Arithmetic
however, as well as other branches of mathematics, took root slowly among the western nations. Up to within a hundred years of Columbus' voyage, the University of Prague was considered progressive for offering a course of lectures on the art of reckoning with the fingers. The scholarship of England was content, in Shakespeare's day, with less mathematical instruction in the great universities of Oxford and Cambridge than is now given in village schools. Now and then a master mathematician, a Napier or a Newton, appeared in private life or among the learned professors, but mathematical lectures were not popular. Gentlemen's sons left arithmetic to "mere shopkeepers," who in turn got on with exceedingly crude methods of casting accounts. A recent writer, referring to the still more recent neglect of arithmetic in the noted preparatory schools of England, as Eton, Harrow, and Rugby, remarks: "We are safe in saying that before the close of the past (18th) century the ordinary school boy of England's famous public schools could not divide 2,021 by 43, though such problems had been performed centuries before by boys brought up on the banks of the Ganges."
In the American colonies, the college of William and Mary included a professor of mathematics in its first faculty, 1688, and thus enjoys the honor of having established the first American chair of mathematics. In 1749 the college faculty granted George Washington a commission as a land surveyor, which we may suppose fairly exhausted the mathematical curriculum of that college. a member of the Yale class of 1714 testifies that common arithmetic and a little surveying were the full extent of the mathematical instruction received by his class. The records of Harvard show that at this date two hours a week in the senior year were given to arithmetic, geometry, and astronomy, while algebra was not introduced until 1726.
Of arithmetic in the elementary schools of the colonies little can be said; for elementary schools existed only in the larger towns and the most favored localities. Arithmetic in the schools was confined to counting and to exceedingly simple combinations of integral numbers, or was not taught at all. Ordinarily the teacher, unless he were some collegian earning a trifle to help himself on his way, could not work in fractions, and indeed he was thought to do his whole duty if he kept order and taught the brighter children to read and write.
After the close of the American Revolution educational facilities improved rapidly. Arithmetic soon gained an acknowledged place in a boy's education, but we were well on into the nineteenth century before arithmetic was considered suitable for girls. Arithmetic for boys, but needle-work and knitting for girls.
As has been said, our earliest arithmetical ideas and our arithmetical texts were brought over from England. One of these early school books was a primer by George Fox, published in England in 1674, and subsequently republished in this country. It contained the alphabet, exercises in reading and spelling, explanations of Scripture names, Roman numerals, lessons in the fundamental rules of arithmetic and weights and measures, a perpetual almanac, and a Friends' catechism. This book was popular in Philadelphia. Similar texts were used in New England and in Virginia. Hodders' Arithmetic, or That Necessary Art Made Most Easy, published in London, 1661, and republished in Boston in 1719, is said to be the first purely arithmetical book printed in this country. In his Autobiography Benjamin Franklin mentions Cocker's Arithmetic as having been of great service to him. This text appeared in London in 1667 and was reprinted in Philadelphia during the Revolutionary War. It was an authority so long that "according to Cocker" became a proverb. Dilworthe's Schoolmaster's Assistant, the most popular of all English arithmetics used in this country, was published in London in 1744. Several American editions appeared, the last in Albany as late as 1824.
The first American arithmetic was written in 1729 by Isaac Greenwood, the first professor of mathematics in Harvard College. It was designed for the use of his college classes, and had little or no circulation outside. During the forty years which followed the Revolutionary War, a large number of arithmetics appeared in America. Three of these are famous,--The New and Complete System of Arithmetic, by Nicholas Pike, (Newburyport, 1788); The Schoolmaster's Assistant, by Nathan Daboll, 1800; and The Scholar's Arithmetic, by Daniel Adams. Other arithmeticians and their numerous texts have passed from memory, but Pike, Adams, and Daboll were held in affectionate remembrance by the grandfathers of the generation now in school, and the names of these three are sure of a place in the list of American educators.
Some of the characteristics of these early texts may be stated as follows:
1. An imperfect and evasive treatment of fractions as though the author did not understand the subject. 2. Cancellation was apparently unknown. 3. The English system of periods of six places each was followed. A billion was considered a million million. 4. No mental problems were given. 5. The rule of three, proportion, was taught as a mere rule, ignoring ratios and their equality. 6. Certain indirect solutions were sometimes introduced practically based on showing that results other than the right one were incorrect. 7. An attempt was made to introduce pleasing and ethical features, such as puzzles, and problems based on the expensiveness of vice. 8. Explanations were calculated merely to explain the application or working of rules. No attempt was made to give reasons why a step was legitimate or why a certain operation gave a correct result.
As might be expected, recitations were unheard of. Each pupil "ciphered" for himself and crowded up with the others to his instructor's desk to have his "answer" approved, to have a new "sum set," or to be admonished, as the case might be, for inability to "follow the rule."
Of recent years it may be said that a popular knowledge of arithmetic is greater in those sections of North America where public schools have reached their highest efficiency than in any other part of the world; but our contribution has been made to methods of instruction and to business methods. In its theory we follow the Hindu arithmetic practically unchanged.
The earliest mathematical notions of children and of savages are geometrical and physical rather than numerical. Dim graspings of space, distance, time, and mass precede the ability to make a distinction between one and more than one. Even with some command of number the primitive mind clings to other modes of expression. Children speak of a great distance by saying "a l-o-n-g way off," the prolongation of long being proportionate to the fancied distance. The Coeur d'Alene Indians indicate the proximity or remoteness of a lake or a river, by pronouncing the word "syah" with a peculiar upward prolongation of the first syllable so expressive that, to one who understands the customs of the mountains, they convey an accurate idea of whether the lake is an hour, or a day, or a week's journey distant.
Yet we must believe that counting is nearly as old as speech. Travelers have found no tribes so low in the scale of intelligence as to have no numerals. Even domestic animals are thought to have some idea of number. Farmers have a theory that crows can count as far as three. This is only a theory, however, based on a tradition that if a party of hunters enter a cornfield singly or in a group to lie in ambush for the black robbers, the crows will not come near until three men have gone away.
Each language and district has its series of numerals, but nearly all are based on five or some multiple of five. The inhabitants of New South Wales have but four numerical words in their vocabulary--a word each for one, two, and three, and an additional word for an indefinite number, having some such signification as many or plenty. To express five they display the fingers of one hand, and for ten the fingers of both hands. To express a greater number, which we may believe is seldom necessary, the fingers of an additional person are brought into use. In many aboriginal dialects, the word for five is also the word for hand, while ten is equivalent to two hands. Going a step further, certain South American tribes call the toes into requisition. Ten is expressed by a word meaning all the fingers; twenty, by all the fingers and toes. The term for forty is fingers and toes of two men. Other South American numerals with their significance are: five, the hand finished; six, one of the other hand; ten, two hands finished; eleven, foot one; twelve, foot two, etc. The Caribbean words for ten and twenty are quite poetic, signifying all the children of the hands and all the children of the hands and feet.
In the Zulu language the word for five is finish hand; for six, taking the thumb; for seven, pointer; for eight, keep back two fingers; for nine, keep back one finger; while at the word for ten the open hands are clapped together. If the student will begin at the little finger of the left hand and count to the left thumb, then to the right thumb and right forefinger, he will see the pertinence of these numerals.
The Eskimo expression for twenty is man, for forty two men. Illustrations may be given without number to show that counting is based usually on the fingers and toes.
The Aztec numeral system is interesting not only in its formation but also for its system of pictorial representation. A small banner or flag denoted twenty; if divided into corner sections by a vertical and by a horizontal line passing through the center, and one of the sections was colored, the flag indicated five; if two sections were colored, ten; if three, fifteen. Numbers below five were denoted by as many dots. Twenty 20's or 400 were indicated by a feather or quill, the hollow stock of which was commonly used to contain gold dust. Twenty 400's or 8,000 were denoted by a treasure sack or purse. Thus 12,038 in our system would have been denoted in the Aztec system by a running picture of one sack, ten quills, one full flag, one flag three-fourths colored, and three dots.
The Egyptian hieroglyphic system is like the Aztec in principle, differing only in symbols and in scale. One is a straight vertical stroke representing a staff. The next symbol, the significance of which is not known, denotes ten and resembles an inverted U or a croquet wicket. The third symbol denotes 100 and resembles the spiral line to be had by slicing a flat snail shell. One thousand is denoted by an object which for want of a better word we shall call an image. Ten thousand is denoted by a pointing forefinger; 100,000 is denoted by a fish; 1,000,000 by a man holding up both hands in utter amazement, and 10,000,000 is represented by a circle resting on a line, possibly suggestive of the universe or the uttermost bounds of knowledge. The scale is uniformly ten. Thus to write 1,200,042 in the Egyptian system we represent one man in amazement, two fishes, four wickets, and two vertical staves.
In the last two systems we have examined, and the list may be extended indefinitely, we may notice:
1. There is a distinct symbol for each order. In the Aztec system we have dots, flags, quills, and sacks for ones, twenties, four hundreds, and eight thousands. In the Egyptian system we have a peculiar and unmistakable sign for ones, another for tens, another for hundreds, and so on. The sign for one order can never be used for another order. The sign for two tens cannot be used for two hundreds. We must use wickets for tens and spirals for hundreds.
2. The value of a symbol is the same wherever it is placed. A flag, a dot, two quills, three dots, and a sack would signify the same number as a sack, two quills, a flag, and four dots. The value of the number is to be found by adding the values of the various signs regardless of their position.
3. Repeating a symbol repeats its value. To express the value of any number of flags less than enough to make a quill, it is necessary to repeat the symbol flag.
It is clearly evident that systems of counting arose from using the fingers and toes as counters. As to the origin of higher orders we have the germ in the very natural step of setting aside some object as a counter every time the tale of fingers or of fingers and toes was completed. Certain African tribes set aside a pebble for each five, the Aztecs evidently set aside a counter for each twenty. The inhabitants of some of the islands of the South Pacific count with nuts and cocoanut stalks, laying down a small stalk for each ten and a large stalk for each hundred, that is, for ten small stalks. Two large stalks, four small stalks, and six nuts would therefore signify two hundred forty six. The tens and hundreds of our numeral system originated, beyond a doubt, in some such primitive device. Instead of saying one big stalk, little stalk, we say one hundred ten, with this difference, that we forgot centuries ago what our words originally meant. Crude as four big stalks, three nuts may sound, and crude as it might seem to express 403 in South Sea symbols, our system has but three essential improvements over that of the Aztec, the Egyptian, and the South Sea Islander:
1. The Hindu hit upon the plan of representing the higher orders (tens, hundreds, thousands,) by the same characters used to denote the ones. In the primitive systems we have examined, as that of the Egyptians, it made no difference whether we drew four wickets and three staves or three staves and four wickets. In either case the sum of the symbols is to be taken and it is immaterial which stand first; but in the Hindu system place is made essential. There is a difference between 43 and 34. The first place is reserved for ones, that is for numbers from 1 to 9. The place on the left of ones is reserved for tens, and the third place, the second to the left of ones, is reserved for hundreds, etc. A symbol for four may be made to stand for four tens or for four hundreds by the simple device of putting it in the second place or the third place as may be desired. In this way it becomes unnecessary to retain separate symbols for tens, hundreds, etc., and this cumbersome feature of the aboriginal systems falls off. So important is the feature of position or place that the learned Hindu regarded it as a direct revelation from heaven.
2. Another step in advance is that of using distinct characters for each number less than ten. Instead of repeating dots, staves, or nuts, or indeed counters of any kind, the Hindus made a set of characters ranging from 1 to 9, from which our own have been derived.
3. The Hindus also hit upon the idea of using a character without value, a mere space filler, to occupy places not needed by the symbols of the number. Thus in writing 240, they used a cipher, 0, to fill the first place and throw the numeral 4 into the second place where it must be to stand for four tens. Otherwise the number would read twenty-four. This device of a cipher, in itself of no value, obviates the necessity of using ruled columns.
The Roman method of notation, by means of the letters I, V, X, L, C, D, and M, now seldom employed except for paging or sectioning, was at one time the sole reliance of European merchants and mathematicians. As late as the middle of the sixteenth century English shopkeepers kept their books and rendered their accounts in cumbrous Roman numerals. The Roman numerals, however, were used only to record results. Computations were made with the aid of counters or with a numeral frame called an abacus.
The names of our first ten numbers have lost their original meaning. A certain African tribe says bird's foot for four, referring of course to the toes, three forward and one rear, on the foot of a bird. Doubtless our one, two, three, four, five, six, seven, eight, nine, and ten, had some such meaning before they were used for numerals, but, however that may be, all trace of their original force has disappeared. For all that we now know of their history, six, seven, and eight might as well have been used in the reverse order. Eleven and twelve are from old Gothic forms anlif and twalif, in which we recognize the Scottish ane and twa prefixed to lif which is thought to signify ten. Thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, and nineteen are evidently three-ten, four-ten, etc. Twenty is twain-ten. Thirty is three-tens. Hundred is hund-rede, in which rede means a number or account. Thousand has now no other significance. Million is from the Latin word mille, signifying thousand, and means a great thousand. Billion and trillion are from bi and million and tri and million, signifying the second and third powers of a million, from which, however, we have diverted them. Naught comes from ne and aught, meaning not aught, not anything, nothing.
The characters used to express numbers are nine digits or significant figures, a cipher, and a decimal point:
1, 2, 3, 4, 5, 6, 7, 8, 9, 0,.
The first three digits are supposed to be modifications of one, two, and three pencil strokes. Of the first it is unnecessary to speak. An approximate 2 may be formed by making two short horizontal strokes, carrying the pencil on the paper from the right end of the first stroke to the left end of the second so as to form a Z. A three may have developed, it is thought, from three horizontal strokes, the pencil being carried on the paper as before. An examination of old script forms lends plausibility to this theory. 4 to 9 inclusive are said to be modifications of the initial letters of the old Indo-Bactrian names of the numbers they represent. 0 is considered a Brahminic symbol. It may be called zero, cipher, or naught, but never aught. The decimal point is a clerical device of modern origin, due to Simon Stevin, the Belgian inventor of decimals.
Nothing could be more fatal to a scholarly apprehension of our present system of arithmetic than to take the features of the system for granted, as though they were inherent in the principles of civilization and could not be otherwise. We have been so long accustomed to say that 3 x 4 = 12 that the expression 3 x 4 = 22 seems ridiculous; yet if the student will follow patiently he will see not indeed that the product of 3 and 4 is ever other than a dozen, but that a dozen may be written 22 quite as reasonably as it may be written 12. All depends on our understanding of the meaning which attaches to the numerals in their various positions. If our system were based on five and its powers, instead of on ten and the powers of ten, that is if we set aside a counter for each five instead of one for each ten, we should need but four digits. Seven would be written as a five and two ones, thus, 12. A dozen would be written as two fives and two ones, or 22. If 4,312 be a number written on the scale of five, it is composed of 4 one-hundred-twenty-fives, 3 twenty-fives, 1 five, and 2 ones. On the same supposition 20.2 is composed of 2 fives and 2 fifths. Such a system would be called a quinary instead of a decimal system.
See Geometry