by flagmen a rodman and a chain- The Leveler ie assisted by a rodman, sometimes by ciiaiiimen and an axeman- In the office of the Chief Engineer ere Aarlttant Engineers, Draughtsmen (Topographical, (Mechanical, Archite:tural) and Tracers or Copyists. There may be like positions in a Division Engineer's Office. Clerks and Stenographers, although employed in an Engineering department, are not included in the Btagineering nomenclature which 1* summarized in the following table: ENGINEERIiNiG CORPS. Chief Eingineer. Division Engineers. Draughtsmen. Inspectors. Assistant Engineers. Topographical. Mechanical. Architectural. Tracer or Copyist. I Chainmen. Transitmfn. Axemen. Topographers . Levelers . Hodmen. Flagmen . Chainmen , Rodmen . Chainmen . Axemen THE A'SlSIiSTAINT BNGlNEEiR. 4e WHERE POSITIOINS ARE OPENED. GOVERNMENT. All positioBS In the Engineer Departmerit at Large are under the War Department with headquarters at Baltimore, Md. Boston, Maiss. Buffalo, N. Y. Charleston, S. C, Chattanooga, Tenn. Chicago, 111. Cincinoiaiti, Ohio. Cleveland, Ohio. Detroit, Mich. Galveston, Tex. Grand Rapids, Mich. Jacksonville. Fla. Little Rock, Ark. I^uisville, Ky. Memphis, Tenn. Milwaukee, Wis. Mobile, Ala. Nashville, Tenoi New Liondon, Conn, New Orleans, La. Newport, R. I. New York, N. Y. Norfolk, Va. , Philadelphia, Pa. p:ttshurg, Pa. Portland, Me. Portland. Oreg. Rock Island, 111 . St. Louis, Mo. St. Paul, Minn. San Franci-sco, Cal. Savannah, Ga. Sea'ttle, Waeih. Sioux City, Iowa. Tampa, Fla. Vicksburg, Miss. Washington, D. C. Wheeling, W. Va. Wilmington,' Del. Wilmington, N. C. Yellowstone Park, Wyo where examinations niay be taken except that of Assistant Engineer which is taken at Washington, D. C. Draftsmen are on demand in nearly all branches of t)he Federal Service. Civil Etogineen?. are also certified to the Reclamation Ser- vice and the Quartermaster's Department at Large. Inquiry may be made to 'the United States Civil iService Commis- sion at any of the above named towns for dates of examinations, ap- plication blanks, etc. NEW YORK STATE AND OOUNTY; !E3xaminations may be taken in Albany, Amsterdam. Auburn, Bingham ton, Buffalo, Elmira, Hornellsvllle Ithaca, Jamestown. Kingston, LockpOTt, Malone, Newburg. New York, Ogdensburg, Olean, Plattisiburg, Poughkeepsie, Rochester, Syracuse, TTtica, Watertown. The iCommlssion receives applications for any position at any time. Apply to "State Civil Service Commission," Albany, N. Y. NEW YORK CITY. Borough Manhattan. — ^Topographical Draughtsman. Borough The Bronx.— Inspector of regulating, paving and grading; mechanical and topographical draughtsmen. 4d THE ASSIiSTANT BflSFiGIiNIEEiR. Borough Brooklyn. — Aixeman; ohainmaa; rodman; inspector of regu- lating, paving and grading; inspector of seiwers, or construc- tion; transitman and computer; assistant engineer. Borough Queens. — ^Rodman; transflman; topographical draughtsman; assistant engineer. Borough Richmond. — lAxeman; rodman; leveler'; transitman; topo- graphical draughtsman. Department of Water Supply, Gas and Electricity. — Engineer cori)s (all grades). Department of Parks. — ^Engineer corps. Department of Bridges. — ^Etagineer corps. Department of Docks and Ferries. — Engineer corps Department of Sewers. — Engineer corps. Department of Highways. — EIngineer corps. Department of Buildings.— ^Inis.pector of plumbing, light and ventila- tion, of masonry and carpentry, of steel construction, of elevators . Department of Finance. — Engineer corps. Department of Education. — ^Inspector of heating and ventila'tion, of buildings; draughtsmen. Board of Aqueduct Commissioners. — Eng'ineer corps. Board of Water Supply. — 'Engineer corps. The old Board of Rapid Transit is now attached to the Public Ser- vice Commission for the First District and the men of its engineer corps are subject to the State civil Service. For information and blank applications applv 'to "Municipal Civil- Service Commission," 29© Broadway, New York Oity. Notices of coming examinations are posted in the public room or tneir office. These notices, as well as those for the State and Government Service appear regularly in "THE CHIIIF." I'HE ASS'I'STANT ELXiaiNEEU BOOK I THE AXEMAN AXEMAN. — An axeman is o.n.e who carries and uses the axe. WHO HIS SUPERiIORiS ABE.— On reconnaissance he is under the orders of the Assistant Engineer. On survey worlc he is under the orders of the Head Chainman; sometimes under those of the Transitman. CIVIL SEiaVIOE RBQUIRIEMENTS. Fedsral Civil Service — Panama Canal. Title: Helper. Age limits: 18 to 40 years. Salary: $50/ per m.ontlh — ^Traveling expenses Written Examination. Relative Su'bjects. weights. 1. Spelling (20 wordis of a/verage difficulty in oomoaaon use) 20 2. Arithmeti'C' (sim(ple tests in adLitiou, siuhtraction, multiipHica. fion, and division of whole numbers, and in common and deci- mal fractions, and "United States money) .' 20 3. Letter-writing (a letter of not less than 12'5 words on somie s.uil)ject of general interest. ComipeltitorB will be permitted to' select one of two sujbjects given) 20 4. Penmanship (tlie handwriting of tlbe competitior in the subject of copying from plain copy v.ill be 'consideTed with spe- cial reference to the elements of legibility, rapidity, neatness, general appeairance, etc.) . . . ., 15 5. Copying from plain copy fa simple test in copying accurately a few printed lines in the comipetiltor's handwriting) . . .-. 15 6. Training and experience 10 Total 100 Applicants must not have lesis ithan one year's practical experi- ence in similar work, provide'd tihat two years' study- in a school of civil engineering will be accepted as equ'valenit to tlhis experience. Persons appointed to the position of iheliper will be eligible tor promotion to the grade of chainman witihiout examination. One day will be required for this examination. e THE AXEMAN New York State and County Civil Service. Silent .'New York IVIunicipal Civil Service. Title: Axeman. Age limit: Not less than 18 years of age. Grade: Group 3 — ^First grade. Salary: Annual compensation of $900 or less. Written Examination: Schedule D — Part I. Relative Sulbjeots. weighits. 1. Handwriting (as shown in examination papers) 1 2. Arithmetic, viz.: Addition, subtraction, multiiplication and division , 1 3. Questions relating to the technical knowledge required for the position of axeman tJ 4. Experience tending to qualify him for that position 2 Tiotal of weights 10 QUESTIOlNS GIVEN AT CIVIL SEiR\aCE EXAMINATIONS. Arithmetic. 1. Write in figures the whole numiber, one million sevenl'y thousand and five hundred and seven, followed by the decimal number fourteen thoiusanthfi. 2. Write in. words the following number, 300007.7009. 3. Wrilte the following numiber (not numibers) decimally, ten thou- sand and se^ren and forty-three ten thouisandthe. 4. Write decimatty «he following: One million, twenty-three thou- sand and eighty and thirty itlhouisandths. 5. Add 7306098, 6121774, 178S3, 970007, 106, 9651. 6. Add 219 fit. 7 in., 847 ft. 9 in., 79« ft. 3 in., 654 ft. 11 in., -"^ ft. . . 5 in 7. Add the following fractions: S-'S, 5-8, 11-16, 7-15, 3-10. 8. Add the following fractions 2-3, 5-8, 4-9. 11-lC, 17-18. , 9. Multiply 690.875 by 78.096. 10. Multiply 3 and 3-8 by 2 and 5-7. 194.5 205.9 11. Find the amount of x J67.3 543.7 THE AXEMANi ' 7 12. Divide 322,022,362 by T8.746. 13. [Divide 37 and 1-7 by 6 and 4-5. 14. Divide 684.007 by 79.6. \ 15 A church is 86.64 ft. long, 51.28 ft. wide and 31.5 ft. high *lo l&e> eaves. Find the cost of x-a'Qt-ing the outsdd^ walls at 34 cents a square yard. 16. What number added to 7-9 of 18 and 3-4 will equal 9-10 of 41: and 2-3. 17. If 3% is 2% times a certain number, what is 3% times the same nuimiber? 18. If a laiborer can reap a field of grain in 4 and 4-5 days, how Ions would it take 4 laborers to reap a field 6 and 1-4 times as large? 19. II a man can walk 200 miles in 9.375 days, how far, at the same: rate, can he walk in 16.626 days? 20. A field is 5>6 rods wide, and contains 2i5 acres 88 square rods. Find the cost of fencing it at 66 and 2-3 cents a yard. . 21. If 9 and 3-8 yards of cloth cost $26 and 1-4, how many yards' could be tooug:ht for $24 and 4-5? 22. Write in words lO'0'6.30€. 23. Write in Jlgiures seventy and three hundredths. 24. iWlhat is the difference in area between 4 square feet and four leet square? 215 How many tenths of a fo6t are there In 30 incflies? 26. Adid 49 feet and 8 inches; 6 feet and 9 inches; 13 feet and ID. iniches; and 7 feet and 8 inches. 27. Add 4.05 feet; 19.982 feet; 17.33 feet and 2.63S feet. 218. Suibtract hy fractions and by decimals 3-8 from 12-16. 29. Divide *2 and 1-2 by 4-5; multiply ijjy 3-8; add 3-4 and sifbtracfe 1-2. What isi the result? 30. Four men working 8 hours each can shovel 128 cubic yards oC earth; how many cubx yards of earth can 8 men s'lovel if they work 12 hours eaab-at the same rate. Techflical. 1. What are the duties of an ax«man as you understand them? 2. How do you fix points in sofit, marshy ground? 3. (a) Wlhat do the letters T. P. mean When used in connectlcia with a set of levels? Ob) What do tihe same leitters mean wlxea. us'ed in an extensive survey of a portion of the city. g THE AXEMAN 4, May a stake drive croaked from improper driving? U so state In what way? 5. What dbher cause's may make a stake drive badly? « (a) How far aipart are center stakes ordinarily placed on the line ' of a street to be graded? (b) How far apart are stations made on sewer work? 7 (a) In what way can an axeman assist a chainman In windy wealtlher? Ob) How can he assist tJhe transitman in such weather? S. Show and describe how you would cut a bench mark on the root of a tree. -9., What does an axeman have to do besides carry and drive stakes? 10. Describe the shaping and driving of a stake so as to secure the best resoillts as bo.. solidity and accuracy of position. 11. (a) Describe fully tihe work of setting a center stake on a transit i;ne. (lb) How are soich stakes marked? 12. In what other ways ibesides the use of stakes are sitatlons marked on a line? 13. Desioriibe the method you would pursue to cut straight on a line thrO'UigIb brush. 14. Describe the operation of seftting reference stakes or otherwise referencing a point wihich may be lost unless so marked. 15. In doing street grading, where sloipes have to be provided for, lEtate the position of all stakes thalt would be set at a cross- section in emibanikment, and also what marks would toe placed on them. 16. How sihould stakes be set for a job of sewer work? 17. What condiilhons control the sizes and lengths of stakes? 18. If called uipon to hold the rod, sitate all you wondd do in case where the rod had to be extended. 19. If called upon to bold the front end of a chain, state all Uie things you must aittend to, in order to get correct measure- ments. iO State the operation of setting Uip a tranisit instrument 'in the field not using the box. :21. Whait other duties ordinarily fall upon an axeman bes.ides those mentioned in previous questions? 22. (a) How are stationsi marked on rocks? (b) How on a stone back pavement? (c) Hiow on an asphalt pavement? THE AXEMAN 9 23 (a) Woat in.arks are placea on center line stakes for grading a street? (b) Are stakes ever placed between regular stations, and if so how are tliey marked? 24. If for any reason you were told' to set an ofl'set stake at tlie end of a line, how would you do it? 25. If called upon to set uip a ,'tran'sit iniitrument over a point, state the imiportant things you Shoul'd pay attenfon to. 2G. In carrying an instrument on ithe tripod where there are brushes and fences, what is the safest way? 27. In what ways can an axeauan ass:sit a transitman while 'he is sieit- ting u,p a transit. 28. In 'i'he transit work (mot the level work) ■of r'unniug ihe center line of a street in tlh.e city, how far apart are the stakes usual- ly placed and what marks are placed upon them? 29. What addifonal marks are placed upon them to iodica'te what grading is to be done? 30. 'When heavy filling or cutting are to be done, what other stakes are 'set at ea'Oh laation besides tho center staikes, and how are they marked? 81. In doing leveling where points have to 'be set on between sta- tions, what are such points called? 32. (a) If called urjon to set up a transit instrum'ent, what two points are to he particuiarly attended to? (b) Would you screw the leveling S'crew very t-ght or how? 33. What is the ddiference beiwten a rod usod for a transit survey and cne used for a level survey? 34. In cutting throivg'h brush, what is the 'best way to keep on line? 35. Assume that the engineer sent you tiO the office to bring every- thing necessary for a transit survey, what would you bring? 3C. Kow is a stake referenced w'hen it is likely to be des'troyed? 37. iHow many stakes would be required tc' give sitations in a transit line half a miie long, allov/ing 7 for plus stations including 'the last stake? 38. Where a railroad cut is to he made, where are the stakes set? 39. Wriite in plain Iciiters (about % inch high) the ahbreviations for the following as written on stakes: (a) Forty-one plus, fifty, center-line; (b) Slope stake, 15 nuimerator, 20 denominator; (c) Bencfti m'ark, plus seventy-one, point, twio, six, three. 4-0. (al iH'ow is a transit point accurately marked on a stake; (b) How when it falls on a rook ledge? 10 THE AXEMAN The following questions were given at an examination for roim^n aad axeman in Buffalo, Fete. 26, 189'S. 1. (a) Make a sketcih and giive descriirition of a level rod, also de- scribe a vernier and ii:s use. ((h) Describe the manner of using a. leivel rod adtl its Telatlon to a level. 2. (a) What is a "ibench-mark?" (lb) What is meant Iby a "turning-point?" (c) What is meanit by a "^back-sight?" (d) Wbat is the misaning of "datum" or "datum-plane?" 3. (a) In measuring any distance over rough or uneven ground^ - describe how the measurement should be made to secure ac- curaxsy. (b) If a tape line is divided into feet and teni;bs of feet, ho Wlhat is the approximate distance in feet from the entrance ot the Elliotit '.Square Building to the Genesee Hotel? (c) Kovr many feet in height would you estimate the EJliott Square Building to he? Experience, 1. What is your age? 2. (a) State whait your education has been, giving dates and places. (b) State particularly whether you have pursued any engineer- ing studies, and if so state the length and character fuC the course. 3. Siiate fuTly tjie practical exiperience you have had with engineers, giving lengtSh. and character of service. 4. Give the names and addresses of two or more persons to whon» application may be made if necessary, for verification of your statements. THE AXEMAN ]1 SCIENTIFIC REQUrKEMlENTS QiR WHAT HE SHOULD KNOW. The scientific Tequirements of an axfcman are good handwriting. correal ispellixi'g, sure figuring. 0ood Handwriting. — A good, plain, broad husiness handwriting, ■without unnecessary flouTishing; the Height of the letters uniform and the capitals in proportion with 'tile small letters. The writing tolerably parallel with tshe top and bottom edges o-f the book or sheet, and the general slanit uniform throughout are the priniciipal conditions of a good 'handrwnting. Attention should be paid to the plain writing of figures and numbers w.hich ihave aometimes to be read or copir.d by others from a distance. Correct Spelling. — The axeman ahculd be not only a lair but a. very good sipeUer and able to catch ttnd correot a mistake in letters, reports, etc., which he may 'have to compose or cihiecte. Under this head it tnay be staited that he should be able to -write a letter or a little com^position on a given subject with facility and accuracy of ex- pression- We here advise tihe student engineer to avoid the use of foreign Iterms twhen, equally good 6quivalents can be found in the domeEtic language and dictionary; however, he is sutpposed to be or become familiar with these terms as toe may find tbem in scientific books, trea/tises, etc. Sure Figuring. — Although cot expected to be a lighitninig calculator he Js in be rsup.d.. but atowve all, sure in his figuring. The four funda- mental operations, addition, subtraction, multiplication and division K-hould be very familiar to him, not only when whole numbei-s are in- volved, 'but also when decimals or fraction® are concerned. Addition of many numlbers is often a stuimbllng 'block and should be practiced by tihie beginner because it is of frequent occurrence. He sihall also itry to become proficient in factoring and in leduc- inig fractions or fractionary expressions to lower terms, this will always simplify his work. A certain knowledge of mensuraition is desirable, and his examination pi^pers may contain questions involving areas of simple figures or volumes of elementary solids. ARITHMETIC. Arithmetic is the Science of Numbers. A Magnitude or Quantity is anything that can be supposed to be greater or smaller than it is. Matheipatii's treat of measurable magnitudes and quantities only. A Ma^hitu'de is a whole tlhe parts of which are not separated, as time, weigiht, length, etc. .,-. ^ A QuaTitity is formed of, separate .parts, as a sathering -of per- soire, anraggregate of , olbjects, etc. A Number is the result of the comparison (also called measure- ment) of a magnitude or quantity with another magnitude or quan- tity ir.f the same kind supposed to be known. 12 THE AXEMAN A Unit is a known msgnitude or qitantity with which, we n^eas'ire unknown magnitudes or quantties of the same kind. Standard. — When a unil is ertablished by law or usage it is call-sd a Standard. Arbitrary Unit. — 'In the mfas.urem'ent of magnitudes, the utiLt may he asiS'umed at will or ic'onvenisnce, greater in one case, smaller in anottbier; it is called 9.r.ed by long usage. An Abstract Number is one of which t'he. nature of the unit is un- known. Generalization of the Term Quantity. — This prefixed and for the saVe of brevity, both magn tudes and quantities are generally includ- ed under the single term of quantities. Further Extension- of the Term Quantity. — The expression quan- tity :s often a/pplied not oiily to the things themselves, but to the rsumbers whic'h are their measurements. When measuring a quantity there may be three cases: 1-. Whole Number. — The quant'ty is equal to or is greater than the unit and contains it an exaict number of times. The resulting number is a whole numibei. 2°. Fraction. — The quantity is smaller than the unit and con- tains onty a portion of that unit. The resulting number is a Fraction. 3°. Mixed Number. — The quaMtity is greater than the unit; it coTitains it a certain numlbor of times and there remans a portion smalkr than the unit The resulting numiber is a Mixed Number The Hyphen. — ^WTien writing numbers in words, place a hj-pihen between the parts of all numbers from twenty-one to ninety-nine, "both in,olusive. when composed of more than one w«rd. ' • Cue Numbers. — Numbers are often written in words in legal docu- m»rt:. Tliey are generally follo'wed by the same number written in Arabic notation and placed between parenthesis as: Three hunCred sixty-flve and three tenths (3GS 3) feet. These Arabic numbers may be called Cue Numbers and are so used for punpo-ses of quick read- Irg and cbcck'ng, THE AXEMAN 13 How to Read Numbers. — Tlie right way to read 101;274, etc., is one hundred one, two hundred seventy-four, etc. The Comma. — The comma is placed between the classes of units. of a mimiaer.. It is often om.tted however. Classes of Units. — The classes are: Simple umits, thous.ands, millions, etc.; also thousandtlhs, millionths, etc. Orders of Units. — Each class contains three (3) orders: Units,. tens and hundreds running from' right to left. The Deeimal Point. — A period, called decimal point, Is- placed in a mixed number between the integral part and the decimal portion wliich foljows. It should' never be omitted. Roman Numbers.— 7 stands for 1, V for 5, X for 10, L for 50, C for 100, D for 500 and M foir l,OiOO. ' '■ Abbreviations. — ^A smaller unit, written to th« left of a greater one, is suibtiacted from i!.:ie latter, as: IV = 4 (IV is marked IIII on. clock end watch dials); IX. = 9; XC = M: CD = 400, etc. Some- times a Roman number is surmounted by a dash or vincnhim; it then expresses thousands as IX =: O.O'OO. Addition. — Oiperaltion which consists in taking in any order all the unit's and portions of units OiE several num/bers and forming with them a s.ncle mimJber called their Sum or Tc'tal. Addition of Long Columns of Numbers. — iWlhen long columns of .nuim'.bers are to be a.d''yed, the stnJeat si'iouikl end'eaivor to add more. than one figure at a time. He may pick those figures which aggregate 10, 15, 20, etc., ani add the intermediate fi.^v.re.d when ccnven tni. ^ridiMon of Denominate Niu-^.i-e-s. — Begin by adding the smallest s-fltsidiaTry units on the right and' divest-. their- sum of all the next higher units it co^'tains writing' drwn rnly the remainder and carry- ing the higher units to those on the leiBt in the njum'bers to he added. Continue until the primary or principal units have bsen added. In the addition of a column, wlheo the sum' becomes greater than the numiber required to make one unft of the next 'higher graide, divest that sum at once of one such unit by marking a dot on the paper, or pad and continue the addition with the remainder; at the end of the column after writ ng I'je remainder down, count the numiber of dots so mmrked and add ti;eir nnmrber to tii'S neyj'. hig-hbr column on 'he left. ■ Sign of Addition. — Jhe sign of ad 'ition as the horizontal-vertical or Romaii' cross -j- plaiced .between all the numihers to be added; it is read' 'Plus. ■■ ■ ■ - ■ ... ... To Prove an Addition. — ^^The shortes!t way to prove an addition is to do it 'ovtr aga.n from bottom to to^p. Sign of Equality. — The s"gn of equality is two short equal hori- zontal i.aTaIleIs ^ ; it is read Eq-^«1. 3 4 THE AXEMAN Equality. — '^B equality is the indicatioa that a quantity, written on the left of the Sign = has the tame value as another quantity -written on th? riffht oif ttiat sign. TSentity.— An identity is an equality between known quantities. Equat!on,»--(An equation is an equality between quant. ties some of wh ch are uniknoiwn. . Solving »n Equa-tiom — To spire am equation is to find the value of ttie unknown -qu^h'titiei. which it couiLa.us. A simile exam.ple will Illustrate this: Tahe for instaaee the sum of the nuflibiers 79, M, 39 and re>presen't it by S ; the relation between these nuiabers is 79 -h 84 + 39 = S; this is an equation; but per- form thp addition and find the sum 202 and the a'bove relation be- comes 79 + 84 + 39 = 202 which is an identity. Thus we may say tiiat to solve an equation is to find the numeneal values whlcn. being suibstituted for the unknown quantities will transform the equaiiiou Jnto an identity. Formula. — ^A foTimola is an equality indicating the series of operations to he performed on certain quantities in order to obtain a certain result. Subtraction. — Am operation which consists in takinig from a num- ber called minuend (m) all tbe units and parts of units contained in another number called subtralier.d (s). The result is called the differencs (d) of the two numlbers or the ramainder of their subtrac- tion. 8'an of Subtraction. — The sign of suibtraction is a horizorttal dash — placed between the minuend written first and (t'he sulbtraiiend. Thu)|: 84 — as = d; 84 — 38 = 4S. Generally m — s = (J. To Prove a Subtraction.— Add from bottom to top the difference and the subtrahend; iMe sum must equ^iL tlieminueud. Multiplication. — An operation which consists in- teipeotlng-, a num- ber call^ed, multipjieand (M) as many times ae there are uai^ts in an- other number called miritlplisr (m) : the result is called th« product (p) of the numbers, aad the numbers theraseives are called factors of th« product. This definition may be extended to the case where tlie factors are not whole niumibers. Sign of Mutiplicatioii.— The sign of multiplication is the olbisque or St. Andrew's crcs X, called multiplied by, and pliced between the factors written one after the other. . - Tbus: 35 X 7 = p; 35 X 7 = 245. Generally M X m = p. Multiple of a Number.— ^Common Multiple. The ppod'uet of a Wihole number by a whole numlfjer is a multiple of either. In 7 X 3 = 21; 21 is a, multiple of 7 and also a multiple of 8- 21 la therefore a common muiuiple of 7 and 3. TPIS AXEMAN I9 To prove a Multiplication. — ^Multiplication may be proved l>y a second mnitip'licat.on in wiadob. the factors are inverted. This 1& tike auTest but the longesit method. A shorter method^i^i given-i)elo;w. Find the residue of a number by 9. — ^Let 3098G57 be a gi^en num- ber. The residue of that number by 9 is the remainder of its d.vlslon hy 9 (see Division). Bui L'h'is rcTnainrler may be easily obtained as follatvs without penforming the divis on. Add all the digits of the number, and when i:i the coarse of that process a partial sum greater than 9 is otitained, add its figures together bg'fc>re continuing, and so ccmtinne until the last d git ftas been used, and the final result la a single- figure. iSkiipall 9's on the way. Thus: 3*8 = 11 (i* I ' z) J a*s = a J a*s =13 (i<-3'4^ ; 4*7, 11(1*1 =z) This final result Z is /A« Rtsialut 6y 3 .S9:3 cu. ft.; What is their, capacity In gal- lons? H only the total caijacity of the 8 tanlts were requred, the quick- est way would be to find the sum of the given eapaicities in cu. ft., a-nd multi.r'';y it by 7.18052 (niiaitoer o€ gals, per cu. ft.), thus reducing the. ir.o1'-©ai to one add tion aad'-one multjp'licailion.; ib«t if tlj-e eaipaclty In gala, of estch tank were also required, it would be advantageous to proceed as follows: Ig ■ THE AXEMAiN Form a table of the first nine (9) multiples of 74S052, adding H to itself then .to the last multiple obtained in order to get the next one. Now, to find Ithe capacity in gals., say of the 4th tanK, write the ca^iacity In cu. ft., and a line under it. Pick from toe taDie 1— 748052 2—1496104 „.^, _ 3— 224415C 9i34cu.ft. 4—2992208 --~ 5-3740200 2992208 fi— 4488312 224415G 7— 523G3C4 523C3G4 8—5984416 C7324C8 9— C732'4C8 72S15.3S1CS gals. Proof. 10—7480620 of multiples as formed the products hy 4, 3, 7 and 9: place them where they rigihtly belong in the multiplication: add for final result and (place the decimal ipoint yhere it sJiould be. Power of a Number. — ^When the factors of a product are equal, the iproduot is called a power of the factor. Square of a Number. — A power is a square wlhen it is the product of two (2) equal factors as 7 X 7 = 49 in wihSch 49 is the square of 7. Tlie tprm square is derived from the fact that the area of a square Is obtained by multiplying the length of its side by itself, or taking it twca as a factor. Cube of a Number. — A power is a cube when it Is the picduot oJ three (3) equal factors as 5 X 5 X 5 = 12S, in wMch 125 is the cube 0* 5. The term cuibe is deriTed from the fact that the volume of a cu'be is otolained by multiplying the lenigth of its s.ide by itself and again by itself, or by taldng it three times as a factor. A product, for Instance, of 4, 9, etc., equal factors would be called the 4tJh or the 9th, etc., power of that numiber. Exponent. — An abbreviation is used. Instead of writing all the equal factors of a power, it i& agreed to write ondy one of them and to place on its right and a little above a small figure indicating how many times it is to be taken as a factor. ^T'his small figure is called an exponent. Product of Powers of a Number _/«fl'o'/SJ< exponenfj £xample . t', -c' = 4''**= -f ' DiviGJon. — An operation toy means of which we find one of two factoTS of a product when that product and the other factor are given. The given iproduct is called Dividend (D) of the division- THE AXEMAN 17 the knoiwn factor is called the Divisor (d), and the unknown factor which is souigiht is called Quctient (q). We knoAf that a quotient is. Beldcon exact an'd that the.re is generally a Remainder (r) or Residue^ Sign of Division. — ^Tlie sign of division is a smuU dash with a. poinit aboive and Cne below -f it is read divided by, is placed afler- the dividend and Is followed by the divisor. For instance, to indicata the divisliori of 72 by 8 which we know gives the quotient 9, we writ* 72 -H 8 = 9 ; genera,lly D h- d = q. Other Sign of Division. — In the study of fractions it is ishcwrL that a fraction expresses the quotient of its numerator by its denomi- 72 nator, so that the preceeding identity may be written — ^ 9 on D 8 more generally — = q, and another sign of division is a horizontal d line separating the dividend written above it from the divisor writtea below it. Proof of the Division. — We prove a division by multiplying the- divisor by t)he quotient and adding the remainder, if there is any; the- result thus oibtained must equal the dividend. When there is a re- mainder, the formula of division is D = dq .-]- r. Another proof much- s'horter consisi'.s in finding the residue by 9^ of the rem-ainder and smbtracting it from any figuire or sum of figure* in the dividend, then, finding the resWue of -u-hatever is left of th® dividend. This must equal the product of ithe resiidues of the divis'cr- and quotient. r Jieji'c/i/e */ ^ Kflitlue 'f Ma Dividtncl atftr 'It - . -"* 1 .#. ata 7f 3 14-0 37a I *I879 75 of ei atie/ ^ . 659 600 S9 7 SZS 7Z8 675 , Si 8 = Residue ef Me Remame/er Abbreviation.— IMany calculators, do not write the partial protlucts 3O0 75 600 5.26 and GTS (in the aibove example), but suMracit thomL mentally from the partial dividends 314, 140, 659, 597 and 728 sa>mg: fo.T in.s.tance at the beginning of the operation: 4 times 5 20; fro^ 24 leaves 4 (which write) anid carry 2; 4 times 7, 28 and 2, 30; from f>S3 597 72» S3 18 THE AXEMAN II leaves 1 (wtich write) and bring dowa 0. etc.; they write the- operation as here illustrated. This process is a saving in time and Epace, as also a drll in mental calculation.. Quotient of Powers of Numbers. — To divide two powers of a ETimber, suhtract the exponents. ; Example : -jp =^' 11^'' - // * Characters of Divisibility. By 2. — ^A number is- divisible by 2 when it is an even number, that is. to say wlhen it ends with 0, 2, 4, G or 8, a& 70, 836. By 3. — A number Is/ divisible by 3 when its residue is zero or is divisible by 3. By 4. — A mimber is divis'tile by 4 when the numiber formed by the last two figures to the rig&t is divisiible by 4; 7628 is divisible by 4 because 28 is divisible by 4. By 5. — A numiber is divisible by 5 when it ends wifh or 5, as 75, 270. By C. — A numiber is d'ivisiibie by 6 when it is divisi'ble by 2 and 3 as 474, because when a number is divisible by several otibers it is di- vis'ible by their product. By 8. — ^A numiber is divisible by S when the number formed hy the last three figures to the right is divisible by 8; 37104 is. divisiblo by 8 because 104 is divisible by 8. By 9. — A n.umlber is divisible by 9 when its residue is 9 or 0. By 10. — ^A fiitfmiber is divisible by 10 when the lasit figure to the right is 0. By 100. — A number is divisible by 100 when the las.t two figures to the right are 00. By [O'^ . Generally a numberis diyuiiU iy lO^it/hen itenals with n zeroj. . By 11. — A .number is divisible by 11 when the sum of the figures of' even irank suibtraoted from the sum of the figures of un- even ran.k (increased by 11 if necessary) is or divisible by 11, &s 958-32,, 33040811. By 12. — ^A number is divisiible by 12 when it is divisible by*3 and 4 as 756. By 15. — A number is divisible by 15 when it is divisible by 3 and 6 as 255. Prime Number. — A iptrime number is one which is divisible only t>y itself and 1; as 1, 2, 3, 5, 7, etc. THE AXEMAN 19 Table of Prrme Numbers. — To form a table of prime nuiabers, write the natural series^ of numlbers 1, 2, 3, 4, 5, G, 7, etc., and cross or ex- ipunge all the nunubers which are not prime; that is to say, all multi- ples of prinie numbers (exceipit 1). To expunge the multlplee of 2 in the series of numbers, cross out all tihre second ttumibers beginning with 4. To exipiunge the multiples of 3, cross out all the third numibera beginning with_ 9 (C was oiossed already as a multiple of 2). To expunge the multiples of 5, cross out all the fifth numbers beginning with 25. Continue in like manner by expunging the multiples of 7, 11, 13, etc. i ^ 3 ^ 5 JK 7 jK ^ yd i\ }i a y( }g ) n/j Table maf be yi \7 j6 \9}^j/^ti^^^pf^Z9^\ antinuei/ as Ar as 31 ^^^^^37^^ ^ M ^43 ^ ^ I desireii. Itiskntwn as^ yi "tl /^ 1^ ^ ^ ^53^ ^ ^ pi ^ 59 ^ ) ERATOSTHtNt's SlE , J tmee ..3' S once ". S 7 once ■■ 7 20 THE AXEMAN Common Multiple (C. iM.)-— A comm'on multiple of -several num- bers is a number exactly divisible Tsy each one. Abbreviation. — We represent a common multiple of seyeral num- bers a, h, c by the abbreviation C. M. (a, b, c). Least Common Multiple (L. C. M.).— The least common multiple of several numbers is the sii.cllest nunbeT exactly divisible by eaih one. Exam.p!es: 1C8 = C. M. (7, S) ; 56 = L. C. M. (7, 8). Find the L. C. M. of Several Numlbers. — Any €. M. of several n.umibers must be divisi'bl© by all the numbers; it must then contain all the prime factors of each one with an exponent at least equal .lS . lOS H a>cis IZO s _ S' lO - s-0 L.CM-(3;S;6;a;IZ) - IZO ■ IZO -r 3 = *0 IZO ^ S = 24 IZO -7 6 = Bo IZO -=■ S IS IZO -r /2 = lO The Least Common Denominator (L. C. D.) of several fractlODS is the L. C. M. of their denominatOT's. Addition of Fractions. — Fractions can be added or su'btraotej only when they have the same Riame which is when they have the same denominator. If the gliven fractions have a common denominator (C. D.), add their numerators- and give to their sum the common denominatoT (C. D.). If they have not, red-uice them ito a C D. and thus revert to the preceeding case. You may or not reduce the sum to a mixed number, according ta the requirements of the particular prolblem you have to solve. E.X. .3. -. 10 ZIP 3x70 5x3S 4x4-0 3xZ8 , ZIP US' ^ 'Ca S4 "4770 " 8^;r35 * 7;f40*'0'^S SSO ZSO ZSO ZSO ■ ns*ieo*84 - 6Z9 _ , S9 — Tab " Z80 zao 24 THE AXEMAN In the case of mixed numbers, find first the sum of the {ractions, then that of the whole aumbers increasing it by the unius which anight be contained in the sium of the fracticas. J « 1 . ZIO Humer J ^4:1 385 3BS M\ IS'}- - 1-ez '-f:} 110 = zzo fiW 70 CD. '770 1 560 carried 2 lSZ7\ 2 lS'^o 87 Ans. Zl +/^^ C.D. = Z80 \ 6Z9 \ Z 560 63 >"'•' ^*ih Subtraction of Fractions. — ^Reduce the two fractions to a com- mon denominator, then suibtraot. the numerator of the subtrahend from that of the minuend; the difference will, be the numerator of a frac- tion having the same denominator as the two given fractions. In case of mixed numbers obtain first the difference of the frac- tions then that of the whole numbers. If the fraction of the minuend is less than, that of Ihe sujbtrahend, increase it by 1 ('by adding its denominator ito its numerator) in order to render the^^ subtractiom possible, then add 1 to the subtrahend be- fore subtractiJig the intagenj (whole numibers accompanying the frac- tions). Examples: I- Z L , 7- Z3l ^ S3 3 /es .■ /■ ' 7 e. a 3 i:]^ Zl l:}* = „/« Ans. 5 Zt IS Ans. SZ Multiplication of Fractions.— IVIultiply together the nimiera;tors of the fractions for the numerator of the renulied product. Reduce the fraction (product to lower terms if ipossible.' Example jLx^x-^x-^= 3« 7x a «r 5 s ■* « 7 6 *««,7.«- ,^M s 3Z ' In the case of mixed nunaJbers reduce them to improper fractions; I)erform the multiplication as by above rule, reduce to lower terms it possible, then reduce back to a mixed number. Sxample slxSl ;, zS. .13. , Jl. ^ _il. = ''^za losJl. 4J« '^ 3 e 7z 7£ 3 Z - -3 X -^ 3«5 /i' 1 ■ 5 " f ' 2 •f «2 8 THE AXE MAIN" 25 Division of Fractions. — iDivision of a fraction by an integer. Eifther divide the numerator or, if not possible, multiply the denonu- nator by the whole numlber. Ex.I .SI. ^ 3 , -ilJ-l^-A. S^. JZ : -I ^ S = .J- =^3. 7 ■ 7 7 ■* ■ -f-xS" SO Division of a WPiole Number by a Fraction. — ^Multiply the whole number by the -fraction inverted. £j( 7 - -^ 7 X J- Division of a Fraction by anotiier Fraction. — IMultiply the frao tion dividend by the fraction divisor inverted. Division of Mixed Numbers. — ^Reduce them to improper fractions and perform the division as on simple fractions. Fx JZJ. - 3— - -^ ^-iil .££ -^ - SI X S SIxZ _ lOZ _ ^ Z^ . ■f- ■ 8 ~ «■ ' 8 * PS 'h'lZB zs ' as zs DElCirM'AiL NUiMBEIRS. Bj' a natural extension of the law of formation of numbers, it Is natural to write the teaths of a numiber to the right of the units, the tenths of tenthisi or hundredltbs to the right of the tenths, the tenths of hundredthiS or thousandths to the right of the hundredths, etc. Decimal Point.^ — ^In OTder to avoid confusion between the integral and flue decimal portionis of a number, it is agreed tO' separate them with a period called Decimal Point. Ex. 3 and 7-10 = 3.7; 18 and 8-1000 = IS.OCS, etc. Addition of Decimal Numbers. — Like the addition of whole num- bers; the decimal point in the sum is placed under the decimal points of the numbers added. Ex. 2701.04 602.904 97.1 3401.044 Subtraction. — ^If the minuend 'has not as many decimals as ihe subtrahend, zeros can be supposed added to the right of it. Make the subtraction as for whole nvitberS and in the difference plajce a deci- mal point under ithose of the numbers subtracted. Ex. 271.06 83.491 187.569 26 THE AXEMAN Multiplication. — iMuItiply the numbers as if they were whole numbers and in the produict separaite by a decimal point as many figures on the right as there are decimals in all the factors multiplied. Ex. 64.03 X 2.6 X 9.801 64.03 4 2.6 8 38418 5 12806 166.47S 5 9.801 9 166478 9 1331824 1498302 1631.650S78 9 Division. — 1. When the divisor is a whole numiber, divide the dividend as if it were a whole numiber, and in the quotient place a deicimal point when the first decimal figuire of ithe diividend is brought down. 2. When the divisor is a decimal number, add to the right oif the dividend or of th© divisor as many zeros as are necessary in or- der that both dividend and divisor shall have the same number of decimals. The decimal ipoint in both is then sujppressed and the case reverts to the division of two whole numbers. Ex. 27.1 -r- 3.0S7 = 27.100 -h 3.0-37 = 27100 -h 3037 wb'ch per- form. Tbis may also be written 27100.000 -f- 3037 which is the first case. Reduction of a Common or an Improper Fraction to a Decimal Number. — Divide the nuimerator of the given fraction by its denomi- £x./: i 7^" s \ 70 \ a. 87S ' 60 o /4/7J. ^ = a. S75 . 8 £x.//: 4-S^y 7 \S0\a.7l4- Anj- £ 0.714-L _Z r/ie Fraction 2 meanj ?:. of O.OOt 30 ^7 2 £x///: T — 3 1 7*1 Z4.ef '-« 1* '2 11 i8 ~2 3 3 THE AXEMiAN 2T nator. If the denorojinator is not contaiaed in the numerator, write in the quotient, following rt with the decimal point; thee add zeros to- the riglit of the dividend and continue the division until t'he de- sired api)iroximat;on is obtained or unitll 'there is no remainder. If there is a remainder, this will ibe the numerator of a fraction having same denom. as the giveffl one and it will have tO' he added to the quotient. That fraction will he a part of unit of the Jasit decimal of the quotient. Reduction of a Decimal N umber to a Fraction. — If the decimal number is not accompanied by a fraction, take it without the decimal ipoint as the nuimerator of a fraction whose denominator stall be I followed by as many zeros as there 'are decimals in the given number. M a frajct'ion accoimpanaes tihe given decimal numiber, suppress the decimal point, reduce the whole exipiression to an imiproper fraction adding to t'he right of the deniominaitor as many zeros as there are decimals in the given number. loo 50 so Ex. II : ISI7~ IS 17X7 *l _ loezo I ogZ _ 531 - 15 S. 7 7x100 ~ TOO TO ' 3S ~ 3S DElNOMINATE NUMBERS. Denominaite Numibers belong tC' on- of the following classes: Lengir) I- Extension \ Area Volume Liquid Z'- Capaoily | ^'^^" 3'- Weight \ZZdupois 4-°- Time Si Angle e- \/filue Extension. — ^A definite portion of space. Length. — The unit of length is the yard (Yd.) . The yard is divided- into 3 feet and the foot into 12 inches. The multiples of a yard are the rod (rd.), ipole m- perch which, equals 5 and 1-2 yds.; the fuirlonlg (fur.) which equals 40 rds., or 22* yds.; the mile (mi.) which equals 8 furlongs, or 1760 yds. il. Fur. RJs. Ydj. Ft. In' [ a -- 3ao = / 760 = s Z.60 = 63 3eo Comparative Table of ] I f-o = Zao 660 7 3zo Lenafh MeaJuresA /= S.s ^ ies= i98 ' ' 1^3=36 I It The surveyor's chain (ch.) is 66 ft. long divided into 100 linlis (1.). Comparative Table of \ / 'f = 100 66 ifurveyor'i Length Measures :\ I tS ^ le.S \ ' ' 0.£6' 7.9a " see... S(|.mi. 3g A. Z3 040 i(ftM. ■jq.f. 1 64-0 1 = SO /o - : f3S6o 1 ^3S6 -28 THE AXEMAN Area or Square Measure. — ^Tlie unit of area is ithe square yard Litre (l.j A cube of O.l rn. a sic/e . Solio/ifv ( \Stere ijf-.) ^ A cube of /m. a side {Mai for .J/ • i X ■/ (^ .- measuring wood for.,fMeI). , z-a^ \ Weight I \&ramme(gr.j = v/eightof Uu.cm.dliiillhd ifiarer (rr^X In engineering, areas aire reckoned in square mieters (sq. m.) and volumes in cubic meters (cub. m.). In Itinerary distances the kilometer (k. m.) is taken as a unit. In common weighing the kilograim (kg.? is taken as a unit. Multiples. — The multiples are the names of the units prefixed with deoa. for 10; hecto. for 100; kilo, for 1,000; myrla for 10,000. Sub-multiples are the same names prefixed with deci. for 10th; centi. for lOOth; milli. for 1 OOOith. ^ s ^ si! 5 ? V ^ile of Lengtft Measures . Mym.Km. HmJ)m.M. O O , OOO Jm.cm.mm. OOO THE AXEMAN 31 TahU vf ( Mym'- Km^Hm^ Dm"- M^ dn,*- cm} mm?- Area Meoisurej :\o0,00,00,oo,00 . O o , o o , o o Table of ' . M? dm? cm? mm? Volume Meajijres:\o 00 , ooo , ooo . ooo, ooo, ooo Table of , nec¥art Am Centiare Agrarian Meajures : \ o o , 00 . o o I ca. = /M? Capcciy Measure,: | 0,000.000 Table of Salidify Meajures ■■ ) Dst St djt (Wood for FueJ) \ O O O E E Table of ,2 13 ~ S £ i i a i. — Weight MeaJures : . T- <3 ' . . and between tlie opening of the V a small fig- ure called \l SI index is written indicating what root is to be taken or how ^ many equal factors had to be multiplied one by the other to produce the number under the radical sign. When no in- dex is written, 2 is to be understood. V/^/ , jcfuare roor of 81 inc/icahs fhat a certain number ivas taken ■twice aJ a factdr fo produce 81 . That number is 9;( 3^= 9n 9 = 81) ar7i/ r/e w rite VS'l = 3 \[3^, cube roof- of 343 mdifates that a certain number was taken 3 times as a factor to produce i43 . Tliat number is7;(7^- 7 it7> 7= 34-3J ani/ we write \/343 = 7 Square of a ^'tn\\^s\gsxss.(^o-'5)i>(6o*s)^6ox60i-eoxi*6oxs*sxf=fo->z/'eo*sUs': of two Numbers : I Rule : The square of a Sum of 2 numbers equals the square of i-he first, j>luj twice the first by the second , plus fhe JCfuare of the second . This may also be stated thus: The square of a number contain- ing tens and units is composed of the square of the tens, plus twice the product of the ten® by the units, plus the square of ithe units. Cube of a Sum of Two Nurtibecs . Si, =_(5o+3jjc(50* ^) x (So*3) = =(So*3/xC50*3) =(^o'-*ZxS0x3-*3^)x(S0+i) = So'* 3x.Sol3* 3xS0x 3^* 3' . Rule. The cube of a sum of 2 numbensi equals the cube of the first plus 3 times the square of the first by th« second, plus 3 times the fiirst by the square of the second, plus the cube of the second. This may also be stated thus: The cube of a number containing ters and units is composed of the cube of the tens, plus 3 times the square of the tens by the units, plus 3 times the tens by the square c.f the units, plus the cube of the units. Involution. Evolution. Table of iquares Ijtfuares : I 4 3 IS tS 36 49 S4 81 Numbers, and S(\uare Rooti .\ ft urn hers ! /Z34S678S Square Roots. Square Root of a Number less than 10 = \oo . It will he found inthe at:veTable Example} . /' ^^4 - 8 ; Z- Vji = 8 within a unit. Square Root of a Number greater than lo = 100. Example ■ \/s £9 si- 04 /Arrangement of He Operation ; /dumber 5 23 92 04 I 2302 Root -± ' J '«.» 43 ii» 3 o$.lo.4 /noz 9io4 2 THE AXFJ.MAN Zl SQUARE ROOT. It is the finding of one of two equal factors the product of which is a siven number. Rule. Sfi'parate the given num'ber into sets of two figures begiE- ning with the units. Talve the square root of the greatest square contained in the last set tO' the left (there may be only one figure in that set), it will 'be ithe first figure to the left of the root required; guibtract the square of that figuire from the left hand set, and to the right of the remainder bring down the next set of two figures from which separate the last one to the right; divide the left hand portion hy the double of the first figure found, the quotient is the seconsl figure of the root or is too large. Veirify by wirtting it to the rigM of the double of the first figure and multiplying the no. thus form-ed by the fis. on tir^Ial; if the product oan be subtracted from the re- mainder followed by the second set, ithe fig. is correct, if not, dimin- ish it until a subtraetioiL' is possitde. By the side of the seccma remainder bring down the 3d set of figs, and separate the last figure a.fc before. Divide the lefit hand portion by the dionble of the aa. formed by the two figs, obta'ned at the root; the quotient is verifies 8« the first. The O'pe.ration is so continuei until all the sets of two figs, have been used, and their number is the same as the nuimber of figs, in the root. If an appiioximation greater than a unit is desired in the root. place a decimal point after the root already oibtained and continue by adding sets of two zeros at ithe right of the successive remainders for each of which a decimal will be cibtaihed in the root. Abbreviation. — When we have more than half the number of fig- ures desired in the square root of a nininlber, the other figures may be had by dividing .the remainder by double the root obtained. Square Root of a Fraction. — The square root of a fraction equals the quotient of the square roots of its two terms. Vi S - s/5 8 v/«" When the denominator is not a perfect square, it is always a3- vantageous to render that denominator rational. Rationalizing the Denominator of a Fraction. — When the denomi- natoT is not rational, multiply both termis of the fraction by the de- nominator. ^ _ v^ 1/ 2 « \/j " _ Vg* .; v^ a/r expression wtiich B ' i/4B ^ ' i^' ' 48 THE AXEMAN 35 RATIO A/ND PROPORTION. Arithmetical Ratio of Two Quantities — It i& generally defined as the number measuring one quantity when the other quantity is taken as the unit of meiasiireiinient; or again as the quotient of the num- bers which express the measures of the two quantities when referred to the same unit. A 'ratio is written as follows: Suppose that A and B measured with a third quantity contain respectively 7 and 11 units, the ratio of 7 A to B is represented by 7 : 11, or hotter by — , and the ratio of B 11 11 to A by 11:7, or 'better iby — . 7 Inverse Ratio. — These ratios are inverse of each other and their product is 1: 7 11 11 7 The terms of a ratio may be multiplied or divided by the same number. _7 _ 7xJ ^ 7-^2 // ~ //X3 ~ II^Z Proportion. — The equality of two ratios as; 7 14 7-11 :: 14:22, or better by — = — , and is read 7 Is to 11 as 11 22 14 is to 22, or 7 on 11 equals 14 on 22. The first iterm of each ratio is an antecedent. The second term of each ratio Is a consea.uent. The first and last terms are also called extremes. The second and third termis are also called means. In a proportion we can mi:l.tiply (or divide) by the same numlber either the antecedents or the ccnsequents. // y^thaviJ-^ 1±. ^e ihall hcvt alio li^ = /*!!,/,«' -?±1 = -L't±± II Zi II il II zz - The product of the extremes is equal to the product of tire means. 7 _ J* 7« ti _ 14' II y , ZZ = I4t II 30 THE AXEMAN The order of the antecedents and consequents mny be r^in-j^i pro'Vided that in each transformation the product of the extremes shall equal the product of the means -2- = J^ ■ "^ - II ■ 2Z /■» . //-'22 . 14 - 7 . II _ 7 lA _ZZ Zi. - IL. II ' ZZ ' 14 ZZ ' '/ ~ 7 ' 7 ~ II- ' ZZ II ' ZZ~ 14 ' 7 /i ' 14 ' ■; Aaain weJiave.l-^ , = J* ^, TjJl _ !±±zz, ,^ JltJJ- ^ JL. ^ J2- . II ZZ ' II ZZ 141-ZZ 22 14 A/joiZ^I - 1±^, 7-11 14-zz ^, 7-11 ^ JJ_ ^J_nl?ichmaybe expressee/W/us. II 22 // 22 14-ZZ 22 "«• The sum, or the difference of the first two terms of a proportion Is to the sum or the diff. of the last two terms as the first is to the third or as the second is to the fourth //ence a/jc:2±a. = ^-/z „r JLliL = J±!^z_ „^ 7-ni . /'>->22 „i,,ch may oe I4-,ZZ 14-ZZ 7-11 14-ZZ 11-7 ZZ-14 ' expresiea/ ihus ■ The Piim of t'.ip first two terms of a proportion is to their differ- ence as the sum of the last two terms is to their difference. TBOHiNICAL REQUIREMENTS Instruments. — The axeman wields an ax in preliminary work over a wco. ed countiy; oltfn a h?tchet is sufficient, hie rirJMrS or erects poles; he dr-ives stakes and plugs, and should be able to cut them h mseif. He arives tacks, into plug's. He helps set stone monuments or to uncover existing ones and 'assiits in chaining, in holding the rod, in caTrying the transit or the level, in cleaning these instruments, in packing them in their respective boxes. Hence he should know what are an ax, a hatchet, a pick, a shovel, a cold chisel, a stake, a plug, a. tack, a pole, a .plumb-bob, a chain, a ro>d, a transit and a level. DEFINITIONS OP SOME INSTRUMENTS. , ^ The Ax. ..Definition. — An ax is a cutting tool or instnmient fo-rmed with two parts: a metallic head and a wooden handle. The head is gen- erally steel. Description.— There are three portions in the head (or poll) of an ax. let. The slightly curved cutting edge; 2d. The re-enforced body through which an oblong socket or eye is left to receive tiie end of the handle, and SU. the back or 'heel which is blunt like a hammer and Is used as such. The wooden hrndie (oi helvt) is sightly curved in length and is composed also of three portions: is^r. The tool end is oblong to conform with the eye or socket of the head and into this opening it is drivea witih great force when- the wood is very dry; this is done in order to secure great adhesion. 2d. The body "or cen- tral P'irtion with ronnded edges for better gras.ping the tool and 3rd. the free end which I's made a, little thicker than the body in order to Bict as a sfcOTj to the slipipin'g of the tool thpougb the hands. THE AXEMAN 37 The Hatchet. — A hatchft isi a small ax with a shOTt straight han- dle. One side of the head has often a slit axd to pull up nails. The student is earnestly advised to write similar deflnitioes of the well known tools, a pick, a shovel, a cold chisel, which he may- have to use. The above definition and description of an ax are given rather as a guide. Stake. — A stake is a piece of wood generally 1" x 2" x 18" (the si.gn ' denotes feet and the sign " denotes inchesO, sharpened aL one end for driving it into the grrv^nd. One of the faces at the other end must be made smooth in order to permit of 'writing on it. Any kind of wood at hand may ibe used, but old chestnut rails' can easily be cut Into very good stakes. The length of the 'Stakes, varies with the practice of different departments. In ibrusiy coimtry, longer sticks may be used with a paper or card stuck in a .slit on the top. Plug or Hub. — A plug (also called hu.b) is a piece of wood gen- erally 2" X 2" X IS", sharpened' at one end like a stake. Tack. — A tack is a short nail with a relatively broad head. Some heads are rounded, others are conical, others have a dent in the cen- ter. BrasiS' tacks are preferable. Pole. — A pole is a lonig and slender piece of wood (generally a straight lim'b of a tree) -with a (Sharpened end for driving it into the ground. Poles are O'ften fitted with a w'hite or red rag or flag at the upper end. Plumb-bob. — ^A plumib-ibo'b is an instrument used to determine the vertical of a place. It is a heavy ibody freely .suspended from a cord or string; it is generally 'top-shaped and a little elongated. It is made of iron or brass with a steel 'point; a cap with a central hole is screw- ed on the neck and through this hole a string is previously 'passed and knotted on the iaiside or secuied to a concealed' reel. The arrangement ins'u.res the center of gravity of the 'bob and its point (being on a line with the string, that Is to say in the vertical, when the instrumen't is freely suspended. How Should Stakes and Poles be driven — Poles and stakes must be placed in a vertical position and the rod must also be held verti- cally. No expl'anal:'nn is thoughl to be necessary for the use of tools such as an ax, a hatchet, a pick, a shovel, etc. ' HOW TO USE THE INSTRUMENT'S. Poles. How to set therii,— When running lomg preliminary lines th'B object of which is to study the ground in order to select the best place for a lo-oation, principally when siiich lines run thrO'Ugh a wooded country, distant points are often marked with long poles surmounted with a bright flag when they would not otherwise be conspicuous and easily discernible. The lower end of a pole is sharpened to secure 38 THE AXEMAN a better hold in the ground. To set a i>ole in position a hole must be dug, with pick and shovel, deep enoug'h to insure the pole against be- ing blown diQiwn by the wind or uipset by accidental shocte. When it is not poBsi'ble to dig a hole deep enough to insure the stability of the pole or when the heiigM and thereifOTe the usefulness of the iwle would be uiiminis'hed by burying too mi,ch of i:, or when rock is struck near the surface, stores anid boulde^rs are pliiced at the foot of the pole and all aro'und it in as secure a way as Tpossible, or it may be found necessary to steady the pole with at least three wood- en braces one end of wh.ch is sharpened and driven obliquely into the ground and the other is nailed to the pole. Surveying Pole. — A surveying pole is a wooden rod 5 to 10 ft. long, painted alternately white and red and with a sharp ferrule at the lower end. The upper end may or not. carry a flag. These poles are intended for temporary use only. How to Drive a Stake. — The staJ^e should be held, over the point to be marked, in a vertical position ■s\4th one hand and driven with a hammer (it may be the heel of the hatchet) held in the other hand, striking gentle blows at first, and inoreasiimg the effort until the to

Precautions Necessary to Drive a Stake Properly. — It nuisr not de- viate too much from the vertical direction by the point passing' be- tween pieces of rock. A slight variation may be corrected by striking gently the sides of the ,stake so as to bring the top to the right point. When a deviation begins to t^ke place, it is better to leave the stake as it is if the point has penetrated the ground sufficiently deep to In- sure security of iinsition. If rock is struck near the grounl, it may be uncovered, a mark such as a crosi? made on it with :he hatchet or witlh a small colri chisel, then covered over with atones and a refer- THE AXEMAN 2'i ence iStalve driven as near the point as posisiible. Tbe distance from the point to the stalte is then marlied on the smooth face of the stake ■which must face the point. How to Drive a Plug. — ^A plug is driven with the same care re- quired for driving staltes; it is driven deeiper or until the top is nearly- flush with the ground, say not more than 1" from it. How to Care for His Instruments. — The axeman should keep his ■ tools and the instruments or implements in his charge clean and in their proiper place. THE AXEMAN'S WORK I^i THE FIELD. What He Carries. — ^He carries an ax, a bundle of stakes and plugs, marking crayon, tacks and some nails. Reconnaissance. — Oxh preliminary work or reconnaissian.ce he ac- companies the engineer or chief of party, who selects the points through whiohi the lines are to run, and he marks these points with poles or stakes. With poles if the 'points are far apart or in a wooded country, with stakes if they are closer together and in relatively open ground. He clears the ground between these points cutting brush, underbrush and sometimes even limlhs of trees so that the points may be easily visible to the transitman and the leveler. Caution. — The cleared space aibout transit sitations must be am- ple, the line free from underbrush, but the axeman' mur.t not injure, trees or shrubs unnecessarily. 1 Location. — On lO'Cation work, that is to say when the line is final- ly determined and marked on the ground, he accompanies the transit- man and the leveler and drives stakes and plugs as directed. Where Stakes are Set. — A stake is- driven at every point on the line where an elevation is taken, generally at each ^station, that is at the end of every chain (l.Ofl ft.), even though no elevation were taken at some of these points. ~ Pius Stake. — Any stake set ibetween stations, for inistance at Sta. 32 + IS. 3, is called a plus stake, and the point where it is set a plus point. How He Acts When Setting Stakes. — When the setting of stakes is directed bv the transitman, he should stand to one side and be at- tentive to signals. When he is through he must move quickly and leave room for the chainmen. Elevation. — Datum — An elevafon is the height of a point above or below anotner known point named datum. Negative Elevations. — The elevations of points below the datum are negative. For instance E. — S2,l indicates that tbe point consid- ered is S2.1 ft. below the datum. 10 THE AXEMAjN Marking a Point Where No Stake Can Be Driven.— When a stake jannot be driven at a point which is to be rpcorded, as on a hard sur- face roadbed, on a sidewalk, etc., a mark such as a cross X, an an-ow point, or ^— crow-foot may be made with either a sharp Sool, red paint, a ^ colored {pencil, or even chalk, and markings wiicli should be on the stake are written at the point of the arrov OT on a near-by sidewalk, tree, house, lamp post, etc., or on a refei- lence stake if practicable. Reference Stake or Guard. — A reference stake is one driven near ahe place or pomt where a stake or plug should be and at rig-ht angle to the alignment; its smooth face is turned towards the point ind on it are written the station of the point with the diiEtance right (R) or left (L) that the reference .stake is from the point. Where Plugs Are Set. — ^A plug is set at every permanent point, ifhat is to say at points which are of imjportance and must nemain on Ihe ground, such as a transit point. A ipoint of curve (where a curve begins) P. C, a point of tangent ijwhere a tangent ibegins), P. T. In leveling, T. P. indicates a turning point which is the point where the rod is placed when two readings, a tack-sight (B. S.) and a fore- sight (F. S.) are taken on it. Transit Point. — ^A/transit point (T. P.) is generally the end of an iftligninent and the beginning of another, or the print of intersection of two lines. A transit point miy be on a long straight line. Many engineers prefer the aibbreviation P. I., point of intersection, instead mf T. P., transit po;nt, because of the similarity between T. P. and P. T. Marking the exact point where a Plug is driven. — A tack or nail is driven into the head of a plug to mark the exact point to be recorded Sy the plug. Referencing a ,Plug. — (Neair each plug reference stakes^ are set with the markings of the iplug on them, such as P. I. 9 + 40. G2 25 R or 25 L. Four reference stakes are necessary to properly reference a point where a plug cannot be placed. The intersection of the diag- •nals of the quadrilateral which they form is the point. (Fig. 1) ''K S$ THE AXEMAlsr 41 Alignment. — An alignment ig the direction of a straight line marked with plugs and stalves driven into the ground. He helps find Points and Line. — ^The axeman helps the transit- man and leveler_in_findiiig tlie alignments, the T. P.'s' and the stakes, Ijecause he is more than they familiar with the ground, having gone over it in the preliminary work. Protects Transitman and Leveler. — The axeman may have to pro- tect the transitman and the leveler trom the rays at the sun which might prevent them from correctly sighting or reading; this he may do by _interpcsjng his uands or his hat between the sun and the in- strume'nt. Likewise in very windy weather, to prevent the snaking ol the transit or level, he may stand on the windy side extending his coat as a protection. Protects Chainmen. — ^In windy weather the chain may sag too much under the pressure of the wind, the axeman may then place his hands or fingers against it with arms extended near the middle ol the chain. Bencn Mark. — A bench-mark is a small level surface prepared on a lock, a stunip or other durable material on wliich to rest the rod, an 1 the elevation of which is carefully taken : it serves to correct linesi of levels nr to start neiw ones: its abbreviation is B. M. Preparing Bsnch Marks. — The axeman will have to cut bench marks (B. M.) where directed, as on a rock, en a wall, on the stump ■of a tree. On rock he may have to use a cold chisel in preparing a level bed for the rod. The elevation is 'paintej near by. On the stump of a tree he u>?es the hatchet in preparing swch a ted, then he cuts away a litUe of the stump above and below ttiat hed and finally blazes a portion of the trunk for a place on which to ■write or paint the elevation. Distance between Bench Marks. — 'Benoh marks should not bn further apart than half a mile, their elevation should be carefully cheicked with a second line of levels and the errors should not be more than 0.01 ft. or 0.015 ft. Another duty^of the axeman is helping to set monuments or un- covering existing monuments set un-Jer the surface O'f the ground. Monument. — A monument is a block of s/tone generally square on -top permanently set in the ground, or a bolt sealed in the rock, which marks a corner as of a street, a property, the intersection of two prin- cipal alignments, or simply a poinjt of an important line such as the center line of a long structure. Setting a Monument. — With pick and shovel he digs a hole a lii- tle larger than the monument to be set and to a depth determined by the engineer; the hottom is carefully pre'pared for the bed of the stone. A few flat stones or spa.wls will 'prop up the monument and make it steady, or a bed may be made, with concrete. When in position the 42 THE AXEMAN remaining apace is filled witb earth and stones uniformly packed on all sides or with concrete, care toeing taken not to disturb the position of the stone by unequal sp'ltl'ing or iby stocks V ten ready, a cross Is cut on the top face or a copper 'plug is sealed in it with a marking nick at the center of it to icificatG the exact point of the ground. If the monument is to be buried, some large stcnii musit first be placed on top to protect it from the blows of the pick in case it should be ne<;essary to uncover it later. Copper Bolts. — On rook, monuments are bometimes replaced by •cofiper bolts sealed and marked at tbo center. Monuments indicate Direction — Bench IVIarks indicate Elevation — Monuments refer to points and lines or indicate direction; bench- marks indicate elevation. Uncovering a Monument. — ^When it becomes necessary ;o uncover a. monument, great care must be taken so as not to injure its top with the pick or to shake it from its position. Cross-section St^es. — During construction the axeman sets stakcs- ■Where cross-seotions are taken, one on tlie center line on the B. S. face of which are written the station, as 12 + 18.3, and sometimes the ■depth of cut (excavation) as 0. 5.4 or + 5.4 (wbich means that the- elevation of the ground is equal to the elevation of t'le grade + 5.4 ft.), or the height of fill (em*anikment), as F. Kl.S or — 13. S (which means that the elevation of the ground is equal to the elevation of the grade —13.8 ft.). Markings on Cross-section Stakes.— The figures written on a stake are called markings. In the example just given, the station number 12 4- 18 3 might he called the name of the stake; it indicates its posi- tion whioh is 12 chains (the engineer's chain is 100 ft. long) plus IS -and 3 tenths feet, or 1218.3 ft. distant from the beginning of the line. The figures C. 5.4 or -|- 5.4 indicate that the around i; 5.4 ft. higher than grade and that there will be a cut or excavation 5.4 ft. deep at that point; the figured F. 13.8 or —13.8 indicate that the ground is 13.8 ft. lower than grade and dliat there is a fill or embankment 13. S ft high at that point Slope Stakes. — ^Other stakes are placed on cross-sections; at the point where an excavation is to begin or at the point where an em- bankment will meet the ground. These are called slope stakes and are marked S. S. If such slope stakes are placed at these very points, they are apt to be disturbed by careless workmen, so that they shoiila be referenced with other stakes placed at convenient points on the same cross section line an.l rbearing a nnrking such R. S. 3f. 4 L which means Reference Slope staike 4 ft. to the left, or R. S. S. 3 R, wihich means Reference Slo'ue stake 3 ft. to the right. Some engineers .simply have their slope stakes; set at a uniform distthice ri.cht or left from the point where they should be driven. JLiny of these stakes however will have to be reset from time to time. Acting as Flagman, Poleman.— The axeman may bo required to act as a flagman or poleman in establishing a line. THE AXEMAN ri- In the capacity of a flagman he establishes transit or plug points, under the direction of the engineer or chief of party. These points- should be so selected as to he in positioas. relatively clear from un- cerbruslh as along a ridge Ttey should be visible in both directions, backsight and foresight, and their distance should 'be easily chain- atle. He sets a 'plug as directed, drives a tack int-j its head and sets the- pole with the point of the ferrule on the tack, standing squarely be- hind it and holding it veirtical bybalancing it with the tips of the fin- ders of both hands. When obstructions are on the line, he leaves a sight mark on: them, such as a piece of paper nailed to a fence (top rail), an indenta- ition or a crayon mark on a projecting rock, etc. Ranging in a Line. — It oonisasts 1° in producing an alignment until a good transit point is obtained, and 2^ in interpolating points, in an alignment between two tiransit points (or angle points). Prolongation of an Alignment. — ^Standing