Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Morrison, Duncan Grant
Master of Arts - MA
University of British Columbia
[Abstract not available]
The Development of Symbolism In Elementary Mathematics. by Duncan Grant Morrison A Thesis submitted for the Degree of Master of Arts In the Department of Mathematics. The University of British Columbia April 1930. Table of Contents. page. PART I. Introduction. 1 Egyptian Number Symbols. 2 Babylonian " " 3 Syrian, Phoenician, Hebrew Number Symbols. 6 Grecian Number Symbols. 7 Roman Number Symbols. 9 Peruvian, Aztec, Maya Number Systems. 12 Chinese and Japanese Number Systems. 13 Hindu-Arabic Number Systems. 15 Summary. 19 PART II. Symbols for addition and Subtraction. 21 Symbols for Multiplication and Division. 26 PART III. Symbols for Equality. 32 Symbolism used in Representing Ratio, Geometric and Arithmetic Proportion. 37 PART IV. Symbols used for representing the Unknown and Powers of the Unknown. 47 Symbols for Representing Powers of Known Quantities. 66 Conclusion. 67 Bibliography. List of Charts. page. Chart I. Number Symbols. 19a. Chart II. Symbolism for Addition and Substraction. 25a. Chart III. Symbols for Multiplication and Division. 31a. Chart IV. Symbols for Equality. 36a. Chart V. Symbols for Ratio, Geometric and Arithmetic Proportion. 41a. THE DEVELOPMENT OF SYMBOLISM IN ELEMENTARY MATHEMATICS. This paper outlines the genesis and development of mathematical symbolism. The treatment will be divided into four main sections as follows: fart I - Symbolical Representations of Number. Part II - Symbols Used to Represent the "Four Fundamental Operations." Part III - Symbols Used to Represent Equality, Ratio and Proportion. Part IV - Symbolical Representations of the Unknown and rowers of the unknown; of Powers and Roots of Known Quantities. - PART I -SYMBOLICAL REfRESENTlON OF NUMBER. The first topic to consider in the study of mathematical notation must necessarily be the symbolism used in number systems for had there not been developed symbolical represen-tations of numbers, the need for further notation would never have arisen. This section, therefore, contains a brief review of number systems. It will not be possible in this enumeration to begin with the "first number system," or to speak in positive terms of the "first use of a certain principle or symbol." This is due to the absence of authentic records of early people, and also to some extent to the mysticism surrounding early mathematics. Especially in the urient, the latter had a very great influence for as late as the 17th century Seki Kowa, a great mathematic-ian of Japan, revealed his secrets to only a few of his disciples and then made them promise never to make public these discoveries. This discussion is limited by emphasizing only those phases which seem to have the most direct bearing on future developments. A great many theories regarding the origin of the various symbols have therefore been omitted and the space devoted to form of symbols, time and place of first recorded appearance, principles used in the formation of number systems, and in some cases a brief comparison of different symbols and systems. As the majority of the systems considered used the decimal scale, when no reference is made to scale this may be inferred. With this brief word of introduction, a consider-ation will be made of the following systems: A. Egyptian F. Roman. B. Babylonian. G. reruvian, Aztec, Maya. C. Phoenician and Syrian. H. Chinese and Japanese. D. Hebrew. 1. Hindu-Arabic. E. Grecian. A. Egyptian Number Symbols. A system of number symbols v^hich is thought to be one of the oldest in the world is that found on Egyptian tablets of wood and stone dating 3,300 B.C. The picture symbols then used are shown on Chart I, the symbol for I, being a staff, for 1,000 a lotus plant, 10,000 a pointing finger, and 1,000,000 a man with both arms raised in astonishment. These first Hieroglyphic signs were angular in form, but gradually this type yielded to the Hieratic, which was merely the rounding off of the earlier symbols through the rapid , manipulation of the reed pen. About the 8th Century B.C., a still more abbreviated form the Demotic came into use. F. Cajori in "The History of Mathematical Notation," Vol.1, p.12 gives these forms. Traces have been found of systems using as base the numbers 5, 12, 20 or 60, but in the three forms mentioned above the decimal scale is employed. In the actual writing of the symbols, the additive principle was used, the symbol for 2 being the symbol for 1, once repeated. Not more than four of the same kind were grouped together - the symbol for 5 (see Chart) was thus written with three unit symbols followed by two unit symbols with a space between the groups. It is interesting to note from this that the Egyptians realized the disadvantage of having a large number of repetitions of unit symbols to represent a higher number. The Romans avoided this difficulty by writing V in place of five strokes. Although the Egyptian symbols are listed first, it is not to be inferred from this that they were the first symbols. Indeed it would seem that a system as well developed as this could not come into being out of nothingness, but must have been preceded by inferior ones. That is that there must have been a gradual development and not a sudden creation of such a splendid system. It can merely be said that these are the earliest symbols on record. B. Babylonian Number Symbols. One of the most valuable number systems ever developed was that first used in Babylonia around 2,400 B.C. The characters, as can be seen from Chart I, were very easily formed, being merely marks made with the old writing implement, the "Stylus." The imprint of this implement was made upon clay tablets, the tablets were baked and so there exists a perfect record of this early symbolism. The marks being mostly wedge-shaped in form, caused the name "cuneiform" to be applied to this system. The origin of the symbols seem veiled in doubt. Quite possibly the Babylonians used their fingers for counting, and the first signs may have been merely pictures of fingers. Mr. Grotefend believes that the symbol for ten originally represented two hands in prayer, the palms being pressed together and the thumbs out. The scale used was the sexagessimal, that is the base was sixty; the symbol for one thus represented 60, 3, 600, or the n-th power of sixty. A consideration of the principles used in the formation of the Babylonian number system is interesting in that several rules are followed which do not appear in latter systems. There may be noted the following: 1. In numbers up to 100, the additive principle was used. That is, a number of symbols placed in juxtaposition represen-ted the sum of the symbols (See symbol for 2). 2. The Subtractive principle. This principle is, when two symbols are in juxtaposition, the value of the symbol is equal to that of the larger minus the smaller symbol. As a rule the order is changed, that is, the larger symbol appears in the place usually occupied by the smaller and vice-versa. In Babylonian notation this principle was seldom used. Ocassionally, however, 19 was written 20 lal 1. The lal indicated 'minus' and so this may be regarded as a crude example of this principle. 3. Multiplicative principle. This principle is, instead of using a new symbol to represent large numbers, one of the smaller valued symbols is used with some distinctive mark to 5 indicate that the value represented is to be obtained by multiplying the original symbol by value of this mark. In the Babylonian system, in numbers over 100 a smaller number symbol placed to the left of a larger indicated that the product of the larger by the smaller was to be taken as the value of the combined symbol. (See symbol for 1,000). 4. The Local Value principle. Quite briefly this principle was, that the value of the symbol was indicated by its position relative to other symbols. In Babylonian notation some use was made of this principle for as was noted above, the symbol for one also could represent higher values, the context being used to determine which was intended. These principles have been given a general definitions, and in the consideration of other systems they will be referred to merely by name. In the system just considered, the chief disadvantage is found in the fact that such a large number of different symbols required. In some cases also there seems to be some difficulty in being certain of the value to assign to symbol. In many ways, however, the numerical notation was very splendid, representing a very great acuity in the formation of symbolism. This is exemplified in the use of the local value principle. It is interesting to note that the Babylonians also had to some degree the concept of zero, in that they possessed a symbol for "absence of number." (This was not used for computations). On the whole it may be said that the Babylonians possessed an adequate notation in that it met their needs. 6 C. Syrian and Phoenician Number Symbols. A brief note on Phoenician and Syrian symbolism will suffice. Each used the additive principle, and in the Syrian, multiplication by ten was indicated by a dot over the symbol. Chart I shows the forms; but as no great advance was made in symbolic representation by these, further investigation will not be attempted. The slowness with which improvements in notation have been made and accepted is clearly exemplified in the fact that the Syrian system was used in the 7th century A.D. four thousand years after the Egyptian first used symbolic representation for numbers. The Syrian system shows traces of a binary scale, based on the number two. D. Hebrew Number Symbols. The Hebrews had an unique contribution to make in that they were the first to use an entire alphabet for designating numbers. The numbers 1, 2, 3.... 10, 20.... 100, 200, 300, 400 were represented by means of the 22 characters of their alphabet. All numbers between 1 - 400 were thus possible. Over 400, two schemes were used. At first two symbols placed side by side. Later "end forms" (See Cajori, Hist, of Math. Nots.) of five letters of alphabet were used for 500, 600, 700, 800, and 900. Thousands were indicated by two dots over symbol. As in the Egyptian, the Hebrew system was influenced by cursive writing and so passed through three stages, the final, being the Rabbinic. Since the Greeks also used the alphabetical system of number representation, a discussion of the relative advantages, of this system and the hierogylyphic will be postponed until Grecian symbolism has been analyzed. E. Grecian Number Symbols. Ancient Greece produced many great mathematicians and one would expect that Grecian number symbolism would represent a great advance. An investigation will reveal whether or not the expectation is realized. Around 1500 B.C. a very progressive civilization existed on Crete, an island near Greece. Here was developed a number system, but as the island was under Egyptian control, and the symbols showed the influence of Egypt, it would not be correct to call this system Grecian. The first truly Grecian symbols were those developed about 600 B.C. 'ihese are known as the "Herodianic Signs," being named after Herodianus who described them in 200 B.C. In form these "Signs" were quite simple being merely the initial letters of the numerical adjectives. By using combinat-ions of the symbols for 1, 5, 10, 100, 1,000 and 10,000, any number up to 100,000 could be expressed. As Chart I shows, fractions were included. Ihese "Herodianic Signs" were used for several centuries when they were replaced by the alphabetical symbols. Cajori suggests that "the commercial intercourse of the Greeks with the other nations brought about the change," but that leaves unanswered the question, why did the Greeks follow the idea of the Hebrews rather than of the Phoenicians or Syrians? The new Grecian symbols were the twenty-four letters of the Greek alphabet with four additional antique characters; old 8 van, koppa and sampi; these three being used for 9.90, 900 respectively. By means of these twenty-seven characters any number up to 1,000 could be represented. The thousands from one to nine were indicated by a dash to left of symbol. The influence of the Hebrews may be seen in the use of the mark for multiplication by 1,000. Ten thousand was represented by M and 20,000, 30,000 etc. were indicated by writing OL , etc. above M. Fractions in this system were indicated by writing the numberator first with an accent followed by the denominator written twice with two accents after each. Unit fractions were indicated by writing the denominator once with two accents after it. Partiality for unit fractions is further indicated by the fact that 1/2, 1/3, 1/4, had special symbols. It has been noted that the Grecian system was quite similar to the Hebrew, using alphabetical symbols, and the principles of addition and multiplication. Neither used the principle of local value although the Greeks approached the idea in their use of the abacus. On the whole the study of the Grecian system is quite disappointing in that so few improvements are found over existing quotations. A brief note on the relative merits of the early "Signs" and the later alphabetical system concludes this discussion of Grecian Symbolism. 1. The one and only advantage that can be seen in the alphabetical representation is that for a number such as 1739, only 4 symbols are required, while in the older system 12 are needed. 2. The early system is superior in that a certain analogy exists between the addition of tens and the same hundreds, in the alphabetical scale this disappears. See chart. 3. The greatest superiority seen however in the Herodianic signs is in the fact that only six symbols were needed to represent numbers up to 100,000 whereas in the alphabetical system 27 symbols were needed for numbers below 1,000. Despite the fact that the new Grecian system was a backward step it was used for a very long time, indeed in the 16th Century A.D. a book appeared using these alphabetical symbols. Note. Fink in his Geschichte der Elementar-Mathematik speaks of a third Grecian system appearing around 500 B.C. This also used the alphabet for symbols but the letters were chosen at random and the order determined arbitrarily. F. Roman Number Symbols. The Romans never used their alphabet for numeral purposes and so avoided some of the difficulties of the Hebrews and Greeks. The earliest forms were those used by the Etruscans, who ruled Rome until 500 B.C. (see Chart I). It is said that these numerals resemble the letters of the Etrurian alphabet. The old Roman forms are quite similar to these and so have not been listed. The origin and development of the Roman numerals as they now exist are subjects about Which few facts are shown and many hypotheses made. D.E. Smith in "History of Mathematics" Vol.11, lists among others, the following suggestions: 1. "Decussare theory" for the origin of X as symbol for ten. This theory is that the symbol X came from crossing out ten single strokes by a line (decussare - to cross or cut) and then abbreviating this to X. Half of the X would be a V or an inverted V and so these two came to be used as signs for five. 2. Karl Zangemeister in 1887 advanced the theory that the entire Roman system was based on the "decussare principle." A crossing line would thus multiply any number by ten. From this the symbols for 1,10 and 20 would be I, X, and XX, respectively, the XX being abbreviated or changed rather fromK made by two vertical lines with a common crossing one. The symbol for 100 was obtained by drawing a line through X the sign for ten, and then altering this until it resembled a C. This last hypothesis seems very fanciful. 3. A great many writers fall back on the old Roman and Etruscan forms, and attempt to explain the later symbols from these, for example, they claim that the symbol for 100, the C, came from the Etruscan sign resembling "theta." 4. Other historians claim the signs for five and ten came from old picture symbols of hands, the picture symbol of one hand developing into a V and the symbol of two hands into two V's or anX. As each theorist rejects all previous theories, a further investigation would not be fruitful. Much more worth while will be a consideration of the principles used in the formation of the Roman number symbols. There are noted the following: 1. The additive principle is used in earlier systems, but is here used very often. In fact some writers speak rf it as being the dominating principle in the Roman system. 2. The subtractive principle is also used considerably, if a letter is placed before another of larger value, the value of the symbol is obtained by subtracting the smaller from the larger. Cajori finds that this subtractive principle was seldom used by early Roman writers, and not used to any great extent until after the middle ages. 5. There are occasional uses of multiplication in the formation of this system. Thus VM sometimes means 5,000 and not (1,000-5). Most authors state that the horizontal bar or viniculum was used over a number to represent multiplication by one thousand. D.E. Smith however claims that the viniculum was never used for this purpose. Fractions were included in the Roman system, special partiality being shown to duodecimal ones, it is suggested as a reason for this preference, that most common fractions such as 1/2, 1/5, 1/4 etc. are more easily expressed as twelfths than as tenths or twentieths. The Roman numerals represent a great advance in notation being easily formed, distinctive, and not leading to ambiguity. Because of these facts the system has been used very widely, indeed, as late as 1275 the Roman notation was the one commonly used in Europe although the Hindu-Arabic numerals were known long before this date. The chief disadvantages of the Roman system are: 12 1. The difficulty of expressing large numbers. Local value principle is not used. 2. System not well adapted to computations. G. Peruvian, Aztec and Maya Number Systems. Before considering the nindu-Arabic numerals a brief review will be made of certain other number systems. 1. Among many Indian tribes, knots have been used to represent numbers, several schemes have been used, the most common being the "quipu." This is merely a twisted woolen cord the strands of which are of different colours. Knots are tied on the various strands and the value of the number represented by the knot is determined by the colour, size and relative positions of the knots. Among Peruvian and North American Indians the quipu described above has been very commonly used for represent-ing numbers. 2. From the chart a fairly good idea can be obtained of the Aztec system of numbers. A dot symbolized a unit. Dots were repeated until ten or twenty were reached, the symbols for ten being a lozenge, for twenty a flag, une hundred was a feather with only quarter its barbs, two hundred was a feather with half its barbs, or more exactly half a feather. For three and four hundred, the symbols were feathers with three-fourths and all their barbs respectively. (See Chart). The systems considered above are interesting in that many different bases are used. There are traces found of ternary quaternary, octonary and vigesimal systems, the bases being 3. 4, 8 and 20 respectively. The latter, that is the system ! 13 with 20 as base, is as D.K. Smith says "a reminder of the barefoot times, when men counted on their toes as well as their fingers." 3. The vigesimal system of numerical hieroglyphics used by the Maya tribes of South America about the beginning of the Christian era, can be seen on the chart. It is especially interesting because it is one of the very few systems using the local value principle. A zero symbol resembling a half closed eye is also introduced. H. Number systems of China and Japan. 1. There seems to be great difficulty in discovering the early number forms of China. This difficulty is caused quite largely by the fact that in 213 B.C., the Emperor of China, Shi Huangti ordered all books and manuscripts to be burned and thus nearly all early records were lCBt. It is agreed, however, that the earliest form of numerical representation used was that by means of knots, the colour, size and relative position giving the value. (C.F. Indian "quipu"). From the chart can be seen the forms of the numerals as used in three different Chinese systems. The first, (Old Chinese) numerals were written vertically from the top of the page to the bottom. In the Scientific or "Rod Numeral" system vertical lines ware as the symbols for one to five, five itself being indicated by five parallel vertical lines. For the wymbols six to nine, the one to five symbols were used with a horizontal line above. A single horizontal line represented ten. The hundreds again were indicated by vertical lines. So the same sign represented units and hundreds, and the symbol for tens also represented thousands. In many ways this system shows similarity with the Hindu-Arabic, being not greatly inferior to the system now in use. These Chinese "rod" numerals were used in 403 A.D. and are found in writings of the 19th Century, giving an idea of the persistence of notation. The modern Chinese symbols as given by D.E. Smith are used exactly as the Hindu-Arabic numerals 1-9, the value being determined by the relative positions. The main difference is that in the Chinese notation the numbers may be written from left to right or from top to bottom. The excellence of the Chinese system has been a very great advantage to Oriental mathematicians. The influence of the Chinese system on the Japanese will be noted in the next section. 2. Japanese Number Symbols. It seems very peculiar that historians have not recorded more fully the developments of the Japanese number symbols. For it seems reasonable to expect that a people who in 1670 had developed a native calculus, would also have many contrib-utions to make in the field of notation. F. Cajori however merely remarks, "the Japanese make use of Old Chinese numerals, but have two series of names for the symbols one indigenous and the other derived from the Chinese language." This appears to be a very inadequate treatment, but other historians, Ball, Fink and others neglect the subject entirely. A book by D.E. Smith and Yoshio Mikami, on "The History of Japanese Mathematics"adds a few interesting facts. Speaking of the introduction of the Chinese rod numerals into Japan, Smith and Mikami say, "The date of the introduction of the Rods into Japan is unknown, but at any rate from the time of Empress Suiko 593-628 A.D., the "Chikusaku (bamboo rods) were used. These were thin round rods about 2mm. in diameter and 12 cm. in length, but because of their liability to roll, they were in due time replaced by "sangi" pieces, square prisms about 7mm. thick and 5cm. long. By means of these "sangi," numbers were represented in the same manner as in the earlier Chinese system, the units hundreds, and ten thousands being represented by vertical rods and the tens, thousands, hundred thousands etc. by horizontal rods. Later pictures of these rods or merely lines were drawn on paper to indicate the numbers. In the 17th Century, Japan produced one of her greatest mathematicians, Seki Kowa. This splendid scholar invented the native calculus known as "yenri," or the "circle principle." In representing numbers he used the Chinese ideograms, but wrote the numbers at the side of a vertical line, and thus introduced "bosho shiki" or side notation. Fractions were represented by writing the denominator, a vertical line and then the numerator. His representation of the unknown and powers of the unknown will be referred to later. I. Hindu-Arabic Number System. There have been many volumes written concerning these numerals but in this treatment only a very brief summary can be attempted. There will therefore be no effort made to list the many theories for the origin of the symbols; and the various forms used will be referred to, with very little elaboration. The aim will be to discover those features of 16 the system in which improvements were made over contemporary systems, and those features which were merely borrowed. There will also be noted some of the developments made since the ninth century B.C. Avoiding the mooted problem of origin, the question of greater importance is "when did the perfected Hindu numerals first appear?" Although Brahmagupta (b.598 A.D.) gives rules for computing with a zero, thus indicating the existence of a system with this important symbol, there is little to indicate that his system was the Hindu Arabic, in fact most historians place the time of the first appearance of the time number system in the year 876 A.D. Cajori lists the following characters with the date of appearance. 1. Sanskrit letters 2nd Century A.D. 2. Apices of Boethius end of mil die ages. 3. Gobar numerals of West Arabs. 4. Numerals of East Arabs. 5. Numerals of Maximus Planudis 14th Century. 6. Devanagari numerals. 7. From Mirrour of world, Published 1480. 8. From arithmetic of Wagner 1488. 9. From De Arte Supputandi by Tonstall 1522. A great many historians have used the resemblances seen among these above symbols to prove that the systems were directly connected, that is that they all belonged to a "royal" lineage with the Sanskrit letters as the progenitor. Several authorities including Cajori, doubt this assertion however and show that the resemblances seen between the ancient Sanskrit alphabet and forms of the Hindu Arabic are not sufficient to validate this theory. 17 One form of numbers, the Gobar dust numerals (sometimes spelled Ghubar) H? very interesting. This system was discov-ered by Mr. Silvestre de Sacy on an old manuscript and has been further explained by Alexander von Humboldt. In these number symbols the group signs are dots or zeros, the two being identical. Quoting from Karpinski, a book on Hindu Arabic numerals "A dot above a character indicates tens, two dots hundreds, and so on, i.e., 5 with one dot above indjcates 50, ti with two dots above 500, three dots 5,000. It is suggested that these dots sprinkled like dust above the numerals gave rise to the word"Gobar". A more possible origin of the name is that the symbols were first written on the old Brahmin counting table, a slab covered with dust, and that the name Gobar arose from this connection. Regarding the time of appearance this brief note from Vol. I Introduction to History of Science. "By middle of 10th Century a special form of Hindu numerals was in use in Muslim Spain. These numerals were developed (without zero) in Spain and were explained quite fully by Gerbert who was Pope Sylvester (930-1003)." The Devanagari numerals, (the name meaning townscript of the Gods) resemble the present Arabic numbers to a much greater degree than the Gobar. In this Devanagari system the symbols for one, two, three, six, and seven are indeed very similar to 1, 2, 3, 6, 7 respectively. Without considering the various dates suggested for the first appearance of the present "Hindu Arabic" system a brief vK.Hb*-matte note^on the forms used for zero. The earliest form found is the dot used by the Hindus. In the 12-13 centuries the symbol 0 was used with a horizontal or starting line drawn through it. The Greek letters "theta," symbols resembling T and "phi" and a circle with a horizontal line above it were all used at some time. Michael Rolle (1652-1719) used "theta." It is not known when or where the present symbol for zero 0 was first used. A symbol which was used at first with the Roman numerals and later with Arabic is the Spanish "calderon." This symbol, which resembles the letter U was used in Mexico Uity in 1649, VllUCCXClps or 7u291c standing for 7,291 pe^os. This letter symbol was used before 1600, only with the Roman numerals and in this notation it supplied to some extent the need for the local value principle. The same symbol is found in manuscripts from Spain and Northwestern Italy, in every case representing "thousands" as in manuscript of 1649. In Portugal a symbol resembling $ called "cifrao" was used for the same purpose. The historical development of the Hindu-Arabic numerals has been treated very fully by many historians, an exceptionally fine review being made in book referred to above (D.E. Smith and L.C. Karpinski "The Hindu-Arabic numerals"). As a further investigation of the subject is here impossible this section will be concluded by listing some of the advantages of the Hindu-Arabic over previous systems as evidences of superiority must be included. 1. In Hindu-Arabic only 10 symbols needed to represent any number. For comparison note Grecian system. 2. Computation much easier than in any previous system. Compare multiplication of 27 by 8, with XXVII by VIII. 3. Multiplication table smaller than other systems using local value principle. For Babylonian system a table extend-ing up to "60 times 60" would be needed. Summary. It has been noted that the early number forms were merely bars or strokes and that the only principle used in the form-ation of the early systems was the additive. For indicating large numbers this proved very inadequate and so the princip-les of multiplication and division came to be used. The Hebrews and Greeks introduced a new idea using their alphabets to represent numbers. The many difficulties attached to such a method have been noted previously. The most valuable scheme of all seems to be that of using the local value principle. In this as has been stated, the value of the symbol depends on its position relative to other symbols. This principle was employed to some extent in the Babylonian and Maya systems, but the bases of these scales were sixty and twenty respectively and so required many symbols and large multiplication tables. The Hindu-Arabic system however, was based on the decimal scale and as the local value principle was used throughout only ten symbols were required. Thus it is seen that by means of this valuable principle any number can be expressed clearly and definitely by means of a very few symbols. ) 9 3 . CHART I. SYSTEMS OF NUMBER REPRESENTATION PEOPLE TRIE NUMBER SYMBOLS EGYPTIANS 33oo B.s. 1 2 5 10 20 100 1,000 0 36 n F y H * a Î BABYLONIANS 2400 B.C. Y YY YYYYf < « Y^-PHOENICIANS 250 B.C. ! H ! ! Ni — — SYRIANS 6-7 Cent. A.D. ! M - A 0 HEBREWS 1400 B.C. p n t ^ B GREEKS 600 B.C. f t/ T T A A A H X later (X <3 € L K ÇL ROMANS before 500 B.C. < ! ! Vor/\ X X X e later 1 n V X c M PERUVIAN 11-lb Cent.A.D. SEE NOTE ON "QUIPU" * ' ! AZTECS 11-12 Cent.A.D. - 1 - 0 F MAYA First Cent. ? * * - — — CHINESE A three systems B C ?? J i + ?? ! !f 3 f ?? / N m n — — ! — JAPANESE 393 A.D. — ^ % HINDU-ARABIC 9 th Cent. 1. 5 1 0 I O O . i . o o o While it is possible that the future may produce some system, which by its ease of number representation and suggestion of the number in the symbol may supplant the Arabic system, for the present the system in use seems to meet all the requirements and therefore may be said to be adequate. 26 - PART II -SYMBOLS USED TO REPRESENT TEE FOUR FUNDAMENTAL OPERATIONS. The number systems discussed in fart I were valuable in themselves, but had it not been possible to manipulate these number symbols according to certain rules, the science of mathematics would never have progressed very far. This section, therefore, will deal with the signs used in representing what are known as "the four fundamental operations" - viz. addition, subtraction, multiplication and division. It has seemed advisable to group these, considering first, the symbols for addition and its inverse operation subtraction; and then the symbols for multiplication and its inverse division. Accompanying the discussion will be charts indicating briefly the symbols used, time and place of introduction, and originator if known. Cajori lists a number of unsuccessful attempts to introduce other symbols for these operations. These seem to have no influence so have been omitted from this treatment. A. Symbols used for addition and subtraction. 1. The earliest symbol found representing minus, has been referred to in Part I. This ideogram, see Chart 2, was used by the Babylonians over 4,000 years ago. 23 2. In the Ahmea Papyrus (c. 1700 B.C.) "a pair of legs walking forward" was the symbol for addition, "a pair walking away" the symbol for subtraction. To represent addition of unit fractions,the fractions were placed side by side with the larger one first, the others being arranged in descending order of magnitude. 3. Diophantus a Greek living in the 3rd century A.D. used the Egyptian method of juxtaposition of symbols to represent addition, but extended it to integers as well. The chief disadvantage of the scheme was that it necessitated grouping positive and negative quantities. For subtraction Diophantus used the symbol seen on Chart which in form resembles "an inverted psi with the top shortened." Symbols used occasionally in Greek mathematics are also shown in Chart 2, the slanting line for addition, and the semi-elliptical curve for subtraction. A combination of the two was sometimes used for total result. 1 4. In Bakshali Arithmetic (8-9-10th Century)"YU" was used as a symbol for addition a n d + for subtraction. The latter sign is exactly like the modern sign for addition but appears after, rather than between the quantities; thus 12 7_+. 1 represents 12-7. Cajori refers to the subject as follows. "This is a sign which I have not met with in any other Indian arithmetic, The sign now used is a dot placed over the number 1. Transliteration. to which it refers, here therefore is a mark of great antiquity. As to its origin I am not able to suggest a satisfactory explanation." It is interesting to note that the Hindus used the dot to represent a negative quantity. This seems to be the first appearance of a symbol to represent the qualitative rather than the quantitative concept of number, i.e., the dot did not stand for subtraction but merely indicated that the number was negative. 5. In the 15th Century a rather unusual sign for "minus" was used by A1 Qalasadi an Arabian mathematician. The symbol is shown on Chart 2. For addition however there was no symbol, the older method of juxtaposition of quantities being used. 6. Before considering i*he origin of the present signs for addition and subtraction we must note that in 1484 a Frenchman Chuquet introduced the symbols (p or p) and (m or m) for plus and minus. In 1494 Pacioli an Italian followed this example, and from then on the notation enjoyed considerable popularity, being used by most of the 16th century Italian mathematicians. These signs were used also by some Portuguese and Spanish writers of the early 17th century. 7. The s y m b o l s a n d -. a. A German algebra published in 1481 contains the first recorded use of The symbol is explained by saying that -4 is "4 das ist -." No symbol occurs for addition, the German "and und" being used throughout. The sign -)- used for addition occurs first in print in J. Widmans book published 1498. There are however manuscripts of an earlier date which contain both the symbols + and -. One of these a latin manuscript now in the Dresden library contains both the symbols, but in some cases -+* was also used for "et" not meaning addition, but the general "and." b. It is thought that the sign for addition + originated in one of the florescent forms of "et" used in latin manuscripts. J.L. Walther gives an extract from a manuscript dated 1417 where the "et" resembles 4 , the downward stroke however not forming a perfect right angle with the horizontal one. There are several theories suggested for origin of the symbol -. These will merely be listed here. 1. Suggested that the mark - originated in a simple bar used by merchants to indicate tare. 2. L. Rodet, Act Soc. Philol. Alencon Vol.111, 1879, believed the sign came from old Egyptian hieratic sign. 3. Belief that the symbol originated in old sign used by Heron and Diophantus. Some historians believe this became T and then -. 4. W.W.R. Ball in "History of Mathematics" suggests that the symbol arose from a bar written over a word where letters are omitted. None of these theories supported by much concrete evidence. c. Spread of symbols and -. The signs + and - were adopted very speedily in Germany, France and other Northern European countries. In the south the p and m continued to exist for some time, gradually losing out in the 17th Century. t 25 d. Forms of-t- and -. There have been several different forms used for these signs. Some of these appear on Chart 2, so only a brief note is necessary. 1. Plus Signs. a. The Greek + as used by Widman in 1489 and also found in early German manuscripts. Rudolff, Stifel, Recorde, Oughtred, Harriot, Rahn, Descartes and the great majority of mathematical writers have used this form. b. The Latin cross, see chart 2, was used by a Frenchman, Francis Vieta, in 1591. A great number of his countryman and some other mathematicians followed his example, the more notable being, Hunt, Herigone, Ruygens, Permat, Rolle, L'Hospital, Swedenborg and De Moivfe. ^he majority of these are 17th Century writers; in the next century this form became less common and gradually very rare. c. A third form was used in 1563 by a Spaniard, De Rortega, and is found also in writings of Descartes, halley, Wolff and a few others. Geographically this symbol had a very wide distribution, but it did not threaten the supremacy of the Greek cross at any time. * Some other forms such as the Maltese cross were also used. 2.' Form used instead of -. used A symbol which was/for minus and which competed with the - sign for four hundred years is the ^ which is now used for division. In 1525 an arithmetic by Adam Reise, German, contained this alternate sign. Reise also used the early sign 2 S a CHART T& SYMBOLS USED FOR ADDITION AND SUBSTRACTION SYMBOL FIRST USED BY TIME REMARKS ADD. SUBT. Y ^ BABYLONIANS 2300 B.C. ?? Sign meant "minus" EGYPTIANS. (AHME'S PAPYRUS) 1700 B.C. To add fractions, juxtaposition used. / 0 GRECIAN 300 A.D. For total result Y U + BAKHSHALI ARITHMETIC 8-9-10 CENT.A.D. Note Subtraction Sign-H r AL QALASADI (ARAB) 15 CENT. A.D. Also used juxtaposition. — GERMAN ALGEBRA 1481 Used "und" for add. p or p m or m CHUQUET (FRENCH) 1484 Sign very popular. + ? J. WTDMAN (GERMAN) 1489 See notes. A. REISE. (GERMAN) 1323 Alternate sign. . . . R. DESCARTES (TRENCH) 1637 3...7 r 7-3 DE HORTEGA (SPANIARD) 1363 MALTESE CROSS AND LATIN CROSS j* or — h 32 for minus. in this he is followed by several later writers, and the idea seems to be to overcome confusions due to the use of the - sign for so many different things. In the 17th Century this sign occurs more frequently, being used in many merman, Swiss and Dutch books. It would be interesting to make an investigation of other methods suggested for representing addition and subtraction, but space will not permit. The chief difficulty in the present scheme is that the signs + and - have each two uses. That is the signs indicate positive and negative quantities as well as addition and subtraction. There is therefore some danger of ambiguity, and to overcome this, many new signs have been offered. None have however met with universal acceptance, and so for the present the double use of the signs is commonly accepted. B. Multiplication and Division. 1. The Babylonians used ideograms to represent multiplication 1 and division "A-DU" signifying times and "IGI-GAL" division. The Egyptians had no symbols for multiplication or division, although making use of both these operations. They used the term "wshtp," which meant "to incline the head" when referring to either of these processes. 2. The Greeks had no special symbols but represented division as a fraction the dividend being written in the numerator and the dividend in the divisor. 1. Transliteration. 3. Among ilindu mathematicians multiplication was represent-ed in several different ways:-a. The first, was by writing the quantities to be multiplied in juxtaposition, in the Bakhshali arithmetic is found 5 32 6 1 phalam for 5/8 of 320 = 20. 20 b. Bhaskara, 12th century A.D., used a dot in representing multiplication. As there was no explanation given it is not known exactly what was meant by this dot and most historians seem to be of the opinion that it was not a symbol but merely a mark used in writing the numbers. c. A third method was using the letters "bha," which was a contraction of the word "bhavita" meaning product. This scheme was used quite consistently by Baskara, who placed the letters after the factors to be multiplied. The symbol used for division, in the Bakhshali arithmetic was merely the letters "bha," abbreviated from "bhaga," mean-ing part. The numbers immediately preceding these letters were considered as the denominator. Quite frequently however the Hindu writers indicated the operation fractionally as the Greek had also done. 4. An Arab A1 Hassari in the ll-12th Century introduced the use of the fractional line in representing division. The dividend was written in the numerator of a fraction, the divisor in denominator as in earlier forms, but a line was drawn separating numerator and denominator. Leonardo of Pisa also used this notation about the same time and some believe he was the originator and not A1 Hassari. 28 5. One of the earliest suggestions of a symbol for multipli-cation and division was offered by Michael Stifel around 1545. This was the use of the German capitals M and D (see forms). It seems very peculiar that neither the German people or Stifel himself accepted these signs. Stifel used the English capitals M and D, being followed in this by Stevin and others. Stifel also used a lunar sign )or( to represent division . 6. John Rahn introduced in 1659 the sign ? for division. In doing this he cleared up considerable ambiguity, for it will be remembered that there were two signs in use at this time for subtraction, and by appropriating one of these for division, he improved the symbolism for both. Rahn's book "Teutsche Algebra," (in which the sign first appeared as a symbol was not adopted in that country, in England however, both the book and the symbol were well received and as a result the sign ? was soon adopted. Some historians indeed, claim that John rell, an Englishman introduced this sign for division. F. Uajori finds "no evidence to support this assertion." The sign itself has enjoyed much popularity, being used by John Wallis and many other eminent mathematicians, and has gradually come to be used by all the English speaking world. On the European continent however the Leibnitzian notation is used. 7. Before considering the Leibnitzian signs it is necessary to discuss briefly the use of the symbol known as the "St. Andrews Cross." In 1631, W. uughtred in his book "Clavis Mathematicae," lists this cross as a symbol for multiplication. Some historians have found that it was used much earlier. In fact F. Cajori, Hist, of Math. Note. Vol.1, p.252-266, discusses eleven uses made of St. Andrews'Crass in medieval manuscripts and early books. These are: a. In solution of problems by the process of two false positions. (Leonardo of risa, and others). b. In solving problems in compound proportion involving integers (Leonardo of Pisa, Chuquet). c. In solving problems in simple proportion involving fractions. (In earliest printed arithmetic Treviso Italy 1478). d. In addition and subtraction of fractions. (Chuquet). e. In division of fractions. (Pietro Borgi). f. In checking results of computation by process of casting out 9's, 7's and 11's. (Facioli). g. As part of group of lines drawn as guides in multiplication of one integer by another. (Pacioli). h. In reducing radicals of different orders to same order. (Stifel, 1554). i. In computing on lines, to mark the line indicating "thousands." (English and Continental texts). j. To take the place of multiplication table above 5 times 5. (Recorde 1543). K. In dealing with amicable numbers (N. Chuquet). So it is evident that the St. Andrews Cross, X, had a wide variety of usages in 14-15-16th Centuries. It seems doubtful if it was ever used for multiplication before 1631. Cajori states, "Our own studies have failed to bring to light a clear and conclusive case where, before Oughtred^-was used as symbol for multiplication." John A ' a l l i s defended the use of this sign. 8. Symbols used by Leibnitz for multiplication and division. a. Marly symbols. In 1666 Leibnitz in his first mathematic-al publication used a capital C, placed as shown on Chart 3, for multiplication and division. For thirty years he used these symbols and the Oughtred X. b. I. In 1698 Leibnitz proposed the dot, as a symbol for multiplication, giving as his reason that the symbol X was very easily confused with the unknown X. The dot was used at an earlier date by Hindu mathemat-icians and also by Harriot and Gibson, but as there is no explanation offered, it is not known whether these intended multiplication or if the dot was merely a separatrix. II. In 1684 Leibnitz proposed the use of the colon : to represent division. In explanation he said that it was customary to indicate c divided by d by writing the c over d with a fractional line between. very often however "it is desirable to avoid this and to continue on the same line by the interposition of two points; so that c:d means "c divided by d." In Germany, Christian Wolff used in his text the signs : for division and the dot . for multiplication and this assisted in the spreading of these symbols. At the present time, contin-ental Europe and Latin America countries with hardly an except-ion use these Leibnitzian forms, in the English speaking countries however the sign 4 is used with few exceptions. In fact it has been said that "no symbols are so completely observant of political boundaries as are f and : the symbols for division. This is a source of confusion and ambiguity and it seem desirable to have for such an important operation as division one symbol used throughout the entire mathematical world - a world symbol. In this brief discussion of the signs used for the "four fundmental operations" it is noted that the sign + has been given world recognition as the symbol for addition, and the sign - is used most commonly for subtraction. The great conflict seems to be in the symbols for multiplication and division. The future may produce new symbols for these last operations, but if not, it would seem advisable to accept one of the above mentioned notations, and to insist on the universal usage of these symbols in expressing multiplication and division. 3 Î d . CHART Iff. SYMBOLS USED FOR MULTIPLICATION AND DIVISION SYMBOL FIRST USED BY TIME REMARKS MULT. DIV. A-DU IGI-GAL. BABYLONIANS 2300 B.C. TRANSLITERATION bha bha BAKHASHALI ARITHMETIC 8-9-10 CENT.A.D. TRANSLITERATION - AL HASSAR (ARAB) 11-12 A.D. SEE NOTES. 3 R y M. STIFEL (GERMAN) 1545 SOMETIMES USED ^ FOR DIV.,AS, 8)24 s 3 X W. OUGHTRED (ENGLISH) 1631 STKN SMAT.T.-m? T^AT? *+* EITHER OR — y P. HERIGONE (FRENCH) 1644 LEIBNITZ SOMETIMES USED THE COMA — J. H. RAHN (SWISS) 1659 SEE NOTES G. W. LEIBNITZ (GERMAN) 1668 ALSO USED FRACTIONAL LINE : G. W. LEIBNITZ 1684 POPULARIZED BY WOLFF G. W. LEIBNITZ 1698 POPULARIZED BY WOLFF 32 - PART III -SYMBOLS USED TO REPRESENT EQUALITY, RATIO AND PROPORTION. This section will be divided into two main headings. A. Symbols for Equality. B. Symbols for Ratio and Proportion. In Proportion both arithmetical and geometrical will be considered. As in previous sections the forms of the signs used, the originators, the date of first appearance, and any other pertinent information will be considered. There will also be an attempt made to compare and contrast signs, and to show developments. The struggle between various symbols will be commented upon. A. Symbols used to represent Equality. 1. The first symbol shown on Chart 4 is the Egyptian mark to represent equality. In the Ahmes papyrus (c 1700 B.C.) this was used, the meaning being "it gives." According to Ball, the symbol = was also used by Ahmes. 2. In the third Century A.D. Diophantus, a Greek, used an "iota" with a sigma written as an exponent (see chart) to represent equality. According to Heath a later Greek sign resembled M 33 3. In the Bakshali Arithmetic as has been noted previously, "phalam" or its contraction "pha" was used for equality. 4. An Arab, A1 Qalasadi, in the 15th Century used an equality sign which resembles the Grecian symbol used by Diophantus. The form of this sign may be seen on Chart 4, and when one remembers that the Arabs wrote from right to left the resemblance becomes more pronounced. In each case the curved stroke faced the second member of the equation. 5. Regiomantus, a German, in 1473, used the dash - to represent equality and was followed in this by Luca Pacioli, Ghaligai and other Italian writers. 6. H. Cardan, an Italian, 1539, followed the usual custom of writing aequale or aequator to represent equality. In some cases however he used a blank space to represent this. 7. Before speaking of the sign for equality introduced by Recorde, it must be noted that a great many mathematical writers have written in full some word meaning "equals" rather than using a special symbol. Some of the words used for this purpose were; aequales, aequantur, (mentioned above) esgale, faciunt, ghelych, gleich and sometimes an abbreviated form aeq. Kepler, Galileo, rascal, Napier, Briggs and Fermat all used one of these forms, and the slowness with which symbols have been adopted is seen from the fact that for fully one hundred years after the introduction of Recorde's equality sign, eminent mathematicians continued to write this relation-ship in words instead of expressing it symbolically. 8. In 1557 Robert Recordd's book the "Whetstone of Witte" appeared, in which two parallel horizontal lines were used as a symbol for eauality. The sign received some attention, but not until the middle of the 17th Century was it used to any extent. At this time it was popularized by three mathematic-ians, William Oughtred, Thomas Harriot and Richard Norwood. Daniel Webster defines a symbol as "a visible object representing to the mind a semblance of something which is not shown but realized with it." If this definition is accepted, the Recordian sign for equality is a very fine symbol for "noe 2 thyngs can be moare ecualle" than two parallel lines. W. Ball rather spoils this nicety by discovering that the sign was used as a common abbreviation for the Latin "est." The sign itself came in grave danger of being discarded during the 16-17th Centuries. There are two reasons for this. A. Many different meanings for sign. Besides representing equality the following uses are found: a. Vieta, French, 1591 used = for arithmetical difference, h. R. Descartes, French, 1638 used = for plus or minus. c. J. Caramuel, 1670 used = as a separatrix in decimal fraction 102=857 being 102.857. d. Dularens, French, used = for parallel lines. e. G.H. Paricius, 1706, used = to separate numbers in arithmetical problems. So the symbol = had five meanings other than equality. B. Symbols competing with = . a. J. BruteO, French, 1559 usedj^ for equality. b. W. Holzmann, German, 1571, better known as Xylander, used two vertical parallel lines. M. cantor suggests that possibly the Greek word "equals" t<rcro<- was abbreviated into two "iotas" and this through further abbreviation became Xylander's sign, several Italian, Dutch and French writers, used this sign in the next century. c. An unusual sign used by Leonard and Thomas Digges, English, in 1579 is seen on Chart 4. d. Herigone, French, 1634 used the sign shown on Chart 4. Signs for greater than, and less than are also exhibited. e. F. Dulaurens, French, in 1667 used sign shown on Chart 3. He also had symbol for greater and less than. f. S. Reyher in 1698 used a single vertical line. He attributes this notation to the Dutch astronomer Jacob Golius. g. R. Descartes, French, in 1637 introduced a symbol representing in form the present sign for infinity. This symbol for equality will be considered more fully later. The excellence of the Recordian symbol for equality was tested very severely by these two conflicts, the first of usages, and the second of competing symbols. 9. All of the symbols listed above were competitors of the Recordian sign for equality, but only the one introduced by Descartes ever threated the supremacy of the parallel lines. The Cartesian sign therefore merits some consideration. Descartes' sign is supposed to have been suggested by the appearance of the combined ae in the word aequalis meaning "equal", The genesis of the sign however is a much debated question, some believing it to be formed by a contraction of oe, or oo pressed together and the left part of the first o being erased. Cajori however believes the sign is merely the astronom-ical symbol for Taurus. This seems a possible explanation as the signs of the zodiac have been in use for a long time, and the type available in any printers office. After 1637pDescartes' renown spread far and wide, for his Geometrie oroved to be the first Analytical Geometry, and also gave for the first time the perfected exponential notation a^, (n a positive integer). As a result of his fame, and also since his symbol for equality was introduced in this Geometrie mathematicians the world over were soon following his example and using the Cartesian sign for equality, in ttolland the most influential mathematicians of the 17th Century copied the symbol, also in France it was widely used. Although occasion-al uses of it are found in England (Samuel roster's Miscellan-ies 1659) the greatest popularity of the sign was in France and Holland in the 17th and early 18th Centuries. The struggle between the Recordian and cartesian symbols was waged chiefly on the continent, in angland Recorde's = had gained complete supremacy during the 17th Century; John Wallis, isaac barrow and Newton all assisting in its promotion During the 18th Century the Recordian symbol became more popular on the continent, and the fact that Leibnitz favored 3 6 a. CHART TK SYMBOLS USED FOR EQUALITY SYMBOL FIRST USED BY TIME REMARKS EGYPTIANS AHME'S PAPYRUS 1700 B.C. W.W.R. BALL GIVES AS SYMBOL USED BY AHMES. c DIOPHANTUS (GREEK) 3RD CENT. A.D. LATER USED SIGN Lj pha BAKHSHALI ARITHMETIC '8-9-10 CENT TRANSLITERATION AL QALASADI (ARAB) 13TH CENT. SEE NOTE ON DEVELOPMENT — REGIOMANTUS (GERMAN) 1473 20Ô& i 800 - 40 INDICATED 200x t HOO ^ 40 — R.RECOHDE (ENGLISH) 1537 SEE NOTES [ J. BUTEO (FRENCH) 16TH CENT. SEE NOTES XYLANDER (GERMAN) 1579 SEE NOTE ON POSSIBLE ORIGIN M T. DIGGES (ENGLISH) 1579 L. DIGGES ALSO USED THIS SIGN P.HERIGONE (FRENCH) 1634 ALSO USED 3/2 FOR GREATER AND 2/3 FOR LESS ! S. REYHER 1698 REYHER ATTRIBUTED THE SIGN TO J. GOLIUS R. DESCARTES (FRENCH) 1637 SEE NOTE ON ORIGIN AND IMPORTANCE n F. DULAURENS (FRENCH) 1667 ALSO USED T FOR GREATER AND 1 FOR LESS THAN it, assisted in its universal acceptance. Indeed, Cajori states that the final victory seems mainly due to Leibnitz, for had he favored the Cartesian sign, Germany and the rest of Europe would have followed him, and the Cartesian symbol would have been accepted in England, at the same time that the calculus notation of Leitnitz replaced that of Newton. Thus the acceptance or rejection of a notation frequently has depended on its acceptance or rejection by some one eminent man. Conclusion. It is possible to find different notations or different usages of the same notation for nearly every branch of mathematics. This fact makes the Recordian sign for equality stand out very prominently in its unique position of being universally accepted. B. Symbolism used in Representing Ratio, Geometric and Arithmetic Proportion. 1. In an early Hindu (Bakhshali) arithmetic there is an attempt made to represent Geometric Proportion symbolically. Chart 5 shows how (10:163/60=4:163/150) was written using this early method. The "pha" is merely an abbreviated form of "phalam"which meant equals. 2. A1 Qalasadi, an Arab of the 15th Century used three dots written between each two numbers to represent Geometric Proportion. On the chart may be seen 7:12:: 84:144 written using this scheme, and it will be noted that the largest number appears first. 3. Regiomantius, a German mathematician used the single dot as a sign of separation and around 1460 A.D. wrote a.b.c. for the present a:b:c. 4. An arithmetic written by Boethius, and appearing around 1488 contained symbolical expressions for both arithmetical and geometrical proportion. As may be seen from Chart 5 the numbers of the proportion were written in squares, and lines were drawn to indicate those that were to be taken together. The number 8 in the arithmetical oroportion seems to have no significance, for the symbol merely indicates that 6, 9 and 12 are in arithmetical proportion. 5. Michael btifel in 1553 used the vertical lines of separat-ion, and wrote geometrical proportion as is indicated on Chart 5. 6. Nicolo Tartaglia, whose writings appeared about the middle of the 16th century, used two slanting lines to separate the three terms of a proportion. On the chart is shown the geometrical oroportion (9:5=100:x) represented in this manner. 7. C. Clavius, German, 1608 used the single dot to represent geometrical proportion. This notation enjoyed considerable popularity, and may be found in use as late as 1699. 8. Rene Descartes, French, 1619 used vertical lines to separate the numbers of a geometrical proportion, the only difference between his notation and that of Stifel being that Stifel used the single lines throughout, while Descartes wrote A]EtjjcjD. This notation was copied by a few continental writers, but soon died out. 9. D. Schwenter in 1625 wrote 65-51-85 to represent the Geometric Proportion 65:51=85:x. He then indicated, how to find the fourth member namely by dividing the product of 51 and 85 by the number 65. Galileo follnw^d Schwenter's practice of using horizontal lines. Sometimes however, Galileo merely used dots or blank spaces to separate the numbers of the proportion. 10. All of the above schemes for representing Geometric proportion were used prior to 1631. Besides these symbolical methods indicated there were many of a purely rhetorical nature, or a combination of symbolical and rhetorical. As an example of the latter, John Stampioen, a merman, in 1639, indicated a:b=b:c by writing a,,b gel :b,, c. In considering the St. Andrews cross, x, as a symbol for multiplication, it was found to have many other uses as well. One of these was in representing proportions involving fractions. Its earliest use for this purpose seems to be in the first arithmetic ever printed (Treviso. Italy 1478). One of the problems calls for the solution of the proportion 7:250 — 63/l6:x. The arrangement given in this early arithmetic is 7 ^/250 - 63 The lines served to indicate the numbers to be 1 ^ 1 - 16 multiplied together. The 7,1,16 are multiplied together and their product, 112 was written above the 7. The number 250 was then multiplied by 63, and this product divided by 112, giving the answer. Many mathematical writers especially in France, Italy and Germany followed this device. In England, Recorde introduced a symbol of his own. This resembled a large Z and the proportion 3:18= I6:x was written 3-716. 8^-11. William oughtred an English mathematician in 1651, introduced two symbols, the dot to represent ratio and the four dots to represent proportion. These symbols will be considered, more fully later. C. Sturm, in 1685, wrote a,b:;c,d: 12. In 1651 the English astronomer vincent Wing introduced the colon (:) as a symbol for ratio. A great conflict ensued between the notation of Wing and that of Oughtred - a struggle which lasted until the middle of the 18th century. For propor-tion Wing used the four dots writing a:b::c:d. 13. The struggle noted above was just finishing when a new notation was suggested, and this was soon in conflict with the winner of the previous engagement. This notation was introd-uced by Leibnitz (or Leibniz). He had previously used the colon :, to represent division and so in writing proportion he now did not introduce a new symbol, but showed the equality of ratios by means of the colon and the Recordian equality sign thus! a:b=c:d. 14. Other symbols for Ratio. a. Richard Balaam, English, 1653 used a colon in a horizontal position for Ratio 1:3 was written 3..1. A German, F. Schmeisser, 1817, adopted this notation. b. John Alexander, English, 1693 used a.b or to indicate the ratio a:b. c. To many writers, ratio was something more than division. Ratio might involve "incommensurable magnitudes, express ible by two numbers, one or both irrational," but division was associated only with rational numbers. a CHART SYMBOLS USgD FOR RATIO rHOrcRTISH. SYMBOLS FIK5T UlKi BT TIME RATIO aSOM.rKOP. ARITH.PRQP. iO ) 8AKHSHALI ARITHMETIC 8-9-10 CERT. 44 84 12 ? AL QALASABI (ARAB) A3 CENT. BOSÏHIUS 1488 ÏKCIOMAHTUS (GSRMA8) 1460 .00 [ ziiOOz! PACIT zz g. STI7SL (C3KMRM) 1353 ;//3//ioo//? t. TABTAGLIA (ITALIAN) 1336 P.JL26.3.T CLAVIU3 (03RHAH) 13S3 R. D3SCARTBS (FRENCH) 1619 A .B h.B::C.D 3:4:6 OUCHTRSD tEHOLISH) 1631 -B-C-? 6. 6AMLN0 (ITALIAN) 1633 V. HING (EHCHTH) 164? :B = C:B G.W. LSIBHITZ (CERHAH) W 3 (Oughtreds ratio a.b did not indicate division). In 1696 Samuel Jeake (English) distinguished ratios from fractions by using "instead of the intervening 3 line, two pricks" and so 2:3 was written Emmanuel Swedenborg, Upsala, 1716 used the symbol ** for ratio of quantities. Peter Herigone, French, 1834 used the Greek letter to represent ratio. Other symbols used for Arithmetical Proportion. Richard Balaam speaks of "arithmetical disjunct proportionals" and writes 2..4=3..5. John Kirby 1735 followed the latter device. E. Weigel, German, 1693 wrote(O) 3/'.'4.7 in representing that these numbers were in arithmetical Proportion. L. Abbe Deidier, French, 1739, has the following proportion, 20.2 .*. 7.60. The three dots for arithmet-ical proportion proved very popular and were used by C. Wolff in 1710, by W. Emerson in 1763, and by others exactly the same way as is seen in Deidier's notation. P. Cossali, Italy 1857, used the colon for arithmetical ratio and expressed proportion in the following manner a:b .*c:d. C. Wolff sometimes wrote arithmetical proportion in the form a-b=c-d. There are several other less important forms, which will not be listed here. Some writers used for arithmetical proportion, the symbolism introduced by Oughtred for geometrical proportion. 43 16. Symbols used for arithmetical and geometrical proportion. In considering the symbols used for arithmetical and geometrical proportion, notice must also be taken of the symbols for arithmetical and geometrical progression, for a progression is merely a continued proportion. Some of the symbols used are listed below: A. Symbols used for Geometrical progression. a. W. Oughtred used the symbol ** to indicate geometrical progression, as an examole 2,6,18,54,162 represent-ed that these numbers were in continued geometrical proportion. This symbol was used very extensively in the 17th, 18th and 19th centuries. Among the English writers using the sign for this purpose may be included John Wallis, 1670, William ^histon, 1725, Alexander Malcolm 1750, Sir Jonas Moore 1688, and John Wilson 1714. b. Colin Maclaurin 1748 indicated geometric progression as follows I:q:q2;q3. The only difference between . * his notation and that of Oughtred is that uughtred used a comma for separating the numbers and Maclaurin used a colon. E. Bezout, Paris, 1797, and L. Despiau both used Maclaurin's notation in expressing geometrical progression. B. Symbols used to represent arithmetical progression. a. John Wallis, English, 1670, expressed arithmetical progression as A,B,C,D or a,b,c,d,e,f. b. Bezout and Despiau, mentioned above, used the sign for arithmetical progression, writing J. 1,2,3,4,5 etc. A great number of French writers including Lamy, Suzanne and Fournier, and a smaller number of Spanish-American writers adopted the notations of Bezout and Despiau for arithmetical and geometrical progression and the s i g n s a n d ^ - a r e found quite frequently. c. John Kirby, London, 1735 and Emerson, 1763 used for arithmetical progression. d. Gilbert Clark, London, 1726, used the sign * * and J.J. Blassier 1770 the sign_L in expressing arithmetical progressions. Although there were many different symbols used, this does not mean that arithmetical progression has been expressed symbolically more frequently than has geometrical progression. As a matter of fact arithmetical progression has been but rarely expressed by means of a symbol. 17. Among the many different symbols for ratio and proportion, the majority seem to be of a very transient nature. Only a very few seem to have had any great influence, and among these must be listed the notations of Oughtred, Wing and Leibnitz. A brief consideration will be made of these three. A.The symbols used by Oughtred for ratio and Geometric propor-tion. It has been noted that there were in existence many symbolical and rhetorical representations of proportion around 1631. About this time however, the science of algebra became more developed and there arose the need for more adequate symbols for ratio and proportion. To meet this need Oughtred introduced the dot . to express ratio and the four dots :: for proportion. Despite the fact that the dot was used for many different purposes at this time, oughtred's notation was less ambiguous than any then in existence, and accordingly were accepted quite widely. Some English mathematicians using these signs were, John Wallis, ¿amuel Foster, Sir Jonas Moore, Isaac Barrow and Vincent Wing. B.The notation of Oughtred was gaining popularity in England, but in 1651 as has been seen, Vincent Wing introduced the colon : as a symbol for ratio and wrote proportion a:b::c:d. The struggle between the colon and the dot thus commenced in the last half of the 17th century, in the period up to 1700 the dot managed to maintain its hold on English mathematicians and as been noted the two mast influential mathematicians in this pBriod Wallis and barrow both used the notation of Oughtred. in the early part of the eighteenth century the dotv^s still found in use in English books, but the colon was gaining in popularity. On the continent the dot as a symbol of geometrical ratio and the four dots for proportion were introduced later than in England. The dot is found, however, in writings of Dulaurens, De L'Hospital, Jakob Bernoulli, Johann Bernoulli, Hermann, Rolle, parent and others, and even the middle of the 18th century it was used occasionally. Gradually however the colon was accepted in Europe, Johann Bernoulli, Parent and others changing over from the Oughtred notation to this later one. Leibnitz, Clairaut and other great mathematicians adopted this notation, and its use seemed assured. In England, the United States, Spain, Portugal and Latin American countries the colon was the symbol commonly used in the 19th century, in the first two countries mentioned wing's notation is still found occasionally. C. For a time it seemed as if king's notation was to be accepted universally in representing ratio and proportion. It has been noted that even Leibnitz had used the : colon in representing geometrical ratio and four dots for proportion. In 1693, however, he expressed his disapproval of the use of special signs for ratio and proportion, and suggested using the division sign (colon) and the Recordian equality sign for representing proportion. This symbolism was evidently much superior to the one in use at that time, but over a century passed before the Leibnitzian notation was used to any great extent. The introduction of the equality sign proportion is commonly accorded to Leibnitz, but it seems quite possible that he may have been influenced by the Dutch mathematician. Stampioen. This writer anticipated by nearly half a century Leibnitz' use of the equality sign, the only difference in the two notations being that Stampioen wrote a,b,c,d, and Leibnitz a:b= c:d, or a/b= c/d. The investigation of the symbols for ratio and proportion has proved very interesting, especially the account of the struggles for supremacy that took place between various symbols. It is indeed strange however that the Leibnitzian notation for ratio and Geometric Proportion has not been accepted more readily. - PART IV -SYMBOLICAL REPRESENTATIONS OF THE UNKNOWN AND POWERS OF THE UNKNOWN; OF POWERS AND ROOTS OF KNOWN QUANTITIES. The symbols used for representing powers and roots have in some cases been quite complicated, and so in this section the different forms will not be indicated by charts but will be presented as they occur in the discussion. A. Symbols used for representing the unknown and Powers of the unknown. 1. A very complicated symbol for the unknown may be seen in the Rhind Papyrus ^Ahmes c 1550 B.C.). This symbol indicated "a heap" which shows that the Egyptians had a fairly definite idea of an unknown quantity. There does not seem to be any symbol representing powers of the unknown, except the pair of walking legs signifying "make in going." This indicated the square of a number. 2. Diophantus^a ureek living in the third century A.D.^used the Greek letter S or S° or S to represent the unknown. In the plural the symbol was sometimes doubled and case endings were added. Many theories are advanced as to the origin of this symbol, the one of G.H.F. Nesselmann seeming quite possible "the symbol was the final sigma" and was chosen because it was the only letter not already used in representing numbers. In representing powers of the unknown, the square and the cube had special signs, and the higher powers were formed by combinations of these two, the exponents being added. The diagram shows the representations of powers up to the sixth. The Greek letters 'delta" and "Kappa" were likely chosen because they were the initial letters of the Greek words -48 and power K - cube. Symbols. Present Grecian Name ¿ J Square x3 K ' cube X* square square x5 square cube x6 cube cube 3. The Chinese mathematicians (See ChuShih Chieh 1303 A.D.) used a positional method for indicating unknowns and their powers, it was superior to the Grecian system in one way namely - it included representations of more than one unknown. As can be seen from the representation of the equations x2+y2 _ g2 - Q and 2x + 2y-u = 0 the manipulation with such symbols would be very tedious -1 -1 4. In the Bakhshali arithmetic mentioned previously, the dot was used to indicate an unknown quantity, and in india is still used for this purpose. In earlier Hindu writings the symbols for the unknown are merely contraction of adjectives, the first unknown being "pa" a contraction of the word for white, the second "ca" a contraction of the word for black, and thus many different unknowns may be used. Bhaskara, a Hindu of the 12th century A.D. speaks of raising to powers, and seems to be the first mathematician of his race to express these powers. The words varga and g'hara were taken for the square and the cube respectively, and higher powers were indicated by combinations of these. Unlike the (¿reeks the principle of involution was used, that is the exponents were multiplied. In representing powers where the index was a prime the additive principle was resorted to. An Arab,Abu Kamil (c.900) used instead of the names of colors, the names of coins in representing unknowns; later still Al-Karkhi called one unknown "thing" and a second "measure." He did not however use any contracted sign and so his represent-ation was not entirely symbolical. 5. A1 Qalasadi in the 15th century used different letters to represent the unknown and powers of the unknown. F. Calori, Hist, of Math. Nots. Vol.1, p.95, gives these very complicated signs. 6. Regiomantus, a German mathematician writer of the 15th century had a symbol ^ ^ for the unknown and also a symbol for the square of the unknown. Calori, Hist, of Math. Not. Vol.1, p.95, shows the solution of a quadratic equation using these symbols. 7. An Italian, Luca Paciola, 1494-1523, used the word "cosa" (Italian for "thing") to represent the unknown quantity, and from this word the terms "Coss and Cossic art" were used in Germany and England as synonyms for algebra. This choosing of the first power of the unknown for the "principal unknown" differed from the Arabic. In the Arabic the principal unknown called "mal" (assets) was the square x and the word "jidr" (plant root, lowest part) was used for the present x and was therefore the square root of mal. To return to the notation of racioli for the unknown, as has been shown, cosa or the abbreviated form co. was used. Abbreviations were also used in representing powers ce.(censo) cu.(cubo) representing the square and the cube of the unknown. Pacioli used the multip-licative principle of the Hindus in forming higher powers by combining the square and the cube. Some powers such as the fifth and the seventh could not be thus represented and for these he followed the practice of contemporary writers calling the fifth power "primro relato" p? r? the seventh power "secundo relato" 2? r? the eleventh power "terzo relato" 3.°r.° and thus could represent all powers of the unknown. As a rule Pacioli used & to represent roots, but some cases have been found where he used it for indicating powers. (thus^30^ was the 29th power). 8. Somewhat later than Pacioli a German, Chris toff Rudolff published an algebra, (the first printed algebra) in which the unknown and powers up to the ninth were expressed symbolic-ally. The symbol (meaning radix) represented the unknown; for the square a symbol similar to was used, and for the cube the symbol resembled the letter c with a final loop like a deformed letter e. This last symbol would seem to indicate that Rudolff had been influenced by the notation of Pacioli and early writers, higher powers were expressed by combinations of these symbols using the multiplicative principle. Rudolff introduced distinctive symbols for the fifth and seventh power the latter being ( b ^ ). 9. Michael Stifel, German, 1544 used the notation of Rudolff in representing the unknown and its powers up to and including the ninth power. Stifel was led to the idea of extending the notation by noticing the form for the seventh power. Instead of using the "primato relato" notation of facioli, he used the following; all powers which could not be expressed by a combination of the symbols for the unknown, its square and cube (using multiplicative principle) were to be indicated by writing the Greek letter (3 with a letter of the English alphabet as its coefficient. These English letters were arranged in their natural order; b[3 indicated the seventh c^ the eleventh, d the thirteenth powers respectively. Stifel did not always use the symbols which Rudolff had introduced. Sometimes he represented the unknown by q, as Cardan also did, and sometimes used the letters A^B,ÇD along with the original symbol for the unknown. 10. In the first nine sections, the method for representing the unknown and its powers has been, in the majority of cases, means of abbreviation. In the next few sections will be found examples of a different type of symbol, namely, the use of numerals to indicate the powers of the unknown. Before considering this symbolism a brief consideration will be made of the two fold use of the "R" and "JL". As will be discovered later R from "radix" meaning root, and -(¿.from "latus" meaning side, were used quite frequently as symbols for the root of a number. Pacioli's use of the R in expressing powers has been noted, and Ramus in 1593 used 5^. to represent 5x, although J^5 indicated This double significance of R and ^ w a s very confusing, but was employed by a large number of writers. 11. A Frenchman, John Buteon in 1559 introduced a rather unusual notation. Rejecting the use of the words "res" "census" he introduced the latin words for "line," square and cube" using these to represent the unknown and its square and cube. As symbols he used p , (fjP; the last two being very suggestive. When more than one unknown was required Buteon used A, B, C, D-12. In tricing the development of the present exponential system of representing powers, the various steps can be referred to but briefly. The first to be noted will be the symbolism of a French bishop, Nicole Oresme, 14th century. If we omit the rhetorical notation of rsellus and that contained in the Scholia of Neophytus, Oresme's notation was the first formed on the index plan. His notation represented a very promising beginning, but the exponental concept which it contained was lost soon after and not rediscovered until the 15th century. 13. Nicholas Chuquet, 1484, a French writer seems to have led the way in the discovery of this exponental symbolism, he wrote the exponent of the number without any reference to the base, a Thus 12 represented 12 x. For Chuquet this notation did not lead to ambiguity for 5 was never used for 9. This notation which "was nothing less than the use of positive exponents written without reference to the base," was also used by the Italian Rafael Bombelli, 1572. In the notation of Bombelli, however, the exponent was written within curved brackets, (2) representing x. 14. A Dutch writer Gielis Van Der Hoecke, 1537, used the symbols N, pri, se, 3*, 4^, 5^ and for the unknown and its powers. This notation is quite similar to that used by the Austrian, Grammateus, 1518. 15. Simon stevin, a Belgian mathematician in 1585 introduced a new notation for the unknown and its powers. This was similar to the notation of Bombelli only that a circle enclosed the numbers;^ and (2) representing the unknown and its square. 16. Francis Vieta, French 1591, used the vowels to represent the unknowns and the consonants to represent Known quantities. Powers of the unknown were indicated rhetorically. As will be noted later, this introduction of literal coefficients had a great Influence in future symbolism for the unknown and its powers. Albert Girard, Loraine 1629, used vowels and consonants as Vieta had done, but expressed powers of the unknown as Bombelli had done. 17. a. Several writers, in expressing the powers of the unknown^used what is known as the "Galea" or scratch method. This was merely drawing a line through numbers representing the exponents, the symbols ^ ^ $ ^ and Indicating x^ x^ * x^ x^ respectively. As an example of this F. Uajori gives the notation of P. Cataldi of Bologna, 1610. b. The notation of Thos. Digges, English, 1579, was quite similar to the above. In his notation however the 54 numbers were not crossed out, but were deformed. Neither of these schemes vv*3 very widely accepted. <z 18. A Frenchman, rierre Herigone, 1634, wrote a5 for a , ab4 4 for 2b . This scheme of putting the coefficient before and the exponent after the letter proved quite popular. Among those using Herigone's symbolism may be included Cardinal Michangelo Ricci, Christian Huygens, John Craig and Joseph Moxon. Although it was used by both Huygens and Craig in 1684, this notation is seldom found in writings of the 18th century. 19. James hume, a Scotchman residing in raris, introduced a (it 3 very superior notation. In 1636 he wrote A for A , thus placing the base or unknown, on the line and elevating the exponent to a position above the line, and somewhat to the right of the letter. Except for the fact that Hume used Roman numerals in expressing the exponents, his notation was exactly the same as the one used at the present time. 20. The following year 1637 witnessed the introduction of the modern notation. Rene Descartes the originator really made very little change in the notation of James Hume in writing powers of the unknown. He merely expressed the exponent in Arabic rather than Roman numerals. Johann Stampioen in 1639 suggested the same form, and did not acknowledge any indebted-ness to Descartes. F. Cajori however doubts that Stampioen originated this notation independently of Descartes. Descartes had not always followed this device; indeed it was through using the notation of earlier mathematicians that he came to 55 realize the need of more perfect symbolism. In expressing the 2 square he quite frequently wrote aa rather than a . A second innovation of Descartes' was in using the first letters of the alphabet to represent known quantities and the last letters to signify unknowns. Thus x,y,z became symbols for the unknowns and A.B.C.D or a,b,c,d the knowns. Where only one unknown was needed Descartes used x. There has been considerable discussion as to why x should be chosen in preference to y or z. Many writers believe that Descartes was influenced by the resemblance between x and the early German symbol . Indeed some authorities claim that Descartes interpreted this sign as x, and used it accordingly. The Cartesian notation was not immediately accepted. A few writers preferred Hume's or Herigone's. J.H. Rahn in 1659 suggested the use of small letters for the known and capitals for the unknown. In Holland, where Descartes' Geometrle first appeared^ the exponental notation was soon accepted, J. Stampioen using it in 1639 and Fr. van Schooter in 1646. The gradual spread of this symbolism makes an interesting study, but it must suffice to say that C. Huygens, F. Mersenne, J. ..allis, i Barrow and the majority of later mathematicians used this exponential symbolism. Fermat, at first favored the algebraic notation of Vieta, but in his last book, published 1679 he used the Cartesian. 21. There were several different suggestions for expressing the powers of the unknown, made after 1637, a few of these will be listed as follows: p a. J. Stampion, Dutch 1639, wrote [3 A for A . b. Leibnitz in 1698 wrote {m} y + a for (y + a)*". c. Samuel Foster, London 1659, expressed powers by vertical and horizontal lines —s ,__/, T , representing the square cube and fourth powers respectively. d. In Maandelykse Mathematische Liefhebberye 1754-69 TT) signifies the mth power and v the mth root. e. Repetition for factors for designation of powers was also used by Hariot, Collins and others. 22. Negative, Fractional, Literal, imaginary exponents. Oresme, Chuquet and others suggested the negative and fractional notation, but of course did not use the exponental symbolism. J. Wallis in 1659 speaks of negative and fracti^na indices but does not use them, in fact Newton seems to have been the first to introduce and use the modern x"^ and xs and at the same time introduced general exponents such as a^. W.W. R. Ball gives in his "Short history of Mathematics", p.394, an interesting note on the further use of this form. "To the best of my knowledge Newton had been the first to employ literal exponental notation, and Euler, using the form a^ had taken a as the basis of a system of logarithms, it is probable that the choice of e for a particular base was determined by its being the vowel consecutive to a." 23. In the 19th century the values of a*\ where a and n may be negative or complex numbers, came to be studied. From this study there arose the need of a symbol representing the "principal value." A.L. Cauchy, raris, 1821, designated all the values that a^ may take for given values of n ¡L<3-^ *oj by the symbol ((a)) so ((a)) = e .e where ^ means the tabular log. of (a), e = 2.718... ^ 7T- 3.141*** k = 0,11. +2 In a*, when a and x may be both complex numbers, log a has an infinite number of values. Martin Ohm represented some partic-ular value say by the symbol /a//cO/ F. Cajori in Hist, of Math. Not. vol.1, pp.358-360, gives the notations used for expressing complicated powers such as a-th power of the b-th power of x. These must be omitted. Conclusion. In this very brief account of the symbols used for expressing the unknown and its powers there have been noted three general schemes. a. The Rhetorical, where the actual names of the unknown and its powers were used. b. The Abbreviated rlan. This was merely the use of initial letters, or the first two letters of words for unknown and the square and cube. in forming higher powers, such as the fourth, combinations of the lower symbols were used, sometimes by adding and more often by multiplying the exponents. In the latter case special symbols were required for prime powers. c. The Index rlan. This was by indicating the exponent of the number, although in the earlier systems formed on this plan, there seemed no definite understanding as to what was meant by the exponent. At first the number 1,2,3 & were written without designating the base. This scheme was quite adequate for a time as there was no need of representing the powers of known 58 quantities symbolically, and such powers could be computed and the result written. After the innovations of vieta and others whereby literal coefficients came to be used, and also several unknowns, this scheme was 2 2 very inadequate, for 15x - 16y could not be represen-ted by 15"-16." In attempts to overcome the defects of this symbolism, the exponential notation of Descartes finally evolved. This notation proved very splendid being adequate and elastic. This latter quality is seen in the ease with which the symbolism was extended to include negative, fractional, literal, imaginary and complex powers. The development of the science of algeora has been hastened very greatly by this notation. In conclusion a quotation from F. Cajori "nowhere is the importance of good notation for the rapid advancement of a mathematical science exhibited more forcibly than in the exponental symbolism of algebra." B. Symbolism used in Representing Roots of Known and Unknown Quantities. Symbols for rowers of Known Quantities. It may seem strange that the symbols for Roots of unknown quantities are included in this, rather than the preceding section. The reason is that the symbols for roots of the unknown and roots of the known quantities have been so closely connected, that had they been treated in section A, they would have to be considered in this section. To avoid this needless repetition they have therefore been omitted from Section A. After considering briefly the symbols for roots, the notation used in expressing powers of known quantities will be invest-igated. a. One of the earliest signs used for representing the root of a number was the right-angled symbolLT^ used for square root by the Egyptians (c 1550 B.C.). b. In the writings of Brahmagupta (seventh century A.D.) the letter c was used to indicate square root. For example c 450 was 1/450. c. A1 Qalasadi (Arab 15th century) used 13 the initial letter of the word "jidr" to indicate roots. This sign was written above the number whose square root was to be found and usually was separated from it by a horizontal line . d. John Seville in translating the Arabian works into Latin choose for "jidr" the latin word radix - root. Leonardo of Pisa and Luca Pacioli both used this word or its initial letter, but as has been mentioned in a previous section, this R came also to be used in representing the unknown. The R for root proved a very popular symbol however, and was used by a great number of mathematicians. In some cases a line was drawn through the lower part of the R and the sign became B . These two forms were used for several centuries, gradually yielding to the newer forms. The 16th century witnessed their greatest popularity, and although several other symbols were suggested (such as John Widmans' "ra"), the forms mentioned above were in this century commonly accepted as symbols for square root. In England the newer signs soon gained a foothold, but in Spain and J-taly the mathematical writers adhered very tenaciously to the R. The last use of this sign is in a Spanish text in 1784. There w a p a number of different methods for indicating cube root, fourth root and higher roots. A few of these will be listed. 1. Johannes Scheubel, German 1545, used Widmans' ra for square root, but indicated cube root by ra.cu or/VV\/and fourth root by ra.ra or /V\/ 2. Rafaele Bombelli, Italian 1572, expressed roots as follows: R.q indicated square root, R.c cube root, R.R.q fourth root, R.p.r fifth root and R.q.c sixth root. In expressing the cube root of a polynomial, Bombelli used a symbol which resembled two L's the second being written in a backward position. 3. J.P. de Moya, Spain, indicated square root by r. The expressions for cube root and fourth root were rrr and rrr , respectively. 4. James Hume, 1635, used. Roman numerals to indicate the order of the root. (3%vl was the sixth root). e. The sign J^L from Latin word "latus" meaning "the side of a square." This symbol was used to signify root by Junius Nipsus, a Roman surveyor of the second century A.D. Several later writers accepted this symbol, the one using it the most extensively being Peter Ramus (Basel, 1569). In his writings -4LJ& stood for the fourth root. Lazurus Schoner in a later edition of Ramus' arithmetic used J2c 4 for V ^ (Schoner, Pacioli and others, however, sometimes used indicating unknowns and sometimes roots). After the invention of logarithms the symbol was used very seldom to indicate roots. It seems very peculiar that Henry Briggs, who devoted a great part of his life to the study of logarithms, should have used for indicating roots, f. John Napier (c. 1839) invented a notation for expressing roots, but throughout all his writings used the cumbersome notation employed by his contemporaries. The notation which Napier suggested was derived from the figure ! f being the symbol for square root the fourth root and thus any root up to the ninth could be expressed symbolically. 1 2 3 4 5 6 7 8 9 7 g. The symbol The sign l/ was first used in Germany, but no satisfact-ory explanation has as yet been offered as to its origin. L. Euler believed this sign was merely the initial letter r of the word radix, with the letter somewhat deformed. This view was held for many years, but lately a new theory has come forward namely that the sign originated in a dot. This would seem rather fanciful, but the study of four manuscript al&ebrag have convinced German mathematicians at least, that this is the correct theory. As a further investigation of the origin of this sign is not possible here, the next consider-ation will be the popularity and use of the sign from the time of Christoff Rudolff onward. In 1525 Rudolff used"/ for square root/yV\/for cube root, and/VV^ for fourth root. The reason that the cube root had more strokes than that for fourth root being that the latter sign was merely two square root signs written together. These clumsy signs may be found in some writings of a later date, of which but two will be mentioned. In Recordes' "Whetstone of Witte", 1557, the forms appear, and Samuel Jeakes in 1696 used a modification of these early symbols. In 1545, about twenty years after Rudolff's symbols first appeared, another German, Michael Stifel, introduced symbols for roots, which seem to be superior to those of Rudolff. In his notation and represented the square root and the cube root. Stifel made the downward stroke longer than Rudolff did and thus his sign came closer to the modern form. A very important advance in the symbolism used in representing powers may be found in the notation of Simon Steven. This Belgian writer expressed the cube by writing a small three enclosed in a cifcle, after the sign, thusl/^) Square root was indicated by the sign without any additional mark, while the fourth root and roots of a higher order were easily expressed by the numbers (4.5 &.). This was the first notation in which roots of an order higher than the fourth could be expressed in compact symbolism. Romanus used somewhat the same notation but enclosed the numerals in two round parentheses, rather than the circle of Stevin. John Wallis in 1655 made a further advance, in writing the root indices in numerals without enclosing them in circles or parentheses, he writes for the cube root % of R . In 1685 Wallis also introduced general indices and wrote = R. The plan of writing the root index within the opening of the radical sign was suggested by Albert Girard in 1629, but was not used for a long time afterward. In the 18th century V this notation was gradually adopted and the symbol y was almost universally used. Rene Descartes in 1637 expressed the cube root by writing the radical sign with a horizontal line (veniculum) and thus obtaining the form V Speaking of the service rendered by Descartes, toward perfecting algebraic symbolism F. Cajori, Hist, of Math. Not. Vol.1, p.370, "If Descartes, instead of extending the application of the radical sign ^ by adding to it the veniculum, had discarded the radical altogether, and had introduced the notation for fractional as well as integral exponents, then it is conceivable that the further use of radical signs would have been discouraged and checked; it is conceivable that unnecessary duplication in -i/4 4/ 3 notation as illustrated by b ' a n d y b would have been avoided; it is conceivable that generations upon generations of pupils would have been saved the necessity of mastering the operations with two difficult notations when one alone, the exponential, would have answered all purposes." Descartes 64 did not avail himself of this opportunity, nor did Isaac Newton, for although Newton introduced fractional exponents, he also used the radical notation. h. Aggregation used with the radical sign. In the notations of Rudolff and Stifel, dots were used in expressing aggregation under the radical sign, for example -^z 15 p. Vz 8 is ^L5 + but V z .15 p. Vz 8 is ^15 1^87 Oughtred adopted the dot for expressing ratio, and so was-not able to use the German notation, in place of the dot he used the colon, and this modified notation was later used by Wallis Kersey and other writers. There were numerous other notations used in expressing this aggregation, but possibly the most interesting of all is the one used by John Kersey in 1673. in writing l/l m - iil n2 - 5 he)rtasW2) : 1 m - lil nn - s 1 2 ^4 2 14 Here may be seen the use of two symbols for aggregation, the colon and veniculum, of which either one wculd be sufficient. i. One of the more recent developments in expressing roots is found in the decision that""\^ a will represent only the "principal value" of the radical. G. Peano in "Revisa de Matematica," was the first to use this notation, and when he desired a symbol to represent all of the roots he placed a star between the radical sign and the a. j. There have been noted above, a great number of notations for expressing roots, not all of these, however, have different basis symbols, so in classif-ying the notations there will be need of only five groups. 1. Those notations in which signs are chosen arbitrarily far roots and where the knowledge of the symbol for square root and cube is not sufficient for expressing roots of a higher order. As an example of this may be considered the notation invented (but never used) by Newton. 2. Those notations using R - radix as a basic element. There have been a great number of these, Chuquet and racioli being the originator of two. As alternate forms^R- aTid r \A/eee sometimes used. 5. Notations based on The symbolism used by Ramus has been mentioned previously. 4. Notations using the radical sign l/ . These have been treated quite fully. 5. The use of Fractional Exponents. There is need for a more universal adoption of the last system mentioned, for instead of having separate notations for expressing roots and Dowers, the use of fractional exponents makes possible the expression of both by the same notation. In this connection may be noted the recommendation of the United States National Committee on Mathematical Requirements (1923): "With respect to the root sign^/ the committee recognizes that the convenience of writing assures its continued use in many cases instead of the fractional exponent It is recommended, however, that in algebraic work involving 66 complicated cases, the fractional exponent be preferred." II. Symbols for Representing rowers of Known Quantities. In the early development of algebraic symbolism, no signs were used for representing powers of a given number. Before the time of vieta, literal coefficients were not used, and given quantities were stated explicitly in numerals. Thus there was no need for symbols representing powers of known quantities, for such powers could be computed immediately and the result written. Because of this fact symbolical represent-ations of powers of the unknown preceded by over one thousand years the introductions of symbols for powers of known quantities. The influence of the introduction of literal coefficients upon algebraic symbolism was very extensive, and resulted first, in the development of more adequate notation for expressing powers, and secondly, in the subsequent discarding of all earlier symbols. The notations formulated by Hume, Herigone, Descartes and Leibnitz were applicable to the expression of both known and unknown quantities. The raising of a known quantity a to a power m was not expressed symbolically, however, until quite recent times. Although there are many other interesting features which might be considered in relation to the symbolical representat-ion of the unknown and powers and roots of known and unknown, the above must suffice. It has proved to be the most fascin -ating investigation so far attempted. CUNCLUSiON. in the four preceding sections there been considered a large number of symbols. The notations used in elementary mathematics have been treated quite fully, the only ones omitted being the symbols for aggregation, and the symbolical representation of fractions. The inclusion of these notations or the attempt to discuss the symbols of higher mathematics would have resulted in a very sketchy treatment; and to avoid this it seemed advisable to include only those phases found in Parts I, II, III and IV. All of the notations were attempts to meet some definite need. Thus, the number systems were developed when oral symbolism was no longer adequate, and the changes made from time to time were occasioned by dissatisfaction with the existing systems. The great number of changes in notation seem to have been caused by the fact that so few symbols have been found permanently adequate. Through some new discovery or through changes in usage, many symbols have been found insufficient. As an example of this may be considered the notation used by Uhuquet in expressing powers of an unknown quantity, ais symbols were quite adequate as long as only one unknown was used, but with the introduction of several unknowns his method of representation became useless. At the present time there is a great lack of uniformity in the usage of symbols. A dot placed between two integers, to represent according to its position either multiplication or a decimal fraction has not the same significance in England and the United States. It would seem therefore that the greatest need is not the introduction of new symbols, but rather is uniformity of usage of existing symbols. BIBLIOGRAPHY Cajori, Florian, History of Mathematical Notations, 2 Vols. Smith, D.E., Ball, W.W.R., Fink, K., History of Mathematics, 2 Vols, Short History of Mathematics. A Brief History of Mathematics. Cajori, Florian, History of Elementary Mathematics, Smith, D.E. and Mikami, Yoshio, A History of Japanese Mathematics. Smith, D.E. and Karpinski, L.C., Hindu Arabic Numerals. Sarton, George, Introduction to the History of Science, Vol. I. Carnegie Institute of Washington. Chase, A.B. Translation and Photographs. Gow, James, The Rhind Mathematical Papyrus British Museum 10057 and 10058, 2 Vols. History of Greek Mathematics.
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Mathematics
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The development of symbolism in elementary mathematics
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