The history of mathematics: A brief course (2013)

Part II. The Middle East, 2000–1500 BCE

In the five chapters that constitute this part of our study, we examine the mathematics produced in two contemporaneous civilizations, in Mesopotamia and Egypt, over a period from about 4000 to 3500 years ago. We shall look at the way each of these societies wrote numbers and calculated with them, and we shall discuss the uses they made of their calculations in geometry and applied problems.

Contents of Part II

1. Chapter 3 (Overview of Mesopotamian Mathematics) sketches the archaeological and mathematical background needed to appreciate the mathematical achievements recorded on the Old Babylonian tablets.

2. Chapter 4 (Computations in Ancient Mesopotamia) discusses some of the arithmetical and algebraic problems solved on the cuneiform tablets.

3. Chapter 5 (Geometry in Mesopotamia) looks at the area and volume problems solved by Mesopotamian mathematicians and their use of the Pythagorean theorem.

4. Chapter 6 (Egyptian Numerals and Arithmetic) introduces the numbering system used in ancient Egypt and the idiosyncratic method of multiplying by repeated doubling that is characteristic of this culture.

5. Chapter 7 (Algebra and Geometry in Ancient Egypt) discusses the applications of these numerical techniques made by the Egyptians in the areas of surveying, commerce, and engineering.

Chapter 3. Overview of Mesopotamian Mathematics

Some quite sophisticated mathematics was developed four millennia ago in the portion of the Middle East that now forms the territory of Iraq and Turkey. This mathematics, along with a great deal of other lore, was written on small clay tablets in a style known as cuneiform (wedge-shaped), each tablet devoted to a limited topic. Nothing like a systematic treatise contemporary with this early mathematics exists. Scholars have had to piece together a mosaic picture of this mathematics from a few hundred clay tablets that show how to solve particular problems.

3.1 A Sketch of Two Millennia of Mesopotamian History

The region known as Mesopotamia (Greek for “between the rivers”) was the home of many successive civilizations. The name of the region derives from the two rivers, the Euphrates and the Tigris, that flow from the mountainous regions around the Mediterranean, Black, and Caspian seas into the Persian Gulf. In ancient times this region was invaded and conquered many times, and the successive dynasties spoke and wrote in many different languages. The long-standing convention of referring to all the mathematical texts that come from this area as “Babylonian”—a term used as early as 450 BCE by the Greek historian Herodotus—gives undue credit to a single one of the many dynasties that dominated this region. Nevertheless, the appellation does fit the present discussion, since the tablets we are going to discuss are written in Old Babylonian.

Although many different peoples invaded this region over time, occupying different parts of it, we are going to oversimplify this history and divide it into eight different civilizations, as follows:

1. Sumerian. The Sumerians were either the original inhabitants of the region or immigrants from farther east. They spoke a language unrelated to the Semitic and Indo-European groups. They held sway over this region for several hundred years, starting about 3000 BCE. It was the Sumerians who invented the cuneiform writing, made by pressing a stylus into wet clay. Many of the small clay tablets containing such records dried out and have kept their information for over 4000 years.

2. Akkadian. These people were conquerors who spoke a Semitic language and adapted the Sumerian cuneiform writing to their own language. One consequence was the compilation of Sumerian–Akkadian dictionaries, very useful for the later deciphering of these documents. The Akkadians established a commercial empire under King Sargon (ca. 2371–2316 BCE), which eventually collapsed and was replaced by a system of city–states in which the city of Ur at the mouth of the Euphrates was dominant.

3. Amorite. The Amorites, like the Akkadians, spoke a Semitic language. They invaded the area just before 2000 BCE and established a number of small kingdoms, of which Assyria was the first to become prominent, soon to be succeeded by Babylon under Hammurabi (1792–1750 BCE). This was the time when the Old Babylonian mathematical tablets were written. These tablets are the ones that will be discussed in the present chapter and the two following.

4. Hittite. The Hittites expanded from the west, the region now called Turkey. They spoke a language of the Indo-European family (the family to which English belongs). By 1650 BCE they had established a kingdom to rival the Amorites, and in 1595 they sacked the city of Babylon. The Hittite civilization collapsed around 1200 BCE due ultimately to pressure from the west exerted by the “Sea Peoples,” among whom were the Peleset, a people known to us from the Bible as the Philistines. They are the source of the name Palestine.

5. Assyrian. The Sea Peoples, although they caused the collapse of the Hittite Empire, did not occupy the portion of Mesopotamia that had been part of that empire. Instead, an empire based in the old city of Assyria began to grow and expand as far as its very well organized army and clever diplomacy could sustain it. The Assyrians eventually controlled a large portion of the region between the Mediterranean and the Persian Gulf, including present-day Palestine and parts of northern Egypt. Since this empire included the city of Babylon, it absorbed a great deal of the culture associated with that city. The Assyrian Empire was finally conquered by the Chaldean King Nebuchadnezzar (605–562 BCE).

6. Chaldean. This empire, although very short-lived (ca. 625–539 BCE), is well-known in the West because of Nebuchadnezzar, who is mentioned in the books of Kings, Jeremiah, and Daniel in the Bible. It was Nebuchadnezzar who conquered Jerusalem in 597 BCE and took the King of Judah and his followers into exile in Babylon. This civilization exerted a great influence on the writers of the Bible, especially the customs of the Chaldean court, where astrology was taken seriously.

7. Persian. As is well known from the Book of Daniel, the Chaldean empire was conquered in 539 BCE by the Persian king Cyrus the Great. Cyrus repatriated the exiles from Jerusalem and allowed the rebuilding of the Temple. The Persians, who speak an Indo-European language, have had an unbroken civilization since that time, although one subject to many changes of dynasty and religion. We shall see them coming into the story of mathematics at various points.

8. Seleucid. The high period of culture in mainland Greece coincided with the rise of the Athenian Empire in the middle of the fifth century BCE. The Athenian Empire was perceived as a threat by the Spartans, who brought it down through the Peloponnesian War (431–404 BCE). By that time, however, Greek scholarship and the Greek language were well established as intellectual forces. When the Macedonian kings Philip and Alexander conquered the territory eastward from mainland Greece to India and westward along the African coast of the Mediterranean, they consciously attempted to spread this culture. As a result, intellectual centers grew up in widely separated places where scholars, not all Greek by birth, wrote and argued in the Greek language. The three best-known Greek mathematicians, Euclid, Archimedes, and Apollonius, lived and worked in Egypt, Sicily, and what is now Turkey.

When Alexander died in 323 BCE, his empire was divided among three of his generals. Besides the original Macedonian kingdom centered at Pella just north of Greece, there were two other regions with centers in Egypt and the Fertile Crescent. Egypt was ruled by the general Ptolemy Soter (the last of his heirs was Cleopatra, who presided over the incorporation of Egypt into the Roman Empire under Julius Caesar) while the regions around the Fertile Crescent were ruled by general Seleucus and thereby became known as the Seleucid Kingdom.

Unfortunately, in an introductory course, we cannot provide full details of the development of mathematics over this vast period of time. The reader should bear in mind that the mathematical examples in this chapter and the two following are a limited selection, wrenched out of their context. We are focusing on just a few salient features of one or two periods in this long and complicated history and are cherry-picking only the mathematics that seems most likely to interest the modern reader.

3.2 Mathematical Cuneiform Tablets

Of the many thousands of cuneiform texts scattered through museums around the world, several hundred have been found to be mathematical in content. Deciphering them was made simpler by mutilingual tablets that were created because the cuneiform writers themselves had need to know what had been written in earlier languages. A considerable amount of the credit for the decipherment must go to Sir Henry Rawlinson (1810–1895), who spent several years transcribing a trilingual inscription carved in a cliff at what is now Bisutun, Iran. This inscription, in Old Persian (an Indo-European language), Babylonian (a Semitic language derived from Akkadian), and Elamite (a “language isolate,” having no close relatives), tells of the reign of the Persian king Darius, who was successor to Cyrus the Great and reigned from 522 to 486. Its decipherment led to the recovery of the Akkadian language, which had gone extinct in the first century CE; and Akkadian led to the recovery of the Sumerian language, the language of the earliest civilization in Mesopotamia. Sumerian and Akkadian were freely mixed over a period of centuries and nearly melded into a single language.

By 1854, enough tablets had been deciphered to reveal the system of computation used in ancient Mesopotamia, and by the early twentieth century a considerable number of mathematical texts had been deciphered and analyzed. A detailed analysis of the ones known up to 1935 was presented in a two-volume work by Otto Neugebauer (1899–1992), Mathematische Keilschrifttexte, republished by Springer-Verlag in 1973. A more up-to-date study has been published by the Oxford scholar Eleanor Robson (1999).

Some of the tablets that have been discussed by historians of mathematics appear to be “classroom materials,” written by teachers as exercises for students. One clue that points toward this conclusion is that the answers so often “come out even.” As Robson (1995, p. 11, quoted by Melville, 2002, p. 2) states, “Problems were constructed from answers known beforehand.” See also the more recent book of Robson (2008, p. 21). Melville provides an example of a different kind from tablet YBC 4652 of the Yale Babylonian Collection in which the figures are not “rigged,” but a certain technique is presumed. Although there is an unavoidable lack of unity and continuity in the Mesopotamian texts compared with mathematics written on media more amenable to extensive treatises, such as papyrus and paper, the cuneiform tablets nevertheless contain many problems similar to problems studied in other places such as India, China, and Egypt.

The applications that were made of these techniques must be conjectured, but we may confidently assume that they were the same everywhere: commerce, government administration, and religious rites, all of which call for counting and measuring objects on the earth and making mathematical observations of the sky in order to keep track of months and years. According to Robson (2009, pp. 217–218), the education of a surveyor, which required both numerical and geometric skill, was of crucial importance in keeping public order:

When I go to divide a plot, I can divide it; when I go to apportion a field, I can apportion the pieces, so that when wronged men have a quarrel I soothe their hearts and [. . .]. Brother will be at peace with brother.

3.3 Systems of Measuring and Counting

The systems of numeration still used in the United States, the last bastion of resistance to the metric system, show that people once counted by twos, threes, fours, sixes, eights, and twelves. In the United States, eggs and pencils, for example, are sold by the dozen or the gross. Until recently, stock averages were quoted in eighths rather than tenths. Measures of length, area, and weight show other groupings. Consider the following words: fathom (6 feet), foot (12 inches), pound (16 ounces), yard (3 feet), league (3 miles), furlong (1/8 of a mile), dram (1/8 or 1/16 of an ounce, depending on the context), karat (1/24, used as a pure number to indicate the proportion of gold in an alloy),¹ peck(1/4 of a bushel), gallon (1/2 peck), pint (1/8 of a gallon), and teaspoon (1/3 of a tablespoon). The strangest unit of all in the formidable English system—no longer the English system since the UK became part of the European Union—is the acre, 1/640 of a square mile. In the United States, a square mile was called a section, and farms commonly consisted of a quarter of a section, 160 acres. (In metric units, an acre is about 0.4 hectares.)

Even in science, however, there remain some vestiges of nondecimal systems of measurement inherited from the ancient Middle East. In the measurement of both angles and time, minutes and seconds represent successive divisions by 60. A day is divided into 24 hours, each of which is divided into 60 minutes, each of which is divided into 60 seconds. At that point, our division of time becomes decimal; we measure races in tenths and hundredths of a second. A similar renunciation of consistency came in the measurement of angles as soon as hand-held calculators became available. Before these calculators came into use, students (including the present author) were forced to learn how to interpolate trigonometric tables in minutes (1/60 of a degree) and seconds (1/60 of a minute). In physical measurements, as opposed to mathematical theory, we still divide circles into 360 equal degrees. But our hand-held calculators have banished minutes and seconds. They divide degrees decimally and of course make interpolation an obsolete skill. Since π is irrational, it seems foolish to adhere to any rational fraction of a circle as a standard unit; hand-held calculators are perfectly content to use the natural (radian) measure, and we could eliminate a useless button by abandoning degrees entirely. That reform, however, is likely to require even more time than the adoption of the metric system.²

3.3.1 Counting

A nondecimal counting system reported (1937) by the American mathematicians David Eugene Smith (1860–1944) and Jekuthiel Ginsburg (1889–1957) as having been used by the Andaman of Australia illustrates how one can count up to certain limits in a purely binary system. The counting up to 10, translated into English, goes as follows: “One two, another one two, another one two, another one two, another one two. That's all.” In saying this last phrase, the speaker would bring the two hands together. This binary counting appears to be very inefficient from a human point of view, but it is the system that underlies the functioning of computers, since a switch has only two positions. The binary digits or bits, a term that seems to be due to the American mathematician Claude Shannon, are generally grouped into larger sets for processing.

Although bases smaller than 10 are used for various purposes, some societies have used larger bases. Even in English, the word score for 20 (known to most Americans only from the first sentence of Lincoln's Gettysburg Address) does occur. In French, counting between 60 and 100 is by 20s. Thus, 78 is soixante dix-huit (sixty-eighteen) and 97 is quatre-vingt dix-sept (four-twenty seventeen). Menninger (1969, pp. 69–70) describes a purely vigesimal (base 20) system used by the Ainu of Sakhalin. Underlying this system is a base 5 system and a base 10 system. Counting begins with shi-ne (begin-to-be = 1), and progresses through such numbers as aschick-ne (hand = 5), shine-pesan (one away from [10] = 9), wan (both sides = both hands = 10), to hot-ne (whole-[person]-to-be = 20). In this system 100 is ashikne hotne or 5 twenties; 1000, the largest number used is ashikne shine wan hotne or 5 ten-twenties. There are no special words for 30, 50, 70, or 90, which are expressed in terms of the basic 20-unit. For example, 90 is wan e ashikne hotne (10 from 5 twenties). Counting by subtraction probably seems novel to most people, but it does occur in Roman numerals (IV = 5 − 1), and we use subtraction to tell time in expressions such as ten minutes to four and quarter to five.³

3.4 The Mesopotamian Numbering System

As the examples of angle and time measurement show, the successive divisions or regroupings in a number system need not have the same number of elements at every stage. The Mesopotamian sexagesimal system appears to have been superimposed on a decimal system. In the cuneiform tablets in which these numbers are written the numbers 1 through 9 are represented by a corresponding number of wedge-shaped vertical strokes, and 10 is represented by a new symbol, a hook-shaped mark that resembles a boomerang (Fig. 3.1). However, the next grouping is not ten groups of 10, but rather six groups of 10. Even more strikingly, the symbol for the next higher group is again a vertical stroke. Logically, this system is equivalent to a base-60 place-value system with a floating “decimal” (sexagesimal) point that the reader or writer had to keep track of mentally. Within each unit (sexagesimal rank) of this system there is a truncated decimal system that is not place-value, since the ones and tens are distinguished by different symbols rather than physical location.

Figure 3.1 The cuneiform number 45.

3.4.1 Place-Value Systems

Since we take our familiar place-value decimal system for granted, it is worth remembering that several advanced civilizations, including those of Egypt, ancient Greece, and ancient Rome, did not have such a system. The Egyptians and ancient Greeks (who probably copied the Egyptians in this matter) had, as we do, individual symbols for the numbers 1 through 9, but they had nine more symbols for 10 through 90 and another nine symbols for 100 through 900. (The Greeks used their 24-letter alphabet, along with three obsolete letters, to get these 27 symbols.) Even the Chinese system, which was decimal, used separate symbols for each power of 10. For example, the numbers 1, 2, and 3 are symbolized as –, =, and ≡, while 10 was represented by a cross shape (+ or †). But 20 was written as =+, that is, a 2 whose value was shown by being attached to the symbol for the corresponding power of 10. This extra symbol, we can now see, was not needed if you have a symbol for an empty place, since the physical location of the 2 suffices to show its value. Thus, the Chinese system was “just short of” a full place-value system.

It is therefore somewhat surprising that a pure place-value sexagesimal system arose as early as 4000 years ago in Mesopotamia, with which Egypt, Greece, and Rome were in contact. Somehow, the advantages of the system penetrated only Greco-Roman science, not commerce and other economic activity.⁴ In its original form, this system lacked one feature that we regard as essential today, a symbol for an empty place (zero). The later Greek writers, such as Ptolemy in the second century CE, used the sexagesimal notation with a circle to denote an empty place.

Although it obviously began as a decimal system, since there are distinct symbols for 1 and 10, the special symbol for 100, which one would expect in such a system, does not occur in the clay tablets from which most of our knowledge about Mesopotamian mathematics is derived. The reason is that at some point the Mesopotamians developed a true place-value system of writing numbers, using 60 as a base. This system was taken over, but for scientific purposes only, by the Greeks, who passed on to the modern world the idea of dividing a day into 24 hours, an hour into 60 minutes, a minute into 60 seconds, and a circle into 360 degrees. To get a picture of the way the writers were thinking that doesn't require the use of non-standard symbols, historians of the subject have invented a way of transcribing the numbers into easily recognizable forms. We shall now describe this transcription.

3.4.2 The Sexagesimal Place-Value System

You are familiar with the fact that the number 3926 means 3 × 10³ + 9 × 10² + 2 × 10¹ + 6 × 10⁰, that is, 3000 + 900 + 20 + 6. We are working in base 10 here, and each digit will be an integer between 0 and 9.

If we were interpreting it in base 60, the symbol 3926 would mean 3 × 60³ + 9 × 60² + 2 × 60 + 6, that is, 648000 + 32400 + 120 + 6, or 680,526 in decimal notation. We could write this equality as 3926₆₀ = 680526₁₀. However, we shall normally omit the subscripts, since it will be obvious which of the two bases is meant.

In sexagesimal notation, each digit is between 0 and 59, and that fact gives rise to some ambiguity: How can we be sure the number we just looked at was not meant to be, for example, 39 × 60 + 26? The distinction would be clear in authentic cuneiform notation, since there is a special symbol for 10. To make it clear in our transcription, we will use a comma to separate the digits of all sexagesimal numbers. In that way, we can distinguish between 3, 9, 2, 6 and 39, 26. Between the integer and fractional parts of the number, we shall write a semicolon in our transcription. Thus 35, 6; 12, 9 means , which in decimal notation would be 2106.2025. As another example, the number that we write as 85.25 could be transcribed into this notation as 1, 25; 15, meaning .

Converting a number written in sexagesimal notation into decimal notation is a very easy matter. Just insert the appropriate power of 60 (positive for digits left of the units digit, negative for digits right of it) in each place and multiply by the digit in that place; then add the products. This skill requires hardly any practice. You should be able to verify easily that 13,7;21 converts to and that 2,29;15,11 converts to , where the 5's repeat forever. As you see, a terminating sexagesimal number may fail to terminate when translated into decimal notation. A terminating decimal number, however, will always terminate when converted to sexagesimal notation, because 10 divides 60. Our first task is to spend a little time converting between the hybrid notation just introduced for the sexagesimal system and the decimal system we are familiar with, so that numbers written in this system will appear less strange.

3.4.3 Converting a Decimal Number to Sexagesimal

The procedure for converting from decimal notation to sexagesimal requires separate handling of the integer and fractional parts of a number. We begin by discussing how to convert integers. The general procedure is sufficiently well illustrated by the conversion of the decimal number 3,874,065 into sexagesimal notation. The rule is to do repeated long division with 60 as the divisor, using the remainder at each stage as the sexagesimal digit and the quotient as the next dividend, until finally a quotient less than 60 is obtained and used as the leftmost digit:

with a remainder of 45. Hence the units place is 45. Now we divide the quotient (64567) by 60:

with a remainder of 7. Hence the 60-digit is 7. We then divide 1076 by 60:

with a remainder of 56, so that the 60²-digit is 56, and now the 60³-digit is immediately seen to be 17. Thus

All you have to remember is that you are working leftward from the “sexagesimal point” (the semicolon). You can verify that this is correct by converting in the opposite direction: 17 × 60³ + 56 × 60² + 7 × 60 + 45 = 3, 874, 065.

We next show how to convert a fractional number from either common-fraction or decimal-fraction form into sexagesimal form. Since multiplying a number by 60 is equivalent to moving the sexagesimal point to the right, the basic principle is that these successive sexagesimal digits reveal themselves as the integer part of the product when the number is repeatedly multiplied by 60. This principle is completely obvious and trivial if the fraction is given in sexagesimal form to begin with. For example, suppose the number is N = 0 ; 43, 12, 19. Then 60N = 43 ; 12, 19, so that the first digit of the original fractional number N is the integer part (43) of 60N. After discarding that integer part, we can get the second digit by multiplying what is left again by 60, and obviously that will be 12 in the present case. This procedure works whether or not the fraction is given in sexagesimal form. All we have to remember is to carry out the procedure “dual” to the procedure just described for converting integers. In this dual procedure, you repeatedly multiply by 60 and take the integer parts of the products as the successive digits. This time, you are working rightward, again away from the “sexagesimal point.”

We illustrate with the fraction . We find

Thus, the first digit right of the sexagesimal point is 4. To get the second one, we need to convert the fraction , which is done exactly the same way:

Thus, we have found that

And you can verify that this is correct:

Decimal fractions can be converted by following this procedure, using a hand calculator to facilitate the computation, or by changing it to a common fraction. For example, we could convert 0.337 to and proceed:

The first digit is thus 20. To get the next one we continue:

Thus second digit is now seen to be 13, and we get the third and final digit by converting to . Hence

Peculiarities to Watch For

If you have a repeating decimal for which you know an exact value as a common fraction, for example, 0.33333 . . . , which you know is , it is best to convert it to the common fraction before converting it to sexagesimal. The procedure given above will correctly convert into 0 ; 20. But if you work with its infinite decimal expression 0.3333 . . . , you will first of all have difficulty multiplying it by 60, since the multiplication has to start at the right-hand end, which is infinitely distant. Even if you do the obvious thing and say that 0.333. . . × 60 = 19.99999 . . ., you will get 19 as the first digit and then have to convert 0.99999 . . . , which will similarly yield 59.999999. . .when multiplied by 60. Hence you'd find that 0.333 . . . ₁₀ = 0 ; 19, 59, 59, 59, . . . ₆₀, which is correct, but clumsily expressed.

Some fractions do have nonterminating sexagesimal expansions. For example, will repeat with period 3:

3.4.4 Irrational Square Roots

Some of the tablets contain the sexagesimal number 1; 24, 51, 10, which is . It is clear from the context that this number is being used as an approximation to . The tablet YBC 7289 in the Yale Babylonian Collection exhibits the number 0 ; 42, 30 in a context where it clearly means . There are various ways in which these approximations might have been arrived at. The simplest conjecture amounts essentially to what eventually became generalized as the Newton–Raphson method of approximating the root of a functional equation f(x) = 0. In the limited context of the equation x² − 2 = 0, the method has a much simpler explanation than it gets in calculus courses.

We begin by noting that must be between 1 and 2. Hence, let us begin with as an approximation. This number happens to be too large, but we do not have to know that in order to improve the approximation. If an approximation a is too large, then will be too small, and hence we are likely to improve the approximation by averaging a and :

Starting with , we find the next approximation to be

This is our new a, and we continue from there. If is taken as an approximation of , then is an approximation for , and that is the approximation actually used in the tablet YBC 7289. The next approximation to is , and its sexagesimal expansion begins 1 ; 24, 51, 10, which is the approximation used in the tablets.

This explanation is only a conjecture; we don't really know how square roots were found. In the next chapter, we shall discuss how computations were done within the sexagesimal system. You can imagine that it was not so easy as it is with our base-10 system.

Problems and Questions

Mathematical Problems

3.1. Convert the sexagesimal number 11, 4, 29 ; 58, 7 into decimal notation.

3.2. Convert the decimal number 4752.73 into sexagesimal notation.

3.3. Convert the improper fraction into sexagesimal notation. (Get the first four digits on the right of the sexagesimal point. This sexagesimal expansion does eventually repeat. However, it has a powerfully long period!)

Historical Questions

3.4. What special circumstances made it possible to decipher the cuneiform tablets?

3.5. Why was it important for a government in ancient times to have a cadre of competent surveyors?

3.6. What are the differences in the counting systems used in ancient China, Egypt, and Mesopotamia?

Questions for Reflection

3.7. Why would it be more difficult to use a sexagesimal place-value system than a decimal place-value system? How would you overcome this difficulty if you had only the sexagesimal system to use?

3.8. What other bases, besides 10 and 60, have you heard of being used? Suppose one person was using base 7 and another base 12. What advantages would each have over the other?

3.9. How might a sexagesimal system have originated in the first place, since by far the commonest bases used throughout the world are 10, 5, and 20?

Notes

1. The word is a variant of carat, which also means 200 milligrams when applied to the size of a diamond.

2. By abandoning another now-obsolete system—the Briggsian logarithms—we could eliminate two buttons on the calculators. The base 10 was useful in logarithms only because it allowed the tables to omit the integer part of the logarithm. Since no one uses tables of logarithms any more, and the calculators don't care how messy a computation is, there is really no reason to do logarithms in any base except the natural one, the number e, or perhaps base 2 (in number theory). Again, don't expect this reform to be achieved in the near future.

3. Technology, however, is rapidly removing this last vestige of the old way of counting from everyday life. Circular clock faces have been largely replaced by linear digital displays, and ten minutes to four has become 3:50. This process began long ago when railroads first imposed standard time in place of mean solar time and brought about the first 24-hour clocks.

4. Perhaps, considering the cries of outrage whenever any attempt is made to use the metric system in the United States, we should not be surprised.