IN MEMORIAM FLORIAN CAJORl leath's Mathematical Monographs Issued under the general editorship of Webster Wells, S. B. :i)fcssor of Mathematics in the Massachusetts Institute of Technology GRAPHS BY ROBERT J. ALEY, A.M., Ph.D. Professor or Mathematics in Indiana University . Heath & Co., Publishers New York Chicago Number 6 Price, Ten Cents Heath's Mathematical Monographs Number 6 GRAPHS ALEY Dr. Aley has prepared a valuable Chapter on Graphs for Wells's Essentials of Algebra and Wells's New Higher Algebra. The editions containing this chapter will be supplied when specially ordered. HEATH'S MATHEMATICAL MONOGRAPHS Number 6 GRAPHS BY ROBERT J. ALEY, A.M., Ph.D. ■A PROFESSOR OF MATHEMATICS IN INDIANA UNIVERSITY T c- BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1902 (an ' / Copyright, 1902, By D. C. Heath & Co. CAJORI INTRODUCTORY STATEMENT At the present time the Graph is used so extensively in many lines of work that it is necessary for the non- technical reader to know something of it. When we note that the fluctuations in the price of wheat, the changes of temperature for a month, the age of conver- sion in children, the advancement in learning a trade, the strain on a girder under different loads, the death rate at different ages, and the solutions of numerical and algebraical problems have alike been subjected to graphi- cal methods, we conclude that elementary mathematics should take some note of the subject. Graphical methods permeate so many subjects and may be used so freely in the different parts of elemen- tary mathematics that it seems that there may be use for a brief treatment outside of the text-book. Such a treat- ment in the hands of the teacher gives him the power to use it whenever the occasion demands. Such a treat- ment may also help the general reader to an understand- ing of the graphical treatment now given to so many subjects. GRAPHS. Definitions. A Graph is a representation by means of lines, straight or curved, of some set of measured or numerically represented facts. Axes. Two lines intersecting at right angles, as in Fig. I, are called the Axes of Coordinates. OXy the horizontal one, is the /-axis, or Axis of Abscis- II I -l-ar,-f2/ —x.-y III + a?,-2/ IV Y' Fig. I. sas ; O Y, the vertical one, is the K-axis, or Axis of Ordinates. O, the intersection of the axes, is the origin. Quadrants. The axes divide the plane into four parts, called quadrants. These quadrants are num- bered from I to IV, as in Fig. i. I 2 Graphs. Coordinates. A po,int is located when its distance ,ujid direcliori fr6m each of the axes is known. The distance from OV is the ;r-distance, or abscissa. The distance from OX is the ^-distance, or ordinate. The two distances constitute the Coordinates of the point. A point is denoted by the symbol (x, }^), where ;r is the abscissa and j^ the ordinate. Convention as to Signs. In the representation of points, distances to the right of the F-axis are posi- PJ-3.,3) P3(-3,-l) i ! Pi(M) KC 2.-3) Fig. 2. tive, to the left negative. Distances above the JT-axis are positive, those below negative. An x in the first or fourth quadrant is + , in the second or third it is — . A _7 in the first or second quadrant is +, in the third or fourth it is — . These are indicated in Fig. i. Plotting Points. To locate the point /*i(3, 4), we measure 3 units to the right of (9Fand then measure 4 units up from OX. (See Fig. 2.) The point -PaC— 2, 3) is 2 units to the left oi OV and 3 Temperature Curve. 3 units above OX. The point Pjy— 3, — i)is 3 units to the left of (9Fand i unit below OX. The point P4(+ 2, — 3) is 2 units to the right oi OY and 3 units below OX. Temperature Curve. The temperature at noon on twenty successive days was as follows: 60, 62, 64, 63, 61, 6^, 73, 75, 74, 72, 70, 68, 69, 70, 74, 70, 67, 65, 63, 64. To show this graphically we take as our X-axis (Fig. 3) a line which we let represent a temperature of 60 degrees. Each unit along this line represents one day, and each unit above the line one degree of heat. The temperature on the first day, 60 Y Fig. 3- degrees, is at the origin, O. The temperature on the second day is 62, and is shown at the point (i, 2); that is, I to the right of o and 2 above the X-axis. The temperature of the other 18 days is shown in a similar way. If a smooth curve is drawn through these 20 points, the result is the temperature curve, or the graph of the noon tem- perature for 20 days. 4 Graphs. In the same way the graph may be used to show the fluctuations in the price of wheat, the increase in skill in learning a trade, or in fine anything that may be represented by the combination of two series of facts. Solution of Problems by Graphs. 1. A travels 4 7niles an hour, B 6 miles an hour. If A has 2 hours the start, when and where will B overtake hint f In this problem let each space along the JT-axis (Fig. 4) represent i mile, and each space along the F-axis represent one-half hour. At the end of the first hour A is evidently at the point A. At the end of the second hour he is at B, and so on. His path in time and space is readily seen to be OP. B does not start until 2 hours after A starts, so his path begins at L, four spaces (2 hours) above O. At the end of first hour B is at C, and his path is LP. B overtakes A when his path crosses that of A. This occurs at P. A perpendicular from P to Solution of Problems by Graphs. 5 OX intersects it at M, 24 spaces to the right of O. B, therefore, overtakes A 24 miles from starting- place. The length of the perpendicular PM is 12 spaces. Hence B overtakes A 6 hours after A starts, or 4 hours after he himself starts. 2. Two towns P and Q are 48 miles apart. A walks from Y to (^ at the rate of 3 miles an Jiotir a7td rides back at 12 miles a7i hour, B starts from, Q two hours after A starts fvm P, a7td rides to P at the rate of 8 miles an hour, and walks back at 4 miles a7i hour. When and where do A and B meet the seco7id time ? Figure 5 shows the solution to this. The paths of A and B are marked and can be easily under- Fig- 5. stood from what has preceded. Their second meeting place is M, which is seen to be 36 miles from P and 12 miles from Q. The line ML is 17 6 Graphs. spaces long, and so they meet 17 hours after A starts, each vertical space here having been chosen to represent one hour. 3. A, B, and C, travelling at 6, 8, and 12 miles an hour^ start at the same time around aji island 48 miles in circumference. When and where are they again all together ? The graph in Fig. 6 shows the result at once. The two perpendiculars OS and IR are 48 spaces apart. Since the road is a circle, any point in IR simply represents the completion of one circuit and Fig. 6. really represents a new starting-point in OS. A's path is OPj PQ, QR, three complete circuits. B's path is OL, LM, MN, NR, four complete circuits. C's path is OD, DP, PM, MQ, QT, TS, six com- Solution of Problems by Graphs. 7 plete circuits. At the end of these circuits they are all together. The time IR is 24 hours. 4. A man walking from a town A to another B at the rate of 4. miles aii hour, starts one hour before a coach which goes 12 miles an hour and is picked up by tJie coach. O71 arriving at B, he observes that his coach journey lasted 2 hours. Find the distance from A /^ B. Let spaces (Fig. 7) to the right be miles and spaces up be quarter hours. AP is the path of the man while walking. The carriage path is CP. The intersection P is 6 spaces to right and 6 spaces up. The carriage picks the man up 6 miles from A Fig. 7. and i\ hours after he has started. The destina- tion is reached 2 hours from P ; that is, the line CP is continued until it cuts a time line 8 spaces above P at B. AM equals 30 miles and is the distance between the towns. Graphs. A Linear Equation in Two Variables. 3x-h4J^= 12. We say in algebra that such an equation is inde- terminate, for we can get an indefinite number of values of x and j^ that will satisfy it. To get a solution we need only to assign arbitrarily \.o x 3. value and then solve for j. The following is a set of solutions : x= o y= 3 X— I 7=2i X — 2 7=li ^=3 J=i ;ir=4 y^o X 5 7 = - The list could be extended indefinitely. We can write these solutions as the points (o, 3), Fig. 8. A Linear Equation in Two Variables. 9 (I, 2i), (2, 1 1), (3, I), (4, o), and (5, - |)- These points may be plotted as in Fig. 8. It will now be noticed that these points are in a straight Hne. The line is called the graph of the equation. If the line be produced indefinitely and the x and y of any point found by measurement from the graph, the values thus found will satisfy the equa- tion. For example, if we select the point Q, we find that its x, OM, is - 4, and its y, MQ, is + 6. These values satisfy the equation ^x + ^y— 12, for 3(- 4) + 4(6)= 12. An equation of the first degree in two variables always has a straight line for its graph. ax -{- by = c is a general linear equation in x and y. A set of solutions is as follows : X = o y x— I y x= 2 y ^=3 y b _c — a b _c — 2a ~~~b b _ c — A,a etc. etc. lO Graphs . We see that a change of i in the value of x makes a change in y. If these points were plotted, they would appear (Fig. 9) very much like the side view of a uniform straight stairway, in Fig. 9. which the width of the steps is i, and the height - • The points are readily seen to be in a straight line. A shorter way of getting the graph. Since the linear equation always represents a straight line, we can draw its graph if we know two points upon it. In general, the two points most easily determined are those where the graph cuts the coordinate axes. The point on the Jf-axis is found by putting 7 = and solving for x. The point on the F-axis is found by putting x — o and solving for^. A Linear Equation in Two Variables, ii Example. 2x— ^y— lo. If J = o, .r = 5 ; and if ;r = o, y — — 2. Y (6.0) (0.-2) Fig. 10. The require(i graph cuts the X-axis at (5, o) and the F-axis at (o, — 2). Plotting these two points, the line is easily drawn as in Fig. 10. Simultaneous linear equations. (1) r5;r + 4;j/=22l (2) \lx-\- 7= 9J Draw the graphs of these two equations on the same diagram as in Fig. 11. It is found that the two lines intersect at a point P whose coordinates are (2, 3). The x and y (2 and 3) of this point is the solution of the equations. Two simultaneous linear equations in x and y have but one solution. Each equation represents 12 Graphs. a straight Hne. The solution is the point common to both hnes ; that is, the intersection of the hnes. Fig. II. Two straight Hnes can only intersect in one point, so there is but one solution. (1) r ^+ j=i (2) L2.r + 2^= 7 If we undertake to solve the above equations, we encounter a difficulty. We find that we cannot Y Fig. 12. eliminate x without also eUminating y at the same time. The Quadratic Equation. 13 If we draw the graphs of these equations, we find they are represented as in Fig. 12. The graphs show at once where the difficulty is. The Hues are parallel and so do not intersect at all. In the language of mathematics, they intersect at in- finity, which is just another way of saying that they never intersect. The Quadratic Equation. ax^-\- bx + c =y is an equation which, whenj = o, is the type form of the quadratic in a single variable. If the quadratic in x is thought of in the above form, it readily yields to graphical representation. Graph of x'^ — 2 x — 2, = o. We write x'^ — 2x — ■^= y. Solving this for x in the usual way, we get x= I ± V4 -\-y. The following Hst of values for y and x are readily found : 1. y = o x= I and — i 2. y = — I X = 2.y and — .7 Z. y — — 2 X— 2.4. and — 4 4. j/ = — 3 X = 2 and o 5. J/ = — 4 X = I and I 6. / = I ,r = 3.2 and — 1.2 H Graphs. 7. y = 2 X = 3.4 and - 1-4 8. y = 3 ;r = 3.6 and - 1.6 9. y=. 4 ;r= 3.8 and - 1.8 10. y^ 5 ;r = 4 and — 2 11. etc 12 X — ^ and etc. -3 Plotting these points carefully and connecting them by a smooth curve, we get the result shown in Fig. 13. It is seen that the graph in this case is a curve, and that it cuts the axis of X in two points. These points are at distances of 3 and — I from the origin. 3 and — i are the two roots of the quadratic x'^ 2X — ^ — o. A quadratic always represents a curve that can be cut in two places by one straight line. Write X^ — 2X ■\- I =0. 2x •\- \ —y. The Quadratic Equation. 15 Solving for x, we J have X = I ± Vj. ^ = ;r = I 7= I X — 2 and 7 = 4 ;r = 3 and — I y = 9 ;r = 4 and - 2 etc. etc. Plotting these points and drawing a smooth curve through them, we have the curve shown in Fig. 14. This curve does not cross the axis of X, but touches it at the point (i, o). The first member of the given equation being a perfect square, the equation Fig. 14. has two equal roots. The graph of a quadratic having equal roots always touches the axis of X at a distance from O equal to one of the equal roots. If we consider the equation x'^—6x-\- 10 =j/ and treat it as the above, we get a graph shown in 1 6 Graphs. Fig. 15. This curve does not touch the axis of X. If in the equation x^ — 6x -\ 10 = j/ we put 7 = Fig. 15. and solve, we get imaginary roots for x. The graph of a quadratic having imaginary roots does not touch the axis of X. Simultaneous Quadratics. 1. X +y = 2. 2. ;rj/ = — 15. Square (i), subtract 4 times (2), and extract the square root, and we have {l)x-y= 8 {^) x-y = -^ In Fig. 16 the various Hues of the graph are numbered to correspond with the numbers of the equations. Simultaneous Quadratics. 17 Equations (i) and (2) give a straight line and the double-branched curve known as the hyper- bola. These intersect at the points P, Q, whose 0\ i^ < cs-r ^^s-r- / Fig. 16. coordinates are (-3, 5) and (5, - 3). These are the only solutions to the system of equations. The auxiliary lines (3) and (4) intersect line (i) in P and Q, and hyperbola (2) in R and 5. \. xy = 12. 2. x^ -\- y^ = 40. The auxihary equations appearing in the solution are : 3. jr-f j/ = + 8. 4. X -^y = -^. 5. x—y = ^A,. 6. A'-j = -4. i8 Graphs. The graphs of all these equations are shown in Fig. 17 by the corresponding numbers. The solutions are at the points -P, Q^ R, and 5. Y Fig. 17. The Complex Number. In making general the treatment of quadratics, the complex number becomes necessary. The imaginary unit or z( = V— i) appears in the extraction of the square root of a negative quan- tity. E.g., V — 4 = V4 ( — I ) = V4 • V — I = ± 2 V— I =±22. a + bi is the type of all complex numbers. The imaginary and complex numbers are graphically represented by means of Argand's diagram. Two axes intersecting at right angles are used just as in ordinary graphic work. The horizontal one is the axis of reals, and the vertical the axis of imaginaries. The Complex Number. 19 In Fig. 18 CA and BD intersect at right angles at O. 0A(— i) = - OA =0C. Hence we may regard — i as an operator which reverses OA, or which turns it about O through an angle of 180°. We might then think of 2 = V— i as an operator which turns OA through an angle of 90°, or OAz = 0B. Then OAz^ = OBi ^0C\ OAi^= OBi^=^ OCi ^0D\ OAi^ = OBP = Oa^= ODi = OA. These results merely show that when we regard i as an operator which turns a quantity through an angle of 90°, we get results consistent with the known algebraic set of facts : •3 -^ _ V I \.i I i — I I. 20 Graphs. To represent any complex number as ^ + iy, we measure on OX a distance OM=x and a perpen- dicular distance PM = y. The point P, or as is iy Fig. 19. frequently more convenient, the line OP, is said to represent x + iy. OP = V,t'2 + j/^ = r, radius vector, and when taken with the positive sign is called the modulus. The angle MOP = ^ is called the amplitude. X = OP cos 6 =^ r cos d ; J = (9/^ sin ^ = r sin ^. Hence ;r + 2^7 = r(cos ^ + ^ sin ^). If OS represents x^ + /;/j, and 6^7" represents x^ + /^/g* then OP represents {x^ + iy^ + (;r2 4- ^3^2)' OP being the diagonal of the parallelogram of which OS and (97" are two adjacent sides. The Complex Number. 21 The diagram shows at once that if OP repre- "''"*" ^ + iy' (say), then y = jTj + x^ and y = y\ + ^2' and hence OP = OS + OT. Fig. 20. The diagram may be used to iUustrate nearly all the principles of the complex number. Enough has been given to show its adaptability. FACTORING AS PRESENTED IN WELLS' ESSENTIALS <?/ALGEBRA I. Advanced processes are not presented too early. II. Large amount of practice work. No other algebra has so complete and well-graded development of this important subject, presenting the more difficult principles at those stages of the student's progress when his past work has fully prepared him for their perfect comprehension. The Chapter on Factoring contains these simple processes : CASE I. When the terms of the expression have a common monomial factor — thirteen examples. II. When the expression is the sum of two binomials which have a common binomial factor — twenty examples. III. When the expression is a perfect trinomial square — twenty-six examples. IV. When the expression is the difference of two perfect squares — fifty-five examples. V. When the expression is a trinomial of the form x^-{-ax-i-l? — sixty-six examples. VI. When the expression is the sum or difference of two perfect cubes — twenty examples. VII. When the expression is the sum or difference of two equal odd powers of two quantities — thirteen examples. Ninety-three Miscellaneous and Review Examples. Further practice in the application of these principles is given in the two following chapters — Highest Common Factor and Lowest Common Multiple. In the discussion of Quadratic Equations, Solution of Equations by Factoring is made a special feature. Equations of the forms ;^*— 5.^ — 24 = 0, 2X^ — x = o,x^-\-4x^ — x — ^ — o, and ;^J— I =0 are discussed and illustrated by thirty examples. The factoring of trinomials of the form ax^+lx-\-c and ax'^+I^x^-^c, which involves so large a use of radicals, is reserved until Chapter XXV, where it receives full and lucid treatment. The treatment of factoring is but one of the many features of superiority in Wells* Essendals of Algebra. Ha/f Leather, 338 pages. Price $1.10. D. C. HEATH & CO., Publishers BOSTON NEWYORK CHICAGO Wells's Essentials of Algebra is the best book that can be placed in the hands of a preparatory student. The one essential fea- ture that I admire in the book is its simplicity. So many authors now write algebras and geom- etries to see how hard they can make them, and the consequence is that only the brightest students acquire any real knowledge of the subject. Prof. Wells leads the student step by step, and his book is well named The Essentials, as it contains all that is essential to a student entering any of our great universities. EDWIN W. RAND, Principal Princeton University Academy. The definitions ; the problems ; the chapter on factoring ; the factoring of trinomials ; the discussion of the principles of fractions ; the so- lution of equations by factoring ; all combine to make Wells's Essentials of Algebra a book of su- perior worth. Such a book I can recommend. E. MILLER, Professor of Mathematics, University of Kansas. Wells's Essentials of Geometry is far superior in the arrangement of propositions, the choice of proofs, and in the method of presenting the subject. The student is stimulated to help himself, to make his own proofs, to gain logical power. L. B. MULLEN, Ph.D., Dept. of Math., Central High School, Cleveland, 0. In the order of theorems, the proof of corollaries, the grading of the original exercises, and the opportunity for original work, Wells's Essentials of Geometry is notably superior. C. A. HAMILTON, Dept. of Math., Boys' High School, Brooklyn, N. T. The Vital Difference lies in this: Wells's Essentials Plane and Solid Geometry is characterized by the inductive method of demonstration, and always requires the student to do for himself the maximum amount of reasoning and thinking of which he is capable, while most geometries require of the pupil in his demonstrations little personal power except that of memo- rizing. They give the reasons for statements ; while Wells, whenever the student should be able to give the reason for himself, always asks why the statement is true. No other geom- etry develops so strongly the power of vigorous independent thought. D. C. HEATH & CO., Publishers BOSTON NEW YORK CHICAGO Wells's Mathematical Series. ALGEBRA. Wells's Essentials of Algebra ..... $i.xo A new Algebra for secondary schools. The method of presenting the fundamen- tal topics is more logical than that usually followed. The superiority of the book also appears in its definitions, in the demonstrations and proofs of gen- eral laws, in the arrangement of topics, and in its abundance of examples. Wells's New Higher Algebra ..... 1.3a The first part of this book is identical with the author's Essentials of Algebra. To this there are added chapters upon advanced topics adequate in scope and difficulty to meet the maximum requirement in elementary algebra. Wells's Academic Algebra ..... 1.08 This popular Algebra contains an abundance of carefully selected problems. Wells's Higher Algebra ...... 1.32 The first half of this book is identical with the corresponding pages of the Aca- demic Algebra. The latter half treats more advanced topics. Wells's College Algebra ...... 1.50 A modem text-book for colleges and scientific schools. The latter half of this book, beginning with the discussion of Quadratic Equations, is also bound sep- arately, and is known as Wells's College Algebra, Part II. $1.33. Wells's University Algebra . . • • • 1.3a GEOMETRY. Wells's Essentials of Geometry — Plane, 75 cts.; Solid, 75 cts.; Plane and Solid . . . . . . .1.25 This new text offers a practical combination of moxe desirable qualities than any other Geometry ever published. Wells's Stereoscopic Views of Solid Geometry Figures • .60 Ninety-six cards in manila case. Wells's Elements of Geometry — Revised 1894. — Plane, 75 cts.; Solid, 75 cts.; Plane and Solid . . . . .1.25 TRIGONOMETRY. Wells's New Plane and Spherical Trigonometry (1896) . $1.00 For colleges and technical schools. With Wells's New Six-Place Tables, $1.25. Wells's Plane Trigonometry . . . . • '75 An elementary work for secondary schools. Contains Four-Place Tibles. Wells's Complete Trigonometry . . . . .90 Plane and Spherical. The chapters on plane Trigonometry are identical with those of the book described above. With Tables, $1.08. Wells's New Six-Place Logarithmic Tables . , . .60 The handsomest tables in print. Large Page. Wells's Four-Place Tables . . • • • .25 ARITHMETIC. Wells's Academic Arithmetic ..... $i.oo Correspondence regarding terms for introduction and exchange is cordially invited. D. C. Heath & Co., Publishers, Boston, New York, Chicago HEATSi S MATHEMATICAL MONOGRAPHS ISSUED UNDER THE GENERAL EDITORSHIP OF WEBSTER WELLS, S.B. Professor of Mathematics in the Massachusetts Institute of Technology It is the purpose of this series to make direct contribu- tion to the resources of teachers of mathematics, by pre- senting freshly written and interesting monographs upon the history, theory, subject-matter, and methods of teach- ing both elementary and advanced topics. The first five numbers are as follows : — 1. FAMOUS GEOMETRICAL THEOREMS AND PROBLEMS AND THEIR HISTORY. By William W. Rupert, C.E. i. The Greek Geometers, ii. The Pythagorean Proposition. 2. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert. ii. The Pythagorean Proposition (concluded), iii. Squaring the Circle. 5. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert. iv. Trisection of an Angle, v. The Area of a Triangle in Terms of its Sides. 4. FAMOUS GEOMETRICAL THEOREMS. By William W. Rupert. vi. The Duplication of the Cube. vii. Mathematical Inscription upon the Tombstone of Ludolph van Ceulen. 6. ON TEACHING GEOMETRY. By Florence Milner. Others in preparation. PRICE, 10 CENT5 EACH !;t;f!!!;! D. C. HEATH & CO., Publishers BOSTON NSW TORE CHICAOO Photomount Pamphlet Binder Gaylord Bros., Inc Makers Stockton, Calif. PAT. JAN. 21, 1908 77 ■*^ M7^> n THE UNIVERSITY OF CALIFORNIA LIBRARY U. C. BERKELEY LIBRARIES CDbl3^^a^