QA a/? A* UC-NRLF $B 543 472 g 3JEMENTARY COURSE IN GRAPHIC MATHEMATICS BY MATILDA AUERBACH INSTRUCTOR IN MATHEMATICS, ETHICAL CULTURE HIGH SCHOOL NEW YORK CITY A LLYN and BACON Boston Neto |forfe Chicago AN ELEMENTARY COURSE IN GRAPHIC MATHEMATICS BY MATILDA AUERBACH INSTRUCTOR IN MATHEMATICS, ETHICAL CULTURE HIGH SCHOOL NEW YORK CITY ALLYN and BACON Boston jfteto pork Cbtcap AS Copyright, 1910, By MATILDA AUERBACH. Norton on Press : Berwick & Smith Co., Norwood, Mass., U.S.A. PREFACE, The object of this little book is threefold: — first, to show the pupil some practical uses of the graphic method; second, to plan a course in graphic algebra that will lead naturally and along interesting paths to the work in the solution of equations; and finally to save both teacher and pupil time and energy needed to hunt up suitable material. Every type of work outlined in the book has been tested and found suitable for classroom use. The writer has done a considerable amount of work in this line with her classes for the past nine years, and has never failed to find it a spring by means of which she has been enabled to arouse an interest in the mathematics. Though elementary in its form, it is believed the monograph will be found to be thoroughly scientific. It endeavors to introduce in simple form ideas which the pupil will come to deal with in more advanced work and in no case introduces an idea which must sooner or later be unlearned. In the Appendix at the end of the book may be found a number of statistical tables, obtained chiefly from the Bureau of Statistics at Washington, from which teacher and pupil may freely draw without waste of time. The writer has aimed to cover a wide variety of topics and at the same time to select those in which figures were not too large for convenient use. MATILDA AUERBACH, Ethical Culture High School. 259726 Digitized by the Internet Archive in 2008 with funding from Microsoft Corporation http://www.archive.org/details/elementarycourseOOauerrich CONTENTS CHAPTER I Introductory: The Meaning of the Graph . . i CHAPTER II Some Practical Uses of the Graph ... 6 a. IN SURVEYING ....... 6 b. IN KEEPING STATISTICS, RECORDS, AND AS READY RECKONERS 8 C. IN REPRESENTING FORMULAS . . . • r 3 d. IN THE SOLUTION OF PROBLEMS INVOLVING THE ELEMENT OF TIME l8 CHAPTER III Study of the Function and Equation a. THE FUNCTION . . . b. THE EQUATION I. SINGLE LINEAR . II. SIMULTANEOUS LINEAR III. SINGLE QUADRATIC AND THOSE OF HIGHER DEGREE ..... IV. SIMULTANEOUS LINEAR AND QUADRATIC V. SIMULTANEOUS QUADRATIC 22 28 28 29 31 33 34 APPENDIX Appendix to Chapter ii 35 CHAPTER I INTRODUCTORY: THE. MEANING OF A GRAPH We all have had the experience of wishing to place a point somewhere definitely upon a sheet of paper, upon the blackboard, or upon some flat surface. How have we done it ? What have we really done when we have said the point is to be three inches from the lower edge and two inches from the right edge? We have done practically what we do when we say New York City is 74° West longitude and 41° North latitude. We have drawn two lines (either real or imaginary) in the first case, one three inches above the lower edge and the other two inches to the left of the right edge of the paper, and have found the point at their crossing — in the second case we have drawn one line through a point on the equator just 74° to the left of the meridian through Greenwich, and another line parallel to the equator just 41° above it. Their point of intersection has again given us the desired point. In the same manner we could construct any map — one of the city, showing points of interest — one of a piece of ground that has been surveyed, or anything of the sort, just by referring each of the points in question to two intersecting lines. These lines are known as axes, and in all elementary work are drawn at right angles to each other. 1 2 GKA PHIC MA THEM A TICS EXERCISES 1. If West longitude is reckoned to the left of the Greenwich axis, how will East longitude be reckoned? If North latitude is reckoned up from the equator, how will South latitude be reckoned ? 2. Using the Greenwich meridian and equator as axes, locate the following cities: (1) New York (74° W., 41° N.) (2) St. Petersburg (30° E., 60° N.) (3) Buenos Ayres (58° W., 35° S.) (4) San Francisco (122° W., 37° N.) (5) Zanzibar (49° E., 6° S.) (6) London (0°, 51£° N.) 3. Using any two streets that run at right angles to each other as axes, locate at least a dozen points of interest in the city in which you live. In locating points in general with respect to two axes, matters may be greatly simplified by using positive and negative numbers. EXERCISES 1. List the following words and phrases under the two heads " positive" and "negative": — right, wrong; debit, credit; right, left; below, above; above zero, below zero; B. C, A. D.; East, West, North, South; sane, insane; pauper, tax-payer; time to come, time past; increase in population, decrease in population. 2. Which of the above might be considered as lying to the right of a vertical axis? Which to the left? Which above a horizontal axis ? Which below it ? We have seen that to locate a point on a plane surface, reference must be made to two axes, for there are in- numerable points that lie four inches to the right of a THE MEANING OF A GRAPH 3 vertical axis, while there is but one that lies at the same time 5 inches below a horizontal axis. EXERCISES Suppose we take the turning point from the year 1907 to the year 1908 as our zero point on the horizontal axis in this diagram, (Fig. 1), and the temperature 0° Fahren- ~W _ _ .£- & _ __ _ X |Jt ±~ -v- _iTim>Axis Fig.l heit as our zero point on the vertical axis: — 1. Where will all points representing time previous to Jan. 1, 1908, be located? Where all those representing time after that date? Where all those representing temperature below zero? and where all those repre- senting temperature above zero ? 2. Through what point would you draw an imaginary line to represent mid-day, Jan. 5, 1908, if each day of 4 GRAPHIC MA THEM A TICS 24 hours is represented by 12 small divisions on the diagram? Dec. 25, 1907, 6 p.m.? Jan. 10, 1908, 8 a.m.? 3. Through what point would you draw a line to represent the temperature 5° above zero (that is, +5°)? 7° below zero? 12° below zero? 4. Look up the temperature for each day of the past week, and record it by means of a diagram. For more complicated problems of this type see Ap- pendix to Chapter II. As you may already have observed, we can in general locate points in the four quadrants into which our surface is divided by the two axes in the following manner. Suppose the distance of all points to the right or left of the vertical or yy' axis in the diagram (Fig. 2) be denoted ~ Ti i)~ v°~ " , 3 J -J YT Fig.2 by x, and the distance of all those above or below the the horizontal or xx' axis be denoted by y. Then when x is positive the distance is measured so many units to THE MEANING OF A GRAPH 5 the right, and when it is negative, so many units to the left of the yy' axis. When y is positive the distance is measured so many units above the xx' axis, and when negative so many below it. For instance, suppose the the point (x, y) = (7, 12) be given. It will be in the first quadrant, (1, Fig. 2), on an imaginary line 7 units to the right of yy / and parallel to it, and on another such line 12 units above xx / and parallel to it — namely point P. If a point is described as (x y y) = (—7, 12) it will lie in quadrant II, 7 units across to the left, and 12 units up, namely point P x . {x, y) = (—7, —12) will lie in the third quadrant 7 units across to the left, and 12 down, point P 9t and finally the point (x, y) = (7, — 12) lies in quadrant IV, 7 units across to the right, and 12 units down, point P 3 . EXERCISES 1 . Locate the points (9, 11), (7, 6), (—15, 17), (—19, —20), (-2, 6), (8, -15), (7, -13), (-11, -9), (-2, 15). 2. Locate the points (1, 5), (3, 7), (5, 2), (9, —3), (12, —6) and draw a line connecting them. Any line (curved, broken or straight) drawn through a series of fixed points as in the last exercise is called a graph. EXERCISE 1. Draw the graph determined by the points ( — 3, — 2), (-1, 0), (0, 1), (2, i), (5, 7), (8, -11). CHAPTER II 50ML OF THL PRACTICAL U5E.5 OF THL GRAPH Now that we have learned to locate points in this simple manner, we are ready for a few simple practical applications in addition to the above. IN SURVEYING EXERCISES 1. In surveying- a hexagonal field a surveyor notes the following points as its vertices: A = (6, 7), B = (20, 20), C=(40,'20), Z>= (35, 0),E= (10,-20) andi^ = (0,-10). Plot the points, and draw the outline of the field. Find the number of square units in the area of the field in two ways: — (1) By breaking the diagram of the field into figures of which you can find the areas and adding them, (2) By a process of subtraction, using the square whose vertices are denoted by the points (0, 20), (40, 20), (40, —20), (0, —20). 2. It is customary among surveyors to have the polygon lie eventually entirely in the first quadrant. Can you see any reason for this ? 3. Through how many units will you have to move the polygon indicated in Ex. 1, so that it shall just lie wholly in the first quadrant ? 4. Will all the values indicating the vertices be changed? 5. Describe the new positions of A, B y C\ D, £ y and F. 6 PRACTICAL USES OF THE GRAPH 7 6. The vertices of a pentagonal field are located by the following points, A = (—20, 15), B = (10, 20), C = (23, —20), D = (—10, —30), E = (—30, —10). (1) Draw the outline of the field. (2) Give new values to A, £, C, D y E, so that the area shall remain the same but the diagram lie wholly in the first quadrant with E on the North- South axis, and D on the East- West axis. (3) Find the area of the field. 7. From the accompany- ing diagram (Fig. 3), find the approximate area of the pond. Eig.3 8. The accompany- ing diagram (Fig. 4), represents the survey of a field with curved boundary. Find the ap- proximate area of the field. ,'^L ^l9 ^ ^%r _ __ _ y ef * i- v^ it j — \-^ ~t"At X s,st ^JJ5 Tt-t'. ""^p9 si- - Jit t + 7 + t Ij -,22- £Z- - ^hT" 1 s *'" 'El. 1 r Fig.4 GRAPHIC MA THEM A TICS IN KEEPING STATISTICS AND AS READY RECKONERS 9. The following table gives the highest and lowest prices in New York, for Middling Uplands Cotton from Jan. 1 to Dec. 31 of the years named. Show the graph of the highest in red ink and that of the lowest in black ink on the same pair of axes, and correct to the nearest half. YEAR HIGHEST LOWEST YEAR 1864 HIGHEST LOWEST YEAR 1872 HIGHEST LOWEST 1826 14 9 190 72 m 18| 1835 25 15 1865 120 35 1873 m 13| 1840 10 8 1866 52 32 1874 18* 14| 1850 14 11 1867 36 15* 1885 13± mi 1860 11* 10 1868 33 16 1890 12| »A 1861 38 Hi 1869 35 25 1895 9ft h\ 1862 69^ 20 1870 25f 15 1863 93 51 1871 21$ 14| 10. What facts does the graph of the table in Ex. 9 bring out clearly before you? 11. Calling one the time axis, and the other the popu- lation axis, draw graphs indicating the following sets of data: (1) The population of the United States per square mile: YEAR POP. YEAR POP. 1800 6.41 1900 25.22 1850 7.78 1904 27.02 1870 12.74 (2) The population of England, Ireland, Scotland, and Wales correct to the nearest 10,000: (Draw the graphs using a single pair of axes, a different kind of line for each, and correct to the nearest 100,000.) PRACTICAL USES OF THE GRAPH YEAR ENGLAND IRELAND SCOTLAND WALES 1831 13,090,000 7,770,000 2,360,000 810,000 1841 15,000,000 8,200,000 2,620,000 910,000 1851 16,920,000 6,570,000 2,890,000 1,010,000 1861 18,950,000 5,800.000 3,060,000 1,110,000 1871 21,500,000 5,410,000 3,360,000 1,220,000 1881 24,610,000 5,180,000 3,740,000 1,360,000 1891 27,500,000 4,710,000 4,030,000 1,500,000 1901 32,530,000 4,460,000 4,470,000 * * After 1891 merged into England. 12. Answer the following questions from the graphs drawn in Ex. 11, (2): (1) In approximately what year was the popu- lation of England 17 million ? (2) What was the population of England in 1835? in 1845? in 1865? in 1875? (3) In which of the four countries has the popu- lation increased least rapidly ? Most rapidly ? (4) In which has there been a decrease ? (5) In what year was the population of two of them practically the same ? In which countries was this the case ? (6) Roughly speaking, when will the population of England be 38 million? (/. e. considering the increase to continue uniformly.) (7) What will be the population of each of the others at that time ? (8) When will that of Ireland and Wales be the same ? What will it be at that time ? (9) Will this happen apparently in the case of Scotland and Wales? For other problems of this type see Appendix to Chapter II. The graphic method of recording the readings of a thermometer and barometer has been adopted by many newspapers. 10 GRAPHIC MA THEM A TICS EXERCISES 1. Observe the readings of the same thermometer at the same hours daily for a week, and record the results of your observations graphically. 2. Record graphically the readings of the barometer as taken from the same newspaper daily for a week. 3. Record graphically the scores of the captains of the girls' and boys' basket ball teams in your school. (One in red and the other in black ink, or one by means of a solid and the other by means of a dotted line.) 4. The Harvard Eights from 1852 through 1905 had rowed 39 races. The records are as follows: TIME TIME DATE WON BY WINNER LOSER DATE WON BY WINNER LOSER 1852 Harvard 1884 Yale 20 31 20.46 1855 a 1885 Harvard 25.15 26.30 1857 ll 19.18 20.18 1886 Yale 20.41 21.05 1859 Yale 19.14 19.16 1887 i< 22.56 23.11 1860 Harvard 18.53 19.05 1888 u 20.10 21.24 1864 a 19.01 19.43 1889 II 21.30 21.55 1865 Yale 17.42 18.09 1890 « 21.29 21.40 1866 Harvard 18.43 19.10 1891 Harvard 21.23 21.57 1867 a 18.13 19.25 1892 Yale 20.48 21.42 1868 a 17.48 18.30 1893 u 25.01 25.15 1869 a 18.02 18.11 1894 a 22.47 24.40 1870 a Foul Disq. 1895 ci 21.30 22.05 1876 Yale 22.02 22.33 1899 Harvard 20.52 21.13 1877 Harvard 24.36 24.44 1900 Yale 21.13 21.37 1878 t« 20.45 21.29 1901 it 23.37 23.45 1879 u 22.15 23.58 1902 u 20.20 20.33 1880 Yale 24.27 25.09 1903 (I 20.20 20.30 1881 u 22.13 22.19 1904 II 21.40 22.10 1882 Harvard 20.47 20.50 1905 II 22.33 22.36 1883 a 24.26 25.59 Show this graphically. PRACTICAL USES OF THE GRAPH 11 As seen above in plotting population curves, valuable surmises might be made in regard to probable increase or decrease in populations during specified periods, or rough estimates could be made as to the probable popu- lations at any stated time, and so forth. Likewise, there is another use of the graph in the way of a "ready- reckoner" where price lists do not include, for instance, all sizes of articles or numbers of articles of the same kind for sale. This will be made clear by the following set of problems: 1. The single ticket by railway costs $2.50. If 10 such tickets be purchased the average cost will be reduced to $2.25. If 50 be purchased the cost per ticket will be only $1.80; if 100, the cost per ticket will be $1.50; and if 200, the cost per ticket will be $1.25. Draw a graph showing this, and answer the following questions by the aid of it: (i) What will be the probable cost per ticket if an excursion of 75 be formed? If one of 125 be formed? One of 175? (2) About how many tickets must be used to reduce the expense per head to just $2.00 ? to $1.60 ? 2. If a certain kind of desk be sold to the individual it will cost $30.00. If ordered by the dozen it will cost $28.50, if 6 dozen are ordered it will cost $22.50, and if 150 are ordered the cost will fall as low as $20.00. Draw a graph showing this, and answer the following questions: (1) What will be the probable cost per desk when 36 are ordered? When 100 are ordered? (2) How many must be ordered so that each shall cost about $25.00? 3. Ordering ink by the gill it costs $ .10. By the pint it costs $ .30, by the quart $ .50, and by the gallon 12 GRAPHIC MA THEM A TICS $1.75. According to this, what should it cost approxi- mately when ordered by the half-gallon? By the half- pint? By the quart and a pint? 4. The average annual premiums (P) for whole life insurance of $500 for the age (A) at entry is given as follows: A = 21 25 30 35 40 45 50 P = - $8.00 $8.66 $10.00 $11.66 $14.00 $16 75 $20.10 What are the probable premiums for ages 23, 27, 33, 37,42, 48? 5. It is found by testing, that the barometer stands at 30 inches at sea level, at 23.5 inches at a height of 6,000 feet, at 18.2 inches at a height of 12,000 feet, at 12.2 inches at 24,000 feet, and at 7.3 inches at 36,000 feet above sea level. Plot the graph indicating these facts, and from it answer the following questions: — (1) How high (approximately) is a place in which the barometer stands at 25 inches? At 20 inches ? (2) How high should the barometer rise in a spot which is 20,000 feet above sea level ? At one which is 30,000 feet above sea level ? 6. In a price list the following table appears: Measuring-tins of capacity P (pints) = 1 2 3 4 6 8 12 Cost in cents C = 10 16 21 24 30 35 42 What will tins of a capacity of 5 pints, 7 pints, 9, 10, 11 pints respectively, probably cost ? PRACTICAL USES OF THE GRAPH 13 7. The cost of fitted lunch baskets is given in the following table: Arranged for number of persons N = 1 2 4 6 Cost in dollars D = 10 18 30 40 What will be the probable cost of baskets for 3, 5, 7, 8, and 10 persons respectively ? IN REPRESENTING FORMULAS In the last set of applications of the graph we have seen that by joining successive given points by straight lines, we may surmise approximate results for inter- mediate points. However, • there has been no law governing the statements thus made, and the results obtained may or may not satisfy existing conditions. In short, it was only a surmise on our part when we drew conclusions. There is, however, another type of problem which may be represented or approximately solved graphically — namely those which rest upon a formula. For instance, we are told that the circumference of a circular is always equal to it times its diameter, or approximately 3^ times its diameter. That is, if C stands for the number of units in a circumference, and D for the number of units in its diameter, C = 7T D. EXERCISES 1. Given C = tyD, where C = number units in the circumference of a circle and D = number units in its diameter: — 14 GRAPHIC MA THE MA TICS (1) Find the values of C for those given in the following table for D. z> = i 14 3* 21 28 c = (2) Call one axis {DD f ), the diameter axis, and the other {CO), the axis of circumferences, and ) plot the points corresponding to the values found in Ex. (1). (3) Connect these points and state on what kind of line they lie. (4) How many of these points would have been needed to enable you to draw that line ? (5) From the line you have drawn find answers to the following questions: {a) When the diameter of a circle is 10 units how many units are contained in its circum- ference ? (b) When D = 10J ft., C =? (c) When C = 100, D =? (d) When C = 75, D =? (e) If the circumference of a wheel is 92 inches, what is the length of its diameter? 2. We are told that an inch contains 2.54 centimeters. Answer the following: (1) The number of centimeters in a given length is then always how many times the number of inches in that length ? (2) Write a formula stating this fact. (3) As in Ex. 1 (1), select any six lengths in terms of inches and make a table showing the PRACTICAL USES OF THE GRAPH 15 number of centimeters in the corresponding lengths. (4) Call one axis {IP), and the other (CO), and plot the points corresponding to the values found in (3). (5) On what kind of line do these points lie? Draw it. (6) How many of these points would have been needed to enable you to draw that line ? (7) From the graph just plotted, answer the following questions: (a) About how many inches in 30 cm. ? (b) About how many centimeters in 20 in. ? (c) About how many inches in 40 cm.? (d) About how many inches in a meter ? 3. The formula for the reduction of Fahrenheit scale to Centigrade scale is C = | (F — 32) where C = the num- ber of degrees Centigrade corresponding to F = any given number of degrees Fahrenheit. (1) Give six values to F, and as. in Ex. 1 (1), show in a table the corresponding values of C. (2) Call the axes of Fahrenheit and Centigrade FF f and CO respectively, and plot the points shown in this table. (3) Connect these points and tell on what kind of line they lie. (4) How many of these points would have been needed to enable you to draw that line? (5) From the lines you have drawn find the approximate number of degrees on a Fahrenheit thermometer when a Centigrade thermometer registers (a), 10°, (b), 100°, (c), 50°, (d) } 120°, ( + 5 x -7 = —3 —2 —1 1 2 3 4 5 6 7 f(x) (2) Plot the points found above, and draw as steady a line as you can through them. (3) For what values of x does the function equal zero? 2? 3? 5? 10? —6? (4) For what values of x is the function negative ? (5) For what values of x is the function positive ? (6) When x = — 2.5, + 2.5 what are the approxi- mate values of f(x) ? (7) How many times does the graph cut the ;r-axis ? (8) How many factors has the expression x* -f 5 x —7 ? (9)* What are they approximately? (10) Could you find the factors exactly? (11) If you were to plot the graph of f(x) eeb-t 2 + 5 x + 6 where would you expect it to cut the .r-axis ? 2. By means of the method employed in the last exercise, plot the graph of: — (1) f{x) =3j 5 + 8^—4. (2) f{x) =4.r 2 — 8.r— 7. THE FUNCTION AND THE EQUA TION 25 (3) f(x) = x* + 3 x + 1. (4) f(x) = 3 x 3 + 4 „r 3 — 8 * — 7. 3. Draw the graph of the parabola f(x) = x* using values of .r between + and — 5 inclusive. 4. Draw the graph of the circle f(x) = ± V 36 — x\ (Use integral values of x between ± 6 inclusive.) 5. Draw the graph of the ellipse f{x) = ± ■£■ 1/ 3 (4 — .r 2 ). 6. Draw the graph of the hyperbola f (x) s± V 2x* + 7. x* 7. Draw the graph of f(x) = x + 2. Those of us who know the trigonometric ratios can now plot the graphs of functions containing them. One example will be sufficient to make this clear. Given x = \ir or 30° \tt or 45° \tt or 60° fix) = Sin see •5 X_2_=707 ^ =.866 i7r or 90° \ir or 120° |tt or 135° 77 or 180° fir or 210° 1 4 3 .866 ^or .707 —.5 |7T ^7r 3.77- 2" fir ITT 2 TT —.707 —.866 1 —.866 —.707 — .5 —.707 etc. — 4 TT -in etc. I -r s ^eE„. -ZZ 5* Oil ^v ^ 5r \ R ~^ "R~ -p - - X r ig,«_ 2 r ± -R o w N ^ I 1 i r f= 1 " -H- 26 THE FUNCTION AND THE EQUA TION 27 It is readily seen that / (x) = sin x has as its limiting values + 1 and — 1. Therefore we shall use the shorter axis as the /(;r)-axis, and the longer one as the x- axis. On the .r-axis the unit n is divided into sixths, fourths, thirds, and halves, therefore we shall use 12 divisions to the unit on that axis (or a multiple of 12). In order to be able to measure tenths on the /(;r)-axis we shall use 10 divisions to the unit on that axis. Finally, so that the graph may be more easily drawn, we shall use the scale 24 to it on the .r-axis. Plotting the points found in the table we obtain the graph shown in Fig. 6. Note. — Sin x is an example of what is called a periodic function — i. e., a function which repeats the same values in the same order after a certain period. From the figure it is readily seen that sin (x + 360°) will be the same as sin x. Therefore the period of sin x is 360° or 27T. EXERCISES 1. If sin x = .7 what will be the sine of (a) (720° + x)? ,(£) (—360° +x)l 2. Plot the graph of cos x =.f(x). 3. Plot the graph of f (x) = tan x. 4. Plot the graph of/(x) = cot x. 5. Plot the graph of f(x) s sec x. 6. Plot the graph of f(x) = cosec x. 7. Plot the graph of f(x) = sin x + 2. 8. Plot the graph of f (x) = sin x + cos x. 9. Plot the graph of f(x) = sin x — cos x. 10. Plot the graph of f(x) = 3 — cos x. ii. Are the above graphs those of periodic functions? If so, determine the period of each. 28 GRAPHIC MA THEM A TICS THE EQUATION From what has been said in the beginning of this chapter it is easily seen that if y = 3 x + 4, x would be the independent, and y the dependent variable and therefore a function of x. If then we call our axes xx' and yy' in place of ^r-axis and f(x) -axis, we may plot the graph of y = 3 x + 4 just as above we plotted that of f(x) = 3 x + 4. SINGLE LINEAR EQUATIONS EXERCISES 1. Draw the graph of y = 5 x — |, and from it find: (1) The value of x when y = 0, 8, 10. (2) The value of y when x = 2, 1, — ^. 2. At what points will the line y == 4 x + 6 cut the axes ? What is the easiest way to find these points ? What then is a simple way to plot an equation of the first degree ? (Such equations are called linear.) Why? 3. Plot, by joining the points where the line cuts the axes: (1) y = x f ;$. (5) x = —y + 4. (2) y = x — 5. (6) 5 x + 2y = 7. (3) y = — 3 x — 2. (7) 9 ;r + 7 J — 8 = 0. (4) y = — 3 x + 2. 4. Can you plot .r = — jK by the method suggested in ex. 3 ? Give reason for your answer. 5. Plot (1) x = — y (2) x = 5 (3) y = — 8 (4) ^'=37 (5)^ = | (6).*-=^ 6. Give the equations stating that: (1) A point is always 10 units from a given line xx'. (2) A point is always 10 units from a line yy'. (3) A point is always at the same distance from each of two lines which intersect at right angles. THE FUNCTION AND THE EQUATION 29 SIMULTANEOUS LINEAR EQUATIONS 7. On a single pair of axes draw the graphs of the following equations: (1) 3 x + 4 7 = 18 (3) \x — 9 = — 2y (2) 5 y — 2 x = 11 (4) x + \y'm 12 8. From the graphs in Ex. 7 what can you say about equations (1) and (2) ? (1) and (3) ? (1) and (4) ? 9. Two straight lines in the same plane in general intersect how often? May they do otherwise? Explain your answer. 10. What can you say of the equations of two straight lines whose graphs intersect once? What kind of equations must they be to give such result ? The line or group of lines that fulfills a given condition is termed the locus of that condition. For instance, the locus of the condition expressed in the equation x = 3 is the line drawn parallel to the yy' axis at a distance 3 units to the right of it. Two loci are said to be coincident when every point in one lies on a corresponding point in the other, or in short, when they have all points in common. Two loci are said to be parallel when they have no point in common, and they are said to intersect when they have a finite number of points in common. 11. What can you say of the conditions expressed by (1) and (2), Ex. 7 above? by (1) and (3) ? by (1) and (4)? Two equations in the same variables are said to be consistent when they do not contradict each other, and inconsistent when they do. 12. Select pairs of consistent equations from Ex. 7. 13. Select pairs of inconsistent equations from Ex. 7, 30 GRAPHIC MATHEMATICS 14. From Ex. 7 can you tell whether all consistent equations can be solved simultaneously? Give a reason for your answer. 15. Do you suppose that inconsistent equations can be solved simultaneously ? 16. How was equation (3), Ex. 7, derived from equation (1)? Are they consistent then? Would you say they were independent of each other ? 17. How would you then define two consistent inde- pendent equations ? Select two such equations* from Ex. 7. 18. Arrange answers to the following questions just as the questions are arranged and underline the cor- responding words and phrases in the two columns. The Linear Equation 1. A linear equation in two variables is satisfied by how many pairs of roots? 2. The graph of a linear equation may be fixed by how many pairs of its roots? 3. In general two linear equations involving the same two variables have how many pairs of roots in common? 4. May two linear equa- tions in the same two variables have more than one pair of roots in com- mon ? What kind of equa- tions are they then ? The Straight Line 1. A straight line con- tains how many points? 2. The straight line is fixed by how many of its points? 3. In general two co- planar straight lines have how many points in com- mon ? 4. May two coplanar straight lines have more than one point in common ? What kinds of lines are they? THE FUNCTION AND THE EQUA TION 31 5. May two linear equa- tions in the same two variables have no pair of roots in common? What kind are such equations ? 5. May two coplanar straight lines have no points in common ? What kind of lines are they ? 19. Solve the following equations graphically, using a new pair of axes for the solution of each pair: (1) \x—y = i Z> ( A\ j ^ — 25 or = 13 K 1 ) \ x +y=% {*) \ y + 62 = 50 x m j* + »«'-2 ,«, {5x + 2y = 8 V) \ y = %x I s ' \2x — 3y = —12 (3) j x + 2y = 7i 2 x + y = 7i SINGLE QUADRATIC EQUATION AND THOSE OF HIGHER DEGREE Suppose we were now asked to solve the equation x* + 5 x + 6 = 0. Factoring, we see at a glance that (x + 3) (x + 2) = 0, and therefore that x = — 3 or — 2. Let us now see how we might have found these values by the graphic method. From what we have learned of functions of a variable and of the single linear equation we can readily plot the graph of y = x* + 5 x + 6. Here we are not interested, however, in all the values of x, but just those which will make y = 0. Therefore, having drawn the graph of f(x) = x* + 5x-\-6 ovy~x 2 + 5x + 6, we run our eye along it until we find the points at which y = 0, or in short, at what points the graph cuts the 32 GRAPHIC MA THEM A TICS ,t'-axis. At these points we find the values of x to be — 2 and — 3 if the graph is accurately drawn. 2 A (-'3,0) 2,5-25)^ mffTtT (-2,0). /- ■my Fig.7 In a similar manner all quadratic equations — also those of higher degree — may be solved. EXERCISES 1. Solve graphically the equations: (1) ** + 11 x + 18 = 0. (2) x* — 7 x + 12 = 0. (3) 2 x* + x + 1 = 0. (4) 4 x* + 4 x + 1 = 0. (5) x 2 + x = 6. (6) ;r 2 + 3 = G x. (7) 9 .r 2 — 5 x — 2 = 0. (8) .9 ,r 2 — 4.68 x = — 4.36. (9) 3 x % + 10 .r 2 + 4.25 x — 5 «- 0. (10) ;r 3 — 4.1 .r 2 — 1.05 # + 11.025 = 0. 2. How many times does the locus of a quadratic equation in x cut the .r-axis? THE FUNCTION AND THE EQUA TION 33 3. Show graphically the character of the roots of the equations: (1) j 2 -3j-4 = 0. x~ (2) '— — ;r -f 2 = 0. (Plot using values between + 3 and — 3.) (3) x* + 4 x + 4 = 0. 4. How does the graph of a quadratic equation indi- cate the fact that the roots of the equation are: (1) Real and unequal ? (2) Real and equal? (3) Imaginary? SIMULTANEOUS LINEAR AND QUADRATIC EQUATIONS Without any further preparation we may now solve the following sets of simultaneous equations. EXERCISES 1. In what points does the straight line 3 x + y = 25 cut the circle x* + y 2 = 65 ? 2. The equation of a circle is x* + y 2 = 49, and the equation of a chord of the circle is 13^+2j = 49. Find the extremities of the chord. 3. Solve graphically the following pairs of equationS{ (3) Ux-iy (j,-6) 2 = 25 4 x + % y + 3 = Find the points common to the following parabolas and straight lines: 4. y* = 9 x, 3 x + 30 = 7 y. 5. y = 3 .f, or — 4 j + 12 = 0. 6. y = 4 4r, * = 6, j = — 8, x = 0, .r = — 4. 34 GRAPHIC MA THEM A TICS 7. y 2 = Sx r x + y = 6. 8. y 2 — 4 # — 8 j/ + 24 = 0, 3 y — 2 x == 8. Find the points of intersection of the following ellipses and straight lines: 9. 2 x 2 + 3 j/ 2 = 14, j — 2 * = 0. 10. 2 ;r 2 + 3 y 2 = 35, 4 4f + 9 j = 35, 4 * — 9 y = 35. 11. 9 ^ 2 + 64 J 2 = 576, 2 J/ = ;r + 10, 2 J = .r + 1. Find the points common to the following hyperbolas and straight lines: 12. x 2 —f = 9, 4 # + 5 y = 40. 13. 16 A' 2 — 9j* = 112, 9.r + 16j = 100, 16* — 9y = 28. SIMULTANEOUS QUADRATIC EQUATIONS Find approximately the points of intersection of the following loci: 14. 2 x 2 + 3 y 2 = 14, f = 4: x. 15. .r 2 + y 2 = 10, ** -f 7 j 2 = 16. 16. x 2 + y 2 = 25, .rj/ = 5. MISCELLANEOUS EXERCISES 1. Find the two square roots of 6. (Hint: Plot the graph of f (x) = x\) 2. Find the three cube roots of 8. f(x) = x % . 3. Find the six sixth roots of 1. 4. Which of the above roots cannot be shown graphi- cally ? 5. Write the equations of two parallel lines and construct them. 6. Write the general equations of two parallel lines. 7. The equation of the circle ax* + ay 2 = C differs in what respect from the equation of the ellipse ax 2 + by 2 = C? What is the shape of the ellipse when a and b differ THE FUNCTION AND THE EQUTAION 35 greatly in value? When a and b are nearly equal? When a and b are equal ? 8. Draw a graph by means of which American money may be changed to: — (1) English money. (3) French money. (2) German money. 7 9. Solve graphically j ?£ff * {[ 10. Two bodies 140 feet apart move towards each other, the first at the rate of 10 feet per second, the second four-fifths as fast. How long before they are 44 feet apart ? APPENDIX TO CHAPTER II Draw graphs to represent the statistics given in the following tables: 1. The monthly mean maximum temperature Fahren- heit in the cities noted for the years 1872 to 1901: d J3 ft Q. < a 43 •— > ►—1 < 26 26 32 46 5Q 6q 75 73 66 53 39 35 36 42 54 66 76 81 78 71 60 49 3i 3i 37 50 62 72 77 76 70 58 45 3i 33 4i 54 64 74 80 78 72 60 45 40 43 5i 63 74 83 87 84 78 66 52 33 35 41 54 66 76 80 78 72 61 47 74 7 6 77 80 84 87 89 89 87 83 78 24 28 39 57 6o 78 83 80 71 59 41 57 6i 67 76 84 90 02 90 86 76 66 37 3« 44 57 68 78 82 80 74 63 51 48 5i 56 65 75 84 88 85 79 69 59 3i 3i 37 50 63 73 78 76 70 57 45 20 24 36 56 68 77 83 80 7i 57 38 Alpena, Mich Boston, Mass. . . . Buffalo, N. Y.... Chicago, 111 Cincinnati, Ohio.. Cleveland, Ohio.. Key West, Fla. . . La Crosse, Wis.. . Montgomery, Ala New York, N.Y.. Norfolk, Va Oswego, N. Y St. Paul, Minn... 30 36 36 43 38 74 30 58- 41 ^6 27 2. The monthly mean minimum temperature Fahren- heit in the cities noted for the years 1872 to 1901: G at 0) Pt4 rt s < >> a C 3 >> "3 bo < in 12 10 16 31 41 52 57 55 49 39 28 19 20 27 37 48 ^ 63 62 55 45 34 18 17 24 35 46 58 63 61 55 44 33 16 19 27 39 4Q 59 65 65 58 46 32 25 27 34 45 5? 65 69 67 60 48 37 20 20 27 38 So 59 64 62 56 45 34 65 67 68 7i 7S 78 79 79 78 75 7i 7 10 22 38 So 60 64 61 53 4i 26 39 43 48 55 63 70 73 72 67 56 46 24 24 30 40 52 61 67 66 60 48 38 33 35 39 47 58 66 7i 70 65 54 44 17 17 24 36 46 56 62 61 54 43 33 2 7 18 36 48 58 -62 60 5i 39 22 Alpena, Mich Boston, Mass Buffalo, N.Y Chicago, 111 Cincinnati, Ohio.. Cleveland, Ohio.. Key West, Fla... La Crosse, Wis.. . Montgomery, Ala New York, N. Y.. Norfolk, Va Oswego, N. Y. . . . St. Paul, Minn.... 19 24 24 23 30 25 66 15 40 28 36 22 11 APPENDIX TO CHAPTER II 37 The preceding material as well as what follows should be made use of in various ways as may be suggested by both pupils and teacher. For instance, on a single sheet of cross-section paper make a diagram showing the mean maximum and the mean minimum temperatures of Baltimore, Md., using a dotted line to show the mean maximum, and a solid line to show the mean minimum. Using red ink draw a line showing the probable mean temperature. 3. Average amounts of precipitation for the year 1904: San Fran- cisco, Cal. Atlanta, Ga. 4.75 5-2 3-3 1 4 02 3-23 i.8b .41 5 3 3 94 69 26 .19 .02 4 4 03 86 .01 4 52 • 44 1.32 3 2 55 26 2.70 3 44 4.21 4 35 Lincoln, Santa Fe, Neb. New Mex. .67 .58 .87 •74 I. 21 •71 2.67 •75 4-59 1. 15 4.36 I.04 4-13 2.7 3-39 2-43 2.14 I.64 2.07 1.05 •77 .68 .76 .72 Salt Lake City, Utah Yellow- stone Park, Wyo. Jan Feb March. . April... May June July.... Aug Sept Oct Nov Dec 1-33 1 4 1.99 2.13 1.97 •73 •52 •74 .80 1.5 1-4 1.43 4 92 3 23 94 65 23 07 99 09 59 86 4. The population of New York City to the nearest 1,000 for the years indicated: TEAR POPU- LATION YEAR POPU- LATION YEAR POPU- LATION 1790 1800 I8IO 1820 33,000 60,000 96,000 124,000 I83O 1840 i8so i860 203,000 313,000 5I6 000 8o6,000 I87O 1880 I89O 1900. .... 942,000 1,206,000 1,515,000 3,437,000* * All Boroughs. Plot the above correct to 10,000 only. 38 APPENDIX TO CHAPTER II 5. Immigration into the United States, correct to the nearest 1,000: YEAR IMMI- GRANTS YEAR IMMI- GRANTS YEAR IMMI- GRANTS 1820 8,000 i860 133,000 I89O 455,000 1825 10,000 1862 72,000 I892 623,000 1830 23,000 1865 l8o,000 1898 229,000 1835 45,000 1870 387,000 I9OO 449,000 I84O 84,000 ,1875 227,000 1902 649,000 1845 114,000 ! i88o 457,000 I903 857,000 I85O 370,000 1882 789,000 1904 813,000 1855 201,000 1885 395,000 6. Income and Expenditures of the United States Government, 1876-1905. (Record to the nearest $1,000,000) : YEAR REVENUE EXPENDITURES 1876 $287,482,039 333,526,611 323,690,706 403,080,983 313,390,075 567,240,852 543,423,859 $258,459,797 267,642,958 260,226,935 3l8,040,7H 356,195,298 487,7I^,7Q2 1880 1885 I89O 1895 I9OO 1905 567,411,611 7. Public Schools in the United States: 1871. 1876, 1880, 1885. 1890, ;2s W 12 3 13 7 is 1 16 7 18 5 5.62 6.06 5-17 6-6i 7.60 1895, 1899 1900, 1905, 1906 e ^£ 20.4 21 .9 21.4 23.4 23.8 •>*3 5 0.2 JH N vO O vO _ f* VO CO t^ - O N N co •-• ~ N r« CO t^ - o o VO N 00 VO h> "d- *»■ ^ rf VO vO m 00 O^ in t>» ^r N 00 CO CO «»■ o O ri- O t^ (N <* ^t in O ON ~ _ CO fH HH *"< N N VO vO O NO N vo VO 00 O^ r^ in co _ r>* tJ- N vo 00 w in m m CO 00 in CO O N 00 ">* o^ N oo N M _ N M M •" M N VC N o N CO O t^ o O* O^ in VO in O ^f oc vo CO O CO VO N N in -i- in o CO O N M HI N h« M '- ri CO VC CO 00 ON 00 O r> cn VO _ 00 "**■ N t^. OO rt ITj VO CO tv. 00 00 o CO OO t^ M _ o »n CO VC VO '^ > N — ~ ** N in M m oo 00 O N $ vo o co m co O M iv. O cc CO t^ oo CO l^. Tj- O N in VO M o in N N ■-« '"* M CO m 5 Tj- 1^ W co <* in 00 00 vo co VO N m *t O CO 00 O tv rf vo tv CO CO 00 N VO CO in oo N OO t^. N CO — 1-1 N ** ** 15 VO VO C* CO >«* a N rj- lo ^ r^ o o •*< o o O in t^ 00 vo co C* N -t- in VO oc VO VO w T^- m •""' N CO N in vo co t^ VO o 00 $ OO H_ c* ft m O CO a* co VC o ^ VO t^ 00 00 t^ >"■ « 2 in 00 CO t^ f CO 00 N W o tv. in O in in O- ►* CO 00 oc 00 <•• in tv 00 00 O u~> N in n- N O vo vo M LT) — i *"" "* N co N in >-« m CO O S O M ^ N t^ fv OO VO CO rj- r & O in vO oo 00 VO N o -* «♦ rv. VC t>. M i-i in t-i l - 1 M N CO CO c/5 :t: : kM :£ . • 3 • O— • to "w" 3 • o • .S '. £ 'a V u • a . Q a o o U a . >* §^ g : 5 : T3 5 • ■ '5- : o . J-l * 0- ; g 2 ^ a M e.22 fl : a °"3,o"3 ^§Ort fa •— > CD — > 3J co T3 o < Q § h > O^ u itt rt co O •- U i2^ — ' H 5 o V V y ^ O c^ S-S -diunsuoo joj paui }U3UlJJBd3( 3Dt£CHS0 d 1 [ 40 APPENDIX TO CHAPTER II 9. Density of population per square mile, of States and Territories, 1790-1900: e \4 03 C rtf G M a a > O O ^ >• ,C en a > a o z z CL, 7-1 8.1 97 12.4 9.8 134 20.1 11.4 18.0 28.8 13.2 23-3 40.3 15.2 30.0 51.0 15.5 3»-3 65.0 17.9 51.4 81.5 20.4 64.6 92.0 22.1 78.3 106.7 28.8 QS.2 1 26. 1 33-3 1 16.9 152.6 39-o 1 40. 1 1790 1800, 1810. 1820, 1830, 1840, 185a 1 86a 187a 1880, 1890, 1900 49-1 51.8 54.1 56.8 61.4 64.0 76.5 95.0 1 10.9 128.5 154.0 187.5 30.2 32-8 37.i 37-1 39-2 39.8 46.7 57-3 63.8 74.8 86.0 943 1.4 2.8 4.3 5.8 8.8 11.7 154 17.9 20.1 26.1 31.2 37-6 1.8 5-5 10.2 14.1 17.2 19.5 24.6 28.9 33-o 41.2 46.5 537 3-2 5-1 77 1 0.0 134 16.8 19.5 21.0 21.0 21.7 22.1 23.2 47.1 52.6 58.7 65.1 75.9 91.8 123.7 I53.I 181.3 221.8 278.5 348.9 15.8 20.4 23.8 27.1 29.9 31.6 35-3 36.2 35-3 38.5 41.8 45-7 634 637 70.9 76.6 89.6 100.3 136.0 160.9 200.3 254.9 318.4 407.0 10. Native and Foreign born population of various cities, correct to the nearest 100 : CITY \ 1870 1880 1890 1900 Washington, D. C: Native born 95,400 I33.IOO 211,600 258,600 Foreign born 13,800 1 4,200 18,800 20,100 Buffalo, N. Y.: Native born 71,500 103,900 166,200 248,100 Foreign born 46,200 51,300 89,500 104,300 San Francisco, Cal.: Native born 75,800 129,800 172,200 225,900 Foreign born 73,800 104,200 126,800 116,900 Portland, Oreg.: Native born 5,700 11,300 29,100 64,600 Foreign born 2,600 6,300 17,300 25,900 Atlanta, Ga.: Native born 20,700 36,000 63,700 87,300 Foreign born 1,100 1,400 1,900 2,500 Savannah, Ga : Native born 24,600 27,700 39,800 50,800 Foreign born 3700 3.000 3,4oo 3400 Hoboken, N. J.: Native born 10,000 18,000 26,300 38,000 Foreign born 10,300 13,000 17,400 21,400 APPENDIX TO CHAPTER II 41 11. The population of a few States, by color at each census: MAINE South Carolina Georgia White Colored White Colored White Colored 1790 96,002 538 140,178 108,895 52,886 29,662 1800 150,901 818 196,255 149,336 102,261 60,425 1810 227,736 969 214,196 200,919 I454I4 107,019 1820 297,406 929 237440 265,301 189,570 15^419 1830 398,263 1,192 257,863 323,322 296,806 220,017 1840 500,438 i>355 259,084 335,3H 407,695 283,697 1850 581,813 i,356 274,563 393,944 521,572 384,613 i860 626,952 1,327 291,388 412,320 591,588 465,698 1870 625,309 1,606 289,792 415,814 638,967 545,U2 1880 647,485 1,451 391,245 604,332 817,047 725J33 1890 659,896 1,190 462,215 688,934 978,538 858,815 1900 693»H7 1,319 557,995 782,321 1,181,518 1,034,813 12. The areas of Indian Reservations for the years indicated given in square miles: YEAR ARIZONA IOWA NEBRASKA N. CAROLINA 1880 I89O. ... I9OO 1907 4,832.5 IO,3l7.5 23,673 26,532.7 I 2 4-5 4.63 682 214 Il6 23.08 102 102 153.5 98.77 13. Departures of passengers from seaports of the United States for foreign countries 1868 to 1907, correct to the nearest 100: YEAR TOTAL YEAR TOTAL YEAR TOTAL 1868..... 32,SOO 1879 51400 I898 94,600 I87O 33,600 I885 87,800 I9OO 155,900 1872 39,900 I89O 105,900 1905 201,200 1873 52,lOO 189I I07,IOO 1907 224,900 1876 46,400 1893 95,100 I878 55,200 1894 1 2 1 ,900 42 APPENDIX TO CHAPTER'II 14. Records of Cereal Crops, 1866 to 1907 Wheat— Average Per acre ^ RS O Oats— Average Per acre Barley— Average Per acre Bushels [i.6 [2.1 13.6 12.4 [1.6 [1.9 12.7 12.3 0.5 3-9 f{ 3-1 0.2 1.6 3.0 04 2.4 2.1 1.1 2.9 1.1 5.3 34 14 3-2 37 2.4 34 5.3 2.3 2.3 5.0 4-5 2.9 2.5 45 5-5 4.0 Dollars 16.83 13.17 10.38 11.73 13-24 1335 13.56 10.65 9.91 10.09 14.65 10.15 15.27 12.48 12.12 12.02 IO.52 8.38 8.05 8.54 8.25 IO.32 8.98 9.28 12.86 8-35 6.16 6.48 6.99 8.97 10.86 8.92 7.17 7.61 9-37 9.14 8.96 11.58 1083 10.37 12.26 Bushels 30.2 25.9 26.4 30.5 28.1 30.6 30.2 277 22.1 297 24.O 31.7 31-4 28.7 25.8 247 26.4 28.1 274 27.6 26.4 254 26.0 27.4 19.8 28.9 244 23-4 24.5 29.6 257 27.2 28.4 30.2 29.6 25.8 28.4 32.1 34.0 31.2 237 Dollars I0.6I 11-53 II.OO II.58 IO.97 II.07 9-03 9-59 10.38 9.52 777 9.01 772 9.50 9.28 11.48 9-89 9.20 7.58 7.88 7.87 774 7.24 6.26 8.40 9.08 773 6.88 7.95 5.87 4.81 575 7.23 7.52 7.63 10.29 10.60 9.68 10.05 9.88 9.89 10.51 Bushels 22.9 22 7 24.4 27.9 237 24.O I9.2 23.I 20.6 20.6 21.9 21.3 23.6 24.0 24.5 20.9 21.5 21. 1 23-5 21.4 22.4 I9.6 21.3 24.3 21.4 25.9 23.6 21.7 194 26.4 23.6 24.5 21.6 25.5 20.4 25.6 29.0 26.4 27.2 26.8 28.3 23.8 APPENDIX TO CHAPTER II 43 15. Value of gold and silver produced in the United States. (Plot correct to the half-million dollars, showing on separate sheets the gold and silver production, and on one sheet the amount of gold produced in California, other States and Territories, and the total amount pro- duced.) i860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 California Dollars 45,000,000 40,000,000 34,700,000 30,000,000 26,600,000 28,500,000 25,500,000 25,000,000 22,000,000 22,500,000 25,000,000 20,000,000 19,000,000 17,000,000 17,500,000 17,617,000 17,000,000 15,000,000 15,300,000 16,000,000 17,500,000 l8,2OO,0OO 16,800,000 14,120,000 13,600,000 12,700,000 14,725,000 13,400,000 12,750,000 13,000,000 Other States and Territories Dollars 1,000,000 3,000,000 4,500,000 1 0,000,000 19,500,000 24,725,000 28,000,000 26,725,000 26,000,000 27,000,000 25,000,000 23,500,000 1 7,000,000 19,000,000 15,990,900 15,850,900 22,929,200 31,897,400 35,906,400 22,900,000 18,500.000 16,500,000 15,700,000 15,880,000 17,200,000 19,101,000 20,144,000 19,736,000 20,417,500 19,967,000 Total Dollars 46,000,000 43,000,000 39,200,000 40,000,000 46,100,000 53,225,000 53,500,000 51,725,000 48,000,000 49,500,000 50,000,000 43,500,000 36,000,000 36,000,000 33,490,900 33,467,900 39,Q29,200 46,897,400 51,206,400 38,900,000 36,000,000 34,700,000 32,500,000 30,000,000 30,800,000 31,801,000 34,869,000 33,136,000 33,167,500 32,967,000 Dollars 156,800 2,o62,000 4,684,800 8,842,300 11,443,000 11,642,200 10,356,400 13,866,200 12,306,900 12,297,600 16,434,000 23,588,300 29,396,400 35,881,600 36,917,500 30,485,900 34,919,800 36,991,500 40,401,000 35,477.100 34,717,000 37,657,500 41,105,900 39,618,400 41,921,300 42,503,500 39,482,400 40,887,200 43,045,100 46,838,400 44 APPENDIX TO CHAPTER II 15. Value of Gold and Silver Produced in the United States — Continued. GOLD YEAR California Other States and Territories Total SILVER 1 89O Dollars 12,500,000 12,600,000 12,000,000 12,080,000 13.570,000 14,929,000 15,235,900 I4,6l8,300 15,637,900 15,197,800 I5,8l6,200 Dollars 20,345,000 20,575,000 21,015,000 23,875,000 25,930,000 3I,68l,000 37,852,400 42,744,700 48,825,100 55,855,600 63,354,800 Dollars 32,845,000 33,]75,000 33,015,000 35,955,000 39,500,000 46,6lO,000 53,088,000 57.363.000 64,463,000 71,053,400 79,171,000 Dollars 57,242,100 57,630,000 55,662,500 46,800,000 31,422,100 36,445,500 39,654,600 32,316,000 32,Il8,400 32,859,000 35,741,140 189I 1892 I893 I894 1895 I896 I897 1898 I899 IQOO 16. Anthracite and bituminous coal production in the United States. (Show record on a single pair of axes and correct to one million.) YEAR Total Total Total Total Anthracite Bituminous Anthracite Bituminous Tons Tons Tons Tons 1880 25,580,180 38,242,641 I9OI 60,302,264 201,572,572 189O 41,489,858 99,377.073 I902 37,024,582 232,252,596 1897 47,036,389 131,739,681 1903 66,678,392 252,389,837 I898 47,705.125 148,702,257 1904 65,382,842 248,738,941 1899 54,030,536 172,524,099 J 905 69,405,958 281,239,252 I900 51,309,214 189,480,097 17. Number of employees thrown out of work because of strikes. Correct to nearest hundred. (Plot correct to 1,000.) YEAR NUMBER YEAR NUMBER l88l 129,500 I89O 352,000 1882 154,700 I89I 299,OCO 1883 149,800 I892 206,700 1884 147,100 1893 265,QCO I885 242,700 1894 660,400 1886 508,000 I89S 302,400 I887 379,700 I896 241,200 1888 147,700 1897 408,400 I889 249,6C0 i I898 249,000 1899 1900 1901 1902 1903 1904 1905 417,100 505,100 543.400 659,800 656,100 517,200 221,700 APPENDIX TO CHAPTER II 18. Number of strikes: 45 CALENDAR YEAR Ordered by labor organ- izations Not ordered by labor or- ganizations CALENDAR YEAR Ordered by labor organ- izations Not ordered by labor or- ganizations l88l 1882 1883 I884 1885 1886 I887 1888 I889 189O 189I I892 1893 223 220 271 240 357 763 952 616 724 1,306 1,284 918 906 248 234 207 203 288 669 483 288 351 525 432 380 399 I894 1895 I896 1897 1898 1899 1 900 I9OI 1902 1903 1904 |1905 847 658 662 596 638 1,115 1,164 2,218 2,474 2,754 1,895 1.552 501 555 363 482 418 682 6l l 706 688 740 412 525 19. Number of Post Offices in the United States, correct to the nearest 500. YEAR ENDED JUNE 30TH 879 880 88l 882 883 884 885 886 887 888 889 890 891, 892 893 POST OFFICES 41,000 43,000 44,500 46,000 48,000 50,000 51*500 53,500 55,000 57,500 59,000 62,500 64,500 67,000 68,500 YEAR ENDED JUNE 30TH I894 1895 1896 I897 I898 I899 I9OO 1 901 1902 1903 1904 1905 1 906 1907 POST OFFICES 70,000 70,000 70,500 71,000 73,500 75,000 76,500 77,000 76,000 74,000 71,000 68,000 65,500 20. Number of offices of the Postal Telegraph Cable Company, correct to the nearest 100. 46 APPENDIX TO CHAPTER II YEAR OFFICES YEAR OFFICES YEAR OFFICES YEAR OFFICES I885.. 300 189I.. 1,200 I897.. 9,900 1903.. 20,000 1886.. 400 I892. . I,400 1898.. II,IOO I9O4.. 2I,IOO T887.. 600 1993- • 1,600 I899.. 12,700 1905.. 23,100 1888.. 700 1894.. 1,800 1900. . 13,100 I906. . 25,300 I889.. 800 1895.. 2,100 1901. . 14,900 1907.. 25,500 I89O.. 1,000 1896.. 9,IOO I 902 . . l6,200 21. Table showing the increase in mileage of railroad in operation in the United States. Given correct to nearest unit: -a a "be 03 a "3 'a 3 ■d5« Hi a Oi GRAND YEAR W J3 £t S £ s •- ■ ■:-■■ ■■ ■' - ' • V — ' . : .: -,■ RECD LD DEC 9 '69 -3PM LD 21-100m-6,'56 TT . Gen . eral ¥£**? . (B9311S10 ) 476 UmveM^crf Cdif orma ;> <■■ )..