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. 
 
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