ill! i ; ill! PHI I If 1 J ! |i iiisi iiiii ill!: ! I 111! , h i; I.,,!, WW i Hi ! I 11 ! Hi 111! Hill II IIIIUIIII I I Ilium III I I Hi" llli I (Eorwll Itttarmtg Etfrrarg BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF Henry W. Sage 1891 A,..ifc.1.^.A..U ,..:... ......:.S.M •fl 9306 Cornell University Library arV17948 Recreations in mathematics, 3 1924 031 221 504 olin.anx Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031221504 RECREATIONS IN MATHEMATICS BY H. E. LICKS With 60 Illustrations NEW YORK D. VAN NOSTRAND COMPANY 25 Park Place 1917 COPYEIGHT, I917, BY D. VAN NOSTRAND COMPANY Stanbope ipreas H. GILSOK COMPANY BOSTON, U.S.A. PREFACE The object of this book is to afford recreation for an idle hour and to excite the interest of young students in further mathematical inquiries. The topics discussed have therefore been selected with a view toward interesting students and mathematical amateurs, rather than experts and professors. The Table of Contents is logically arranged with respect to chapters; but it will be found that within the latter, the topics are subject to no regular law or order. Some of these are long, others short; some are serious, others are frivolous; some are logical, others are absurd. It is feared that many things which might have been included have been omitted, and that still others which should have been omitted, have been included. The indulgence of readers is craved for this seeming lack of consistency and it is sub- mitted in extenuation that the very character of the sub- ject, partaking as it does somewhat of the nature of the curio collection, renders a more orderly treatment practi- cally impossible. The subject matter has been collected from many and divers sources and it is hoped that in spite of the complex nature of the work, the selection will appeal to the readers to whom it is addressed. H. E. L. December, igi6. CONTENTS Chapter Page I. Arithmetic i II. Algebra 22 III. Geometry 44 IV. Trigonometry 63 V. Analytic Geometry 76 VI. Calculus 87 VII. Astronomy and the Calendar 104 VIII. Mechanics and Physics 1 24 IX. Appendix 140 RECREATIONS IN MATHEMATICS CHAPTER I ARITHMETIC 1 OUNTING a series of things and keeping tally of the tens on the fingers were processes used by primitive peoples. From the ten fingers arose ultimately the decimal system of numeration. Recording the results of counting was done by the Egyp- tians and other ancient nations by means of strokes and hooks; for one thing a single stroke I was made, for two things two strokes 1 1 were used, and so on up to ten which was represented by fl. Then eleven was written I fl, twelve 1 1 H, and so on up to twenty, or two tens, which was repre- sented by flfl. In this way the numeration proceeded up to a hundred, for which another symbol was employed. Names for 1 1 , 1 1 1 , 1 1 1 1 , fl fl , etc., appear in the Egyptian hieroglyphics, but a special symbol for each name is not used. Probably the Hindoos first invented such symbols, and passed them on to the Arabs, through whom they were introduced into Europe. 2 GREEK NOTATION The Greeks used an awkward notation for recording the results of counting. The first nine letters of the Greek alphabet denoted the numbers from one to nine, so that a 2 RECREATIONS IN MATHEMATICS represented one, two, y three, and so on. Then the follow- ing nine letters were used for ten, twenty, thirty, etc., so that k represented ten, \ twenty, m thirty, and so on. Then '• the remaining letters r, v, etc., were used for one hundred, two hundred, etc., but as the Greek alphabet had only twenty-four letters, three symbols were borrowed from other alphabets. This was an awkward notation, and there seems to have been little use made of it except to record results. A number having two letters was hence between ten and one hundred, and one having three letters was between one hundred and one thousand; thus, X5 was twenty-four and tkS was one hundred and fourteen. The Greeks were good mathematicians, as appears from their work in geometry, but only a few writers used this arithmet- ical numeration in computations, saying, for example, that the sum of tax and Aj3 was ny. In those days the abacus or swan pan, similar to that seen in Chinese laundries in the United States, was employed to make arithmetical com- putations. From very early days this simple apparatus has been used throughout the East, and it is said that compu- tations are made on it with great rapidity. ROMAN NUMERATION The Romans represented the first five digits by I, II, III, IIII, and V, a V prefixed to the first four gave the digits from six to nine, while ten was represented by X, fifty by L and one hundred by C. This notation is still in use for a few minor purposes, it being modified by using IV for four, IX for nine, etc.; when a watch face is lettered in this nota- tion, however, IIII is always used for IV, because Charles V said that he would allow nothing to precede a V. The ARITHMETIC 3 Roman notation was employed only to record numbers, and it does not appear that arithmetical operations were ever conducted with it. Perhaps this awkward notation re- tarded the development of mathematics among the Ro- mans. Frontinus, a Roman water commissioner, wrote in 97 A.D. a treatise on the Water Supply of the City of Rome, a translation of which, with an excellent commentary by Clem- ens Herschel, was published at Boston in 1889. A long list of the dimensions of the water pipes then in use is given, these being expressed in digits and fractions. The fraction 1/12 was denoted by a single horizontal stroke — , 2/12 by two strokes m, 3/12 by three strokes =: , and so on up to 5/12. Then 1/2 was represented by S, while the fractions from 7/12 to n/12 were represented by adding strokes to the S, thus, S~ ~, indicated 1/2 + 4/12 or 5/6. The fraction 1/24 was indicated by X. The smallest fraction used was 1/288 which was represented by 3. The follow- ing is the description of the pipe No. 50 given by Frontinus: Fistula quinquagenaria: diametri digitos septem S~=: £9quinque, perimetri digitos XXV £3 VII, capit quinarias XLS rr JC3V. Of which Herschel's translation is as follows: The 50-pipe: seven digits, plus 1/2, plus 5/12, plus 1/24, plus 5/288 in diameter; 25 digits, plus 1/24 plus 7/288 in circumference; 40 quinarias, plus 1/2, plus 2/12, plus 1/24, plus 5/288 in capacity. "•' The digit was one-sixteenth of a Roman foot and the quantity of water flowing through a pipe of i\ digits in diameter was called a quinaria. Frontinus takes the quan- tities of water flowing through pipes as proportional to the squares of their diameters, for he says that pipes of 2J and 3§ digits in diameter discharge four and nine quinarias respectively. 4 RECREATIONS IN MATHEMATICS 4 THE ARABIC SYSTEM OF NUMERATION The Arabic method, by which the symbols, i, 2, 3, etc., were used for the first nine integers, seems to have first originated in India, from whence it was carried by the Arabs to Europe, about the year 1200. Long before this time Greek and Arabic astronomers had used the sexagesimal system in the division of the circle, and this, with Arabic numerals, was employed about 1200 in Europe fqr expressing numbers not at all connected with a circle. Thus, 28, 32' 17" 45'" 20^ meant 28 units plus 32/60, plus 17/3600, plus 45/216000, plus 20/1296000. This method of expressing fractions was certainly more convenient than the Roman method as used by the^water commissioner Frontinus. How. the Arabic method of numeration was introduced into Europe is told by Ball in the following interesting account of one of the early Italian mathematicians. LEONARDO DE PISA From Ball's Short Account of the History of Mathematics. Fourth Edition (London, 1908), pages 167-170. Leonardo Fibonacci (i.e., filius Bonacci), generally known as Leonardo of Pisa, was born at Pisa about 11 75. His father Bonacci was a merchant, and was sent by his fellow- townsmen to control the custom-house at Bugia in Barbary; there Leonardo was educated, and he thus became acquainted with the Arabic or decimal system of numeration, as also with Alkariami's work on Algebra. It would seem that Leonardo was entrusted with some duties, in connection ARITHMETIC 5 with the custom-house, which required him to travel. He returned to Italy about 1200, and in 1202 published a work called Algebra et almuchabala (the title being taken from Alkariami's work), but generally known as the Liber Abaci. He there explains the Arabic system of numeration, and remarks on its great advantages over the Roman system. He then gives an account of algebra, and points out the convenience of using geometry to get rigid demonstrations of algebraical formulas. He shows how to solve simple equations, solves a few quadratic equations, and states some methods for the solution of indeterminate equations; these rules are illustrated by problems on numbers. The algebra is rhetorical, but in one case letters are employed as alge- braical symbols. This work had a wide circulation, and for at least two centuries remained a standard authority from which numerous writers drew their inspiration. The Liber Abaci is especially interesting in the history of mathematics, since it practically introduced the use of Arabic numerals into Christian Europe. The language of Leonardo implies that they were previously unknown to his countrymen: he says that having had to spend some years in Barbary he there learnt the Arabic system, which he found much more convenient than that used in Europe; he therefore published it "in order that the Latin race might no longer be deficient in that knowledge." Now Leonardo had read very widely, and had travelled in Greece, Sicily, and Italy; there is therefore every presumption that the system was then not commonly employed in Europe. The majority of mathematicians must have already known of the system from the works of Ben Ezra, Gerard, and John Hispalensis. But shortly after the appearance of Leonardo's book Alfonso of Castile (in 1252) published 6 RECREATIONS IN MATHEMATICS some astronomical tables, founded on observations made in Arabia, which were computed by Arabs, and which, it is generally believed, were expressed in Arabic notation. Alfonso's tables had a wide circulation among men of science, and probably were largely instrumental in bringing these numerals into universal use among mathematicians. By the end of the thirteenth century it was generally assumed that all scientific men would be acquainted with the system; thus Roger Bacon writing in that century recommends algorism (that is, the arithmetic founded on the Arab notation) as a necessary study for theologians who ought, he says, " to abound in the power of numbering." We may then consider that by the year 1300, or at the latest 1350, these numerals were familiar both to mathe- maticians and to Italian merchants. So great was Leonardo's reputation that the Emperor Fredrick II stopped at Pisa in 1225 to test Leonardo's skill, of which he had heard such marvellous accounts. The competitors were informed beforehand of the questions to be asked, some or all of which were composed by John of Palermo, who was one of Fredrick's suite. This is the first time that we meet with an instance of those challenges to solve particular problems which were so common in the sixteenth and seventeenth centuries. The first question propounded was to find a number of which the square, when either increased or decreased by five, would remain a square. Leonardo gave an answer, which is correct, namely 41/12. The other competitors failed to solve any of the problems. (See No. 33 for a problem in Algebra.) ARITHMETIC 7 6 EARLY ARITHMETIC IN ENGLAND The earliest book on arithmetic printed in England was "The Grounde of ^rtes, by M. Robert Recorde, Doctor of Physik." First issued in 1540 it was republished in numer- ous editions until 1699. The following extracts from the edition of 1573 give an idea of the method of instruction. Master. — If numbering be so common that no man can doe anything alone, and much less talke or bargain with other, but still have to doe with numbre; this proveth not numbre to be contemptible and vile, but rather right excellent and of high reputation, sithe it is the grounde of all mens affaires, so that without it no tale can be told, no bargaining without it can dully be ended, or no business that man hath, justly completed. . . . Wherefore in all great workes are Clerkes so much desired? Wherefore are Auditors so richly feed? What causeth Geometricians so highly to be enhaunced? Because that by numbre suche things they find, which else would farre cxccll mans minde. Scholar. — Verily, sir, if it be so that these men by numbring their cunning doe attaine, at whose great workes most men doe wonder, then I see well that I was much deceived, and numbring is a more cunning thing than I take it to be. Master. — If numbre were so vyle a thing as you did esteem it, then need it not to be used so much in mens communication. Exclude numbre, and answer to this question: How many years old are you? Scholar. — Mum. Master. — How many daies in a week? How many weeks in a yeare? What landes hath your father? How many men doth he keep? How long is it sythe you came from him to me? Scholar. — Mum. Master. — So that if numbre wante, you answer all by Mummes. The master then goes on to show how useful numbers are in "Musike, Physike, Law, Grammer" and such like, and then proceeds to teach him numeration, addition, sub- traction, and so on. The master explains and illustrates the process and then tests the scholar by requiring him to 8 RECREATIONS IN MATHEMATICS perform an example, the latter explaining as he goes on and asking questions on doubtful points. Thus, in addi- tion, after having explained the process of carrying, the master gives the scholar the numbers 848 and 186 to be added. Scholar. — I must set them so, that the two first figures stand one ouver another, and the other each ouer a fellow of the same place. And so like- wise of other figures, setting always the greatest numbre highest, thus, as followeth: 848 186 Then I must add 6 to 8 which maketh 14, that is mixt numbre, therefore must I take the digit 4 and write it under the 6 and 8, keeping the article 1 in my mind, thus: 848 186 4 Next that, I doe come to the second figures, adding them up together, saying 8 and 4 make 12, to which I put the 1 reserved in my mind, and that makethe 13, of which numbre I write the digit 3 under 8 and 4, and keep the article 1 in my mind, thus: 848 186 34 Then come I to the third figures, saying 1 and 8 make 9, and the 1 in my mind maketh 10. Sir, shall I write the cypher under 1 and 8? Master. — Yea. Scholar. — Then of the 10 1 write the cypher under 1 and 8 and keep the article in my mind. Master. — What needeth that, seeing there followeth no more figures? Scholar. — Sir, I had forgotten, but I will remember better hereafter. Then seeing that I am come to the last figures, I must write the cypher under them, and the article in a further place after the cypher, thus: 848 186 1034 Master. — So, now you see, that of 848 and 186 added together, there amounteth 1034. Scholar. — Now I think I am perfect in addition. ARITHMETIC g Master. — That I will prove by another example. There are two armies : in the one there are 106 800 and in the other 9400. How many are there in both armies, say you? In those old days it seems that the multiplication table was learned only as far as five times five, and hence a process was necessary for multiplying together two numbers like 6 and 8. The following is the process as given by Recorde. The numbers 6 and 8 were placed on the left- hand side of a large letter X, thus: Then each was subtracted from 10, the remainders placed directly opposite on the right-hand side, and a line was drawn under the whole, thus: Next the units figure of the product was found by multiply- ing together the remainders 2 and 4, and the other figure of the product by subtracting crossways either 2 from 6 or 4 from 8; thus, 2 times 4 is 8, and 2 from 6 (or 4 from 8) is 4; therefore, 8x /2 48 and hence six times eight makes forty-eight. Napier's bones, used in England in the seventeenth cen- tury, consisted of nine sticks numbered at the top 1 to 9 inclusive, each stick having on its side the first nine mul- tiples of the number at the top. These bones were hence IO RECREATIONS IN MATHEMATICS merely a multiplication table. | When it was desired to multiply a number by 57, the sticks headed 5 and 7 were taken and their multiples used. Thus suppose that 89 was to be multiplied by 57. First, looking on the stick 5,, the 89 u 40 45 56 63 5073 multiples of 5 by 8 and 9 were taken off and set down as shown, then looking on the stick 7 the multiples of 7 by 8 and 9 were taken off and set down; then the addition gave the product of 89 by 57. Thus were arithmetical opera- tions performed in England less than four hundred years ago. 7 THE SIGNS OF ARITHMETIC The signs + and — are supposed to have been first used in Holland in the fifteenth century, to denote excess or deficiency in weight of bales of goods. The normal weight of a certain bale being, say, 4 centners, it was marked 4 c. + 5 lb. if it weighed 5 lb. more than the normal, and 4 c. — 5 lb. if it weighed 5 lb. less. These signs were used in a similar sense in Widman's Arithmetic published at Leipzig in 1489. It was not until about 1540 that they were used as signs of operation, that is, as directions to perform addition or subtraction. The sign = was first used in works on Algebra, the earliest ARITHMETIC II mention being in Recorde's Whetstone of Wit issued in 1557, his selection of that sign being because "no 2 thyngs can be moare equalle." The decimal point came much later, for fractional num- bers were generally written in the duodecimal or sexagesi- mal form prior to the fifteenth century, as has already been explained. Napier and Briggs, the inventors of logarithms seem to have been the first to use, about 1620, the decimal method and the decimal point, although at first there was no point, but a line was drawn under all the decimals. It is scarcely more than a hundred years since the decimal point came to be generally used in the United States. For example, Willett's Scholar's Arithmetic, used in the public schools of New York City was issued in a fourth stereotype edition in 1822. On page 23 it is said that in adding sums of money, the dollars, cents, and mills should be kept separate by placing a point between them, but the point used is a colon. On page 24 in subtraction, it is said that dots must be used to keep these units separate, but the dot used is a comma. Under multiplication the colon is used in some examples and the comma in others. Under divi- sion (page 27) the comma is used, and also numbers like $56.43 are written $56 43cts.. Under "Reduction of Fed- eral Money" on page 55, the single parenthesis is used as a decimal point; the problem being to reduce 387652 mills to dollars, the number is first divided by ten and the result stated as 38765(2; then this is divided by 100 giving 387(65:2, and finally the answer is given as 387 65cts 2m. Nothing more is said about decimals until we come to "Decimal Fractions" on page 151, and there the period is formally introduced as the decimal point, and the opera- tions on numbers containing decimals are well explained; 12 RECREATIONS IN MATHEMATICS even then it seems necessary to mention that a quantity like $590,217 means 590 dols. 21 cts. 7 m. A large part of the time of the children who used this book was devoted to intricate problems concerning pounds, shillings, pence, and farthings. The signs X and -j- to indicate the operations of multi- plication and division were not in common use before 1750. Prior to this date parentheses were not used in a case like a (b + c + d), but a straight line was drawn over the b+c + d. The use of the shilling mark / to indicate division is comparatively recent, it having been first employed about i860. In this country it was rarely used until after 1890, but is how very commonly found in algebraic notation, and it will generally be used in the later chapters of this volume. Thus, 4/261 has the same meaning as -7- or 4 4- 261. 201 This new division mark is of especial advantage in sim- plifying printed work, either in setting algebraic expressions in type or in writing fractions with a typewriter. ARITHMETIC AMUSEMENTS 8 Multiply 37037037 by 18; also multiply it by 27. Multiply 13 7 1 742 by 9; also multiply it by 81. Multiply 98765432 by 9; also multiply it by 1 1/8. Think of any number, multiply it by 2, then add 4, multiply by 3, divide by six, subtract the number you thought of, and the result will be 2. ARITHMETIC 13 10 Think of any number, and add 1 to it, then square these two numbers and subtract the less from the greater. Now if you will tell me this difference, I can easily know the number you thought of, for I merely subtract 1 from the number you give me, then divide by 2, and the result is the number of which you thought. 11 To find the age of a man born in the nineteenth century. Ask him to take the tens digit of his birth year, multiply it by ten and add four; to this ask him to add the units figure of his birth year and tell you the result. Subtract this from 124 and you will have his age in 1920. Thus for a man born in 1848, 4 X 10 = 40, 40 + 4 = 44, 44 + 8 = 52, 124 — 52 = 72, which will be his age in 1920, if then living. 12 Ask a person to multiply his age by 3, add 6 to the prod- uct, then divide the last number by three and tell you the result. Subtract two from that result and you have his age. 13 A woman goes to a well with two jars, one of which holds 3 pints and the other 5 pints. How can she bring back exactly 4 pints of water? 14 By the help of the following table the age of a person under 21 can be ascertained: I 2 4 8 16 3 3 5 9 17 5 6 6 10 18 7 7 7 11 19 9 10 12 12 20 ii 11 13 13 13 14 14 14 iS iS iS iS 17 18 20 19 J 9 14 RECREATIONS IN MATHEMATICS A B C D E Ask the person to tell you in which column his or her age occurs. Then add together the numbers at the tops of those columns and the sum will be the age. Thus, if a person says that his age is found in qolumns B and E, then 2 + 16 = 18 which is his age. 15 All integral numbers are either prime or composite. A prime number is one which has no integral divisors except itself and unity. There is no simple method of ascertain- ing what numbers are prime except that of the "sieve of Eratosthenes." By this method the odd integral numbers are written in ascending order, thus, 3, 5, 7," 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, etc., then every third number after three, every fifth number after 5, every seventh number after 7, and so on, are crossed off, and those remaining are primes. Thus from the above numbers, 9, 15, 21, and 27 are first crossed off, then 15 and 25, and then 21; the remaining numbers 3, 5, 7, 11, 13, 17, 19, 23, 29, together with 2, are the prime numbers less than thirty. This process becomes very laborious when the numbers run into millions. 16 A perfect number is one which is equal to the sum of its divisors. Thus, 6 equals 1+2+3 an d i> 2 > an d 3 are the divisors of 6. The next perfect number is 28, the third one ARITHMETIC 15 is 496, and the fourth one is 8128. Beyond this there are no perfect numbers until 33 550 336 is reached. Then come 8589869056, 137438 691^28 and 2 305 843 008 139952, 128. This seems to be the largest perfect number thus far found. All of the above numbers end in 6 or 28. It is not known whether or not an odd number can be perfect, but the indications are against this being the case. The above numbers are taken from Ozanam's Recreations Mathematiques, published at Paris in 1750. Other num- bers which he gives are not perfect, because he unfortu- nately made errors in computing them. Ozanam's book was first published in 1698; it passed through many edi- tions and was also translated into English. 17 At rare intervals natural calculating boys come to public notice. One of these was Zerah Colburn who was born in New England in 1804 and taken to London when eight years old to exhibit his powers. He could mentally multiply any number less than 10 into itself successively nine times and give the results faster than they could be written down. He was asked what number multiplied by itself gave 106 989 and he instantly replied 327. With equal prompt- ness he stated that the number which multiplied twice by itself gave 268336125 was 645, this being a problem in cube root for which an ordinary computer would require several minutes. He was asked to name a number which would divide exactly 36083 and he immediately replied that there was no such number, in other words he recog- nized this as a prime number just as readily as we recognize 29 or 37 to be one. He could very quickly multiply to- gether two numbers of four or five figures, and perform 3 6 RECREATIONS IN MATHEMATICS many other remarkable mental feats. These natural cal- culators are rarely able to explain their processes, and their powers fade away and disappear as they grow up and become educated. 18 How many people know that the square of 3 plus the square of 4 equals the square of 5? All surveyors and draftsmen know it, also most machinists and carpenters, but to those in other trades it is probably quite unknown. Another interesting arithmetical theorem is that the cube of 3 plus the cube of 4 plus the cube of 5 equals the cube of 6. Probably few students who read this book have ever before heard of this important relation. 19 In one hand a person has an odd number of coins or pebbles and in the other hand an even number, the knowl- edge of the same being unknown to anyone except himself. Ask him to multiply the number in the right hand by 2 and the number in the left hand by 3. Then ask him to add together the two products and tell you their sum. If this sum is odd the left hand has the odd number of coins, but if the sum is even, the left hand has the even-number of coins. 20 One tumbler is half full of wine and another tumbler is half full of water. A teaspoonful of wine is taken from the first tumbler and put into the other one. Then a teaspoon- ful of the mixture is taken from the second tumbler and put into the first one. Is the quantity of wine removed from the first tumbler greater or less than the quantity of water removed from the second tumbler? Ball in his Mathe- ARITHMETIC 17 matical Recreations and Essays says that the majority of people will say it is greater, but that this is not the case. H. E. Licks, who has studied this problem, claims that the two quantities are exactly equal. 21 A stranger called at a shoe store and bought a pair of boots costing six dollars, in payment for which he tendered a twenty-dollar bill. The shoemaker could not change the note and accordingly sent his boy across the street to a tailor's shop and procured small bills for it, from which he gave the customer his change of fourteen dollars. The stranger then disappeared, when it was discovered that the twenty-dollar note was counterfeit, and of course the shoe- maker had to make it good to the tailor. Now the question is, how much did the shoemaker lose? 22 At an humble inn where there were only six rooms, seven travellers applied for lodging, each insisting on having a room to himself. The landlord put the first man in room No. 1 and asked one of the other men to stay there also for a few minutes. He then put the third man in room number two, the fourth man in room No. 3, the fifth man in room No. 4, and the sixth man in room No. 5. Then returning to room No. 1 he took the seventh man and put him in room No. 6. Thus each man had his own room! 23 An Arab merchant directed by will that his seventeen horses should be divided among his three sons, one-half of them to the eldest, one-third to the second son and one- ninth to the youngest son. How to make the division was 1 8 RECREATIONS IN MATHEMATICS a serious problem, for the eldest son claimed nine horses, but the others objected because this was more than one-half of seventeen. In this dilemma they applied to the Sheik who put his white Arabian steed among the seventeen horses, directed the eldest son to take one-half of the eight- een or nine, the second son to take one-third of the eighteen or six, and the youngest son to take one-ninth of the eighteen or two. Thus, since nine plus six plus two are seventeen, the horses were divided satisfactorily among the three sons. "Now," said the Sheik, "will I take away my own horse," and he led the Arabian steed back to his peg in the pasture. 24 To add 5 to 6 in such a way that the sum may be g. Make six marks at equal distances apart, thus //////. Between the first and second marks draw a slanting line so as to form the letter N; then do the same between the fourth and fifth marks; finally add to the last line three horizontal marks so as to form the letter E. Then the problem is solved, for the five marks added to the given six marks have made NINE. Another interesting problem in this line is to add three marks to a given five so as to make a quotation from Shakespeare. The added three marks give KINI, and you ask, where in this is found the required quotation. After a few minutes silence I reply, "A little more than kin but less than kind." 25 Two impossible problems: (i) If 3 is one-third of 10, what is one-quarter of twenty? (2) A man who had a bale of cotton sold it for $50, bought it back for $45, and then sold it again for $65. What was the net gain to the man? ARITHMETIC 19 26 The Indian mathematician Sessa, the inventor of the game of chess, was ordered by the king of Persia to ask as a recompense whatever he might wish. Sessa modestly re- quested to be given one grain of wheat for the first square of the board, two for the second, four for the third, and so on, doubling each time up to the sixty-fourth square. The wise men of the king added the numbers 1, 2, 4, 8, 16, etc., and found the sum of the series to sixty-four terms to be. 18446 744073 709 551 615 grains of wheat. Taking 9000 grains in a pint we find the whole number of bushels to be over 32 000 000 000 000, which is several times the annual wheat production of the whole world. 27 H. E. Licks once had a class of students well versed in arithmetic, algebra, trigonometry, and calculus, but not one of them could solve the following simple problem, as they knew nothing about bookkeeping. The problem is hence here given for other young people. A Coal Company appointed an agent, agreeing to pay him a salary of $265 for six months, all of the coal at the end of that time and all of the profits to belong to the Company. The Company furnished him with coal to the amount of $825.60 and in cash $215.00. The agent received for coal sold $1323.40, paid for coal bought $937.00, paid sundry expenses authorized by the Company $129.00, paid his own salary $265.00, paid to the Company $200.00, delivered to indigent persons by order of the Company coal to the amount of $13.50. At the end of the six months the Company took possession and found coal amounting to 20 RECREATIONS IN MATHEMATICS $616.50. The agent then paid to the Company the money belonging to them. How much did he pay? Did the Company gain or lose by the agency and how much? 28 THE FIFTEEN PUZZLE About the year 1880 everyone in Europe and America was engaged in the solution of this interesting puzzle. A square shallow box contained fifteen blocks numbered 1 to 15 inclusive and these could be moved about one block at a LiiLgjyyfcij E1SE3E 51E113EI Fig. I. Fig.e. Fig. 3. time, on account of the blank space. The blocks being placed in the box at random, say as shown in Fig. 1, the problem was to arrange them in regular order in the manner shown in Fig. 2. It was a fascinating exercise to shift these blocks until 1 was brought to the upper left-hand corner, then to bring 2 next to it, and thus keep on until the regular order of Fig. 2 was secured. But sometimes it happened, when the lowest row was reached, that the order of Fig. 3 resulted; for this case mathematicians proved that it was impossible to cause the blocks to take the regular order of Fig. 2. Mathematical analysis also showed that for many random positions of the blocks (Fig. 1), one-half of them would result in the order of Fig. 2 and one-half in the order of Fig. 3. ARITHMETIC 21 There is, however, a way by which the arrangement of Fig. 3 can be brought into regular order. Move the blocks until the upper left-hand corner is blank and the blocks i, 2, 3 fill the other spaces of the upper row, then continue until the blocks 4, 5, 6, 7 fill the second row and 8, 9, 10, 11 the third row, then the lowest row can be arranged in the order 12, 13, 14, 15. This is a solution of the puzzle, if the statement of the problem is merely that " the blocks are to be arranged in regular order." This puzzle comes under that branch of arithmetic known as permutations and combinations, and much mathematical thought has been expended upon it. The number of ways in which the fifteen blocks can be put at random in the box (Fig. 1) is 1 307 674 368 000. In the early stages of the craze it was not recognized that half of these combinations lead to the result of Fig. 2 and half of them to the result of Fig. 3. Hence when Fig. 3 was reached, the player usually kept on trying to obtain the arrangement of Fig. 2. Finally, after mathematicians had proved that it is impossible to bring Fig. 3 to agree with Fig. 2, the craze abated. It has been stated that this interesting puzzle was in- vented in 1878 by a deaf and dumb man as a solitaire game. In the height of the craze persons in public conveyances could be seen every day attempting to solve the puzzle. Some physicians thought that this work was a beneficial mental exercise, but others claimed that it led to nervous disorders. A poet well expressed the latter opinion as follows: Put away his crack-brain puzzle, He has climbed the asylum stair; Numbers thirteen, fifteen, fourteen, Turned his head and sent him there! CHAPTER II ALGEBRA 29 UCLID used algebra in a geometric farm, ex- pressing the equations always by words. For J example: if a straight line be divided into two ' parts, the square on the line is equal to the sum of the squares on the two segments plus twice the rectangle of those segments; this in modern algebra is the theorem (a + b) 2 = a 2 + b 2 + 2 ab. Until the sixteenth century all algebraic equations were generally expressed in words; for instance, Omar Kayyam wrote about 1100, "Cubus, latera et numerus aequalis sunt quadratis,'' meaning x? + bx + c = ax 2 . Cardan about 1550 wrote "Cubus p 6 rebus aequalis 20," meaning x? + 6 x = 20. Ramus about the same time wrote 7 = 7 358 060 d = 3 515 820 F = 4i49 3 8 7 y = 5 439 2I 3 are the least numbers satisfying the conditions of the first problem. The total number of cattle is 50 389 082, not too many to graze upon the island of Sicily, the area of which is about 7 000 000 acres. The second or complete problem includes the determina- tion of numbers which not only satisfy equations (1) to (7), but also W + B = a square number, (8) D + Y = a triangular number, (9) and this is to be done by finding an integer N to multiply into each of the results of the first problem, or 17 826 996 N = a square number, 11 507 447 N = a triangular number. A number N that will satisfy one of these conditions can be found without difficulty but to determine N so that both conditions will be satisfied is a task involving an enormous amount of time and labor. In fact, this required number N has never been completely computed. A solution which satisfies (8) as well as (1) to (7) is easily made. Since W + B is 17 826 966 N or 4 X 4 456 749 N, and since 4 456 749 contains no number that is a perfect square, it is plain that N must be 4 456 749. Accordingly, each of the numbers found in the first solution must be multiplied by this value of N in order to satisfy (1) to (8) inclusive; the number W + B is then 79 450 446 596 004 36 RECREATIONS IN MATHEMATICS which is a perfect square, but D + Y is 51 285 802 909 803 which is not a triangular number. It is now time to explain what is meant by a triangular number. The number ten is triangular because ten dots can be arranged in rows in the form of a triangle, the first row having one dot, the second two, the third three, and the fourth four dots. The next higher triangular number is 15 and the next 21, and in general J n (n + 1) is a triangular number whenever n is an integer, n being the number of rows in the triangle. The number 51 285 802 909 803 is shown not to be a triangular number by equating it to § n (n + 1) and computing n from the quadratic equation thus formed when it is found that n is not an integer. Now since 51 285 802 909 803 is the number of yellow and dappled bulls which results from a solution which satisfies equations (1) to (8) inclusive, it is plain that the ninth condition may be expressed by 51 285 802 909 803 x 2 — I n (n + 1), in which x and n are to be integers. When x 2 has been found each of the numbers of the first solution is to be multi- plied by 4 456 749 x 2 in order to give the number of bulls and cows in each herd which satisfy the nine imposed conditions. These numbers were readily seen to be so great that the island of Sicily could not contain all the cattle the prob- lem seems to demand. This requirement, however, was understood to be only figurative, and mathematicians agreed that the numbers might be found altho no useful purpose would be attained by computing them. Thus the question rested until i860 when Amthor demonstrated that 206 545 figures are necessary to express the total number of ALGEBRA 37 cattle and that 766 X io 206542 gives their approximate number. This is an enormous number, and it is not diffi- cult to show that a sphere having the diameter of the milky way, across which light takes ten thousand years to travel, could contain only a part of this great "number of animals, even if the size of each is that of the smallest bacterium. It would be thought that, after this investigation of Amthor, the cattle problem would have been finally dropped but such was not the case. The Way to solve it was well understood from the theory of indeterminate analysis. Let the preceding equation be multiplied by 8, unity be added to each member and let 2 n + 1 be called y; then it becomes y 2 — 410 286 423 278 424 x 2 = 1 which is of the form y 2 — Ax 2 — 1, and it is known that when A is an integer there can always be found integral values of x and y which satisfy the equation. The method of solution cannot well be explained here, but it was de- vised many years ago by Pell and Fermat and is well known to those skilled in higher mathematics. For example, take the simple case where A = 19, or y 2 — 19 x 2 = 1, then the smallest integral values of y and x are 170 and 39. In 1889 A. H. Bell, a surveyor and civil engineer of Hillsboro, Illinois, began the work of solution. He formed the Hillsboro Mathematical Club, consisting of Edmund Fish, George H. Richards, and himself, and nearly four years were spent on the work. They computed thirty of the left-hand figures and twelve of the right-hand figures of the value of x without finding the intermediate ones. This value is x = 34 555 906 354 559 370 ... 252 058 980 100 in which the dots indicate fifteen computed figures, which it is unnecessary to give here, and 206 487 uncomputed 38 RECREATIONS IN MATHEMATICS ones; the total number of figures in this number is 206 531. The final step is to multiply each of the numbers of the first solution by 4 456 749 and by this value of a?, thus giving: White bulls = 1 596 510 34 1 800 Black bulls = 1 148 971 178 600 Dappled bulls = 1 133 192 894 000 Yellow bulls = 639 034 026 300 White cows = 1 109 829 564 000 Black cows = 73s 594 645 400 Dappled cows = S4 1 4°° 3*8 000 Yellow cows = 837 676 113 700 Total cattle = 7 760 271 081 800 in which each line of dots represents 206 532 figures, the total number of figures in each line being either 206 545 or 206 544. In each of these lines there are omitted twenty- four figures at the left end and six at the right end which were computed by the Hillsboro Mathematical Club. This solution is published in the American Mathematical Monthly for May, 1895, where Bell remarks that each of these enormous numbers is "one-half mile long." A clearer idea of its length may be obtained by considering the space required to print it. Each page of this volume contains 32 lines and in each line about 50 figures may be printed, so that one page could contain about 1750 figures. To print a number of 206 245 figures would require 129 pages, and to print the nine numbers indicated above a volume of over 1000 pages would be needed. It is known that Archimedes speculated regarding large numbers, for his book Arenarius is devoted to showing that a number may be written that will express the number of grains of sand in a sphere of the size of the earth. It cannot be proved that Archimedes was the author of the cattle problem, but as Amthor remarks, the enormous ALGEBRA 39 numbers in its solution render it worthy of his genius and proper to bear his name. Its closing challenge still remains open, for the complete solution has not yet been made; and the investigations of Bell show that this would require the work of a thousand men for a thousand years. The little prairie town of Hillsboro, may, however, well exult as a conqueror, for its mathematical club has made the most complete of all solutions of the cattle problem and has proved itself to be highly skilled in numbers. 55 MAGIC SQUARES Fig. 4 shows the well-known magic square containing the nine digits, the sum of each row, column, and diagonal being 15. These numbers may be arranged in other ways, for instance, by taking the left-hand column as the top row, the middle column as the middle row, and the right-hand column as the lowest row. Altogether there are eight different arrangements for this simplest of all magic squares. 4 9 z 3 5 7 a 1 e 1 14 •15 4 IZ 7 6 9 a' 11 10 5 15 z 3 IS 16 Z II 5 3 13 8 10 6 IE 1 IS 9 7 14 4 Fig. 4 FVg-5 Fig- 6 A magic square of sixteen numbers is shown in Fig. 5, the sum of each row, column, and diagonal being 34. Here also there are many other arrangements. Fig. 6 shows another magic square which has the same properties and there are others which will be explained later. A true magic square should have 1 for its smallest number and contain all the natural numbers up to w 2 , where n is the 4° RECREATIONS IN MATHEMATICS number of cells in one row or column of the square. The sum of the first n? natural numbers is S =\rP (n 2 + i), and hence the sum of the numbers in one row, column, or diag- onal is N = J n (n 2 + i). Accordingly, for magic squares of 9, 16, 25, 36, 49, 64 cells the fundamental data are: Cells in one row n = 3 4 5 6 7 8 Sum of all numbers S = 45 136 325 666 1225 2080 Sum of one row N = 15 34 65 in 175 260 ' A magic square with 25 cells has the sum of the numbers in each row, column, or diagonal equal to 65. To form such a square write the numbers 1, 2, 3, 4, 5 in each row of the square in Fig. 7 so that the mean number 3 always comes 3 A 5 1 2 e 1 A 5 1 1 Z ■5 A 5 5 1 2 3 4 A 5 1 Z 3 15 zo 5 10 20 5 IO 15 5 IO 15 ZO 5 10 15 ZO O IO 15 ZO 5 18 Z4 S 6 12 22 3 9 15 IS 1 1 13 19 25 10 II 17 23 4 14 20 21 2 8 Rg.7 Rj.8 Fig. 9 in one of the diagonals, then write the numbers o, 5, 10, 15, 20 in the rows of Fig. 8 so that the mean number 10 always comes in the opposite diagonal. For Fig. 7 the sum of each row, column, and diagonal is 15 and for Fig. 8 it is 50, the total being 65. Then add the numbers in corresponding cells and put the results in Fig. 9 thus forming a magic square of 25 cells where the sum of each row, column, and diagonal is 65. Another magic square of 25 cells may be formed by tak- ing the first five natural numbers in the order 4, 1, 3, 2, 5 and placing them in each row of a square in this order, 3 always coming in a diagonal cell; then in another square writing 15, o, 10, 5, 20 so that 10 comes in each cell of the ALGEBRA 41 opposite diagonal; and finally adding the numbers in corresponding cells. Other squares may be formed by writing the first five natural numbers in different orders, keeping always 3 in the middle, and arranging the auxiliary numbers correspondingly. Altogether twelve magic squares may be formed in this way, and from each of these still others may be formed by interchanging rows and columns. Any magic square with an odd number of cells n in one row, can be formed in a similar way, by writing the first n natural numbers in each row so that \ (n + 1) comes in a diagonal cell, then writing the numbers o, n, 2 n, 3 n, . . . (n — 1) n, so that J (n — 1) n comes in the cells of the opposite diagonal, and finally adding the numbers in corre- sponding cells. Thus for n = 7, the first set of numbers might be 5, 3, 1, 4, 6, 2, 7, when the second set must be 28, 14, o, 21, 35, 7, 42; here 4 must be written along one diagonal and 21 along the other diagonal. The formation of magic squares having an even number of cells is not so easy and it seems that a general rule has not been given. For 16 cells, however, the following rule is applicable. Write in the upper and lower rows of a square the numbers 1, 3, 2, 4; then in the two middle rows write them in the reverse order. Again in the top row put the numbers o, 12, 12, o, and in the lowest row write 12, o, o, 12; in the upper middle row put 8, 4, 4, 8 and in the one below it 4, 8, 8, 4. The addition of these numbers will give a magic square of 16 numbers which will be slightly different from those shown in Figs. 5 and 6. A magic square of 64 cells may be seen on page 163 of Ball's Mathematical Recreations and Essays (London, 191 1), which possesses the wonderful property that if each number be replaced by its square the resulting square is 42 RECREATIONS IN MATHEMATICS also magic, the sum of the numbers in each line being ii 180. We shall now briefly mention the magic squares some- times called "diabolic," or more commonly "Nasik," this being the name of the town in India where A. H. Frost invented them. Fig. 6 shows a Nasik magic square of 16 cells where the sum of each row, column, and diagonal is 34. The sum of the numbers in each broken diagonal is also 34, a broken diagonal being one which is partly on one side of the main diagonal and partly on the other side; thus the numbers 11, 13, 6, and the number 4 constitute a broken diagonal, as likewise do the numbers 2, 3, and 15, 14. In this magic square the sum of the numbers in any small square formed by four adjacent cells is also equal to 34. Truly, this is a marvellous arrangement of the first sixteen natural numbers. Benjamin Franklin, the famous philosopher and diplomat, amused himself with magic squares. At page 251 of Volume 3 of his Collected Works (Philadelphia, 1808) may be seen a square having eight cells on a side, or 64 cells in all, in which the sum of each row and column is 260, while the sum of the numbers in any four adjacent cells is 130. This, however, is not a true magic square, as the sum of the numbers is 292 for one diagonal and 228 for the other. Franklin also devised a square of 2056 cells which is called the "magic square of squares," and a magic circle having many curious properties. Squares having the sum of the numbers in each line greater than %n(n 2 + 1) may be formed by adding an integer / to each number so that the sum of all the num- bers in each line is N = J n (n 2 + 1) + nl. When N is given, values of n may sometimes be found which satisfy ALGEBRA 43 this equation. Thus for N = 1000 and n = 4 the value of / is integral, namely, 187, so that if 187 be added to each of the numbers in Fig. 6, the resulting numbers have the property that the sum of each row, column, and diagonal is 1000. The greatest and least numbers in such a square may be found from the expression N/n ± \ (w 2 — 1); thus for N = 1000 and n — 5 they are 212 and 188. Among other curious squares are those which are filled with the first n 2 natural numbers by the knight's move in chess, each square being occupied only once by the knight. Leonard Euler, the great mathematician, amused himself with such squares and two which he constructed for squares of five and seven sides may be seen in the Encyclopedia Britannica; these squares, however, are not magic, although they have certain curious properties. It is well known that this problem may be solved on the common chess board in many different ways; and one of these gives a square which is magic. CHAPTER III GEOMETRY 56 HE QUEEN of mathematics is the ancient geom- etry as exemplified by Euclid. Elegant, chaste, and beautiful is its logic, wonderful are its con- clusions. It originated in Egypt and came to its development at the great university of Alexandria where Euclid was the founder of its mathematical school. The Greek words jy and nerpov, which form the name of the science, mean land and measure, respectively, so that geometry was originally the measurement of land. In Egypt where the annual inundations of the Nile easily obliterate the boundaries of parcels of land, perhaps the rules of geometry received their first practical application. 57 Euclid lived about 300 B. C. Tradition says that he was mild and unpretending in manner and kind to all genuine students of mathematics. On one occasion a student com- plained that geometry brought no profit, whereupon Euclid directed that three oboles be given him. To King Ptolemy, who asked if there was not an easier way to understand geometry than through study of the Elements, Euclid gave the reply "there is no royal road to geometry." 58 The Elements of Euclid is the title given to that presenta- tion of plane geometry which Euclid prepared about 300 44 GEOMETRY 45 B. C. Written in Greek, it later was translated into Arabic, then into Latin. Translations have appeared in all Euro- pean languages which have been annotated by many differ- ent editors. More than a thousand editions have appeared since the invention of printing about 1480; in fact, no book except the Bible has passed through so many editions as the Elements of Euclid. The first six books of Euclid were generally used in colleges and schools in England and America until about i860. The logic of the presentation is generally perfect, and the treatise gives important facts of plane geometry. It can be successfully used today, for it is certain that it is much better than some texts now on the market and equally as good as many of them. GEOMETRIC AMUSEMENTS 59 The second proposition of the first book of Euclid affords amusement to some beginners because it appears to them that a much simpler method might have been used. The first proposition is to construct an equilateral triangle upon a straight line of given length. The second is "to draw from a given point a line equal to a given straight line"; to do this Euclid lets A be the given point and BC the given straight line; then joining A and B he constructs upon AB the equilateral triangle ABD by the method of the first proposition. R (o From B as a center with a radius BC the circle CC' is described and DB is produced until it meets the circle in E. Then from D as a center with the radius DE a 46 • RECREATIONS IN MATHEMATICS second circle EE' is described and DA is produced until it meets the circle in F. Accordingly, from the given point A the straight line AF has been drawn equal to the given straight line BC. The reader can easily supply the steps ' of the demonstration, but can he state why Euclid did not at once describe from the center A a circle with the radius BC? 60 The fifth proposition of the first book of Euclid has always been known as the "pons asinorum," and it has been gen- erally implied that the boy who failed to understand it was an ass. The word "pons" or - „ "bridge" perhaps originated from the figure ■ which roughly resembles the rude bridge truss used to span narrow streams. 61 The most important proposition in the first book of Euclid's Elements of Geometry is the forty-seventh, namely: The square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. This truth was known to Hindoos and Egyptians long before the time of Euclid, but Pythagoras, who lived 550 B. C, gave a formal demonstration which has caused his name to be frequently applied to the theorem. 62 There are many right-angled triangles having the three sides expressed by integral numbers. The simplest one has 3, 4, 5 for its sides and this was known to the Egyptians who GEOMETRY 47 used it many centuries before the Christian era in con- structing the pyramid ofCephren at Gizeh. If each of these sides be doubled we have 6, 8, io for the sides of a right-angled triangle which has probably been known to surveyors in all ages and which is now constantly used by them in laying off a right angle with the chain. The follow- ing are all the right-angled triangles with sides expressed in integers for which the shortest side does not exceed 15: 3 2 + 4 2 = 5 2 n 2 + 60 2 = 61 2 5 2 + 12 2 = 13 2 , 12 2 + 16 2 = 20 2 6 2 + 8 2 = io 2 12 2 + 35 2 = 37 2 f + 2 4 2 = 2 5 2 13 2 + 8 4 2 = 8 5 2 8 2 + 15 2 = 17 2 14 2 + 4 8 2 = so 2 9 2 + 12 2 = 15 2 15 2 + 20 2 = 25 2 9 2 + 40 2 = 41 2 15 2 + 36 2 = 39 2 IO 2 + 24 2 = 26 2 63 Let m and n be any two integers; then if 2 mn be taken for one of the legs of a right-angled triangle, the other leg is m 2 — n 2 and the hypothenuse is m 2 + n 2 . Thus let m = 9 and n = 4, then the three sides are 72, 65, 97. Let the reader use this rule to determine three right-angled tri- angles each having 48 for one of its legs. 64 The following right-angled triangles with integral sides, have the same area: first triangle, 24, 70, 74; second triangle, 40, 42,^8; third triangle, 15, 112, i|3. 65 A castle wall there was, whose height was found To be just fifty feet from top to ground; 48 RECREATIONS IN MATHEMATICS Against the wall a ladder stood upright, Of the same length the castle was in height. A waggish fellow did the ladder slide, (The bottom of it) five feet from the side. Now I would know how far the top did fall By pulling out the ladder from the wall? 66 fig- iz To divide a circle into three equal parts. One method is as follows: Divide a diameter AB into three equal parts AC, CD, DB; on AC and CB describe the semi-circles shown, also on AD and DB describe semi-circles; then is the area of the given circle divided into three equal parts. 67 When is the sum of the squares of two successive integers a perfect square? This is answered by Osborne in Ameri- can Mathematical Monthly for May, 1914. The two S 2 . Then come Five larger sets smallest are o 2 + i 2 = i 2 and 3 2 + 4? • 20 2 + 21 2 = 2Q 2 and no 2 + 120 2 = 169 2 are also found the largest of which is 803 760 2 + 803 761 2 = 1 136 68q 2 . 68 To divide a circle into n equal parts an approximate solution is the following: Let AB be the diameter of the given circle and CD a diameter perpendicular to AB. Divide AB into n equal parts. On the diameter DC pro- duced lay off CE equal to one-third of the diameter. From E draw a straight line through the second point of division GEOMETRY 49 on AB and produce it to meet the circumference in F. Then the arc AF is the »th part, very closely, of the cir- cumference. 69 The trisection of the angle is one of the famous problems of antiquity. It can be solved in many ways, but not by plane geometry, for Euclid allowed no instruments but a straight ruler and the compasses. However, if it be per- mitted to mark or graduate the ruler, the problem can be solved in the following manner, as was first shown by Archimedes. Let BAC be the angle to be trisected and from A as a center describe a semi-circle with the compasses. Produce the radius BA toward the right. From one end of the ruler lay off on it a distance equal to the radius AC and mark the point thus found. Place the ruler so that one edge coincides with C while the end moves along the produced line BD. When the mark on the ruler coincides with the semi-circle put there a point E, then the arc EF is one-third of the arc BC and the angle EAF is one-third of the given angle BAC. The reader can easily supply the demonstration by considering the dots that have been placed on the figure. 70 THE VALUE OF v Archimedes deduced, about 220 B. C, a rule for the quadrature of the circle, proving that its area is equal to SO RECREATIONS IN MATHEMATICS the square of its radius multiplied by a number which lies between 3 10/71 and 3 10/70. Or, area = ttt 2 , where r is the radius and ir is the number whose approximate value is 31/7. This number ir is also the ratio of the circumference of the circle to its diameter and it turns up in connection with many problems not at all related to the circle. 71 The value of t to four decimal places is 3.14x6. "Yes, I have a number," is a sentence in which the number of letters in each word corresponds to the integers in this value of x. The following, which appeared in the Scientific American of March 21, 1914, will enable its value to be remembered to 12 decimals: See I have a rhyme assisting My feeble brain its tasks sometimes resisting. t C> •> "i 72 An early statement regarding the ratio of the circumfer- ence of a circle to its diameter is found in the Bible in con- nection with the description of Solomon's temple. The architect of this magnificent building was Hiram, a widow's son, whose father was a man of Tyre. In I Kings, vii, 23, and also in II Chronicles, iv, 2, we find the dimensions of a circular tank or pond which was designed by Hiram. "He made a molten sea, ten cubits from one brim to the other; it was round all about and its height was five cubits; and a line of thirty cubits did compass it about." It should not be inferred from this description, however, that the value X = 3 was used in computations by this distinguished architect. The date of the construction of Solomon's temple was about 1007 B. C. GEOMETRY 51 73 The early Romans are said to have used 3 1/8 for the value of ir but Frontinus, in 97 A. D., used 3 1/7, as is seen from the list of circumferences and diameters of water pipes which is mentioned in No. 3. He also used this value in computing areas, as appears from his statement: "The square digit is greater than the round digit by three- fourteenths of itself; the round digit is smaller than the square digit by three-elevenths." The value of v was computed by William Shanks in 1873 to 707 decimal places, surely a great waste of labor, for the most refined computation requires only seven or eight decimals, and in all usual work 3. 141 6 is close enough. It has been proved that the number «■ is incommensurable, that is, the number of its decimals is infinite. THE PYRAMIDS OF EGYPT 74 • It has been claimed that the great pyramid at Gizeh in Egypt was intended to be so built that the length of the four sides of the base should be the circumference of a circle whose radius was the vertical height. Petrie's measure- ments of 1882 give 9068.8 inches for the length of one side of the base and 5776.0 inches for the height of the pyramid when its sides met at an apex. Now 9068.8/5776.0 = 1.5703, whereas \ t is 1.5708. Probably they used 3 1/7 for the value of w; in this case the ratio of one of the sides of the base to the height would have been 11/7 or 1.5 7 14. Petrie's measures of the angle made by the sides of the pyramid with the horizontal gave the mean value 51 52'; this corresponds to 1.5735 f° r th e above ratio. 52 RECREATIONS IN MATHEMATICS The pyramid of Cephren at Gizeh has its sides inclined to the horizontal at an angle of 53° 10'. This corresponds very closely to a slope of 4 on 3, so that the right-angled tri- angle having sides of 3, 4, 5 seems to have been used in its construction. Here the ratio of one side of the base to the height is 6/4 = 1.5 instead of 1.57 as in the other pyramids at Gizeh. Undoubtedly mathematics and astrology con- trolled the design of these pyramids, altho their final pur- pose was for tombs for the kings. So mighty is the great pyramid at Gizeh and so solidly is it constructed that it will 'undoubtedly remain standing long after all other buildings now on the earth have disappeared. 75 Herodotus said that the area of an inclined face of the pyramid was equal to a square described upon its altitude. What value does this condition give for the angle which the plane of a face makes with the base? 76 The King's Chamber in the great pyramid is 10 cubits wide, 20 cubits long, and 11. 18 cubits high. These figures result from Petrie's measurements made in English inches, 20.612 inches being taken for the length of the old Egyptian cubit. The height was hence made one-half of the floor diagonal, so that the three dimensions of the room are 10, 20, I V500 cubits, and the solid diagonal is 62.5 cubits in length. These numbers are proportional to 1, 2, J V5, 25/4. It can hardly be supposed that these dimensions were accidental; they were probably introduced into the design in accordance with some astrological superstition of a mathematical nature. GEOMETRY 53 77 A magnitude or quantity is anything that can be meas- ured. Can a solid angle, like that at the apex of a pyramid, be measured? No one ever spoke of a solid angle as being twice as large as another one. The only measure of a solid angle that has been proposed is the surface of a sphere described from its apex and included between its sides. The radius of the sphere being taken as unity, j x would be the measure of the solid angle at one corner of a cube. What is the measure of the solid angle at the apex of a right cone whose altitude is equal to the radius of its base? 78 THE PRISMOIDAL FORMULA A general method of finding the volume of any of the solids of common geometry is the Prismoidal Formula. Let A and B be the areas of the two parallel bases and C the are aof a parallel section halfway between them; let h be the altitude between the bases A and B. Then the volume of the solid is V = 1/6 h (A + 4 C + B). To apply this to a cone which has a base of radius r and the altitude h, the upper base A is o, since it is at the apex of the cone, the lower base B is irr 2 , and C, the area of a section halfway between the two bases is |ur 2 . Then the volume is V = 1/6 k (o + vr 2 + irr 2 ) = 1/3 Trr 2 h = 1/3 Ah. To find the volume of a sphere draw two parallel planes tangent to it, giving the two bases A = o and B = o; the area of a section halfway between them is irr 2 , where r is the radius of the sphere; also the altitude h is 2 r. Then volume = 1/6 (2 r) (o + 4 irr 2 + o) = 4/3 irr 3 . To find the volume of a masonry pier 16 feet high, the 54 RECREATIONS IN MATHEMATICS top B being a rectangle 8 X 24 feet, and the lower base being a rectangle 12 X 30 feet inside. The areas of the bases are 192 and 360 square feet. The dimensions of a section C halfway between the bases are \ (8 + 12) or 10 feet and § (24 + 30) or 27 feet, so that the area of C is 270 square feet. Then Volume = 1/6 X 16 (360 + 1080 + 192) = 4352 cu. ft. This is a problem which is difficult to solve by the methods of common geometry, for the sides of the pier when pro- duced do not meet at a point, and hence the rule for a truncated pyramid does not apply. - i The prismoidal formula gives volumes of the ellipsoid, paraboloid, and other solids generated by the revolution of curves of the second and third degree about an axis. It also applies to warped surfaces like the hyperbolic parab- oloid when the areas A, B, C are known or can be found. '■■■ Let the student apply the prismoidal formula to find the volume of a segment of a sphere whose altitude is h and the radius of whose base is a. Here a little analytic geometry is perhaps necessary to find C in terms of a and the radius r of the sphere. 79 GEOMETRIC FALLACIES To prove, geometrically, that 24 equals 25; draw a square on cardboard, 5 inches on a side, having an area of 25 square inches, as shown in Fig. 14; then, cut the cardboard into four pieces as indicated by the three broken lines; these four pieces can then be arranged in the rectangular form shown in Fig. 15, where there are three inches on one side and eight on the other, giving twenty-four square inches ia GEOMETRY 55 all. Hence it has been proved geometrically that 24 equals 25. Where is the fallacy? \ \ \ \ \ \ \ \ \ 3 inohea .5 ,.^d.ea « u E •0 3 inches | 2 inches. Fig. 14 6 inches Fig. is 80 To prove that a straight line can be divided into four parts so that the first point is to the third as the third is to the fifth. Let AE be the given straight line which is so divided that AB : BC :: CD : DE; then AB/BC = CD/DE; now cancelling B out of the first member of this equation and D out of the second, there results A/C = C/E, or A : C :: C : E, which proves the proposition enunciated. 81 The semicircumference of a circle is equal to its diameter. Let the diameter be divided into four equal parts and on each part let a semicircle be described. These four smaller semicircles are equal to the given semicircumference; for let d be the given diameter, then \ ird is the corresponding semicircumference; each of the equal parts of the diameter is | d and the corresponding semicircumference is 1/8 ird; hence the sum of these four small semicircumferences is 4 (i/8 nd) or | ird. Now divide each of the parts of the diameter into four equal parts and describe semicircles on 56 RECREATIONS IN MATHEMATICS them and it is clear that the sura of the sixteen semicircum- ferences is equal to the large semicircumference originally given. Thus continue, and when the number of points of division is infinite the sum of all the infinitely small cir- cumferences is, equal to the large original one. But when this occurs all the small semicircumferences coincide with the diameter of the circle and their sum is hence equal to it. Accordingly, it has been proved that the semicircumference of any circle is equal to its diameter. Where lies the fallacy? 82 To prove that any obtuse angle is equal to a right angle. Let ABC be an obtuse angle and DCB a right angle; it is required to prove that these angles are equal. Make CD equal to BA, join AD, bisect it in E and draw EG perpen- dicular to AD. Also bisect BC in F and draw FG perpen- dicular to BC. The two perpendiculars meet in G. Draw GA, GD, GB, and GC; then GA equals GD, and GB equals GC. Since also CD = BA, the sides of the triangle GBA are equal, each to each, to the sides of the triangle GCD. Hence these triangles are in every way equal; and the angle opposite the side GA is equal to the angle opposite the side GD. Accordingly the angle ABG equals the angle DCG; subtracting from these the equal angles CBG and BCG, the result is that the angle ABC is equal to the angle DCB. Therefore, it has been proved that the obtuse angle ABC is equal to the right angle DCB. The fallacy in this demon- stration is not easy to detect, but nevertheless it is there. GEOMETRY 57 83 The circumference of a small circle is equal to the cir- cumference of a larger circle. Let a wheel roll along a horizontal plane until it has made one revolution; then the line AB is equal to its circumference. The small circle in Fig. 17 the figure has also in the same time made one revolution, since it is drawn on the side of the wheel concentric with the larger circle. Hence the circumference of the small circle rolls out the line CD which is equal to AB. Therefore, the two circumferences are equal. Where is the fallacy? 84 Every triangle is isosceles, or two angles of any triangle are equal to each other. Let abc be any triangle. Bisect the angle a; from the middle of be draw a normal to be; the bi- sector and the normal meet at a point. From this point draw lines to b and c and normals to the sides ab and ac. The tri- angles C and D are then equal; also the triangles A and B are equal, whence ad = ae. Accordingly, the third pair of triangles E and F must be equal, whence cd = be. Hence ad + cd = ae + eb, or ab = ac. Thus it has been proved that any triangle is isosceles. Fig. 16 58 RECREATIONS IN MATHEMATICS 85 The three following propositions are certainly interesting. Are they true or false? (i) Let AB be a straight line and C any point on it. On AC and BC as bases construct the isosceles triangles AbC and BaC so that the equal sides make angles of 30 with the bases. Also on AB construct the isosceles triangle AcB so that the equal sides make angles of 30 with AB. Draw ah, be, ca as shown by the Fig. 19 F, 9- broken lines. Then each of the angles of the triangle abc is 6o°. (2) Let ABC be any triangle. On the sides as bases construct the isosceles triangles AcB, BcA, CbA, so that the equal sides of each make angles of 30 with its base. Draw, ah, be, ca, as shown by the broken lines. Then each of the angles of the triangle abc is 6o°. (3) Describe a square on each of the sides of a right-angled triangle. At the centers of these squares put the points a, b, c, and join these points so as to form the triangle abc. Then each of the angles of this triangle is 6o°. 86 The following remarkable fallacy appeared in the Forum of April, 1914: "A cube will readily present to the eye three dimensions, length, height, and breadth. Four GEOMETRY 59 diagonal lines imagined from the comers of the cube will each be at right angles to the other three; hence we have four dimensions. We should find it difficult to construct anything along the lines of these four dimensions for the simple reason that the work would have to begin at the point where the lines intersect and progress outward through within the four lines. We might call these four lines expansion boundaries for if you would cause a cube to expand and maintain its symmetry or proportions, it would expand along these four lines. Any solid can there- fore be considered a cross section of its greater self. The foregoing is the only practical demonstration that can be given of four dimensions." 87 ON THE AREA OF A CLOSED TRIANGLE From Clifford's Common Sense of the Exact Sciences, Fifth Edition (London, 1907), pages 135-137. Hitherto we have supposed the areas we have talked about to be bounded by a single loop. It is easy, however, Rg. ti fig- £2 to determine the area of a combination of loops. Thus, consider the figure of eight in Fig. 21 which has two loops; if we go around it continuously in the direction indicated 60 RECREATIONS IN MATHEMATICS by the arrowheads, one of these loops will have a positive, the other a negative area, and therefore the total area will be their difference, or zero if they be equal. When a closed curve, like a figure of eight, cuts itself it is termed a tangle, and the points where it cuts itself are called knots. Thus a figure of eight is a tangle of one knot. In tracing out the area of a closed curve by means of a line drawn from a fixed point to a point moving around the curve, the area may vary according to the direction and the route by which we sup- pose the curve to be described. If, however, we suppose the curve to be sketched out by the moving point, then its area will be perfectly definite for that particular description of its perimeter. We shall now show how the most complex tangle may be split up into simple loops and its whole area determined from the areas of its simple loops. We shall suppose arrow- heads to denote the direction in which the perimeter is to be taken. Consider either of the accompanying figures (Fig. 21). The moving line OP will trace out exactly the same area if we suppose it not to cross the knot at A but first trace out the loop AC and then to trace out the loop AB in both these cases going around these two loops in the direction indicated by the arrowheads. We are thus able in all cases to convert one line cutting itself in a knot into two lines, each bounding a separate loop, which just touch at the point indicated by the former knot. This dissolu- tion of knots may be suggested to the reader by leaving a vacant space where the boundaries of the loops really meet. The two knots in Fig. 22 are shown dissolved in this fashion. The reader will now have no difficulty in separating the most complex tangle into simple loops. The positive or negative character of the areas of these loops will be sum- GEOMETRY 61 ciently indicated by the arrowheads on their perimeters. We append an example (Fig. 23). ^2} Fig. 23 In this case the tangle reduces to a negative loop a and to a large positive loop b, within which are two other positive loops c and d, the former of which contains a fifth small positive loop e. The area of the entire tangle then equals b+c+d+e— a. The space marked s in the first figure will be seen from the second to be no part of the area of the tangle at all. 88 MAP COLORING It has long been known that only four different colors are necessary in order to color the most complicated map of a country so that contiguous sides of districts shall not have the same color. About 1850 this fact was brought to the attention of mathematicians but, altho much discussion of it has been made, a rigorous proof that only four colors are necessary has not yet been made. Fig. 24 shows a map of nine districts in which the four colors A, B, C, D are used for eight of the areas and there may seem no way to use one of these colors for the other district unless it adjoins upon the same color. However, by the very slight change shown in Fig. 25 the problem is 62 RECREATIONS IN MATHEMATICS readily solved. Thus in all cases a way can be found to color the map by using only four different colors. Fig-. 24 Fig. 25 The reason that five colors are not required seems to be that it is impossible to draw five areas so that a boundary of each sEalTBe contiguous "to the other four. Fig. 26 shows four areas each of which has its boundary contiguous B ( C Eg3 A Fig. 26 JHq.27 with a boundary of the other three areas, but no way can be found to add a fifth area so that it may be contiguous to the other four. Four colors are sufficient for any map be- cause no map has yet been drawn in which five areas are contiguous to four others. But no proof has yet been dis- covered that it is impossible to draw five such areas. The word "contiguous" means that the areas border along a line, not at a point. Districts sometimes occur so that four or more of them meet at a point; for example, in Fig. 27 the two areas colored C meet at a point. Here more than four colors are needed if it is desired to have the areas on opposite sides of the point of junction different in shade. CHAPTER IV TRIGONOMETRY 89 ^HE SOLUTION of triangles was the original ob- ject of Trigonometry, but it has been extended in modern times to include a vast realm of facts regarding functions of angles. The beginner in trigonometry is first introduced to the sine and cosine which are denned by a right-angled triangle in a manner essentially like the following: Let a be the hypothenuse and b and c the legs, and A, B, C the angles opposite to them, then the ratio b/a is called the sine of the angle B, and the ratio c/a is called the sine of C. Or as sometimes stated, the sine of an acute angle in a right-angled triangle is the ratio of the side opposite the angle to the hypothenuse. Thus sin B = b/a and sin C = c/a. Now it has been questioned by H. E. Licks whether this is the best way to define the sine for the beginner. The beginner is young and immature, to him the word "ratio" is more or less of an abstraction, and the fact that this ratio is called the sine does not appear to him significant. Why not state the definition something like this: The sine of an acute angle in a right-angled triangle is a number which multiplied by the hypothenuse gives the side opposite to the angle. This definition puts the matter in quite a different light for it gives the idea that the primary use of the sine is to solve a right-angled triangle, and it states the rule by 63 64 RECREATIONS IN MATHEMATICS which one of the sides may be found when the sine of the opposite angle and the hypothenuse are known. How the sines are tabulated and used is a matter to be explained later. 90 Fifty years ago a very different method of defining the trigonometric functions was in use. Let AOP in Fig. 28 be an angle less than 90 which is measured from the line AO around in a contrary direction to that of the hands of a watch. Let P be any point on the quadrant AB described with the radius OA or OP; let BB' be a diameter normal to AA'. From the point P let perpendiculars PS and PC be y^c i

(P cos0 = i--+---+,etc. (2) f ? ==1+ + i| +•!+£+, eta. 2! 3! 4! In these formulas the values of are to be taken in radians; thus, for an angle of 30 the value of is 1/6 ir, and for an angle of 0° the value of is 7r0°/i8o. In the last for- mula e denotes the number 2.71828 ... or the base of the Naperian system of logarithms. The symbol (!) denotes the product of the natural numbers; thus: 4! = 1 X 2 X 3 X 4 = 24- TRIGONOMETRY 69 Let i denote the imaginary V — 1, and in the last formula change to id; then it becomes e , = I+ „_i;_^ + £i + , et , 2! 3! 4! This may be written in the form \ 2! 4! 6! /■ \ 3! 5! 7! / or e* = cos 9 + i sin 6. (3) Similarly, replacing Oby —id there is found e~' 9 = cos 6 — i sin 6. (4) Adding these two equations and also subtracting the second from the first, there results cos 6 = J (e ie + e~ ie ) i sin = | (e ifl - e"'" 9 ), (5) which are remarkable expressions for the sine and cosine in terms of the imaginary V — 1. These wonderful formulas are due to the great mathematician Euler. What do these formulas mean? 96 COMPLEX QUANTITIES Equation (3) forms a basis for the extensive branches of vector analysis and quaternions, for cos 6 + i sin is what is called a complex number, the general expression for which is a + ib. To define and understand a + ib it is first necessary to understand i. By the rules of simple algebra it is found that: ? = (V~i) 2 = -1, i s = (V~i) 3 = -V^i = -i, # = (\^7) 4 = +1, i 5 = (V^r) 5 = + V-i = +i. Now the following graphic representation agrees with these 7o RECREATIONS IN MATHEMATICS results. In Fig. 33 let a line be drawn from O toward the right to represent +1 and one of equal length be drawn to the left to represent — 1. Also let a line be drawn upward from to represent +i and one downward to represent — i. Now let multiplication by i indicate turning a line of unit length through 90 degrees about the axis O. Then +1 X i = -\-i } or the line +1 has been turned into the position shown by +* in the figure. Also + i X * = — 1 or the line +i has been turned tOjthe position — 1; also — 1 X i = —i, and — i Xi = +1. Thus with this graphic repre- sentation we see at once that i 2 = — 1, i 3 = — i, i 4 = +1 and i 5 = +*. r v > Fkj. 53 XL. J A3 A. °^^\ e, N. A Fig. 34 Now to explain a + ib, let OA in Fig. 34 be laid off to the right to represent +a, and then at A lay off AB at right angles to OA to represent +ib. Then OA + AB repre- sents a + i6, or in other words a + i& locates the point B. But the shortest way to go from O to B is by the hypothe- nuse OB, or by the vector addition OA + AB = 05. Here OB may be called a vector and be indicated in general by R, so that R = a + ib is the vector which locates a point 5, this point being located either by going directly to it by the shortest distance R, or by stepping off a units from O toward the right and then b units upward. When a is negative and b positive a point B 2 in the second quadrant TRIGONOMETRY 71 is located; when both a and b are negative a point B s in the third quadrant is located; when a is positive and b negative a point Bi in the fourth quadrant is located. The complex quantity a + ib is frequently more con- veniently expressed by R = r (cos + i sin 0), in which r is the length of the vector R, while r cos and r sin represent a and 6. If this R is squared it becomes R 2 ="r 2 (cos 2 + i sin 2 0), and if it be raised to the nth power it becomes R" = r n (cos n + i sin n 0), which is known as the theorem of De Moivre. When r = 1 all the vectors are of unit length, and from the above formula (3) it is seen that R = e a or e iB = cos 6 + i sin 0. Hence e ie may be regarded as any radius in a circle of radius unit, this radius making an angle 6 with the positive axis OA. The line OB in Fig. 35 represents such a vector: if this be squared OB revolves to the left through another angle 6 and takes the position OBi. From the last equation several remarkable algebraic expressions may be derived: For 9 = J t, For d = w, For = 3/2 t, For = 2 T, These are wonderful expressions, for e is the number 2.71828; algebraically or numerically they seem incompre- hensible, but by the above graphic method of representa- tion they are clearly understood. p* M i e \w = +»• e* = — I. gfiir = — i. g2«r _ + 1. 72 RECREATIONS IN MATHEMATICS 97 SPHERICAL TRIGONOMETRY The Greek astronomers developed and used spherical trigonometry long before plane trigonometry was known. These scientists were led to study the spherical triangle because it was necessary in the solution of problems in- volving the altitudes, azimuths, and hour angles of the stars and planets. The first tables were those of the chords of the angles. Later the Hindoos introduced the sine instead of the chord. Then the Arabs further developed the theory of both plane and spherical triangles. The sum of the angles of a spherical triangle is always greater than 180 degrees, and the excess over 180 degrees depends on the area of the triangle. A rough rule for the spherical triangles measured in geodetic surveys is that there is one second of spherical excess for each 76 square miles of area. The same rule applies to spherical polygons. Thus, a triangle or polygon of the size of the state of Connecticut has a spherical excess of about 64 seconds. A trirectangu- lar triangle which covers one-eighth of the surface of the earth has a spherical excess of 90 degrees, for the sum of its angles is 270 degrees. When two plane triangles are equal one can always be made to coincide with the other, either by motion along the plane or by turning it over on one of the sides as an axis. But there can be two spherical triangles which have their sides and angles equal each to each, and yet it is impossible by any kind of motions to bring them into coincidence. They must, however, be called equal, since, every part of one is equal to a corresponding part in the other and then- areas are the same. Here is a case where equal things cannot coincide or be imagined to coincide. TRIGONOMETRY 73 98 HYPERBOLIC TRIGONOMETRY Since 1875 there has been developed the interesting subject called Hyperbolic Trigonometry. This has nothing whatever to do with triangles, but it is intimately connected with a rectangular hyperbola. In the circle of Fig. 36 let P be any point on the circle and from it let the perpendicu- lar PS be dropped upon the radius OA ; also at A let the perpendicular AT be drawn until it meets the radius OP produced. Let AOP be the angle 6, then if the radius OA is unity, OS is cos 0, SP is sin 6, AT is tan 0, and OT is sec 0. In the right-hand diagram of Fig. 36 let OA be the semi- major axis of an equilateral hyperbola, and P any point on Fig. 36 the curve. From P drop PS upon OA produced, and from A draw AT perpendicular to AO until it meets OP. Then if AO is unity, OS is called the hyperbolic cosine, SP the hyperbolic sine, AT the hyperbolic tangent, and OT the hyperbolic secant. Let be double the area of the hyper- bolic sector OAP, then OS = cosh 0, SP = sinh , AT = tanh , OT = sech , where cosh means hyperbolic cosine, tanh means hyperbolic tangent, and so on. In academic slang these are often - pronounced cosh, shin, than, shec. In the circle of equals OS* + sf, or cos 2 6 + sin 2 = 1. In the equilateral hyperbola OP 2 equals OS minus SP or 74 RECREATIONS IN MATHEMATICS cosh 2 — sinh 2 4> = i. The letter denotes an angle or a multiple or submultiple of 71-. The letter denotes a number which may have any value. The formulas (1) to (5) in No. 95 are true whether be real or imaginary. Changing to id in formula (5) and then replacing cos id by cosh and — i sin id by sinh 0, they be- come cosh = \ (^ + «-•), ^nh = \ (e e - e~ B ), (6) which are the values of the hyperbolic cosine and sine in exponential form. Here as always, the letter e denotes the number 2.71728, the base of the hyperbolic system of logarithms. Let these expressions be squared and the second subtracted from the first, then cosh 2 — sinh 2 = 1, which agrees with the equation established in the preced- ing paragraph, being here the double of a hyperbolic sector. The computation of the cosine and sine of an angle cannot be made by the formulas (5) since there is no way of obtain- ing the imaginary power e*. But the computation of the hyperbolic cosine and sine is easily made from (6); for instance let = 2, then e 2 = 7.389057 and e~ 2 = 0.135335, hence cosh 2 = 3.626861. As the quantity varies from o to °o, cosh varies from 1 to 00 and sinh 6 from o to °o, while tanh varies from o to 1. There is no periodic repetition of the real values of the hyperbolic functions as is the case with the circular func- tions, but yet they have imaginary periods. / Enough has now been stated to give the young reader a glimpse of the fundamental ideas at the foundation of hyperbolic functions. But, he may ask, of what use or importance are they? The reply to this is, that such func- tions constantly turn up in the solution of practical prob- TRIGONOMETRY 75 lems. For example, take the catenary, which is the curve assumed by a cord or cable suspended from two points and hanging freely under its own weight. Let y be the ordinate or height of any point of the curve above a certain hori- zontal plane, x the distance of the point to the right or left of the lowest point of the curve and c a certain constant, then the equation of the curve is y = \c {f lc + e~ x/c ) = c cosh x/c. This equation in terms of e was deduced a hundred years or more before hyperbolic cosines were ever thought of, but the second form in terms of cosh has' many practical ad- vantages over it. Hyperbolic functions also turn up in the theory of arches^ in the formula for a beam subject to both flexure and ten- sion, in the construction of charts to represent large portions of the earth's surface, and especially in the electrical dis- cussion of alternating currents. The length and areas of many curves which were formerly stated in terms of Naperian logarithms are now more conveniently expressed by hyperbolic functions. CHAPTER V ANALYTIC GEOMETRY 99 ^s^gJESCARTES, a French philosopher, who lived in BIp'trBI t ^ ie ^ rst k a ^ °^ ^ e seventeent h century, de- ^>^wJW vised the method of coordinates by which curves vV -^rl can be graphically represented and their proper- ties be studied through their equations. In this method, as applied to plane curves, two lines called axes are drawn at right angles. Their intersection O is called the origin of coordi- nates. Values of x are laid off parallel to the X-axis, and values of y parallel to the F-axis. Posi- tive values of x are laid off to the right of the F-axis and negative Fig* 3 ? i r -r* • • i ones to its left. Positive values of y are laid off upward from the X-axis and negative ones downward. Thus if a point has the coordinates x = 3, y = 2, it is located at A ; if it has the coordinates x = — 1, y = 1, it is located at B; if it has x = — 2, y = — 1, it is at C; and if itiias x = 4, y = —3, it is at D. 100 The equation of a curve is an equation which gives the relation between the coordinates of any point on the curve. Thus x/4 + y/3 = 1 gives the relation between x and y for 1 76 +y A B. C -Y 0., ANALYTIC GEOMETRY 77 every point; if y = 3 then x = o, if y = 6 then x = —4, if y = —3 then a; = 8. Plotting these three points by laying off their coordinates from the X- and F-axes, it is seen that they are on one straight line. When y = o the line crosses the X-axis at x = 4; when x = o the line crosses the F-axis at y = 3 ; and its inclination to the X-axis is the slope of 3 to 4. Thus any equation of the first degree between two variables is the equation of a straight line. This line is of infinite length for no matter how great y is taken the corresponding value of x can be found from the equation. The equation of a circle is x 2 + y 2 = r 2 and that of an equilateral hyperbola is x 2 — y 2 — r 2 . Equations of many other curves are known to the student who reads these pages. By discussion of these curves we learn their shape, we draw tangents to them at given points, we find where two curves intersect, and later by the help of the calculus we can find their lengths and also determine the areas included between them and the axes. 101 Coordinates are used also for the graphic representation of statistics and natural phenomena. ' For example, let the following be the means of the mean monthly temperatures at a certain place for a series of years: Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. ,20° 28° 28° 34° 47° 6o° 78° 77 66° 47° 38° 25° If the reader will place twelve points at equal distances apart on the X-axis, and then lay off upward the tempera- tures given, 20 for Jan. at the first point, 28 for Feb. at the second point, and so on, and then connect these points with a curve, he will have a graphic representation of the mean 78 RECREATIONS IN MATHEMATICS monthly temperatures throughout the year at the given place. This curve resembles the curve of sines, that is the curve whose equation is y = sin x. In plotting the points through which the curve is to be drawn, it is best to use paper that is ruled into squares. A point to be plotted is sometimes indicated by the notation (5, 3) where the first number is the coordinate x and the second the coordinate y. Let the reader plot the following ten points, numbering them 1, 2, 3, etc., and then join the points by a icurve in the order of the numbers: (1.2, 2.0), (2.0, 1.1), (3.0, 1.0), (4.0, 1.7), (4.1, 2.6), (3.0, 3.1), (2.0, 3.5), (2.1, 4.6), (3.0, 5.0), (3.8, 4.6). 102 Let it be required to find the intersection of the circle % 2 + y 2 = 16 with the straight line 3 x + 5 y = 15. By roughly plotting the circle and the line there will be found the two points (3.9, 0.6) and ( — 1.4, 3.8). But closer values can be found by finding the values of x and y by combining the two equations; this gives 3.95 and — 1.31 for the two values of x, and 0.63 and 3.79 for the corresponding values of y. Now let it be required to find the intersection of the circle as 2 + y 2 = 16 with the straight line x + y = 8. Plotting the circle and the line it is seen that they do not intersect. But combining the two equations there are found for x the two values 4 + V— 8 and 4 — V— 8, and for y the corre- sponding values 4 — \/^8 and 4 + y/^B. These im- aginary values show, of course, that the straight line does not intersect the circle. But have they no other meaning? Yes, they have a definite geometric meaning which will be explained later. ANALYTIC GEOMETRY 79 Let it be required to plot the curve whose equation is Assume values of y and find the corresponding values of x; thus, when y is negative then x is imaginary, when y = o then x is o, when y is very small then x = ±a, when y = J a then a; = ±f a, when y = a then a; = ±| a, when y = 3/2 a then a; = ±| a, as y approaches 2 a then a; approaches o, when y = 20 then a; = ±a, when y is greater than 2 a then a; is imaginary. Accordingly, Fig. 38 represents the real curve. The key-note to the formation of this equation lies in the quantity (y — a)/ a which is ~- raised to the 80th or any other large even power. When this quantity is numerically n g . m less than unity its 80th power is very small; for example, let y = a/1.0, then the fraction is —0.9 and its 80th power is 0.00021. This subtracted from 1 gives 0.99979 th e square root of which is almost 1. Thus on any practicable scale of plotting, Fig. 38 is a proper representa- tion of the equation. 103 TRANSCENDENTAL CURVES Curves involving circular functions are often called transcendental. Like the sine curve, y = sin x, they have a period 2 t and hence repeat themselves in both directions to infinity. For example, the curve whose equation is sin 2 y = sin x sin \ x consists of the series of points, ovals, and lemniscates shown in Fig. 39. Here both x and y are taken in radians. When x = v or 2 t then sin 2 y = o. 80 RECREATIONS IN MATHEMATICS sin y = o, or y = o, x, 2 it, etc., and this gives the vertical column of points. When y = ir or 2 ir then either sin x = o or sin J a; = o, whence a; = J tt, f ir, etc., or 2 ir, 6 n-, etc. When sin a; sin \ x is negative then y is imaginary. Thus with much labor the curves in Fig. 39 are constructed. ■ CXD O ■ OO OO O • CXD Fiq. OO 39 O • OO Re Rg. 40 When the equation is sin 2 y = sin x sin 1/5 x some parts of the diagram undergo great changes and Fig. 40 results. Many beautiful diagrams of such curves, constructed by Newton and Phillips, may be seen in the Transactions of the Connecticut Academy for 1875. The curve tan 2 x + tan 2 y = 100 gives a series of squares spaced like the plan of a rectangular city. When x and y are taken in degrees, each I II I side of a square is 168 36' and the width of t I the streets between the — ■ 1 ' ■ ■ - , Rg. 41 I squares is n 24 The equation y = sin 80 x gives a diagram like the upper one in Fig. 41 while the equation y = sin 81 x gives the lower one. Both of these extend right and left to infinity. The eq uation sin 2 Vx 2 + y 2 = o is satisfied only wh en Vx 2 + y 2 = o°, 180 , 360 , 540°, etc. When Vx 2 + f = 180 then x = 180 and y = 144 or x = 144° and y = 108 . ANALYTIC GEOMETRY 8 1 When Vx 2 + y 2 = 36o°) du — (u/v 2 ) dv. This may be written in the form (vdu — u dv)/v 2 , that is, the differential of a fraction equals its denominator into the differential of its numerator minus its numerator into the differential of its denominator, divided by the square of the denominator. MAXIMA AND MINIMA 109 The differential calculus enables easy the solution of many problems involving maxima and niinima For ex- ample, a tin cylindrical box of diameter a and height h is to CALCULUS 89 be made to contain Q cubic inches of material. If the thickness of the tin is t what must be the ratio of the height to diameter in order that the least amount of tin may be used? Here the quantity of tin is (irah + \ -koF) t, this including the cover of the box; also Q = \ ira 2 h. Taking the value of h from the second equation and substituting it in the first gives (Q/a + \ it a 2 ) t as the expression for the quantity to be made a minimum. Placing the derivative of this equal to zero and solving, there results a 3 = 4 Q/w or Q =\ wa 3 and equating this to the above value of Q, there is found h = a. Hence, for minimum material the height of the box must be equal to its diameter. 110 As a second example, let it be required to find the length of the longest straight stick AB which can be put up a cir- cular shaft in the ceiling of a room, the height of the room being h and the diameter of the shaft a. Here ? it is convenient to let be the angle / which the stick makes with the floor; — S * f<-a-» then AB = h/sin 6 + a/cos 6. Plac- Fi g . 44 ing the derivative equal to zero, there results tan = (h/a) 1 *. Then expressing sin 6 and cos in terms of (h/a)* there is found for the length of the stick AB = (A* + a*)*. This is a simple way to solve a problem which has proved a stumbling block to many. Ill Hundreds of problems similar to the above may be found in books and mathematical journals, hence H. E. Licks gives one not found in books, namely, to determine the path go RECREATIONS IN MATHEMATICS of a ray of light from a source S to the eye at E, when a transparent glass plate is interposed between them. Let s Fig. 45 show the path by the Sx heavy broken line, the light mov- \e*s a , ing in straight lines both within iiJ I a nd without the plate, as is yi i \ known by experiment. Let a, yS i | b, c, d be the distances between _-/_ I S and E measured normal and parallel to the plate. Let be >q 45 the angle which the ray makes with the normal to'the plate before it enters and after it leaves, and 4> the angle which it makes with that normal within the plate. Let vi be the velocity of the light with- out the plate and % the velocity within it. Then the time required to travel from S to E is i '= a sec d/vi + b sec 4>/vz + c sec 8/vi. Also the quantities are connected by the geometric relation a tan + b tan 4> + c tan = d. Now the path must be such as to make the time t a minimum. Hence, if ./V is a constant to be determined, the quantity t = (a + c) sec 6/vi + b sec /v 2 +N [(a +c) tan 6+b tan — d] is to be made a minimum. Differentiating there is found — = (a + c) sec tan d/vi + N (a + c)/co^ 6 = o, dO — = b sec tan <£/% + Nb/cos 2 = o. From the first of these N + sin 0/i>i = o and from the second N + sin 4>/vi = o, whence by ehrnination of N, there is found sin 0/sin = V\/vi. Hence the ratio of the sines of the angles made by the ray with the normal of the plate is CALCULUS 91 equal to the ratio of the velocities of light without and with- in the plate. Thus the path is completely determined in terms of the velocities vi and v . The ratio of sin 0/sin is, in optics, called the index of refraction and its values have been accurately determined by measurements for different materials. Thus when light passes from air into water this index is 1.33, that is, the velocity of light in water is about three-fourths of its velocity in air. 112 THE CELL OF THE HONEY BEE The cell made by the bee in which to store honey is shown in Fig. 46. The end ABDE is the top of the cell which is closed with a plane cap after the cell is filled with honey. The cross-section of the cell is a regular hexagon formed with thin sides of wax. The bottom of the cell abdefg is ter- minated by three equal planes which meet at the apex c and which are rhomboidal in shape so as to form a depressed cup, for the points a, f, d are further away from the top of the cell than are the points b, e, g, and the apex c is still further away. The angles of these rhom- r ' 9 * boids at b, e, g are equal to the angles at c. If a cross- section of the cell be taken anywhere on its length, there results a hexagon each of whose interior angles is 120 , but the six angles in the bottom of the cell at b, e, g, c are only about 110° owing to the inclination of the three planes. It is evident that there is a certain inclination of these planes which will give less material for the cell than if the lower end were made plane like the upper end. To deter- 92 RECREATIONS IN MATHEMATICS mine this inclination is a problem in maxima and minima which has received much attention because the conclusion deduced agrees closely \ with the actual con- struction of the cell. b Fig. 47 gives end and side views of the cell. o Let k be the mean length of the cell, and h — x the length of the side Bb. Regarding abdefg as the cross-section let each of its sides be called r, then be is also r ; but by virtue of the incli- nation of the rhom- bus dbdc the distance be becomes increased as shown in Fig: 48. Let this increased dis- tance be called t. The plane dbdc in Fi g. 47 then has an inclination such that the distance t is vV 2 + 4 x 2 . Now considering the amount of wax in the cell to be pro- portional to the sum of the areas of its sides and bottom, the expression for the total area in terms of x is made a mini- mum. The area of the two sides shown in the side view is r (2 h — x), the area of the inclined rhombus abdc is \ t X r V3 or § r V3 Vr 2 ,+ 4 x 2 , and hence the total area A to be made a minimum is A = 3 [r (2 h — x) + \ r V37 3 + 12 x 2 \. Differentiating this value of A with respect to x, equating the derivative to zero, and solving, gives x = r Vi/& for the \7 48 CALCULUS 93, value of x which renders the quantity A a minimum. For this value of x the area A becomes A x = 3/ (2 h + § r V2) which is proportional to the amount of wax in the sides and bottom of one cell. If the cell had a plane bottom at right angles to the sides, the area of the sides and bottom is found by making x = o in the above expression for A, whence A = ^r{2h-\-\r V3). The ratio of A to A\ now is s = 2k + %rVs _ 4k/r + Vj 2h+%rV2 4 h/r + V2 and the following are values of this ratio for various values of h/r: For h/r = 0, s = 1.225. For h/r = 1, s = 1.072. For h/r = 2, s = 1.034. For h/r = 4, s = 1.018. For h/r =6, s = 1.013. It hence appears that the saving in wax of the actual cell over a cell with plane bottom is 7.2 per cent when h = r, 3.4 per cent when h + 2 r, and 1.8 per cent when h = 4 r. The height of the cell is usually between h = 2 r and h = 5 r, so that the saving in wax is on the average about 2 per cent. Early writers on this problem paid great attention to the angles abd and acd of the rhombus in Fig. 47. The tangent of the angle 0-(Fig. 48) when x has the value r Vi/8 which renders A a minimum, is | r V3/I t, and since t = r V3/2 this tangent is V2. Then by the help of a logarithmic table it is easy to find 0, and its double 109 28' 16" is the angle abd in the inclined rhombus, and this is also the value of each of the angles at the apex c of the pyramidal cup. 94 RECREATIONS IN MATHEMATICS Statements were made that this angle had been measured and found to be 109 28', from which it was concluded that the cell of the bee agreed most closely with that which theory demanded for the minimum quantity of wax. How- ever, evidence regarding these measurements is wanting, and indeed it would be a very difficult matter to measure this angle to such a degree of exactness. This problem first received discussion in the eighteenth century, and writers on it generally extolled the wonderful instinct of the bee in adopting a form of cell which led to economy in wax. The production of wax is an exhausting mttttrttt ttMtt/tt/// f///Jtu Fig. +9 operation for the bee, and moreover sixteen pounds of honey are needed to produce one pound of wax. Economy is hence promoted by any method which will limit the pro- duction of wax to the least possible amount. According to most writers the bee has solved this problem in a most ingenious mathematical manner, and its instinct should be regarded as one of the most remarkable in nature. In order to judge how far these high enconiums are justi- fied, it is necessary to examine the construction of the honey- comb. An inspection of one shows that it is formed by two tiers of horizontal cells with their bases resting on a vertical midrib in which the pyramidal cups are formed. In Fig. 49 the heavy lines of the right-hand diagram give a front view of the cells on one side of the midrib, and the broken lines CALCULUS 95 show the cells on the other side. These two tiers of cells alternate in a curious manner, the bottom of one cell abutting against the bottoms of three cells of the other tier. The cells themselves are either horizontal ot inclined very slightly upward, and the left-hand diagram in Fig. 49 shows a vertical section before they are filled with honey. Both tiers of the comb are supported by the central midrib which is attached to the ceiling of the hive. In building the cells the bees begin at the top and work downward, the base of each cell being of course built before the cell itself. The examination of such a honeycomb will also show that the midrib forming the bases of the cells is thicker than the walls of the cell itself, this probably being so because it is required to carry all the weight of the cells and honey. In fact it has been stated that the midrib is thicker near its top than lower down. The observations of the writer lead to the rough conclusion that the midrib is ij or 2 times thicker than the walls of the cells. This being the case, the above theory falls to the ground as fallacious. Let n be the ratio of the thickness of the midrib to that of the walls of the cell. Then the above expression for the area A becomes A =$r{2h — x + %n V^r 2 + 12 x 2 ). The value of x which ren ders this a minimum is now found to be x = \ r/V$n 2 — 1. From this the following values are found for the angle abd of the inclined rhombus and for the angles at the apex c: For n = 1, abd = 109 28' 16". For n = 1 \, abd = n6°4o' o". For n = 2, abd = 117° 59' 10". For n = 4, abd = 119 38' 58". 96 RECREATIONS IN MATHEMATICS Here the last value is given in order to show that the bottom of the cell becomes practically flat when the midrib is four times as thick as the walls of the cell. For a perfectly flat bottom this angle is of course exactly 120 . While this variation in thickness of the midrib appears to take the problem outside of the domain of pure mathe- matics, yet such is not really the case. Exact observations of the way in which the bees build the comb are needed, as also measurements of the bases of the cells, and perhaps these may be made in the laboratories of natural history. At present the writer offers the following as conjectures: (1) that the cell of the bee is built according to the rules deduced above for minimum material when the midrib is equal in thickness to the walls of the cell, (2) that this shape of the cell is not due to an instinct for securing the minimum quantity of wax, but is entirely due to a method of con- struction which arises from a necessity that the bees in adjoining cells should crowd together as closely as possible. The first conjecture can only be established by measure- ments made on the same midril at both upper and lower parts of the comb, and on different midribs in different kinds of cells. If the angles of the inclined faces of the apex c of the pyramidal cup can be measured, the writer predicts that these will approximate to the °0 0° va l U e 109 28' 16". 6.-.6.-.6.-.0 Concerning the second conjecture c&rPcPc- ^ snou ^ De noted that the midrib O O O O is built by bees which face each _. c „ other in the work as shown in the Fig. 50 left-hand diagram of Fig. 50.- In order that the midrib between the two tiers of cells may be properly compacted it is necessary that the heads of the CALCULUS 97 bees in one tier should alternate with the heads of those in the other tier. In the right-hand diagram the full-line circles show the heads of the bees in one tier and the broken- line circles the heads of those in the other tier. Here it is seen that each bee occupies a triangular position between three other bees, and with this arrangement it is indis- pensably necessary that the bottom of each cell should be a pyramidal cup having three sides. In the theoretic cell deduced at the beginning of this article the inclination of each of the three planes of the bottom of the cell to a cross-section is such that the tangent of the angle is 2 x/r or V§. This corresponds to 35 45' 52". The first conjecture of the author demands that this in- clination should always be 35 45' 52" whatever be the thickness of the midrib. Further investigation of these three planes will show that the diedral angle between any two is exactly 120 , and this in the conjecture of the writer is always closely the case. Let the reader take four spheres of equal size, lay three of them on a table so that each is tangent to the other two, and then put the fourth spheire upon these three. If these points be located and three tangent planes be drawn, it is a simple matter of computation to find that each plane makes an angle of 35 45' 52" with the horizontal, and that the diedral angle included between any two of the planes is 120 . Thus a series of alternating spheres gives the same planes as are found in the cells of the bee; hence one cause of the inclination of the latter is undoubtedly the alter- nating heads of the bees in forming the midrib shown in Fig- 5°- The reason why the cells are hexagonal has often been discussed. All writers are in agreement that this is due to 98 RECREATIONS IN MATHEMATICS the circumstance that each cell is surrounded by six others, and that if any other form than the hexagonal were adopted vacant spaces would be left, which could not be filled with honey. Moreover it is thought by the writer that any cell wall must be built by bees working upon both sides of it. Now in the hexagonal cell the diedral angle between any two adjacent side walls is 120 . At the base of this cell, as we have seen, the diedral angle between any two of the planes forming the pyramidal cup is 120 ; also each of these planes in intersecting a side of the hexagonal cell makes with it an angle of 120 . Hence in the bee cell every diedral angle is 120 . The angles at the top of the cell, where the cap is put on, are not here included; but as long as the bee is in the cell she has only to deal with diedral angles of 120 . The conclusion of this discussion is that the cells of the bee are not built from any instinct for reducing the pro- duction of wax to a minimum, but rather from the necessity that their heads must alternate in forming the midrib in order to properly compact it. This necessity results in planes inclined to each other at angles of 120 . Perhaps it may be said that the bee has an instinct to build planes inclined at this angle, but more properly, it seems to the writer, it may be said that the work of the bees is more easily done in this way than in any other. Economy in labor rather than in material appears to lie at the foundation of the symmetric form of the cell of the industrious honey bee. An interesting critical article by Glaisher will be found in the London Philosophical Magazine for August, 1873, where the history of this famous problem is set forth in full detail. At that date the belief appears to be undoubted that the form of the cell is due to an instinct of the bees for saving as CALCULUS 99 much wax as possible, and this is referred to as one of the most remarkable instances of instinct in nature. Since the discussion here given indicates otherwise, further investiga- tions are in order to fully solve the problem, and these are only possible after many observations and measurements have been made in entomological laboratories. 113 INTEGRAL CALCULUS Differentiation is a definite process and any given func- tion of a single variable can be differentiated. But there is no way to integrate except from a knowledge of what has been done in differentiation. In this respect the two branches of calculus are analogous to involution and evolu- tion in arithmetic; any given number may be raised to a stated power, but when the power is given there is no way to find the root except by guess work and trial. There are about twenty-five fundamental integrals which are known to be correct because the differentiation of them furnishes the given expression with which we start. All the rest of integral calculus consists in reducing the quantity to be integrated to one of the fundamental forms. For instance, / x n ~ l dx equals x n /n because the differen- tial of the latter is #" _1 dx and for no other reason. Simi- larly f siaX'dx equals — cosx because the differential of cos a; is —smx-dx. In all cases the correctness of an integral is to be determined by differentiating it. 114 To the above statement that any function can be differen- tiated there seems one exception. Weirstrass has devised IOO RECREATIONS IN MATHEMATICS a certain series, expressed in symbolic form; for which a derivative cannot be obtained, because in any interval, no matter how small, there are an infinite number of bends of the curve, so that at any given point it is not possible to draw a tangent to the curve. This expression, however, is little more than a curiosity to a beginner. 115 When y - dx is required, y being expressed in positive integral powers of *, then the integral can be directly found from the formula /: -y*2 a*3 mA /yO y.dx =y X -DA + D 2 ^--D 3 ^- + D i --,etc., 2! 3! 4! s! in which A, A, A, are the first, second, and third deriva- tives of y with respect to x. For example, let y = ax 2 + x 3 , then A = 2 ax + 3 x 2 , A = 2 a + 6 x, A = 6, A = o. Then, substituting in the formula, there is found / (ax 2 + x 3 ) dx = 1/3 ax 3 + 1/4 x 4 . Unfortunately this formula does not seem to apply to other functions which have no D equal to o but in each given case we are forced to consult a catalog of integrals, or to reduce the. given function to one whose integral is known. 116 John Phoenix, the first real humorist of America, was a graduate of West Point and hence well versed in mathe- matics. In his essay called "Report of a Scientific Lec- ture," he alludes to the importance of adding a constant to the result of an integration. He says: By a beautiful application of the differential theory the singular fact is demonstrated, that all integrals assume the forms of the atoms of which they CALCUL.US IOI are composed, with, however, in every case the important addition of a constant, which like the tail of a tadpole, may be dropped on certain occa- sions when it becomes troublesome. Hence, it will evidently follow that space is round, though, viewing it from various positions, the presence of the cumbersome addendum may slightly modify the definity of the rotundity. To ascertain and fix the conditions under which, in the definite considera- tion of the indefinite immensity, the infinitesimal incertitudes, which, homo- geneously aggregated, compose the idea of space, admit of the computible retention of this constant, would form a beautiful and healthy recreation for the inquiring mind; but, pertaining more properly to the metaphysician than to the ethical student, it cannot enter into the present discussion. 117 LENGTHS OF CURVES The lengths of nearly all curves are expressed in terms of circular, hyperbolic, or logarithmic functions. Thus, the length of an arc of a circle is always in terms of ir, and the length of an arc of a parabola is in terms of a hyperbolic logarithm. The story is told that a German professor, lecturing to his class two hundred years ago, said that the length of no curve could be algebraically expressed, and that the next day one of the students brought to him the deriva- tion of the length of an arc of the semi-cubical parabola in algebraic terms. This curve has the equation n?y = #\ The derivative dy/dx is 3/2 n~ *x* and the length of an arc between the limits x = o and x = a is: t/o T 4« 27 lo\4» / 27 L\4« / J For example, let the equation of the curve be 4 y = x*, then n = 16, and the length of the arc between the limits of x = o and x = a = 4 is 122/27 = 4-S I 89- Whether the story is true or not, the length of this curve can certainly be algebraically expressed and be computed by simple arith- metic. 102 RECREATIONS IN MATHEMATICS 118 Another curve whose length is expressed by simple algebra is the cycloid. This curve is generated by a point on the circumference of a wheel which rolls along the straight line DE. Thus the point A in Fig. 51 reaches the horizontal line at E when the cir- cle has made half a revolution and in its progress the semi-cycloid APE is described. Let a be the radius CA of the generating circle, and P be any point on the cycloid whose coordinates are x and y, the latter being meas ured downward. Then the length of the curve AP is V8ay and the length of AE is 8 a, expressions of the greatest simplicity. The area between the cycloid DAE and the straight line DE is three times the area of the generating circle or 3 ira?. The cycloid has also interesting properties which will be mentioned later under Mechanics. 119 The lengths of some curves of pursuit are also alge- braically expressible. The simplest case (Fig. 52) is where the hare starts at and runs with uniform speed v on the axis OY while the dog starts at a point A on the X-axis and runs always directly toward the hare with the speed V. When the dog is at P the hare is at Q and the tangent to the curve of pursuit is PQ. Let a be the distance between the initial positions O and A, and let n be the ratio of the speeds CALCULUS 103 v/V. Then the equation of the curve of pursuit, when n is not equal to unity, is a"x 1 ~ n , s? +n an *y ^ ■ -4- — — — -^— ^^^ _J_ — ^^— • 2 (» — i) 2fl"(» + l) I — « 2 For example, let the dog run twice as fast as the hare, or n =\, then the equation of the curve is y = — 7 — a?x 3 H- - a. 3<* a 3 The length of an element of the curve being dx Vi + p 2 , where p = dy/dx, the length of the curve from A to P is found to be algebraically expressed, thus: [°L+m h dx=U-a^-A J" \ dx 2 / 3 3 a l When x = o then y = 2/3 a, and the length of the curve is 4/3 a. Here the dog has run double the distance that the hare has run, and it catches the hare at the point x = o, y = 2/3 a. When n is equal to or greater than unity, the dog can never catch the hare. When n is less than unity the dog will catch the hare. The student in calculus may find it profitable to solve the following problem: Let the dog run 10 feet per second and the hare 8 feet per second, and let a = 720 feet; prove that the dog will catch the hare in 6 minutes and 40 seconds from the instant when the hare starts at O and the dog starts at A. CHAPTER VII ASTRONOMY AND THE CALENDAR 120 STRONOMY is probably the most ancient of the physical sciences, the first facts being ob- served by shepherds who watched their flocks at night. The historian Josephus, in his An- tiquities of the Jews, begins with the creation of the world and follows closely the biblical narrative. Speaking of Phaleg, fourth in descent from. Noah and of his son Tera, who was the father of Abraham, he says: "God afforded them a longer life on account of their virtue and the good use they made of it in astronomical and geometrical dis- coveries." Speaking of the sojourn of Abraham among the Egyptians, he says, "He communicated to them arithmetic and delivered to them the science of astronomy; . . . they were unacquainted with those parts of learning, for that science came from the Chaldeans into Egypt and from thence to the Greeks." 121 The order of the twelve constellations of the zodiac may be remembered by the following ancient lines: The Ram, the Bull, the Heavenly Twins, Next the Crab, the Lion shines, The Virgin, and the Scales, The Scorpion, Archer, and the Goat, The man who holds the watering Pot, And Fish with glittering tails. 104 ASTRONOMY AND THE CALENDAR 105 In memorizing this it is well to note, that the word shines should rhyme with Twins, and Pot with Goat. The order here is from west toward east; when the Ram is setting in the west the Scales are rising in the east, when the Scales are setting in the west the Fish are rising in the east. This is a rough statement only, for at certain seasons of the year less than one-half of these constellations are above the horizon, while at other seasons more than one- half of them are visible at one time. Unfortunately the artist who put several of the constellations of the zodiac on the ceiling of the grand concourse in the Grand Central Station in New York, reversed this order, for there we see Aquarius in the east while the Crab is in the west. The copy from which he worked evidently had been incorrectly made; perhaps he took it from a celestial globe and then turned it around so as to interchange east and west. 122 The greatest of all optical instruments was the reflecting telescope of William Herschel, which was finished in 1789. The tube was forty feet in length, five feet in diameter, and weighed 60,000 pounds. With this telescope, magnifying 6450 times, he discovered two new moons circling around the planet Saturn, and recorded hundreds of new double stars and nebulae. His sister, Caroline Herschel, was his constant companion in all his astronomical labors. William Herschel died in 1822. In 1839 his celebrated son, John Herschel, took down the great telescope, which had then become a victim to the ravages of time and could no longer be used. The long tube was carefully laid upon three stone pillars where it could be preserved as a relic of the past. In the Christmas holidays of that year, John 106 RECREATIONS IN MATHEMATICS Herschel, his wife, and their six children held a family feast in the great tube, and there they sang a song written by him in honor of the occasion: In the old telescope's tube we sit, And the shades of the past around us flit, His requiem we sing with shout and din As the old year goes out and the new year comes in. Merrily, Merrily, let us all sing, And make the old telescope rattle and ring. Full fifty years did he laugh at the storm, And the blast could not shake his majestic form. Now prone he lies where he once stood high And searched the heavens with his broad bright eye. Merrily, Merrily, etc. Here watched our father the wintry night And his gaze was fed by pre-adamite light; His labors were lighted by sisterly love, And united they strained their vision above. Merrily, Merrily, etc. 123 Galileo was the first man who looked at the heavenly bodies through a telescope. It was in 1610 that he saw four satellites moving around the planet Jupiter, and this demolished the theory that the earth was the center around which the planets revolved. These four satellites of Jupiter were the only ones known until 1892, but since then four smaller ones have been discovered. The earth has one moon, Mars has two, Jupiter has eight, and Saturn has nine or ten. The inner moon of Mars is near the planet and has such a high velocity that it rises in the west and sets in the east, while both new and full moon can be observed in a single night. All other known satellites, like our own moon, rise in the east and set in the west. ASTRONOMY AND THE CALENDAR 107 124 BOTANY AND ASTRONOMY If we examine the leafy stem of a plant we shall find the leaves upon it arranged in a symmetrical order and in a way uniform for each species. If a line be drawn around the stem from the base of one leaf stalk to that of the next, and so on, this line will wind around the stem as it rises, and on any particular plant there will be the same number of leaves for each turn around the stem. In the basswood, the Indian corn, and all the grasses, we have the two-ranked arrangement; the second leaf starting on exactly the oppo- site side of the stem from the first, the third opposite the second and hence directly over the first, so that all the leaves are in two vertical ranks, one on one side of the stem and one on the other. Next is the three-ranked arrange- ment such as is seen in sedges; here the second leaf is one- third of the way around the stem, the third one two-thirds, and the fourth one directly over the first. Then in the apple, cherry, and most of our common shrubs, the leaves are arranged in five vertical ranks, and the spiral winds twice around the stem before it reaches a leaf directly over the first one; here the distance between any two ranks, is two-fifths of the circumference of the stem. Then in the common plantain there are eight ranks, and three turns around the stem, so that the distance between any two ranks is three-eighths of the circumference. Now if we express these arrangements by figures, we have the fractions 1/2, 1/3, 2/5, 3/8, in which the denominator expresses the number of ranks and the numerator the num- ber of turns of the spiral line around the stem before it reaches a leaf directly above the one from which it started. Io8 RECREATIONS IN MATHEMATICS Thus 1/2 stands for the two-ranked arrangement where there are two turns. But we notice that the numerator of any fraction is equal to the sum of the numerators in the two preceding fractions, and that the same is true for the denominators. Then the next fraction after 3/8 will be found by adding 2 and 3 for its numerator and 5 and 8 for its denominator, which gives 5/13. Thus we have the following series, 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, etc., and just such arrangements of leaves are found, and no others. The fraction 5/13 gives the law for the common house leek, the others are found in the pine family and in many small plants. The furthermost planet from the sun is Neptune, then follow Uranus, Saturn, Jupiter, the Asteroids, and Mars, then the Earth, Venus, and Mercury. Neptune makes its revolution around the sun in about 60,000 days, Uranus in 30,000 days or 1/2 the time of Neptune; in like manner Saturn's period is nearly 1/3 of that of Uranus, Jupiter's period is 2/5 that of Saturn, and so on until we come to the earth, following closely the same series as given above for the leaves on a stem. Thus the mathematical expression of the arrangement of the leaves of plants is approximately the same as that of the periods of the exterior planets. These arrangements of leaves ensure to plants a better dis- tribution of the light and heat of the sun; the periods of the planets render them stable under the laws of gravitation. Perhaps the botanist, had he known that these figures apply both to leaves and planets, might have foretold the dis- covery of the asteroids or announced the existence of Neptune. ASTRONOMY AND THE CALENDAR 109 125 THE MOON HOAX In 1833 Sir John Herschel sailed from England for the Cape of Good Hope, carrying a large telescope with which to view the southern stars. This was before the times of steamboats and telegraphs so that more than two years passed away before any definite account of the discoveries of Sir John reached England or America. In 1835 the New York Sun published a series of articles, entitled "Great Astronomical Discoveries made by Sir John Herschel at the Cape of Good Hope." In the first article was given a cir- cumstantial and highly plausible account as to how this early and exclusive information had been obtained by the paper. Then comes an interesting account of the inception and construction of the great telescope which he carried to the Cape. The idea of great magnifying power originated, it was said, in a conversation with Sir David Brewster regarding optics. "The conversation became directed to that all-invincible enemy, the paucity of light in powerful magnifiers. After a few minutes silent thought Sir John diffidently inquired whether it would not be possible to effect a transfusion of artificial light through the focal object of vision. Sir David, somewhat startled at the originality of the idea, paused awhile, and then hesitatingly referred to the refrangibility of rays and the angle of incidence. Sir John continued, 'why cannot the iUuminated microscope, say the hydro-oxygen, be applied to render distinct, and if necessary even to magnify the focal object?' Sir David sprang from his chair in an ecstasy of conviction, and leap- ing halfway to the ceiling exclaimed, 'Thou art the man!'" The interest of the reader being thus aroused by this IIO RECREATIONS IN MATHEMATICS imaginary scientific conversation, the article goes on to describe the great telescope which was shipped to the Cape and there drawn by two relief teams of oxen to the place where it was erected. This place was "a perfect paradise in rich and magnificent mountain scenery, sheltered from all winds and where the constellations shone with astonish- ing brilliancy." Here Sir John observed stars and nebulae, but above all he paid particular attention to the Moon. The magnifying power of his telescope was 42,000 times, so that objects on the Moon could be seen as if only six miles away and an object only 18 inches in diameter could be plainly recognized. Hence he clearly saw on the moon "basaltic rock, forests, and water, beaches of brilliant white sand girt with castellated marble rocks." He beheld herds of brown quadrupeds of the bison kind, each animal having a hairy veil over its eyes, and he conjectured " that this was a providential contrivance to protect the eyes from the great extremes of light and darkness to which all beings on the moon are periodically subjected." He also saw a species of beaver which was acquainted with the use of fire as was evident from the smoke that occasionally rose from their habitations. Finally, of course, his search was rewarded by the sight of human beings with wings and who walked erect and dignified when they alighted on the plain. "They ap- peared in our eyes scarcely less lovely than the representa- tions of angels by our more imaginative schools of painters; their works of art were numerous and displayed a proficiency of skill quite incredible to all except actual observers." This hoax was immediately swallowed by the general public and caused much discussion. The Sun issued in pamphlet form an edition of 60,000 copies which were sold ASTRONOMY AND THE CALENDAR I IX in less than a month, and translations of it were made in Europe. In 1859 a second pamphlet edition was issued in New York with illustrations of the moon and with added notes. The author of this most entertaining and successful hoax was Richard Adams Locke, then editor of the Sun. He was engaged in newspaper work for a large part of his life and died in 1871 at his home on Staten Island. An obituary notice describes him as "a warm-hearted man, well read, enthusiastic, and sometimes very eloquent on paper. His habits were rather convivial, but he was just and fearless, full of the best intentions, and overflowing with original inspirations." 126 The planet on which we live claims, of course, a large share of our attention. In 1878 Americus Symmes pub- lished his "Theory of Concentric Spheres," demonstrating that the earth is hollow, habitable within, and widely open about the poles. It contains arguments drawn from the statements of explorers in the polar regions, from the dip and variation of the magnetic needle, from the migrations of fish, from the spots on the sun and from the rings of Saturn. According to this remarkable theory, there are two openings at the poles into the hollow earth, the diam- eter of the northern one being about 2000 miles while the southern one is somewhat larger; the planes of these openings are parallel to each other but they make an angle of 12 degrees with the equator. Capt. Symmes imagined that the crust of the earth is about a thousand miles in thickness but he wisely refrained from giving any account of what is found within the hollow sphere. 112 RECREATIONS IN MATHEMATICS 127 Perhaps the earliest mention of a sun dial is that found in_II Kings, xx, o-n: 9. And Isaiah said, This sign shalt thou have of the LORD, that the LORD will do the thing that he hath spoken; shall the shadow go forward ten degrees or back ten degrees? 10. And Hezekiah answered, It is a light thing for the shadow to go down ten degrees: nay, but let the shadow return backward ten degrees. n. And Isaiah the prophet cried unto the LORD: and he brought the shadow ten degrees backward, by which it had gone down in the dial of Ahaz. On a properly constructed sun dial, such as is described below, the shadow cannot go backward. But a dial having a vertical style or gnomon when tilted from the horizontal possesses the property that the shadow will travel backward for a short time near sunrise and sunset. A dial with a vertical gnomon is, however, quite useless in telling the time of day. 128 THE SUN DIAL Four or five hundred years ago the only way to tell the hour of the day was by looking at a sun dial, for clocks and watches had not then come into use. Fig. 53 shows such a dial, which indicates 2 P. M. by the edge of the shadow cast upon the graduated surface by an inclined gnomon. Of course the sun dial is Flo. 53 useless on a cloudy day, but when the sun does shine it gives apparent solar time with a probable error of about ten minutes, which is sufficiently close for the purposes of agriculture. A sun ASTRONOMY AND THE CALENDAR "3 dial is usually placed in a horizontal plane, but in olden times they were often put on the walls of churches and public buildings, and many such can be seen in Europe even at this day. The board on which the lines of the sun dial are drawn may be of any shape, but in Fig. 54 it is indicated as rec- tangular. This board is to be placed horizontally with its central line NS coinciding with the meridian of the place and is usually observed from its southern side. The shadow of the gnomon AB falls toward the western side in the morning and toward the eastern side in the afternoon. The lines which radiate from the cen- ter A being properly drawn the observer will see the shadow [coinciding with the line 8 at eight o'clock in the morning, with the line 12 at noon, and with the line 3 at three o'clock in the afternoon. When the shadow is one- fourth of the distance from the line 3 to the line 4, the sun time is 3.15 P. M. The gnomon AB must be inclined to the plane of the board at an angle equal to the latitude of the place. Some- times this is a thin sheet of metal ABC fastened onto the board, the edge AB being the true gnomon; sometimes it is a small metal rod AB, the end B being supported by another rod BC. It is essential that the inclination of AB to a horizontal dial plate must be equal to the latitude of the place, or for a dial in any position AB must point to the celestial pole. ii4 RECREATIONS IN MATHEMATICS How to make a horizontal dial: On a smooth board draw the lines NS and EW. The northern end of the line NS is to be numbered 12 for twelve o'clock noon, and the ends EW are to be numbered 6 for 6 A. M. and 6 P. M. The latitude of the place may be taken from a good map with sufficient precision for the construction of a sun dial, or if great precision is required it may be found by an astronomical observation. This latitude X is to be used for constructing the gnomon, and also for computing the angles which the radiating lines of the dial make with the central line NS. To find the angle a which any radiating line makes with the central line AN, let n be the number of hours before or after noon when the shadow should fall on that line; then tan a = sin X tan n 15°. Accordingly, the values of tan a are as follows: For 1 and n o'clock, n = i and tan a = 0.268 sin X. For 2 and 10 o'clock, n = 2 and tan a = 0.577 s ' n ^» For 3 and a o'clock, n = 3 and tan a — 1.000 sin X. For 4 and 8 o'clock, n = 4 and tan a = 1.732 sin X. For 5 and 7 o'clock, n = $ and tan a = 3.732 sin X. For 6 and 6 o'clock, » = 6 and tan a = «> . For 7 and 5 o'clock, n = 7 and tan a •= —3.732 sin X. Now the values of a will be different for different latitudes. The following table gives values of a for three latitudes which have been computed from the above formulas with the help of a trigonometric table. X = 30 ' X = 40 X = so For 1 and 11 o'clock a = 7° 38' 9 46' 11° 36' For 2 and 10 o'clock a — 16 06 20 22 23 S 2 For 3 and 9 o'clock a = 26 33 32 44 37 27 For 4 and 8 o'clock a = 40 S4 48 04 S3 .00 For 5 and 7 o'clock a = 61 49 67 23 7° 43 For 6 and 6 o'clock a = 90 00 90 00 90 00 For 7 and 5 o'clock a = 118 II 112 37 109 17 ASTRONOMY AND THE CALENDAR 115 When the radiating lines have been drawn, the gnomon put in place, and the board neatly painted, the sun dial is ready for erection. The board must be placed duly level with its NS line coinciding with the true meridian, and then, when the sun shines, delighted spectators may compare apparent solar time with their watches and wonder at the scientific skill of the youth who constructed the sun dial. The largest sun dial ever built is at the royal observatory in Jaipur, India; it was erected about 1750 by the Maha- raja Siwai Jai Singh II. Its gnomon is about 175 feet long and this can be ascended by stairs. The shadow of the gnomon falls on a large stone quadrant of 50 feet radius along which it moves at the rate of 2% inches per minute. Jaipur is in latitude 27 degrees north. 129 Southworth, in his "Four Thousand Miles of African Travel" (New York, 1875) gives a novel method of deter- rnining the true meridian: "The Arab when he prays, kneels toward Mecca. It is said that even the youngest never fails to bend, almost accurately, in that direction. Thus, in the form of living flesh, we had the Arab, by whom to find the variation of the compass; and, with the corrected bearing, we could find, when the sun bore due south or otherwise, the true meridian, and consequently noon." 130 The civil day begins at sunset among the Mahomedans and at midnight in Christian countries and is divided into twenty-four hours. The sun dial has been used from a remote antiquity to indicate apparent solar time. Clocks with wheels were devised about 1250 but they did not come ci6 RECREATIONS IN MATHEMATICS into general use until after 1600. It was found that these clocks at some times of the year were slower and at other times faster than apparent solar time. An accurate clock or watch keeps mean solar time, this being the time which would be indicated on a sun dial if the sun were perfectly uniform in his apparent motion throughout the year. The difference between apparent and mean solar time is called the Equation of Time and its values are given in some almanacs under the headings "clock slow" or "clock fast." The following table shows such values to the nearest minute which are to be added to apparent time (or sub- tracted when marked — ) in order to give mean or clock time. 1 Jan. 3 min. 1 May —3 min. 1 Sept. min. 10 Jan. 8 min. 10 May —4 min. 10 Sept. 3 min. 20 Jan. 11 min. 20 May —3 min. 20 Sept. — 6 min. 1 Feb. 14 min. 1 June — 2 min. 1 Oct. —10 min. 10 Feb. 14 min. 10 June — 1 min. 10 Oct. —13 min. 20 Feb. 14 min. 20 June 1 min. 20 Oct. — IS min. 1 Mar. 12 min. 1 July 4 min. 1 Nov. — 16 min. 10 Mar. 11 min. 10 July S min. 10 Nov. — 16 min. 20 Mar. 8 min. 20 July 6 min. 20 Nov. — 14 min. 1 Apr. 4 min. 1 Aug. 6 min. 1 Dec. — 11 min. 10 Apr. 1 min. 10 Aug. 5 min. 10 Dec. — 7 min. 20 Apr. — 1 min. 20 Aug. 3 min. 20 Dec. — 2 min. This table will be useful when one compares his watch with a sun dial. As all the affairs of life are now regulated by clock time, it also explains why the time of sunset appears to rapidly become earlier in October and to rapidly become later in January. The apparent and mean solar time above described is different for places having different longitudes, and in general may be designated as local time. In recent years, owing to the requirements of railroad operation, most ASTRONOMY AND THE CALENDAR 1 17 clocks and watches keep standard time or the local time on a certain meridian. In the United States there are four standard meridians, those of longitude 75 , 90°, 105 , and 120 west of Greenwich. Eastern standard time is mean solar time of the 75° meridian, central standard time is mean solar time of the 90 meridian, mountain standard time is mean solar time of the 105° meridian, and Pacific standard time is mean solar time of the 120 meridian. In going from one of these meridians to the next one, our watch must be set one hour backward or forward according as we go west or east. When a watch keeping standard time is read at a place which is one degree of longitude west of the standard merid- ian it is four minutes faster than mean local time of that place; when the place is two degrees to the westward the watch is eight minutes faster, for three degrees westward twelve minutes faster and so on. When read at places to the eastward it is four minutes^ slower for each degree of longitude. Hence, this must be taken into account also when comparing a watch with a sun dial. 131 DAYS, MONTHS, AND YEARS Julius Caesar, with the help of the astronomer, Siosene- ges, introduced the method of reckoning known as the Julian calendar. The year being 365.2422 solar days, he took 365 such days for a common year and 366 days for a leap year, so that the average length of a year was 365.25 days. This Julian calendar is still in use in Russia and Greece, but it was supplanted in most of Europe in 1582 by the Gregorian calendar. In the Julian calendar all years divis- ible by 4 were leap years; in the Gregorian calendar years n8 RECREATIONS IN MATHEMATICS divisible by 4 are leap years unless they are divisible by 100 and not by 400. Thus, in the Gregorian calendar the years 1600 and 2000 are leap years, but the years 1700, 1800, 1900 are common years. In 1582 the Julian calendar was ten days slower than the Gregorian, after 1700 it became eleven days slower, and since 1900 it has been thirteen days slower. Hence, Jan. 1, 191 7 of the Gregorian calendar corresponds to Dec. 19, 1916 of the Julian. 1752 September hath XIX Days this Year. First Quarter, the 15th day at 2 afternoon. Full Moon, the 23rd day at 1 afternoon. Last Quarter, the 30th day at 2 afternoon. M D w D Saints' Days Terms, &c. Moon South Moon Sets Full Sea at Lond. Aspects and Weather I 2 f g Day br. 3.35 London burn. 3 A 27 4 26 8 A 29 9 11 SA 1 S 38 n % ? Lofty winds According to an act of Parliament passed in the 24th year of his Majesty's reign and in the year of our Lord 1751, the Old Style ceases here and the New takes its place; and consequently the next Day, which in the old account would have been the 3d is now to be called the 14th; so that all the intermediate nominal days from the 2d to the 14th are omitted or rather annihilated this Year; and the Month con- tains no more than 19 days, as the title at the head expresses. 14 e Clock slo. s m. 5 15 9 47 6 27 ' Holy Rood D. 15 t Day 12 h. 30 m. 6 3 10 31 7 18 and hasty 16 g 6 57 11 23 8 16 showers 17 A 15 S. Aft. Trin. 7 37 12 19 9 7 18 b 8 26 Morn. 10 22 More warm iq c Nat. V. Mary 9 12 1 22 11 21 and dry 20 d Ember Week 9 59 2 24 Morn. weather 21 e St. Matthew 10 43 3 37 17 6 9 S 22 f Burchan 11 28 3) rise 1 6 n y. 23 f? Equal D. & N. Morn. 6 A 13 1 52 6 & 24 A 16 S. Aft. Trin. 16 6 37 2 .39 6 O 3 25 b 1 '5 7 39 3 14 26 c Day 11 h. 52 m. 1 57 8 39 3 48 Rain or hail 27 d Ember Week 2 56 8 18 4 23 6 $ 3 28 e Lambert bp.< 3 47 9 3 5 6 now abouts 29 f St. Michael 4 44 9 59 5 55 * k S 30 2, 5 43 11 2 6 58 ASTRONOMY AND THE CALENDAR 119 In Great Britain and its colonies the change of the Julian to the Gregorian calendar was not made until 1752. In September of that year eleven days were omitted from the almanacs. The above is a copy of the calendar for September, 1752, taken from the Almanac of Richard Saunders, Gent., published in London. All English and American almanacs gave similar statements for that month. The Ladies' Diary or Woman's Almanac indulged in poetry, appropriate to the occasion: The third of September the fourteenth is nam'd, For which British annals will ever be fam'd. For by Wisdom and Art to the House made appear The Sun was reduc'd to attend on the Year; His Julian vagaries long time has he known, But has now got a new bridal Year of his own 132 In both Julian and Gregorian calendars the months are those established by Julius Caesar, namely: Thirty days hath September, April, June, and November, All the rest have thirty-one, Save in February which, in fine, In common years, hath twenty-eight, And in leap years twenty-nine. The time when the year began has been different in different countries. In Cassar's reign it appears that March was the first month; thus September was the seventh and December the tenth, as the names imply. The early English almanacs, however, begin the year with January as at present, but the legal year of the British government began on March 25, although March was called the first 120 RECREATIONS IN MATHEMATICS month. In legal and church records prior to 1752, it is common to find dates like Feb. 20, 1695, or Feb. 20, 1695/6, these being intended for the historical or almanac year 1696. 133 A very convenient rule for determining the day of, the week corresponding to the day of the month in any year was given by Prof. Comstock in Science, Nov. 18, 1898. Let F be any year of the Gregorian calendar and D the day of the year. Divide F — 1 by 4, by 100, and by 400, neglecting the remainder in each case. Then find S from F — 1 F-i F-i S = Y +D + - - + - 4 100 400 and divide Shy 7; the remainder gives the day of the week, o indicating Saturday, 1 Sunday, 2 Monday, and so on. For example, take July 4, 1916; here F = 1916, D = 186, (F-i)/4 = 478, (F - i)/ioo = 19, (F - i)/ 4 oo = 4. Then 5 = 2565, and this divided by 7 gives a remainder 3; hence, July 4, 1916 comes on Tuesday. The reason for this rule is clear, if it be remembered that all years exactly divisible by 4 are leap years except when they are even century years, as 1800, 1900, 2000, etc., when they must be divisible by 400; thus the subtractive term (F — i)/ioo prevents the addition of an extra day during such years as 1800, 1900, and 2100, while it also makes only one extra day to be added during the year 2000. Of all the rules for finding the day of the week from a given day of the month and year, this is by far the simplest. For the Julian calendar the following rule may be used to find the day of the week corresponding to a given date. ASTRONOMY AND THE CALENDAR 121 Let F be the year and D the day of the year. Neglecting the remainder in the fractional term, compute S from S = Y + D+ F ~ I -2 4 and divide 5 by 7; then the remainder gives the day of the week, o indicating Saturday, and so on. For example, Columbus discovered America on Oct. 12, 1492; here F = 1492, D = 286, (F — i)/4 = 372, then 5 = 2148 which divided by 7 gives 6 for a remainder; hence America was discovered on a Friday. Again George Washington was born on Feb. 11, 1732; here F - 1732, D = 42, (F — 1V4 = 432, and then 5 1 = 2204 which divided by 7 gives 6 for a remainder; hence, Washington was born on a Friday. The common opinion that Washington was born on Feb. 22 is erroneous. This originated in the idea of irre- sponsible persons that Gregorian time ought to be extended backward into Julian time. This reprehensible idea is founded on no sound principle, and in celebrating the birth- day of Washington on Feb. 22, we all commit grievous error. 134 J. W. Nystrom of Philadelphia devised about fifty years ago the "tonal system" of numeration in which 16 is the base instead of 10 as in the decimal system. The numerals 1, 2, 3, 4, etc., were called An, De, Ti, Go, etc., and new characters were devised for 10, 11, 12, 13, 14, 15. This system embraced also a new division of the year into 16 months, these having the names Anuary, Debrian, Timan- der, Gostus, Suvenary, Bylian, Ratamber, Mesidius, Nic- torary, Kolumbian, Husander, Victorious, Lamboary, Folian, Fylander, Tonborious, the first two letters of each 122 RECREATIONS IN MATHEMATICS month being the names of the sixteen numerals. Nystrom certainly did his work well. 135 Josh Billings in his almanac said that the name February was derived from a Chinese word which meant kondem cold. Josh was right regarding the temperature. At the head of the calendar for July he gives this verse: Young man, let hornets be And don't go nigh the pizen snake too much, For in the month of July They a'in't healthy to the touch. For another month he gives this excellent advice: He who by farming would get rich, Must plow, and hoe, and dig, and sich, Work hard all day, and sleep hard all nite, Save every cent and not get tite. For the first of April he has the following: April Phool was bom this day, A simpleton, but clever, And though 3000 years of age, He's just as big a phool as ever. 136 Comets in ancient times brought great mental distress upon people, for they were supposed to presage war, famine, or pestilence. Even to astronomers the phenomena of the tail being repelled by the sun backward from the nucleus of the comet has been a great mystery, for it seemed to contra- dict the law of universal attraction. Now, however, we understand that the small particles of the tail are driven away from the head by the pressure exerted by the light of the sun, so that the mystery appears to have been solved. ASTRONOMY AND THE CALENDAR 123 Yet even at this day the appearance of a comet incites a feeling of awe, and the words of the poet Holmes arise in the memory: The Comet! He is on his way, And singing as he flies; The whizzing planets shrink before The spectre of the skies. Ah! well may regal orbs burn blue, And satellites turn pale, Ten million cubic miles of head, Ten billion leagues of tail I CHAPTER VIII MECHANICS AND PHYSICS 137 f T IS a misfortune that physicists and engineers teach to students two different systems of units. A boy comes to a technical school, understanding perfectly, from his experience, what is meant by force and what is meant by a force of ten pounds or ten kilograms. The teacher of physics tells him that forces must be measured in poundals or dynes, not- withstanding that no apparatus for measuring forces in such units has ever been made or used. The result is great mental confusion to the boy, from which he does not recover until he joins the class in engineering where he finds that forces are measured in those units to which he had always been accustomed before he entered upon the in- struction of the physicist. All this might be avoided if mechanics were omitted entirely from courses in physics. Surely the subjects of heat, light, sound, and electricity furnish a sufficient field for the physicist, without encroach- ing on the topic of mechanics, which properly belongs to the engineer. 138 The unfortunate equation F = mf comes early in a course in mechanics as taught by a physicist. Here the mass m is measured in units of a standard lump of metal furnished by 124 MECHANICS AND PHYSICS 125 the government; acting for one second on this lump is a force F, which produces the velocity / at the end of that second. More generally / is called the acceleration, or change in velocity in one second, and its unit is one unit of length per second. Let L represent length in general and T time, while M represents mass, then we have F\= ML/T 2 , or force dimensionally equals mass multiplied by length divided by the square of time. The student tries hard to comprehend this, but finds it impossible, for he knows that force is not ML/T 2 and he knows that there is no way to measure a force except by the number of units of force which it contains. The truth of the matter is that the equation F = mf is not true. Experiments and experience teach that mf is proportional to cF where c is a constant, not that mf equals F. When there are two different forces F and G which act at different times on the same body they produce accelera- tions/ and g. Experience and experiments show that these forces are proportional to the accelerations which they produce, whence F/G =f/g. This is a fundamental equation which is entirely correct. If the teacher starts with this, his students will have no confusion of mind. 139 Into an apple cut two holes inclined like ab and cb in Fig. 55. Into each hole put a small quill so that when the string AC is •* A — inserted the friction may be small. Then pull horizontally upon the string '1 by its ends A and C. As the pull increases, the apple will 126 RECREATIONS IN MATHEMATICS be seen to rise vertically by the upward pressure of the string at b; as the pull slightly decreases, the apple will fall. The spectators, who think that the string passes straight through the apple, are filled with wonder at the strange motion of the apple bobbing up and down. 140 CENTER OF GRAVITY Many amusing mechanical tricks depend upon the principle that the center of gravity of a system of bodies always takes the lowest possible position; thus, a system will be stable if its center of gravity, when slightly dis- turbed, tends to fall to its original position. To balance a cork upon the small end of a cane held vertically with that end upward. Put the prongs of two forks into opposite sides of the cork, letting the forks incline downward at angles of about 30 degrees with the vertical. Then the center of gravity of the cork and forks will be below the bottom of the cork, and thus there will be no danger of its falling off the end of the cane. The cane can be carried around held in a vertical position with the cork thus balanced on it. A cork with two forks thus attached may be made to walk along a horizontal bar. Put two pegs of equal lengths into the bottom of the cork to act as legs, one being slightly in advance of the other. Then place the cork with its forks upon the horizontal bar, and set the forks into oscillation like a pendulum, the oscillations being parallel to the plane of the bar. The cork will then be alternately supported upon one of the two legs, and hence will advance or walk along the bar as long as the oscillations continue. MECHANICS AND PHYSICS 141 127 When the vertical line through the center of gravity lies without the base of support, the body will fall over, but when it lies within the base it will not fall. A toy horse standing with only his hind feet on the edge of a table will not fall if a curved wire attached to his breast runs backward and has a ball of sufficient weight at the free end. The horse may be made to rock to and fro without danger of falling, if the center of gravity of the horse and ball always rises when disturbed and if the vertical line through that center does not fall beyond the edge of the table. 142 INERTIA Take several of the round wooden pieces which are used in playing checkers and put them in a vertical pile on a table. Then with a heavy knife blade strike the lowest block very quickly in a direction exactly parallel to the surface of the table. The lowest block will then move out under the impact of the blow but those above it will not be disturbed except that the whole pile will fall vertically to the table. This is an illustration of the doctrine of inertia, for there is no reason why the pile should move laterally unless it receives some impact from the blow; but this does not occur owing to the slight friction between the wooden pieces and to the suddenness with which the force is applied. The principle of inertia is utilized by the Japanese in a simple device (Fig. 57) for preventing the overthrow of 128 RECREATIONS IN MATHEMATICS their pagodas by earthquakes. From the roof A of the pagoda there is suspended a heavy ball B by a wooden pendulum rod. When the earthquake comes the founda- tion of the pagoda is moved laterally to and fro and with it the lower part of the walls. The ball B, however, does not move until the motion can be communicated to it from the roof through the suspending rod. As this is a slow process the top A of the pagoda suffers only a slight lateral motion, and hence the structure is prevented from being overturned by the earthquake. The seismograph \ised for 'recording vibrations due to earthquakes depends upon a similar principle. A heavy ball is so arranged, usually at the end of a horizontal pendulum, that it remains practically at rest while the ground moves laterally from the earthquake shock. Attached to the ball is a pointer touching lightly a sheet of paper on the recording apparatus which rests on the ground or floor. As this paper moves to and fro, the stationary pencil traces a curve which shows the intensity and duration of the earthquake shocks. B Fig. 57 143 The cause of inertia may be imagined to be a change in size or. shape of the atoms of the body due to action of the ether. Thus when a force puts a body in motion the atoms assume new shapes or sizes and thus store up energy. When the moving body meets resistance this energy is expended in overcoming that resistance, and the velocity of the body decreases. When a body comes to rest it MECHANICS AND PHYSICS 129 cannot move again under the action of a force until the atoms have assumed new forms and thus stored up the energy imparted by the force. GRAVITATION 144 Gravitation is the great unsolved puzzle in the mechanics of the universe. The law of gravitation, namely, that any two atoms of matter attract each other with a force pro- portional to the product of their masses and inversely as the square of the distance between them, states merely ob- served facts and gives no clue as to the cause. The word attraction is perhaps an unfortunate one, for it implies that each body pulls upon the other. This might be true if each atom were joined to all other atoms by stretched elastic threads for the transmission of the force, but otherwise it is difficult to account for the force of pull. In fact, instances of pull are rare in mechanics; we say that the horse pulls the wagon, but in reality the horse pushes by his shoulders against the harness. The more rational explanation of gravity is that two bodies are pushed together by pressure exerted upon them from the space beyond their line of junction. To account for this push, LeSage supposed that multitudes of fine particles are moving in every direction through space. If there was only one body in the universe, these particles would impinge upon it from every direction and hence no motion, of the body could occur. But for two bodies, it is plain that each will intercept particles that cannot fall upon the other, so that the bodies will be pushed together. While this accounts for the law of gravitation, it is of course no proof at all of the correctness of the theory, and there is no evidence at all of the fine moving particles. 130 RECREATIONS IN MATHEMATICS Under the hypothesis of an ether which fills all space, the facts of gravitation require that bodies must be pushed together by the pressure of this ether. When two bodies are separated to a distance by applied forces, energy be- comes stored in the ether; when the forces are removed this energy exerts pressures on each body which causes them to move toward each other. This general statement is about as far as we can go in explaining the cause of gravitation, but this rests upon the hypothesis of a universal ether, the existence of which has not been proved by any experimental facts. 145 Many absurd speculations regarding the cause of gravity have been made, and the following, from a pamphlet of 1893 called "Invisible and Visible," is one of the worst. " Gravitation is caused by the earth moving so fast that it draws everything to it, like a train of cars (when you stand close to the track) as it is passing." j Magnetic or electric action can be prevented from being propagated to a distance by screens of suitable material, but nothing has ever been discovered by which the action of gravitation can be screened off. The attraction of the earth acts with the same power upon a body, whether or not other bodies be interposed between it and the earth. Years ago it was recognized that the problem of flying would be solved if by any means a flying machine could be wholly or partially relieved from the attraction of the earth. In 1847 Orrin Lindsay published at New Orleans a pamphlet entitled "Plan of Aerial Navigation, with a Narrative of his Explorations in the higher Regions of the Atmosphere and his wonderful Voyage around the Moon." MECHANICS AND PHYSICS 131 His "plan" consisted in annulling the force of gravity. Well-prepared steel, after being superficially coated, amal- gamated with quicksilver, and then strongly magnetized, proved to be an impervious screen to gravitation. A hol- low box made of these metal plates, rose from the earth; to cause it to descend, a hole was opened in the bottom; to cause it to move laterally, a hole was opened in the side. It is unnecessary to explain here his voyage to the moon. About 1900 there was published a novel by Simon New- comb called "His Wisdom the Defender," in which nights by a huge machine were made by its property of annulling the force of gravity. The inventor and owner made aerial voyages over the earth, and compelled the nations to dis- band their armies under the threat of dropping bombs which would blow their cities into nothingness. Thus this inventor, who was called "His Wisdom," inaugurated upon the earth a reign of universal peace. 146 One of the most interesting papers on the ether of space is that of DeVolson Wood in the London Philosophical Magazine of November, 1885. It is based on the known facts: (1) that the ether transmits light at a velocity of 186 300 miles per second; (2) that the ether transmits 133 foot-pounds of energy per second from the sun to each square foot of the earth's surface. His discussion leads to the conclusions (1) that the mass of a cubic foot of the ether at the earth's surface is 2 X io -24 pounds, (2) that the ether has an elasticity such that it exerts a pressure of 4 X io -8 pounds on each square foot of the earth's surface, (3) that the ether has the enormous specific heat of 4 600 000 000 000, so that to raise one pound of it i° F. would require as much 132 . RECREATIONS IN MATHEMATICS heat as it would to raise 2 300 000 000 tons of water the same amount. This medium, says Wood, will be every- where practically non-resisting and sensibly uniform in temperature, density, and elasticity. In one pound of it there is io 10 times the kinetic energy of a pound of gas. 147 THE"! DIAPHOTE HOAX From a Pennsylvania daily newspaper of Feb. io, 1880. A special meeting of the Monacacy Scientific Club was held on Saturday evening to listen to a paper by Dr. H. E. Licks on the diaphote, an instrument invented by him after nearly three years of study, and now so nearly perfected that he feels warranted in -bringing some few of the results thus far attained to the notice of the public. There were present, besides many scientists of Eastern Pennsylvania, Prof. M. E. Kannick of the polytechnic school at Pittsburg, and Col. A. D. A. Biatic of the Brazilian corps of engineers, who is now in this country making extensive purchases of iron and steel. The meeting was called to order by the president, Prof. L. M. Niscate, who in introducing Dr. Licks made a few remarks, saying that he had had an opportunity to witness a few experiments with the diaphote, and he felt convinced that it would ultimately rank with the telephone, the phonograph, and the electric light as one of the most remarkable triumphs of science in the nineteenth century. Dr. Licks prefaced his paper by saying that the idea of the invention was first suggested to his mind about three years before by reading accounts of some of the early experiments of Bell's telephone, and that a little later when Edison brought out the carbon instrument, his studies had become so far advanced as to assure him of its theoretic MECHANICS AND PHYSICS 133 possibility. By the telephone the sound of the human voice may be heard hundreds of miles away. Why, then, cannot light be transmitted in a similar manner, so that by the use of a connecting wire one may distinctly see the image of the object far removed? This, said Dr. Licks, was the form in which the inquiry first suggested itself to him nearly three years ago, and he felt gratified to be able to exhibit to the club this evening an instrument called the diaphote in which the practical realization of the idea had been in a great measure satisfactorily obtained. The word diaphote, from the Greek dia signifying through, and photos, signifying light, had been selected as its name, implying that the light travelled through or in the wire. Although popularly this might be imagined to be the case, it was really no more so than with sound in the telephone. There the sound waves strike a diaphragm that is set into vibration, and generates induced electricity in the wire, this causing corresponding vibrations in another distant dia- phragm which reproduces similar sounds. In the diaphote, likewise, the waves of light from an object strike a pecul- iarly constructed mirror or speculum which is joined by a wire with another similar speculum; the image of an object in the first modifies the electric current in the wire and passing quickly onward to the receiving instrument pro- duces there a secondary image. The intermediate wire, as in the telephone, may be hundreds of miles in length, yet such is the delicacy of the diaphotic plates that the trans- mitted image of a simple object is almost as distinct as the original, and Dr. Licks feels confident that after the removal of a few obstacles, of a mechanical nature only, the most complex forms will be reproduced with the strictest fidelity as to outline and color. 134 RECREATIONS IN MATHEMATICS The diaphote consists of four essential parts, the receiving mirror, the transmitting wires, a common galvanic battery, and the reproducing speculum. Dr. Licks gave a detailed account of the experiments to determine the composition of the mirror and speculum. For the former he had finally selected an amalgam of selenium and iodide of silver, and for the latter an amalgam of selenium and chromium. The peculiar sensitiveness of iodide of silver and chromium to light has long been known and their practical use in photog- raphy suggested their application in the diaphote. It was found, however, after many experiments, that their action must be so modified that each ray of light should influence the electric current proportionally to its position in the solar spectrum, and the element selenium was selected as best adapted to this purpose. At first a small mirror was employed with only a single wire, but the images in the speculum were confused and indistinct so that it became necessary to make the mirror of pieces each about one-third of a square inch in area and each having a wire attached. In the diaphote exhibited by Dr. Licks to the club, the mirror was six by four inches in size, and there were 72 wires which were gathered together into one about a foot back of the frame, the whole being wrapped with insulating covering; and in reaching the receiving speculum each little wire was connected to a division similarly placed as in the mirror.- From a galvanic battery wires ran to each dia- photic plate and thus a circuit was formed which could be opened or closed at pleasure. Dr. Licks explained how the light caused momentary chemical changes in the mirror which modify the electric current and cause similar changes in the remote speculum, this causing a similar image which may be readily seen or be thrown upon a screen by a second MECHANICS AND PHYSICS 135 camera. He explained how the proportions of selenium should be scientifically adjusted to the resistance of the electric current so as to avoid any blending of the repro- duced images. This, he said, had been the problem which had caused him the most trouble and which at one time had seemed almost insurmountable. At the close of the paper an illustration of the powers of the instrument was given. The mirror of the diaphote, in charge of a committee of three, was taken to a room in the lower part of the building, and the connecting wires were laid through the halls and stairways to the speculum on the lecturer's platform. Before the mirror, the committee held in succession various objects, illuminating each by the light of a burning magnesium tape, since the rays from gas are deficient in actinic power; simultaneously on the speculum appeared the reproduced images, which for exhibition to the audience were thrown on a screen considerably magnified. An apple, a penknife, and a trade dollar were the first objects shown; in the latter the outlines of the goddess of liberty were recognized and the date 1878 was plainly legible. A watch was held for five minutes before the mirror and the audience could plainly perceive the motion of the minute hand, but the motion of the second hand was not satisfactorily seen, although Prof. Kannick by looking into the speculum said it was there quite perceptible. An ink bottle, a flower, and a part of a theater handbill were also shown, and when the head of a little kitten appeared on the screen the club expressed its satisfaction by hearty applause. After the close of the experiments the scientists con- gratulated Dr. Licks on his invention, and the president made a few remarks on the probable scientific and industrial 136 RECREATIONS IN MATHEMATICS applications of the diaphote in the future. With telephone and diaphote it may yet be possible for friends far apart to hear and see each other at the same time, to talk, as it were, face to face. In connection with the interlocking switch system it may be used to enable the central office to see many miles of track at one time, thus lessening the liability to accident. In connection with photolithography it could be so employed that the great English papers could be printed in New York a few hours after their appearance in London. Our reporter also learned that Dr. Licks will lecture on the diaphote next week before the American Society of Arts, and that he will make definite arrangements for the manufacture of the instrument as soon as the seven patents for which he has applied are formally issued. Within a week after the publication of the above article, it was copied in whole or in part by numerous papers throughout the United States, many commenting editorially on the great possibilities of the marvellous diaphote. Some papers said that sunlight would be transmitted by it from the sunny side of the earth to light the side which was in darkness. The New York Times said "the imagination almost fails before the possibilities of what the diaphote may yet accomplish." The only paper which recognized the article as a fake seems to have been the New York World, which said, "the hoax is a clever one and is interesting also as depending for its success upon the opposite of the mistake which was at the bottom of Locke's famous 'Moon Hoax'; it is the misuse of the word mirror in connection with the new 'invention' which has made the miracle of it so accept- able to the public." Within a month after the publication of the diaphote hoax, items appeared in the papers announc- ing the invention in Pittsburg of an instrument called the MECHANICS AND PHYSICS 137 "telephole" by which two persons at a distance could see each other as they talked over the telephone, and by which any written or printed document could be transmitted in- stantaneously to any distance. The inventors of this in- strument, it was stated, had labored many years in making experiments and now success had been attained. While Dr. Licks used 72 wires, the Pittsburg inventors used but one, and their applications for patents were soon to be granted. News of the diaphote soon spread to Europe, and in due time there came back to us stories of wonderful inventions there made. For instance in 1889, the news came that a young German, named Korzel, exhibited an instrument by which a person in one city could read a newspaper held before a receiving plate in another distant city. The secret of this marvellous instrument, it was said, lay in the sen- sitiveness of selenium to the effects of light, its electric con- ductivity changing with the color and intensity of the light which impinged upon the plate. Very curiously all the inventors of such instruments have used selenium since its properties were first utilized in the diaphote by Dr. Licks. Almost every year similar stories have appeared, the most recent being one which was published in the New York Times of May 29, 1914, in the form of a cable dispatch from London. This article states that on the previous day, Dr. A. M. Low, a well-known scientific investigator, lectured before the Institute of Automobile Engineers on " Seeing by Wire." For five years his experiments had been carried on and now he had attained such success that pictures were reproduced at a distance of four miles. His instrument "has a receiving screen consisting of a large number of cells of selenium, over which a ruler is moved rapidly by a small 138 RECREATIONS IN MATHEMATICS motor worked with a current of high frequency and about 50000 volts pressure. The receiver at the other end is made up of a series of telephone slabs of steel, through which the light passes." Perhaps Dr. Low is on the right track, and if his apparatus becomes a verity, then he should give proper credit to Dr. H. E. Licks by calling it the diaphote. 148 THE ONE-HOSS SHAY The secret of successful engineering construction is to make each part of a structure just as strong as the other parts, so that there can be no weak spot where failure may occur. Oliver Wendell Holmes wrote many years ago a delightful poem on this principle. It begins: Have you heard of the wonderful one-hoss shay That was built in such a logical way It ran a hundred years to a day? The "shay" was supposed to have been built by a Deacon in Massachusetts who was resolved that it should be properly constructed. But the Deacon swore (as deacons do) It should be so built that it couldn't break down, "Fur," said the Deacon, " 'tis mighty plain That the weakest spot must stan' the strain, And the way to fix it, as I maintain, is only jest To make that place as strong as the rest." The wheels were just as strong as the thills, And the floor was just as strong as the sills, And the panels just as strong as the floor, And the whipple-tree neither less or more. And the back cross bar as strong as the fore And spring and axle and hub encore. MECHANICS AND PHYSICS 139 The chaise was designed to run exactly a hundred years, and so it did. When that time arrived a parson was riding in it and the catastrophe came. All at once the horse stood still, Close by the meetin'-house on the hill, First a shiver, and then a thrill, Then something decidedly like a spill, And the parson was sitting on a rock, At half-past nine by the meetin'-house clock. You see of course, if you're not a dunce How it went to pieces all at once, All at once and nothing first, Just like bubbles when they burst. End of the wonderful one-hoss shay, Logic is logic! That's all I say. CHAPTER IX APPENDIX 149 JNCE upon a time a man, after much labor, raised a number of two digits to the 31st power this containing 35 digits. Stating this fact to a lightning calculator, he was about to give the long number, when the calculator said that this was unnecessary and that the root was 13. How did he know this? Simply from having committed to memory a table of two-place logarithms and by making a rapid computation from them. Since the given power has 35 digits its log- arithm lies between 34.00 and 35.00. Dividing these by 31 gives 1.09 and 1.13 as the logarithms of numbers between which the root must lie. Then, remembering that the logarithms of 12, 13, and 14 are 1.08, i.n, and 1.15 the com- puter instantly saw that the required number must be 13. Hence, the man who computed that 34 059 943 367 449 284- 484 947 168 626 829 637 was the 31st power of 13 had his labor for his pains, for there was no opportunity to give a single figure of it to the lightning calculator. In fact 13 is the only number of two digits whose 31st power has 35 digits. 150 MERSENNE'S NUMBERS In 1644 Pere Mersenne made certain statements regard- ing numbers of the form 2 P — 1 where p is a prime. These statements seem to be that the only values of p, not greater 140 APPENDIX 141 than 257, which make 2" — 1 a prime, are 1, 2, 3, 5, 7, 13, 17, 31, 61, 127, 257 and that it is composite for all other values of p. Thus, 2" — 1 = 2047 = 23 X 89, and 2 23 - 1 = 2 388 607 =47 X 178 481. How he arrived at these con- clusions is a mystery, but it is supposed to have been through correspondence with the great mathematician Fermat. There are 56 primes not greater than 257. Mersenne's statement has been verified for 38 of these, namely, for ,10 of the twelve values of 2 P — 1 which he stated to be prime, and for 28 of the 44 values which he stated to be composite. Al- though much acute thought has been spent upon them by great mathematicians like Euler and Gauss, yet 18 values of 2" — 1 are yet unverified, namely, for p = 89, 101, 103, 107, 109, 127, 137, 139, 149, 157, 167, 173, 193, 199, 227, 229, 241, 257. Fermat, in 1679, gave a rule for determining factors of the number 2 P — 1. He said, in effect, that if 2 or 8 or 32 be subtracted from a perfect square, the remainder n will generally divide 2 P — 1 when n is a prime and » - 1 is a multiple of p. Thus, from 25 take 2, the remainder 23 divides 2 11 — 1 since 23 is prime and 23 — 1 is a multiple of 11. From 49 take 2, the remainder 47 divides 2® — 1 since 47 is prime and 47 — 1 is a multiple of 23. From 225 take 2, the remainder 223 divides 2 37 — 1 since 223 is prime and 223 — 1 is a multiple of 37. These three illustrations of the process are given by Fermat. The reason for this rule is unknown to me, but following the same line of procedure I take 2 from 169 and 167 is known to be a factor of 2 83 — 1 ; also taking 2 from 361 the remainder 359 is a factor of 2 179 — 1 ; also taking 2 from 441 the remainder 439 is a factor of 2 73 — 1. Further, taking 32 from 121 the remainder 89 is known to be a factor of 142 RECREATIONS IN MATHEMATICS 2 11 — i. But this method breaks down when 32 is taken from 841 ; here the remainder is 809 and is prime and 808 is a multiple of 101, hence it might be expected that 809 is .a factor of 2 101 — 1, but on trial this is found not to be the case, the division yielding a remainder of 491. Fermat's method gives a factor for some values of 2 P — 1 but it fails in others. The factorization of large numbers is a very difficult subject. Some values of 2 P — 1 have large factors; see Bulletin of American Mathematical Society for December, 1903, where Cole shows that 193 707 721 and 761 838 257 287 are the factors of 2 67 — 1. For a very interesting his- tory of the work done on Mersenne's numbers see Ball's Mathematical Recreations and Essays. 151 In 1850 the Rev. T. P. Kirkman proposed the following problem in the Lady and Gentlemen's Diary, an annual published in England: A schoolmistress takes her fifteen girls out for a walk every day in the week; they are arranged in five rows, each row containing three girls; how can they be arranged for a full week so that no girl will walk with any of her schoolmates more than once? This is generally known as Kirkman's School Girls Prob- lem, and it has been discussed by many mathematicians. The following is Kirkman's solution: Sunday Monday Tuesday Wednesday Thursday Friday Saturday fl'lfl2fl3 ' Ol&lCi d\d\&\ aib^dz aiC2«2 a-ibzez aiCsd3 &1&2&3 A2&2C2 azdvez dibsds 026363 azb\ex ctiCidi C1C2C3 a^dsei O3&3C8 a^Ciei aabidi dzCiP/% 036262 didid 3 badies dsbid b\c%ei Cibsdi baCsdi C2&361 »iejej , Cidtfi ezbzCi dieses e-ibids e^cids o"2&ie2 APPENDIX 143 He also showed that there are four other solutions, so that the schoolmistress might take out her fifteen young ladies every day for five weeks without any girl walking with any of her mates more than once in a triplet. Ball's Mathe- matical Recreations and Essays devotes 31 pages to this problem but gives no clear solution of it. In 1862, Sylvester claimed that the girls could walk every day for thirteen weeks under the final condition of the problem. 162 DETERMINANTS , A determinant is an abridged notation for certain alge- braic operations to be performed. The theory arose from ' the formulas required for solving simultaneous equations of the first degree. Thus, when there are two equations con- taining two unknown quantities, aix + hy = c\, atfc + b 2 y = i-8), (5-2, 2.0), (5.5, 2.5), (7.0, 2.6), (7-5, 2.6)TT7^7T2.9), (7-3, 2.9), (l-o, 3.5), (7-°, 4-S)- When 148 RECREATIONS IN MATHEMATICS the reader has drawn this interesting curve, let it receive careful study. 157 The camber of a bridge is a slight upward curve given to the floor so that the structure may have the appearance of strength and stiffness. In the days of early American engineering an excessive camber was given to a certain railroad bridge; it is said that the superintendent received reports that the piers were sinking so as to leave the middle of the structure higher up. Whether this story be true or not, it is certain that the following was clipped from a news- paper printed in 1879 in the Pennsylvania German region along the Delaware River: "It's warm, Louis, ain't it," said Tod Hartzell to Louis Rapp, as they met on the Delaware bridge yesterday. "Oh, veil, it is," said Louis, "how much you vay now Tod?" "Only 288 pounds," said Tod. "I can beat that," said Louis, "for I vay 294 pounds." The bystanders, by mental arithmetic, added the weights of the men, and then hurried from the bridge which cracked and groaned under the enormous load, while Louis and Tod gracefully moved from the center of the arch which then sprung back to its original position. 158 The following fallacy is taken from an old newspaper where it is dignified by the title "A Scientific Lecture on Glass." A neat, simple, and quick way of punching a hole through a glass plate is, I venture to say, ladies and gentlemen, unknown to most 7 of you. Nothing can be easier than to punch such a hole and at the same time to cause the utter destruction of the glass, but it is not of this that I am to speak, but rather of a simple scientific operation by which anyone may punch or drill a small hole through a plate of window glass, without injuring it or cracking it in the slightest degree. The tools necessary for this purpose are two sets of punches, an old file, and a heavy hammer, which every mechanic possesses. Armed with these, each of you may become skilled in this most interesting and useful accomplishment. APPENDIX 149 The thicker the pane of glass on which you are to operate, the easier is the process. Having selected it, you choose a place not too near the edge, and with the end of the old file scratch two marks upon it crossing each other like the letter *. Then turn the plate over and precisely opposite scratch a similar cross. Next select two set punches of the same size and fasten one of them securely in a vise. Let an assistant hold the plate in a horizontal position with the lower cross resting exactly on the fastened vertical punch, while you with the left hand hold the other punch on the upper cross, and with your right hand grasp the heavy hammer. You then elevate the hammer, but when you strike be careful to give only a moderate blow, for a violent one might cause the destruction of the glass. The effect of the blow, if it be scientifically directed, will be to cause a very slight indentation in the glass. Then let your assistant turn the plate over and again balance it upon the fastened punch, while you with the hammer repeat the careful blow. The indentation will now be more marked than before, and by repeating the process half a dozen times a hole will be made entirely through the glass plate which will be as finely cut as if produced by a swiftly moving rifle ball, while no crack will appear. When I was captured by the Waldamites this accomplishment proved of the greatest benefit to me, and in fact enabled me ultimately to escape. These singular people, although excellent glass workers, knew no way of cutting it, and when they saw how readily I punched such holes, they not only made obeisance before me, but what was better, they gave me boiled rice and roc's eggs to eat, which were very acceptable, as for six weeks I had eaten only roots with now and then an herb. QUOTATIONS 159 I3i» rots opdoyotovutis Tpiytavbis to airo rrjs rr/v b6rp/ ymvuar viro- ruvownp irXevpds rerpaymvov urov tsri rols airo twv rqv opOrjv yoviav 7repiexouow irkevpoiv rerpaydvbis. Euclid, Elements, Book I, 300 B. C. 160 lias kvkXjos wos ejn rpiywvta op6oya>vuif ov rj pkv e/c nivrpov urrj jtxta 7-0)1' wept ri]V SpGr/v, 17 8e irepirjerpos T-q fiaau, Archimedes, Measurement of the Circle, 220 B. C. Translation by Haurer, 1798: Jeder Kreis ist einem rechtwinklichen Dreyek gleich, dessen eine Seite um den 150 RECREATIONS IN MATHEMATICS rechten Winkle dem Halbmesser und die andere dem Umfang des Kreises gleich ist. 161 Sic incertum, ut, stellarum numerus par an impar sit ... . Cicero, Academia, about 50 B. C. 162 This is the third time; I hope, good luck lies in odd numbers. . . . They say, there is divinity in odd num- bers, either in nativity, chance, or death. Shakespeare, Merry Wives of Windsor, 1599. 163 And now we might add something concerning a certain most subtile Spirit which pervades and lies hid in all gross bodies, by the force and action of which Spirit the particles of bodies mutually attract one another at near distances and cohere, if contiguous; and electric bodies operate to greater distances, as well expelling as attracting the neighboring corpuscles; and light is emitted, reflected, refracted, in- flected, and heats bodies; and all sensation is excited, and the members of animal bodies move at the command of the will, namely, by the vibrations of this Spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain to the muscles. But these are things that cannot be explained in a few words, nor are we furnished with that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elastic Spirit operates. Newton, Principia, Book III, 1687; American edition, 1848. APPENDIX 151 164 Vedete hora quanto mirabilmenti si accordano c61 sistema Copernicano queste tre prime corde, che da principio parevan si dissonanti. Di qui potra instanto . . . vedere con quanto probabilita si porsa concludere, che non la terra, ma il Sole sia nel centra delle conversioni de i pianetti. E poiche la terra vien collocata tra i corpi mondani, che indubitatamente si muovono intorno al Sole, cioe sopra Mercurio, e Venere, e sotto a Saturno, Giove, e Marti, comme parimente non sara probabilissimo, e forse neces- sario concedera, che essa ancora gli vadia interno? Galilei, Third Dialogue, 1630. 165 In philosophia experimentali, propositiones ex phaenon- enis per inductionem collectae, non obstantibus contrariis hypothesibus, pro veris aut accurate aut quamproxime haberi debent, donee alia occurrerint phaenomena per quas aut accuratiores reddantur aut exceptionibus obnoxiae. Newton, Principia, Book III, 1687. 166 It is said that the Egyptians, Persians, and Lacedaemo- nians seldom elected any new kings but such as had some knowledge in the mathematics; imagining those who had not, to be men of imperfect judgements, and unfit to rule and govern. Though Plato's censure that those who did not under- stand the 117th proposition of the 13th book of Euclid's Elements ought not to be ranked among rational creatures, was unreasonable and unjust, yet to give a man character 152 RECREATIONS IN MATHEMATICS of universal learning, who is destitute of a competent knowledge in the mathematics, is no less so. ^Franklin, Usefulness of Mathematics, 1735. 167 Dieu parle, et le chaos se dissipe a sa voix: Vers un centre commun tout gavite a la fois. Ge ressort si puissant, l'ame de la nature, Etait enscveli dans une nuit obscure: Le compas de Newton, mesurant l'univers, Leve enfin ce grand voile, et les cieux sout ouverts. Voltaire, Letter to Madame Chatelet, 1735. 168 On s'imagine que toutes ces etoiles, prises ensemble, ne constituent qu'une tres-petite partie $e l'univers tout entirer a l'egard duque(ces terribles distances ne sout part plus grandes qu'un grain >de sable par rapport a la terre. Toute cette imrnensible est l'ouvrage du Tout-Puissant, qui governe e^galement les plus grandes corps, comme les plus petits, et qui dirige le succes des arfnes, auquel nous sommes int6r6sses. Euler, Letters to a German Princess, 1760. 169 Geheimnissvoll am lichten Tag Lasst sich Natur des Schleiers nicht berauben, Und was sie deinem Geist nicht ofienbaren mag, Das zwingst du ihr nicht ab mit Hebeln und mit Schrauben. Goethe, Faust, Part 1, 1790. 170 Der Gebrauch einer unendlichen Grosse als eine Vollen- deten ist in der Mathematik niemals erlaubt. Das Unend- liche ist nur eine Faeon de parler, indem man eigentlich von Grenzen spricht, denen gewisse Verhaltnisse so nahe kom- APPENDIX 153 men als man will, wahrend anderen ohne Einschrankung zu wachsen verstattet ist. Gauss, Letter to Schumacher, 183 1. 171 Then Rory, the rogue, stole his arm round her neck, So soft and so white, without freckle or speck; And he look'd in her eyes, that were beaming with light, And he kissed her sweet lips — don't you think he was right? "Now, Rory leave off, sir, you'll hug me no more, That's eight times today that you've kissed me before." "Then here goes another," says he, "to make sure, For there's luck in odd numbers," says Rory O'More. Lover, Rory O'More, 1839. 172 There are terms which cannot be denned, such as number and quantity. Any attempt at a definition would only throw a difficulty in the student's way, which is already done in geometry by the attempts at an explanation of the terms point, straight line, and others, which are to be found in treatises on that subject. A point is defined to be that "which has no parts and which has no magnitude"; a straight line is that which "lies evenly between its extreme points.'"' ... In this case the explanation is a great deal harder than the term to be explained, which must always happen whenever we are guilty of the absurdity of attempt- ing to make the simplest ideas yet more simple. De Morgan, On the Study of Mathematics, 1831. 173 All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim 3Eg4 RECREATIONS IN MATHEMATICS of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. Maxwell, Faraday's Lines of Force, 1853. Pour les astres en g6neral at pour les grand Cometes en particulier, trois mille ans ne sont pas grand' chose: dans le calendrier de l'eternite" c'est moins qu' une seconde. Mais pour l'homme vous savez comme moi, mathematicien lectfurer, que trois mille ans c'est b'Btocoup, beaucoup! Flammarion, Recits de rinfini, 1892. 175 There still remain three studies suitable for freemen. Calculation in arithmetic is one of them; the measurement of length, surface, and depth is the second; and the third has to do with the revolutions of the stars in reference to one another. Plato, Republic, 350 B. C, Jowett's Translation, 1894. 176 The heavens themselves, the planets, and this centre, Observe degree, priority, and place, Insisture, course, proportion, season, form, Office, and custom, in all line of order; And therefore is the glorious planet, Sol, In noble eminence enthron'd and spherM Amidst the others; whose med'cinable eye Corrects the ill aspects of planets evil, And posts, like the commandment of a king, Sans check, to good and bad: but when the planets In evil mixture to disorder wander, What plagues and what portents? what mutiny? What raging of the sea? frights, changes, horrors, Divert and crack, rend and deracinate The unity and married calm of states Quite from their fixture. Shakespeare, Troilus and Cressida, Act I, Scene 3, 1602. APPENDIX 155 177 Lassune' ke' nipune' ani tis de machir mirive' iche manir se' de evenir tone chi amiche ze forime' to viche tarvine. Flournoy, Des Indes a la planete Mars, 1900. 178 The following is one of the many stories told of "old Donald McFarlane," the faithful assistant of Sir William Thomson: The father of a new student when bringing him to the university, after calling to see the Professor (Thomson) drew his assistant to one side and besought him to tell him what his son must do that he might stand well with the Professor. "You want your son to stand weel with the Profeesorr?" asked McFarlane. "Yes." "Weel, then he must just have a guid bellyful o' mathe- matics!" S. P. Thompson, Life of Lord Kelvin, 1910. 179 Todhunter was not a mere mathematical specialist. He was an excellent linguist; besides being a sound Latin and Greek scholar, he was familiar with French, German, Spanish, Italian, and also Russian, Hebrew, and Sanskrit. MacFarlane, Ten British Mathematicians of the Nineteenth Century, 191 6. 180 The Appendix is the Soul of a Book. Old Proverb, n. d. D. VAN NOSTRAND COMPANY 25 PARK PLACE NEW YORK SHORT-TITLE CATALOG OF Publications f Importations OF SCIENTIFIC AND ENGINEERING BOOKS This list includes all the books published by the M. C. CLARK PUBLISHING CO., Chicago, III., and the technical publications of the following English publishers: SCOTT, GREENWOOD & CO. JAMES MUNRO & CO., Ltd. CONSTABLE & COMPANY, Ltd. TECHNICAL PUBLISHING CO. ELECTRICIAN PRINTING & PUBLISHING CO., for whom D. Van Nostrand Company are American agents Descriptive Circulars sent on request. November, 1916 SHORT=TITLE CATALOG OF THE Publications and Importations OF D. VAN NOSTRAND COMPANY 25 PARK PLACE The books published by the MYRON C CLARK PUBLISHING CO., Chicago, 111. are included in this catalog. Prices marked with an asterisk (*) are NET All bindings are in cloth unless otherwise noted Abbott, A. V. 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