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GRAPBir METT^l)FOR SOLVING CERTA1.>'

ALGEBRAIC PROlJLEMS

GEORGE " . VOPE;PROFESSOR OF CIVIL ENGT'-vertnG IN BO"\V ^11 COLLEGE,

AUTHOR OF 'MANUA: MuROAD EN( INEERS/'

UC-NRLF

NEW YOirr\ VAN ISrOSTRAN^\ MTBLi .

23 Murray Street and 2", V, TvEN S kt.

is ( :).

...I

IN MEMORIAMFLORIAN CAJORI

VAN NOSTEAND'S SCIENCE SERIES.

16.

A GRAPHIC METHOD FOR SOLVING CER-TAIN ALGEBRAIC EQUATIONS. By Prot.George L. Yose. With Illustrations.

IT.

WATER AND WATER SUPPLY. By Prof.W. II. CouFiELD, M.A., of the University Col-

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By Prof. W. II. CoRFiELD, M.A., of the Uni-

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GEAPHIC METHOD

FOR SOLVING CERTAIN

ALGEBRAIC PROBLEMS

GEOEGE L. yOSE,PROFESSOR OP CIVIL ENGINEERING iN BOWDOIN GOLLEOE,

AUTHOR OF "manual FOR RAILROAD ENGINEERS."

NEW YORK:D. VAN NOSTRAND, PUBLISHER,

23 Murray and 27 Warren Street.

1 875.

Copyright 1875, by D. Van Nostrand.

^5-^V4,

PREFACE.

A portion of the following pages firstappeared in Van Nostrand's Engineer-ing Magazine for June, 1875. Themethod was suggested by the commonmode of representing the movement of

railway trains, which was employed as

long ago as 1850, and was first broughtto the writer's knowledge by the lateS. S. Post, the well known Civil Engin-eer. It is, of course, not presented as in

any way taking the place of the farmore elegant and precise methods of

analysis, but only as in some cases a con-

venient mode of obtaining a bird's-eye-view of a problem, and as affording themeans for interpreting certain results.

6which by other processes are not at first

sight quite plain.

The " Cross Section Paper," employedby engineers, will be found well adaptedfor the working of problems by the

graphic method, as it is ruled in squaresof greater or less size.

A GRAPHIC METHODrOR SOLVING

CERTAIN ALGEB1|AIC PROBLEMS

The various methods ordinarily em-ployed for the solution of mathematical

problems are well known to all who arefamiliar with arithmetic, algebra and

geometry. There is however a methodof answering a certain class of questions,and of representing certain results, by adirect appeal to the eye, which is

extremely simple, very effective and insome cases superior to every other mode.This process is, at least in some of its

applications^ by no means new to engin-eers, but it may be both new and in-teresting to some persons, and it is pro-posed therefore without further remarks

to present a few examples of the graphicmethod, the application of which toadditional questions will readily bemade by the reader.

Suppose we have the following ques-tion : If a man travels five miles in one

hour, how far will he go in four hours.This of course is the plainest possiblequestion in simple multiplication. But

suppose instead of the above we havethe problem below. A person walked acertain distance from A to B at the rateof three and a half miles an hour, andthen ran a part of the way back from Bto A, at the rate of seven miles an hour,

walking the remaining distance in five

minutes, and being out twenty-five min-utes in all. A second man walks from Bto A and back again, at a uniform rate^being also out twenty-five minutes in all.At what two times will he meet the firstman, and how far from A will the twopoints of meeting be? Here now is a

question which our simple multiplicationwill not answer ; but by the graphicmethod the second question is nearly ifnot quite as simple as the first.

9To begin with our first question above,draw a horizontal line and divide it into

equal parts as at 1, 2, 3, in Fig. 1-

c

10

by the diagonal line from to A; anyinclined line in the figure representing amovement both in space and time. Ifwe wish to know how far the man willgo in two hours we have only to draw avertical through 2 to cut the diagonal at

B, and from B to draw a horizontal lineto our vertical scale of miles at 10; or if

we wish to know how long the man willbe in going fifteen miles we draw ahorizontal from 15 to cut the diagonalat C, and through C draw a vei^tical tocut the time line at 3. If a second man

goes twice as fast as the first, his pathwill be shown by the more steeply inclin-ed line from 0, on the upper horizontal,to 2 upon the lower one, which passesthrough the intersection of one hour andten miles. Suppose the question was as

follows : Two men start from the samepoint at the same time one going at therate of five miles and the other at tenmiles an hour; how far apart will theybe at the end of two hours ? We see atonce that the vertical distance betweenour two inclined lines, measured upon

11

the perpendicular through 2, is the dif-ference between ten and twenty miles,or ten miles. Let us reverse the ques-tion, thus : Two men start from thesame point at the same time, and travel,one at the rate of five and the otherat the rate of ten miles an

hour; after a certain time they are tenmiles apart; how long have they beentraveling ? Here we have only to takeour distance representing ten miles andfind where it will just go in vertically be-tween the inclined lines, and then pro-duce it upwards till it cuts the time line,which in this case is at 2

;thus showing

that they have been traveling twohours. Suppose again that our firstman starts from a certain point, andthat at the end of four hours hehas gone twenty miles. A second manstarts from the same point at the sametime and reaches the end of the twentymiles two hours sooner than the first

man; how fast did he travel? In thiscase we have only to go back upon thehorizontal line from 4 to 2, and draw a

12line from 2 upon the lower horizontal to

upon the upper one; the inclination ofthis line will give us the rate required^or ten miles an hour.The above questions are extremely

simple, so simple indeed as to be donein the head by any member of a commonschool, but they illustrate the method,which we will apply directly to moredifficult problems.We have seen that differently inclined

lines represent different rates of move-

ment. Let us take another question :X starts from a certain point and tra-vels in a certain direction for a cer-

tain time, his path being representedby the diagonal A B in Fig. 2.

A

13

Y starts an hour later and passingover the same distance arrives an hourearlier. How fast did Y go, and whenand where did he pass X ? The line C Din the diagram represents the movementof Y, its inclination shows his rate, andhe passes X at a distance represented onthe vertical scale by F S, and at the timeshown by F upon the upper horizontal.A third man, Z, starts from the oppositeend of the course at the same time thatX leaves the first end, and goes at therate of the second man, Y; when andwhere will he cross the paths of the twoother men ? It will be seen that whiletwo men may move in opposite directionstime always goes in the same direction,and though a man may stand still, oreven retrace his steps, time always goeson. As a matter or convenience time is

always represented as going from leftto right, in a horizontal direction.The movement of the third man Z istherefore shown by the line E F and hewill pass X at M on the scale of miles,and at the time represented by N. He

14

will also pass Y on the second horizon-tal for distance, and half fway betweenC and F for time. Let us change thequestion with regard to Z, thus: Z leavesthe second end of the route at the sametime that X leaves the first end,*but tra-vels twice as fast until he has gone halfthe length of the course, when he stopsuntil Y overtakes X and then goes onarriving at X's starting point at thesame time that X arrives

^at his (Y's)starting point. What is Z's rate duringthe last half of his course ? In this case

the first half of Z's course is representedby the lineE X; but as he^now stops foran hour we pass along on the horizontalfrom X to S. The remainder of hiscourse is shown by the diagonal from Sto T, the inclination of which is evident-ly the same as that of S B. The rate ofZ therefore during the last half of hiscourse is the same as the uniform rate ofX.

The various algebras and arithmeticsabound in questions like the following :

Edinburgh is 360 miles from Londo:..

15

A starts from Edinburgh and travels atthe rate of 10 miles an hour; B startsfrom London and goes eight miles anhour. If they travel towards each otherwhen and where will they meet ?In this case we lay off EL, Fig. 3,

Fig. 3.

equal by any scale to 360 miles. Nextlaying off any equal parts upon the line

ED, to represent hours, we draw thediagonal E C at such an inclination as toshow the rate of A, viz. ten miles anhour. As B goes in the opposite direc-ion the diagonal showing his movementwill be inclined as by the line L D, theangle of which is of course to represent

16

the speed of eight miles an, hour.The diagonals cross at X, from whichpoint we draw XM and X N. E M byour vertical scale of miles will be thedistance from Edinburgh, and N uponthe time line will show the time at whichthe two men meet.

Let us try the following question. Aprivateer running at the rate of tenmiles an hour sees a ship eighteen milesoff going at the rate of eight miles an

hour; how far can the ship gobefore it is overtaken. Let AB, Fig. 4,

Fig. 4.

represent the eighteen miles which the

ship is in advance of the privateer whenfirst seen. Also let AC represent the

privateer's rate, or ten miles an hour,and let B C represent the rate of theship, or eight miles an hour.The diagonals produced will intersect atC, and drawing CD and C E we haveA D for the time and B E as the distancewhich the ship can go before being over-taken.

Suppose that we have the followingquestion : Two towns are fifty milesapart, A is to leave one of these townsat six o'clock and to arrive at the other at

noon, making four stops of half an houreach at ten, twenty,thirty,and forty milesfrom the starting point. B leaves theother end of the road at seven o'clock,travels twenty miles an hour for one

hour, then turns back and retraces hiscourse for an hour at the rate of tenmiles an hour, then turns around and ad-vances again at such a rate as to meet Aas he is starting from his third halt ;continuing at the same rate B meets athalf past ten a third man, C, who leftthe first end of the route two hours laterthan A did and has been going at a uni-

18

form rate. At what rate has C beentraveling, and where did B meet him ?By the ordinary process this questionwould not be a simple one, but it is

quite so by the graphic method, asseen by the diagram, Fig. 5, in which

Fig. 5.

B is seen to meet C at about 23 milesfrom A's starting point, and C is foundto have been going at the rate of aboutnine and a half miles an hour. Our

figure is too small to give the requiredresult with accuracy. It is to be ob-served in regard to all of these pro-blems that the size and the proportion

19

of the diagram must depend entirelyupon the degree of accuracy which it isdesired to obtain, and also upon thecharacter of the question. Very obliquecuttings of diagonals should be avoided.Todhunter gives the following in his

elementary algebra. A person walkedout a certain distance from A to B atthe rate of three and a half miles anhour, and then ran part of the way backagain at the rate of seven miles an hour,walking the remaining distance in fiveminutes. He was out 25 minutes

;how

far did he run ? Let A B, Fig. 6, repre-

FiG. 6.

sent the whole time, or 25 minutes.Lay off AH equal to any convenientfraction of an hour, and A I equal to the

20

corresponding fraction of three and ahalf miles : the diagonal A K will thenby its inclination represent the rate ofthree and a half miles an hour; producethis diagonal indefinitely toward C.Next lay off B L equal to five minutesupon the time scale, draw the verticalLM, and the diagonal BD inclined atthe same rate as the line A K. Finallyfrom D draw the diagonal D C inclinedat such a rate as to represent seven miles

an hour, upon the same scales of courseas A C represents three and a half milesan hour, and produce it to intersect A Cat C. The whole distance between thetwo points is then shown by B F, and thedistance which the man ran by D M orE F, measured of course by the samescale of miles before employed.

Suppose to the preceding question weadd the following : While the man abovereferred to walks from A to B, and runsand walks back again, a second manwalks from B to A and back again fromA to B, at a uniform rate, being occu-pied in all the same length of time as

21

the man first mentioned; at what pointsand at what times will he meet the firstman ? We will repeat in Fig. 1 the

lines showing the movement of the first

man, viz. A C, C D and D B. A B rep-resents the whole time as before, andA E the distance between the two points;then will E F and F G represent themovement of the second man, and hewill meet the first man on his outward

trip at a distance from his starting pointshown by A I, and after the time A H^and on his inward trip at a distance B K,and at the time A J.The question below is also given in the

work above referred to : A person walk-

22

ed out from Cambridge to a village atthe rate of four miles an hour, and on

reaching the railway station had to waitten minutes for the train, which was thenfour and a half miles off. On arrivingat his rooms, which were a mile from the

Cambridge station, he found that he hadbeen out three and a fourth hours. Findthe distance of the village from Cam-

bridge. In this case we first lay off A B,Fig. 8, equal by any scale to three and a

D EFig. 8.

fourth hours. We next make A L equalto one hour, andAM equal to four miles,when the diagonal A N represents the

23\

rate of four miles an hour, which we

produce indefinitely. Next we go backfrom B to C the five minutes which ittakes the man to go from the Cambridgestation to his rooms, and draw the lineC E, representing the rate of the railway-train, and produce it indefinitely. Ifthe man had not been obliged to wait forthe train we should simply produce thetwo diagonals until they met, when thevertical distance of their intersection fromthe upper horizontal,measuredonthe scaleof miles, would be the distance required.As, however, the man has to wait tenminutes at the station, we take the dis-tance D E equal to that time, and findwhere it will just go in horizontally be-tween the two diagonals, when the ver-tical distance between D E andA B willbe what we require. If the whole time

being thesame theman hadwaited an hourat the station, and we wished to know thedistance, we should apply the line H I,equal to one hour by the time scale, tothe diagonals, and K P would give us the

24

distance;or if the distance K P was

given we should obtain the time H LLet us now pass to a somewhat different

class of questions : Two men start atthe same time to walk round an island ;the first man goes at the rate of five

miles an hour;the speed of the second

man is such as to carry him round theisland in three and a third hours, the dis-tance being ten miles. How long afterstarting will the first man pass the sec-

ond, and how long before he will passhim the second time ? The reader will,perhaps, at first sight not see the relation

between movement on the circular pathand time, as it is a little different fromthe relation between movement on a

straight line and time. He has, however,only to observe that in traveling a circu-

lar path a man while always gettingfarther away from the starting point isat the same time getting nearer to it,or, in other words, he is traveling bothfrom it and towards it at the same time.Our question above thus takes the formshown in Fig. 9, in which the movement

25

of the first man is shown by the diagon-als AB, CD, E F, etc., and that of thesecond man by the dotted diagonals. Itwill be seen that having drawn A B werecommence at C

;this is because in

going from the point represented by theupper horizontal line to the point repre-sented by the lower horizontal, inasmuchas the path is a circular one, we have gotback again to the starting point. Thefirst man it will be seen passes the sec-ond at five hours after starting, and

again at ten hours. K, instead of both

going from A towards N", one of themen goes from N towards A, we haveonly to start from the lower line and in-

26

cline the diagonal in the opposite direc-

tion, and we may vary the rates of speed,and stop the men at any points, for anylength of time, without making thequestion any more difficult. For ex-

ample, the movement of a man whoshould travel in the opposite direction atthe rate of one mile an hour is shown bythe diagonal N O, and he will meet thesecond of the men above referred to at

P, Q and E, from which points we maydraw verticals to the time line, and hor-izontals to the line A N, which will showus just when and where the several meet-ings will take place.We find the following question in Tod-

hunter's Algebra : A and B start to-gether from the same point on a walkingmatch round a circular course. Afterhalf an hour A has walked three com-plete circuits, and B has walked four anda half

; assuming that each walks withuniform speed find when B overtakes A.Let A B, Fig. 10, represent the length ofthe course, and let A C or B H representhalf an hour : then the dotted line

A D E F G H will show the movement ofA, while the four and a half full diagon-als to I will show that of B. Carryingthe two sets of lines on at the samerate we find them together again at J,which by the time scale is ten minutesfrom the time represented by H.

Let us try some of the watch problemsas given in the algebras. In Fig. 11 wehave shown the movement of both thehour and minute hands for twelve hours,and we shall find that the several diag-onals answer a variety of questions.

28

Ml 2 3 4. 5 e 7 8 a 10 II M

Fig. 11.

We may take the distance around theface of the watch as representing time

or distance as we please. It represents

both, and thus we lay off twelve divi-sions upon the upper horizontal line andalso upon the left hand vertical. The

long diagonal represents the course of

the hour hand for twelve hours, and theshort diagonals represent the twelve

revolutions of the minute hand in thesame time. Take now the followingquestion : The hands of the watch are

together at noon, when are they next to-

gether ? We see plainly that the handsare together at noon, at a little after one

o'clock, at a little more after two o'clock,

29

at a still longer time after three, and soon, the precise time being found by car-rying the crossings of the diagonals ver-

tically upwards to the time line. Our

figure is too small to do this accurately.Let us take one hour out of the preced-ing diagram, and enlarge it, as in Fig. 12.

iVI l\l

WF^ig. 12.

Take the following question : Thehands of a watch are at right angles atthree o'clock

;When are they next at

right angles? The hands are at rightangles when they are fifteen minutesapart. The vertical divisions in Fig. 12

30

are each five minutes. The movementof the minute hand for an hour is shown

by the diagonal AB, and that of thethe hour hand by C D. Wherever wecan get a vertical equal to fifteen min-

utes, or to A C, between the two diagon-als the hands will be at right angles, and

by producing this vertical to the timeline, as at M, we get the required time,in the present case between thirty-twoand thirty-three minutes past threeo'clock. Again, let the question be tofind at what time between three andfour o'clock the hands will be diametri-

cally opposite. Diametrically oppositeis thirty minutes apart, and applyingthirty minutes, or six of the vertical

spaces, to the lines AB and CD, andproducing the line upwards, we find thetime line to be cut at N, or about elevenminutes before four o'clock.

It will be evident from an examina-of the preceding figures that the graphicmethod is not confined to questions in-volving time and space alone, but that itis equally applicable to questions of time

/

31

and any kind of work done, whetherlabor performed by men, water discharg-ed by pipes, or the like. Take for ex-ample the following question : A can doa piece of work in five days, and B cando it in three days. In what time willboth working together do it ? Let A B,Fig. 13, be the time in which A can do

Fig. 13.

an amount of work represented by A Cor B D, then will A D represent the rateat which he works. So, too, if B doesan amount of work shown by B E, in thetime AB, AE will represent his rate of

32

work. Make EF equal to B D. BFwill show the amount of work done byboth in the time A B, and A F will bethe rate at which both together work.The several rates being fixed the ques-tion is at once answered. The amountof work being represented by AM, Awill do it in the time represented by A H,B in a time shown by A J, and both to gether in a time A L. The same diagramwill of course answer other questions in

regard to the two men. For example, ifwe know the time in which both mencan do a piece of work, and also thetime in which one man can do it, we findeasily how long the other will be doingit.

Questions like the following are com-mon in arithmetic and algebra: A syphonwould empty a cistern in forty-eightminutes, while a cock would fill it inthirty-six minutes. When it is emptyboth begin to act. How soon will thecistern be filled ? Of course the capa-city of the cistern in this question is im-material

;assume it to be AE, Fig. 14.

Fig. 14.

The syphon can empty it in forty-eightminutes. Lay off therefore by any scaleA C equal to forty-eight. The cock canfill it in thirty-six minutes. Lay offA B equal to thirty-six. The diagonalA H will represent the rate at which thesyphon empties the cistern, wliile the

diagonal A F shows the rate at whichthe cock fills it. The difference betweenthe two is, of course, the rate at whichthe cistern is filled. Producing thereforethe diagonals until they are separated bya vertical distance equal to the assumed

capacity of the cistern, A E, that is, I K,and carrying I K up to the time line atD, we have AD as the time in whichthe cistern will be filled by the joint ac-tion of the syphon and the cock.

34

Suppose that the question, instead of

'being as above, had been as follows :A syphon and a cock acting togetherwill fill a cistern in 144 minutes, whilethe cock acting alone would fill it in

thirty-six minutes? how long would ittake the syphon to empty it ? Lay offA D, Fig. 14, equal to 144 minutes, andA B equal to thirty-six minutes. MakeA E equal to the capacity of the cistern,upon any scale. Draw E H parallel toA D, and B F perpendicular to the same.Through A and F draw a diagonal, andproduce it to intersect a vertical throughD at K. Make K I equal to A E, andclraw I A to intersect the horizontal E Hat H. From H erect a perpendicular tocut the time line at C. AC will then bethe time in which the syphon alone would

empty the cistern.Let us change the question again as

follows : A syphon would empty a cis-tern in forty minutes, while a cock wouldfill it in twenty-two minutes. Both com-mence to act, but after fifty-three min-utes the cock is stopped for twenty-two

35

minutes, and then flows again at a ratewhich would fill the cistern in one hourand fifty-six minutes. When the cockrecommences the syphon stops workingfor sixteen minutes, but after that timethe cistern commences to leak at a rate

which would empty it in one hour and

twenty-five minutes. How long, underthe new conditions, will it be from the

beginning before the cistern will be halffull ? As the cock stops after fifty-threeminutes, which time is represented uponthe upper horizontal in Fig. 15 by the

Fig. 15.

distance S G we draw A B horizontallyand from B draw B C at such an angle

36

as to represent the new rate at whichthe cock supplies water. In the samemanner we draw D E to show the stop-page of the syphon, and afterwards E Fto represent the rate of leakage. The

diagonals B C and E F must then beproduced until the included vertical F Cis by the scale equal to one half of thecapacity of the cistern, or one half ofS T. Finally we produce C F to cut thetime line at K, and S K is the answer tothe question.The various questions in Alligation

may be worked by the graphic method.Suppose for example that a man wouldmix one kind of grain worth thirty centsa bushel with another quality worth

eighty cents, so as to make sixty bushelsworth 50 cents a bushel ; how much ofeach kind must he take ? Lay off B Fin Fig. 16 equal by any scale to thirtycents, B H equal to fifty cents, and B Iequal to eighty cents. The several linesA F, A H and A I will represent therates or values of the different kinds of

grain. Make A J on any scale equal to

37

Fig. 16.

the required number of bushels in themixture. Draw a vertical J K to meetA H, the value or rate of the mixtureproduced, at K. Produce A F indefinite-ly, and from K draw K L parallel to A Ito intersect AF produced inL, and fromL draw the vertical L M. AM will showthe number of bushels at the price B F,and M J will give the number at theprice B I. The result will of course bethe same if we produce A I and drawKN parallel to A L to meet it at N, anddrop the perpendicular NO on to theline P K.

Let us take a question, such as we

38

frequently find, like the following : Aworkman was hired for forty days atthree shillings and four pence for everyday that he worked, but with thecondition that for every day he did notwork he was to forfeit one shilling andfour pence ; he received 3 3s 4d; howmany days did he work. Reduce theseveral amounts to the common unit of

pence for convenience in plotting thework upon paper. Make A B, Fig. 17,

Fig. 17.

by any scale equal to forty days, andmake BC equal to what he would havereceived had he worked all of the time.JVlake A D equal to what he would havelost had he worked none at all, and drawC A and D B. Find where the line E Fwhich is equal by the scale of pence to

39

the whole amount he received, will justgo in vertically between A C and D Band produce it upwards to K. A K uponthe scale of time will then show the num-ber of days he worked, and K B thenumber of days he was idle, the former

being twenty-five and the latter fifteen.If we wish to know how many days heshould work in order that his earningsmay just balance his loss, we have only todraw from the point H, where the twodiagonals cross, the vertical HL whenwe shall have A L as the number of dayshe worked and L B as the number hewas idle.

Todhunter gives the following questionin one of his works. A and B shoot byturns at a target. A puts seven bulletsout of twelve into the bulls-eye, while Bputs in nine out of twelve ; betweenthem they put in thirty-two bullets.How many shots did each man fire ?Lay off A B, Fig. 18, equal upon any scaleto thirty-two, the whole number of suc-cessful shots. Next, lay off seven of the

thirty-two divisions from A to H, and

40

twelve of the same divisions on the ver-tical line from A to I, then will the diag-onal AM represent A's rate of success.In the same way we lay off nine divisionsfrom B to K and twelve divisions fromBj^to L, and B N will represent B's rateof success. Produce AM and B N untilthey meet at C, and from C draw CDperpendicular to AB. CD will showthe number of shots that each man fired.If A made no successful shots, proceed,ing in the same way we should lay off nodivisions upon the line A B, and twelveupon A E, and the line representing the

41

rate of success would be vertical, or inother words he has no success. Insuch case the number of times that Bwould have to fire to put the thirty-twoballs into the target would be found by-producing B C to cut A E produced, thelength of A E thus produced showing thenumber.About the time that the Pacific Rail-

road was opened the newspapers passedaround the following question : Sup-pose that it takes a train just one weekto run the whole length of the road, andthat one train leaves each end of theroad each morning, how many trains willa person meet in going the length of the

road, not counting the train which arrives

just as he starts, nor the train that starts

just as he arrives ? Let the six verticaldivisions in Fig. 19 represent the six

. J^iG, 19.

42

days . The diagonal from the bottomof the left hand vertical to the top ofthe right hand one may show the move-ment of the train running through in one

direction, when the opposite diagonals,eleven in number, will show the numberof trains which he will meet.To what has preceded we add a few

examples for practice from Todhunter's

algebras. In some cases we have giventhe diagrams showing the form whichthe solution will take, while in othercases the construction of the figure is left

to the reader. The ease with which

many questions commonly found in thebooks will be answered by the graphicmethod will depend of course uponthe more or less perfect knowledge of

algebra which may be possessed.Two plugs are opened in a cistern

containing 192 gallons of water ; after

three hours one of the plugs becomes

stopped, and the cistern is emptied bythe other in eleven more hours : hadsix hours elapsed before the stoppage it

would have required only six hours more

43

to have emptied the cistern. How many-gallons will each hole discharge in an

hour, supposing the discharge uniform.In Fig, 20 A B represents three hours,

Fig. 20.

B C three hours, C D six hours, and D Etwo hours. A K shows the discharge bythe two plugs for three hours, and K Ithe discharge by one plug for the elevenhours additional. The inclination of thelines must be such that when A, K andL are in a straight line, L H and K Ishall be parallel. The inclination A Lwill show the rate of discharge of both

44

plugs, and K I or AM will show the rateof one, while the difference between thesetwo or AN will show the rate of theother.

The road from a place A to a place Bfirst ascends for five miles, is then levelfor four miles, and afterwards descendsfor six miles the rest of the distance. Aman walks from A to B in three hoursand fifty-two minutes; the next day hewalks back to A in four hours, and hethen walks half way back to B and backagain to A in three hours and fifty-fiveminutes. Find his rates of walking uphill, on level ground and down hill.A and B are two towns situated 24

miles apart upon the same bank of a river.A man goes from A to B in seven hoursby rowing the first half of the distance,and walking the second half. In return-

ing he walks the first half at three-fourthsof his former rate, but the stream beingwith him he rows at double his rate ingoing, and he accomplishes the wholeof the distance in six hours. Find hisrates of walking and rowing.

45

A and B set out to walk together inin the same direction round a field, whichis a mile in circumference, A walkingfaster than B. Twelve minutes after Ahas passed B for the third time, A turnsand walks in the opposite direction untilsix minutes after he has met him for thethird time, when he returns to his orig-inal direction and overtakes B four timesmore. The whole time since they start-ed is three hours, and A has walkedeight miles more than B. A and B di-minish their rates of walking by one milean hour, at the end of one and two hours

respectively. Determine the velocitieswith which they began to walk.A vessel can be filled with water by

two pipes ; by one of the pipes alone thevessel can be filled two hours sooner than

by the other ; also the vessel can befilled by both pipes together in labours.Find the time which each pipe alonewould take to fill the vessel. The dia-gram given upon a preceding page. Fig.13, is the same as that required for theabove question, by which the answer willbe seen to be three and five hours.

46

A offers to run three times round acourse while B runs twice round, but Aonly gets 150 yards of his third roundfinished when B wins. A then offers torun four times round for B's thrice, andnow runs four yards in the time he for-

merly ran three yards. B also quickenshis rate so that he runs 9 yards in thetime he formerly ran 8 yards, but in thesecond round falls off to his original pacein the first race, and in the third round

goes only 9 yards for 10 he went in thefirst race, and accordingly this time Awins by 180 yards. Determine thelength of the course.A boat's crew row 3^ miles down a

river and back again in an hour and 40minutes. Supposing the river to have acurrent of 2 miles an hour, find the rate

at which the crew would row in stillwater.

A and B start together from the footof a mountain to go to the summit. Awould reach the summit half an hour be-fore B, but, missing his way, goes a mileand back again needlessly, during which

47

he walks at twice his former pace, andreaches the top six minutes before B. Cstarts twenty minutes after A and B,and walking at the rate of two and one-seventh miles per hour, arrives at the

summit ten minutes after B. Find therates of walking of A and B, and thedistance from the foot to the summit ofthe mountain.A and B are set to a piece of work

which they can finish in thirty daysworking together, and for which theyare to receive

48

bridge, and this they have to do againafter two days more marching. Afterhow many days from the beginning ofthe retreat will the retreating force beovertaken? AB, in Fig. 21, represents

A C 1? 15 J 31

Fig. 21.

26 miles, A C one day, C D two days,D E one day, E F two days, and F H oneday. The inclination of the line B Lrepresents the rate at which the retreat-

ing troops march, and the inclinationof CM or M N or N L shows the rate ofthe pursuers. The horizontals M and Nare the two halts of a day each. Thetwo diagonals are to be produced until

they cut, as at L, when A K will give us

49

the time, and AG or K L the distancerequired.A rows at the rate of 8j miles an hour.

He leaves Cambridge at the same timethat B leaves Ely, and is back in Cam-bridge 2 hours and 20 minutes after Bgets there. B rows at the rate of 7^miles an hour, and there is no stream.Find the distance from Cambridge to

Ely. The distance A B, Fig. 22, must

be such that when AD represents 8jmiles an hour, DE 12 minutes, E F 8Jmiles an hour, and B C 7J miles an hour,CF shall be equal to 2 hours and 20minutes.Two workmen A and B are employed

50

by the day at different rates. A at theend of a certain number of days receiv-ed 4 16s., but B, who was absent six ofthe days, received only 2 14s. If Bhad worked the whole time, and A hadbeen absent the six days, they wouldboth have received the same. Find thenumber of days, and what each was paidper day. AC, in Fig. 23, shows the

Fig. 23.

whole time, B C being six days. C E isequal to 4 16s., and B K to 2 14s.A B must be such that T> I being drawnparallel to A C the lines drawn throughE and I and through D and K shall meetupon the line A B.A waterman rows thirty miles and

51

back in twelve hours, and he finds thathe can row five miles with the stream inthe same time as three against it. Findthe times of rowing up and down.A person hired a laborer to do a cer-

tain work on the agreement that for

every day he worked he should receive2s., but that for every day he was absenthe should lose 9d.

;he worked twice as

many days as he was absent, and on thewhole received 1 19s. How many daysdid he work? In Fig. 24, AC is to be

Fig. 24.

twice as great as C B, A D is to representthe man's rate of receipt while he work-ed, and B E his rate of loss while idle.

52

E D is to be equal by the scale to 119s.

A man and a boy being paid for cer-tain days' work, the man received 27

shillings, and the boy, who had been ab-sent three days out of the time, receivedtwelve shillings. Had the man insteadof the boy been absent three days theywould have received the same amount.Find the wages of each per day. In

Fig. 25, A E represents the whole time.

Fig. 25.

A B the man's rate of work, E B whathe received, AF three days, FD theboy's rate of work, and E D what theboy received; A C parallel to F D, shows

, 63

the boy's work for the whole time, andF C, parallel to A B, the man's workomitting three days. The inclinations ofthe lines must be such that F C and A Cparallel, respectively, to A B and F Dshall meet on E B.A railway train after traveling for one

hour meets with an accident which de-

lays it one hour, after which it proceedsat three-fifths of its former rate, and ar-rives at the terminus three hours behindtime

;had the accident occurred fifty

miles further on, the train would havearrived one hour and twenty minutessooner. Required the length of the line,and the original rate of the train.The fore wheel of a carriage makes

six revolutions more than the hind wheelin going 120 yards; if the circumferenceof the fore wheel be increased by one-fourth of its present size, and the cir-cumference of the hind wheel by one-fifth of its present size, the six will be

changed to four. Required the circum-ference of each wheel.A and B can do a piece of work to-

54

gether in 48 days ; A and C can do it in30 days ; and B and C working togethercan do it in 26 days. Find the time inwhich each could do the work alone.A man starts from the foot of a moun-

tain to walk to its summit. His rate of

walking during the second half of thedistance is half a mile per hour less thanhis rate during the first half, and hereaches the summit in 5^ hours. Hedescends in 3j hours, walking at a uni-form rate which is one mile an hourmore than his rate during the first halfof the ascent. Find the distance to the

summit, and his rates of walking. In

Fig. 26, AB represents 5J hours, B C 3fhours, and AD the distance required.A H shows his movement during the firsthalf of the ascent, H E that during thesecond half, and E C his descent.A sets off from London to York, and

B at the same time from York to Lon-don, and they travel uniformly. Areaches York 16 hours and B reachesLondon 36 hours after they have met onthe road. Find in what time each has

55

performed the journey. In Fig. 27, AF

represents A's movement, and D C that

56

of B;EF is sixteen and BC 36 hours.

The intersection of FA with AC andof C D with the lower horizontal mustfall on the same vertical, A D.Two trains of cars, 92 feet and 84 feet

long respectively, are moving with uni-form velocities on parallel tracks. Whenthey go in opposite directions they passeach other in one second and a half

;but

when they go in the same direction thefaster train passes the other in six sec-onds. Find the rate at which each trainmoves.

Two travelers, A and B, start fromtwo places, P and Q, at the same time.A starts from P with the design to passthrough Q, and B starts from Q andtravels in the same direction as A. WhenA overtook B it was found that they hadtogether traveled thirty miles, that Ahad passed through Q four' hours before,and that B at his rate of traveling wasnine hours' journey distant from P.Find the distance between P and Q. InFig. 28, PR is nine hours, T S is fourhours, and P V plus Q Y is thirty miles.

51

Fig. 28.

P Q, the distance required, must be suchthat Q S being drawn parallel to P L,S P shall cut Q M in a point, O, whichshall be four hours back of S.We will conclude by an application of

the graphic method to a question of greatpractical importance, viz. the adjustmentof the running times of railway trains,which, as before stated, has been for a

long time employed by railway mana-

gers, and which first suggested to thewriter the solutions given in the preced-ing pages.

Let the heavy vertical lines, in Fig 29,*represent the successive hours of the

day, and the intermediate finer lines the*Frontispiece.

58,

quarter hours. The horizontal lines rep-resent the several stations along the

road, the vertical distances betweenthem being laid off by scale accordingto the actual distances in miles. Sup-pose that we wish to start a train at sixo'clock A. M., from the station represent-ed by the line A A, so that it shall arriveat the station shown by the line J J atthree o'clock p. m. stopping fifteenminutes at each way station. The num-ber of way stations being eight, thewhole time consumed by stops will be120 minutes, or two hours. From 3 p. m.upon the lower horizontal line we goback two hours, or to 1 p. m. and from6 A. M. upon the upper horizontal wedraw a line which produced would strike1 p. M. upon the lower line. This diagon-al reaches the line BB at 7:23 a. m. Aswe stop at the station 15 minutes, we

pass along on the line B B a distanceequal to fifteen minutes on the time

scale, and from the point thus reachedwe start again parallel to the first

diagonal, ariving at station C at 8:20 a.

69

if. Proceeding in the same way wearrive at station J at 3 p. m., as desired.

The inclination of the diagonal showsthe speed.

If we would start a train from stationA at 8:30 a. m. to arrive at J at 11:15,making no stops, it would pass the trainabove described at station D, and willrun the whole distance in two hours and45 minutes. Trains running in the oppos-ite direction are shown on the diagramby diagonals ascending from left to

right. Thus a train leaving station J at6 A. M., to arrive at station A at noonmaking no stops, will run as by thebroken diagonal from 6 a.m., on the lowerline to 12 on the upper one, passing the6 A. M and the 8:30 a. m. trains, run-

ning in the opposite direction, at stationD. It will be observed that the linefrom 6 to 12 changes its rate of inclina-tion at the horizontal D, by which weunderstand that the train changes its rateof speed at that station, running fasterfrom D to A than from J to D.

If it is desired to work a construction

60

train between the stations E and D from6 A. M. to 6 p. M., the movement ofsuch a train is shown by the short diag-onals between the horizontals D and E,and its time card would be as follows :Leave E at 6 a. m., and arrive at D at1. Leave D at 7:15 and arrive at E at8:15. Leave E at 8:30 and arrive at Dat 9:30, crossing the 6 a. m. and the8:30 A. M. trains from A to J, and beingpassed by the 6 a. m. train from J to A.Leave D at 10 and arrive at E at 11.Leave E at 11:15 and arrive at D at 12:-15, and wait to be passed by the 9 a. m.train from J to A. Leave D at 12:45p. M. and arive at E at 1 :30 P. M., leaveE at 1:45 and arrive at D at 2:45, andpass noon train from station A, and 11:15A. M. train from station J. Leave D at3:15 and arrive at E at 4 p. m. Leave E at4:15 and arrive at D at 5 p. m. LeaveD at 5:15 and arrive at E at 6 P. M.

If a train leaves A at noon and runstowards J, leaving C at 2:05 and reach-ing E at 3:20, and another train leavesJ at 11:15 A. M., and G^ at 1 p. m., run-

61

ning to A as by the diagonal, withoutstopping, the trains will pass at 3 p. m.

at a point betweenD and E,the exact posi-tion of which may be found by the scaleof miles according to which the length ofthe road or the distance A J is plotted,at which place ,a side track must be pro-vided.

In practice the diagram is accuratelydrawn to a large scale, and the severaltrains are represented by differentlycolored elastic lines fastened by pins sothat they may be moved from hour tohour through the day and night as thevarious occurences upon the road may de-mand, some trains being hastened others

retarded, extras put in and all provisionsmade for securing regularity in themovement and freedom from disaster.The grades and curves may if desirablebe shown upon the vertical line A J, bywhich those parts of the road may atonce be seen where from increased resist-ance a lower speed will need to be adopt-ed. Upon a double track road a chart

62

may be prepared for each track, anddiagonals in one direction only will

appear upon each diagram.

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PLYMPTON. The Blow Pipe. A Guide to its Usein the Determination of Salts and Minerals. Com-piled from various sources, by George W. Plympton,C E. A. M., Professor of Physical Science in thePolytechnic Institute, Brooklyn, New York, izmo,cloth I so

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ELIOT AND STORER. A compendious Manual ofQualitative (Chemical Analysis. By Charles W.Eliot and Frank H. Storer. Revised with the Co-operation of the authors. By William R. Nichols,Professor of Chemistry in the Massachusetts Insti-tute of Technology Illustrated, i2mo, cloth $i 50

RAMMELSBERG. Guide to a course of QuantitativeChemical Analysis, especially of Minerals and Fur-nace Products. Illustrated by Examples By C. F.Ramn? alsberg. Translated by J. Towler, M. D.8vo, cloth 2 35

EGLESTON. Lectures on Descriptive Mineralogy, de-livered at the School of Mines, Columbia College.By Professor T. Egleston. Illustrated by 34 Litho-graphic Plates. 8vo, cloth 450

JACOB. On the Designing and Construction of StorageReservoirs, with Tables and Wood Cuts representingSections, &c., iSmo, boards 50

WATT'S Dictionary of Chemistry New and Revised^

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GREENE. Graphical Method for the Analysis ofBridgeTrusses, extended to continuous Girders and DrawSpans. By 0. K. Greene, A. M., Prof, of Civil Engi-neering, University of Michigan. Illustrated by 3folding plates, 8vo, cloth 2 00

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ROGERS. The Geology of Pennsylvania. A Govern-ment survey, with a general view of the Geology ofthe United States, Essays on the Coal Formation andits Fossils, and a description of the Coal Fields ofNorth America and Great Britain. By Henry Dar-win Rogers, late State Geologist of Pennsylvania,Splendidly illustrated with Plates and Engravings inthe text. 3 vols., 4to, cloth, with Portfolio of Maps. 30 00

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BOCJRNE. Treatise on the Steam Engine in its variousapplications to Mines, Mills, Steam Navigation^Railways, and Agriculture, with the theoretical in-vestigations respecting the Motive Power of Heat,and the proper proportions of steam engines. Elabo-rate tables of the right dimensions of every part, andPractical Instructions for the manufacture and man-agement of every species of Engine in actual use.By John Bourne, being the ninth edition of " ATreatise on the Steam Engine," by the " ArtizanClub" Illustrated by 38 plates and 546 wood cuts.4to, cloth $15 00

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SIMMS. A Treatise on the Principles and Practice ofLevelling, showing its application to purposes ofRailway Engineering and the Construction of Roads,&C. By Frederick W. Simms, C. E. From the sthLondon edition, revised and corrected, with the addf-tion of Mr. Laws's Practical Examples for settingout Railway Curves. Illustrated with three Litho-graphic plates and numerous wood cuts. 8vo, cloth. $2 50

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MORFIT. A Practical Treatise on Pure Fertilizers, andthe chemical conversipn of Rock Guano, Marlstones,Coprolites, and the Crude Phosphates of Lime andAlumina generally, into various valuable products.By Campbell Morfit, M.D., with 28 illustrative plates,8vo, cloth 20 3o

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MYER. Manual of Signals, for the use of Signal officersin the Field, and for Military and Naval Students,Military Schools, etc. A new edition enlarged andillustrated. By Brig. General Albert J. Myer, ChiefSignal Officer of the army. Colonel of the SignalCorps during the War of the Rebellion. i2mo, 48plates, full Roan $5 00

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PICKERT AND METCALF. The Art of Graining.How Acquired and How Produced, with descriptionof colors, and their application. By Charles Pickertand Abraham Metcalf. Beautifully illustrated with42 tinted plates of the various woods used in interiorfinishing. Tinted paper, 4to, cloth 10 00

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BOYNTON. History of West Point, its Military Im-portance during the American Revolution, and theOrigin and History of the U. S. Military Academy.Bjr Bvt Major C. E. Boynton, A.M., Adjutant of theMilitary Academy. Second edition, 416 pp. 8vo,printed on tinted paper, beautifully illustrated with36 maps and fine engravings, chiefly from photo-graphs taken on the spot by the author. Extracloth

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PRESCOTT. Outlines of Proximate Organic Analysis,for the Identification, Separation, and QuantitativeDetermination of the more commonly occurring Or.ganic Compounds. By Albert B. Prescott, Professorof Chemistry, University of Michigan, i2mo, cloth... i 75

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McCULLOCH. Elementary Treatise on the* Mechan-ical Theory of Heat, and its application to Air andSteam Engines. By R. S. McCulloch, 8vo, cloth

AXON. The Mechanics Friend ; a Collection of Re-ceipts and Practical Suggestions Relating to Aqua-riaBronzingCementsDrawingDyes Electri-cityGildingGlass WorkingGluesHorology-LacquersLocomotivesMagnetismMetal-Work-ingModellingPhotographyPyrotechyRailwaysSoldersSteam EngineTelegraphyTaxidermy Varnishes Water-Proofing and MiscellaneousTools,Instruments, Machines and Processes con-nected with the Chemical and Mechanics Arts ; withnumerous diagrams and wood cuts. Edited by Wil-liam E. A. Axon. Fancycloth 150

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ERNST. Manual ofPractical Military Engineering, Pre-pared for the use of the Cadets of the U . S. MilitaryAcademy, and for Engineer Troops. By Capt. O. H.Ernst, Corps of Engineers, Instructor in PracticalMilitary Engineering, U. S. Military Academy. 192wood cuts and 3 lithographed plates. i2mo, cloth.. 500

BUTLER. Projectiles and Rifled Cannon. A CriticalDiscussion of the Principal Systems of Rifling andProjectiles, with Practical Suggestions for their Im-provement, as embraced in a Report to the ChiefofOrdnance, U. S. A. By Capt. John S. Butler, Ord-nance Corps, U. S. A. 36 plates, 4to, cloth 7 50

BLAKE. Rejjort upon the Precious Metals : Being Sta-tistical Notices of the principal Gold and Silver pro-ducing regions of the World, Represented at theParis Universal Exposition. By William P. Blake,Commissionir from the State of California. 8vo, cloth 2 00

TONER. Dictionary of Elevations and Climatic Regis-ter of the United States. Containing in addition toElevations, the Latitude, Mean, Annual Temperature,and the total Annual iiain fall ofmany localities; witha brief introduction on the Orographic and PhysicalPeculiarities of North America. By J. M. Toner,M. D. 8vo, cloth . .- 3 75

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