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| ISBN-13: | 9780486161167 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 10/24/2013 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 864 |
| File size: | 60 MB |
| Note: | This product may take a few minutes to download. |
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A History of Mathematical Notations
By Florian Cajori
Dover Publications, Inc.
Copyright © 2012 Dover Publications, Inc.All rights reserved.
ISBN: 978-0-486-16116-7
CHAPTER 1
INTRODUCTION
In this history it has been an aim to give not only the first appearance of a symbol and its origin (whenever possible), but also to indicate the competition encountered and the spread of the symbol among writers in different countries. It is the latter part of our program which has given bulk to this history.
The rise of certain symbols, their day of popularity, and their eventual decline constitute in many cases an interesting story. Our endeavor has been to do justice to obsolete and obsolescent notations, as well as to those which have survived and enjoy the favor of mathematicians of the present moment.
If the object of this history of notations were simply to present an array of facts, more or less interesting to some students of mathematics—if, in other words, this undertaking had no ulterior motive—then indeed the wisdom of preparing and publishing so large a book might be questioned. But the author believes that this history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of the notational problems of the present time.
CHAPTER 2NUMERAL SYMBOLS AND COMBINATIONS OF SYMBOLS
BABYLONIANS
1. In the Babylonian notation of numbers a vertical wedge [ILLUSTRATION OMITTED] stood for 1, while the characters [ILLUSTRATION OMITTED] and [ILLUSTRATION OMITTED] signified 10 and 100, respectively. Grotefend believes the character for 10 originally to have been the picture of two hands, as held in prayer, the palms being pressed together, the fingers close to each other, but the thumbs thrust out. Ordinarily, two principles were employed in the Babylonial notation—the additive and multiplicative. We shall see that limited use was made of a third principle, that of subtraction.
2. Numbers below 200 were expressed ordinarily by symbols whose respective values were to be added. Thus, [ILLUSTRATION OMITTED] stands for 123. The principle of multiplication reveals itself in [ILLUSTRATION OMITTED] where the smaller symbol 10, placed before the 100, is to be multiplied by 100, so that this symbolism designates 1,000.
3. These cuneiform symbols were probably invented by the early Sumerians. Their inscriptions disclose the use of a decimal scale of numbers and also of a sexagesimal scale.
Early Sumerian clay tablets contain also numerals expressed by circles and curved signs, made with the blunt circular end of a stylus, the ordinary wedge-shaped characters being made with the pointed end. A circle [ILLUSTRATION OMITTED] stood for 10, a semicircular or lunar sign stood for 1. Thus, a "round-up" of cattle shows [ILLUSTRATION OMITTED], or 36, cows.
4. The sexagesimal scale was first discovered on a tablet by E. Hincks in 1854. It records the magnitude of the illuminated portion of the moon's disk for every day from new to full moon, the whole disk being assumed to consist of 240 parts. The illuminated parts during the first five days are the series 5, 10, 20, 40, 1.20, which is a geometrical progression, on the assumption that the last number is 80. From here on the series becomes arithmetical, 1.20, 1.36, 1.52, 2.8, 2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4, the common difference being 16. The last number is written in the tablet [ILLUSTRATION OMITTED], and, according to Hincks's interpretation, stood for 4 × 60 = 240.
5. Hincks's explanation was confirmed by the decipherment of tablets found at Senkereh, near Babylon, in 1854, and called the Tablets of Senkereh. One tablet was found to contain a table of square numbers, from 12 to 602, a second one a table of cube numbers from 13 to 323. The tablets were probably written between 2300 and 1600 B.C. Various scholars contributed toward their interpretation. Among them were George Smith (1872), J. Oppert, Sir H. Rawlinson, Fr. Lenormant, and finally R. Lepsius. The numbers 1, 4, 9, 16, 25, 36, and 49 are given as the squares of the first seven integers, respectively. We have next 1.4 = 82, 1.21 = 92, 1.40 = 102, etc. This clearly indicates the use of the sexagesimal scale which makes 1.4 = 60 + 4, 1.21 = 60 + 21, 1.40 = 60 + 40, etc. This sexagesimal system marks the earliest appearance of the all-important "principle of position" in writing numbers. In its general and systematic application, this principle requires a symbol for zero. But no such symbol has been found on early Babylonian tablets; records of about 200 B.C. give a symbol for zero, as we shall see later, but it was not used in calculation. The earliest thorough and systematic application of a symbol for zero and the principle of position was made by the Maya of Central America, about the beginning of the Christian Era.
6. An extension of our knowledge of Babylonian mathematics was made by H. V. Hilprecht who made excavations at Nuffar (the ancient Nippur). We reproduce one of his tablets in Figure 1.
Hilprecht's transliteration, as given on page 28 of his text is as follows:
[TABLE OMITTED]
7. In further explanation, observe that in
[TABLE OMITTED]
We quote from Hilprecht (op. cit., pp. 28–30):
"We observe (a) that the first numbers of all the odd lines (1, 3, 5, 7, 9, 11, 13, 15) form an increasing, and all the numbers of the even lines (preceded by IGI-GAL-BI = 'its denominator') a descending geometrical progression; (b) that the first number of every odd line can be expressed by a fraction which has 12,960,000 as its numerator and the closing number of the corresponding even line as its denominator, in other words,
[TABLE OMITTED]
But the closing numbers of all the odd lines (720, 360, 180, 90, 18, 9, 18, 9) are still obscure to me....
"The question arises, what is the meaning of all this? What in particular is the meaning of the number 12,960,000 (= 604 or 3,6002) which underlies all the mathematical texts here treated ....? .... This 'geometrical number' (12,960,000), which he [Plato in his Republic viii. 546B–D] calls 'the lord of better and worse births,' is the arithmetical expression of a great law controlling the Universe. According to Adam this law is 'the Law of Change, that law of inevitable degeneration to which the Universe and all its parts are subject'—an interpretation from which I am obliged to differ. On the contrary, it is the Law of Uniformity or Harmony, i.e. that fundamental law which governs the Universe and all its parts, and which cannot be ignored and violated without causing an anomaly, i.e. without resulting in a degeneration of the race." The nature of the "Platonic number" is still a debated question.
8. In the reading of numbers expressed in the Babylonian sexagesimal system, uncertainty arises from the fact that the early Babylonians had no symbol for zero. In the foregoing tablets, how do we know, for example, that the last number in the first line is 720 and not 12? Nothing in the symbolism indicates that the 12 is in the place where the local value is "sixties" and not "units." Only from the study of the entire tablet has it been inferred that the number intended is 12 × 60 rather than 12 itself. Sometimes a horizontal line was drawn following a number, apparently to indicate the absence of units of lower denomination. But this procedure was not regular, nor carried on in a manner that indicates the number of vacant places.
9. To avoid confusion some Babylonian documents even in early times contained symbols for 1, 60, 3,600, 216,000, also for 10, 600, 36,000. Thus · was 10, [ILLUSTRATION OMITTED] was 3,600, [ILLUSTRATION OMITTED] was 36,000.
10. Besides the principles of addition and multiplication, Babylonian tablets reveal also the use of the principle of subtraction, which is familiar to us in the Roman notation XIX (20–1) for the number 19. Hilprecht has collected ideograms from the Babylonian tablets which he has studied, which represent the number 19. We reproduce his symbols in Figure 2. In each of these twelve ideograms (Fig. 2), the two symbols to the left signify together 20. Of the symbols immediately to the right of the 20, one vertical wedge stands for "one" and the remaining symbols, for instance [ILLUSTRATION OMITTED], for LAL or "minus"; the entire ideogram represents in each of the twelve cases the number 20 – 1 or 19.
One finds the principle of subtraction used also with curved signs; [ILLUSTRATION OMITTED] meant 60 + 20 - 1, or 79.
11. The symbol used about the second century B.C. to designate the absence of a number, or a blank space, is shown in Figure 3, containing numerical data relating to the moon. As previously stated, this symbol, [ILLUSTRATION OMITTED], was not used in computation and therefore performed only a small part of the functions of our modern zero. The symbol is seen in the tablet in row 10, column 12; also in row 8, column 13. Kugler's translation of the tablet, given in his book, page 42, is shown below. Of the last column only an indistinct fragment is preserved; the rest is broken off.
12. J. Oppert pointed out the Babylonian use of a designation for the sixths, viz., 1/6, 1/3, 1/2, 2/3, 5/6. These are unit fractions or fractions whose numerators are one less than the denominators. He also advanced evidence pointing to the Babylonian use of sexagesimal fractions and the use of the sexagesimal system in weights and measures. The occurrence of sexagesimal fractions is shown in tablets recently examined. We reproduce in Figure 4 two out of twelve columns found on a tablet described by H. F. Lutz. According to Lutz, the tablet "cannot be placed later than the Cassite period, but it seems more probable that it goes back even to the First Dynasty period, ca. 2000B.C."
13. To mathematicians the tablet is of interest because it reveals operations with sexagesimal fractions resembling modern operations with decimal fractions. For example, 60 is divided by 81 and the quotient expressed sexagesimally. Again, a sexagesimal number with two fractional places, 44(26)(40), is multiplied by itself, yielding a product in four fractional places, namely, [32]55(18)(31)(6)(40). In this notation the [32] stands for 32 × 60 units, and to the (18), (31), (6), (40) must be assigned, respectively, the denominators 60, 602, 603, 604.
The tablet contains twelve columns of figures. The first column (Fig. 4) gives the results of dividing 60 in succession by twenty-nine different divisors from 2 to 81. The eleven other columns contain tables of multiplication; each of the numbers 50, 48, 45, 44(26)(40), 40, 36, 30, 25, 24, 22(30), 20 is multiplied by integers up to 20, then by the numbers 30, 40, 50, and finally by itself. Using our modern numerals, we interpret on page 10 the first and the fifth columns. They exhibit a larger number of fractions than do the other columns. The Babylonians had no mark separating the fractional from the integral parts of a number. Hence a number like 44(26)(40) might be interpreted in different ways; among the possible meanings are 44 × 602 + 26 × 60 + 40, 44 × 60 + 26 + 40 × 60-1, and 44 + 26 × 60-1 + 40 × 60-2. Which interpretation is the correct one can be judged only by the context, if at all.
The exact meaning of the first two lines in the first column is uncertain. In this column 60 is divided by each of the integers written on the left. The respective quotients are placed on the right.
In the fifth column the multiplicand is 44(26)(40) or 44 4/9. The last two lines seem to mean "602÷44(26)(40) = 81, 602÷81 = 44(26) (40)."
14. The Babylonian use of sexagesimal fractions is shown also in a clay tablet described by A. Ungnad. In it the diagonal of a rectangle whose sides are 40 and 10 is computed by the approximation 40 + 2 × 40 × 102÷602, yielding 42(13)(20), and also by the approximation 40 + 102÷{2 × 40}, yielding 41(15). Translated into the decimal scale, the first answer is 42.22 + , the second is 41.25, the true value being 41.23 + . These computations are difficult to explain, except on the assumption that they involve sexagesimal fractions.
15. From what has been said it appears that the Babylonians had ideograms which, transliterated, are Igi-Gal for "denominator" or "division," and Lal for "minus." They had also ideograms which, transliterated, are Igi-Dua for "division," and A-Du and Ara for "times," as in Ara-1 18, for "1 × 18 = 18," Ara-2 36 for "2 × 18 = 36"; the Ara was used also in "squaring," as in 3 Ara 39 for "3 × 3 = 9." They had the ideogram Ba–Di–E for "cubing," as in 27-E 3 Ba-Di-E for "33 = 27"; also Ib-Di for "square," as in 9-E 3 Ib-Di for "32 = 9." The sign A – An rendered numbers "distributive."
EGYPTIANS
16. The Egyptian number system is based on the scale of 10, although traces of other systems, based on the scales of 5, 12, 20, and 60, are believed to have been discovered. There are three forms of Egyptian numerals: the hieroglyphic, hieratic, and demotic. Of these the hieroglyphic has been traced back to about 3300 B.C.; it is found mainly on monuments of stone, wood, or metal. Out of the hieroglyphic sprang a more cursive writing known to us as hieratic. In the beginning the hieratic was simply the hieroglyphic in the rounded forms resulting from the rapid manipulation of a reed-pen as contrasted with the angular and precise shapes arising from the use of the chisel. About the eighth century B.C. the demotic evolved as a more abbreviated form of cursive writing. It was used since that time down to the beginning of the Christian Era. The important mathematical documents of ancient Egypt were written on papyrus and made use of the hieratic numerals.
17. The hieroglyphic symbols were [ILLUSTRATION OMITTED] for 1, [ILLUSTRATION OMITTED] for 10, [ILLUSTRATION OMITTED] for 100, [ILLUSTRATION OMITTED] for 1,000, [ILLUSTRATION OMITTED] for 10,000, [ILLUSTRATION OMITTED] for 100,000, [ILLUSTRATION OMITTED] for 1,000,000, [ILLUSTRATION OMITTED] for 10,000,000. The symbol for 1 represents a vertical staff; that for 1,000 a lotus plant; that for 10,000 a pointing finger; that for 100,000 a burbot; that for 1,000,000 a man in astonishment, or, as more recent Egyptologists claim, the picture of the cosmic deity Hh. The symbols for 1 and 10 are sometimes found in a horizontal position.
18. We reproduce in Figures 5 and 6 two tables prepared by Kurt Sethe. They show the most common of the great variety of forms which are found in the expositions given by Möller, Griffith, and Brugsch.
Observe that the old hieratic symbol for ¼ was the cross ×, signifying perhaps a part obtainable from two sections of a body through the center.
19. In writing numbers, the Egyptians used the principles of addition and multiplication. In applying the additive principle, not more than four symbols of the same kind were placed in any one group. Thus, 4 was written in hieroglyphs [ILLUSTRATION OMITTED]; 5 was not written [ILLUSTRATION OMITTED], but either [ILLUSTRATION OMITTED] or [ILLUSTRATION OMITTED]. There is here recognized the same need which caused the Romans to write V after IIII, L = 50 after XXXX = 40, D = 500 after CCCC = 400. In case of two unequal groups, the Egyptians always wrote the larger group before, or above the smaller group; thus, seven was written [ILLUSTRATION OMITTED].
20. In the older hieroglyphs 2,000 or 3,000 was represented by two or three lotus plants grown in one bush. For example, 2,000 was [ILLUSTRATION OMITTED]; correspondingly, 7,000 was designated by [ILLUSTRATION OMITTED]. The later hieroglyphs simply place two lotus plants together, to represent 2,000, without the appearance of springing from one and the same bush.
21. The multiplicative principle is not so old as the additive; it came into use about 1600–2000 B.C. In the oldest example hitherto known, the symbols for 120, placed before a lotus plant, signify 120,000. A smaller number written before or below or above a symbol representing a larger unit designated multiplication of the larger by the smaller. Möller cites a case where 2,800,000 is represented by one burbot, with characters placed beneath it which stand for 28.
22. In hieroglyphic writing, unit fractions were indicated by placing the symbol [ILLUSTRATION OMITTED] over the number representing the denominator. Exceptions to this are the modes of writing the fractions 1/2 and 2/3; the old hieroglyph for 1/2 was [ILLUSTRATION OMITTED], the later was [ILLUSTRATION OMITTED]; of the slightly varying hieroglyphic forms for 2/3, [ILLUSTRATION OMITTED] was quite common.
23. We reproduce an algebraic example in hieratic symbols, as it occurs in the most important mathematical document of antiquity known at the present time—the Rhind papyrus. The scribe, Ahmes, who copied this papyrus from an older document, used black and red ink, the red in the titles of the individual problems and in writing auxiliary numbers appearing in the computations. The example which, in the Eisenlohr edition of this papyrus, is numbered 34, is hereby shown. Hieratic writing was from right to left. To facilitate the study of the problem, we write our translation from right to left and in the same relative positions of its parts as in the papyrus, except that numbers are written in the order familiar to us; i.e., 37 is written in our translation 37, and not 73 as in the papyrus. Ahmes writes unit fractions by placing a dot over the denominator, except in case of 1/2, 1/3, 2/3, 1/4, each of which had its own symbol. Some of the numeral symbols in Ahmes deviate somewhat from the forms given in the two preceding tables; other symbols are not given in those tables. For the reading of the example in question we give here the following symbols:
[ILLUSTRATION OMITTED]
(Continues...)
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Table of Contents
Contents
I. Introduction,II. Numeral Symbols and Combinations of Symbols,
III. Symbols in Arithmetic and Algebra (Elementary Part),
IV. Symbols in Geometry (Elementary Part),
Index,