Fig. 76.—Ancient Egyptian and Assyrian numeration.
Fig. 77.—Mode of calculation by counters and by figures on Abacus.
To understand how the art of ciphering came to be invented, it is necessary to go back to a ruder state of things. In Africa, negro traders may be seen at market reckoning with pebbles, and when they come to five, putting them aside in a little heap. In the South Sea Islands it has been noticed that people reckoning, when they came to ten, would not put aside a heap of ten things, but only a single bit of coco-nut stalk to stand for ten, and then a bigger piece when they wanted to represent ten tens or a hundred. Now to us it is plain that this use of different kinds of markers is unnecessary, but all that the reckoner with little stones or beans has to do, is to keep separate his unit-heap, his ten-heap, his hundred-heap, &c. This use of such things as pebbles for “counters,” which still survives in England among the ignorant, was so common in the ancient world, that the Greek word for reckoning was psēphizein, from psēphos, a pebble, and the corresponding Latin word was calculare from calculus, a pebble, so that our word calculate is a relic of very early arithmetic. Now to work such pebble-counting in an orderly manner, what is wanted is some kind of abacus or counting-board with divisions. These have been made in various forms, as the Roman abacus with lines of holes for knobs or pegs, or the Chinese swan-pan with balls strung on wires, on which the native calculators in the merchants’ counting-houses reckon with a speed and exactness that fairly beats the European clerk with his pencil and paper. It may have been from China that the Russian traders borrowed the ball-frame on which they also do their accounts, and it is said that a Frenchman noticing it in Russia at the time of Napoleon’s invasion was struck with the idea that it would serve perfectly to teach little children arithmetic; so he introduced it in France, and thence it found its way into English infants’ schools. Now whatever sort of abacus is used, its principle is always the same, to divide the board or tray into columns, so that in one column the stones, beans, pegs, or balls, stand for units, in the next column they are tens, in the next hundreds, and so on, [Fig. 77]. Here the three stones in the right-hand column stand for 3, the nine in the next column for 90, the one in the fourth column for 1,000 and so on. The next improvement was to get rid of the troublesome stones or beans, and write down numbers in the columns, as is here shown with Greek and Roman numerals. But now the calculator could do without the clumsy board, and had only to rule lines on his paper, to make columns for units, tens, hundreds, &c. The reader should notice that it is not necessary to the principle of the abacus that each column should stand for ten times the one next it. It may be twelve or twenty or any other number of times, and in fact the columns in our account-books for £ s. d. or cwts. qrs. lbs., are surviving representatives of the old method of the abacus. Such reckoning had still the defect that the numbers could not be taken out of the columns, for even when each number from one to nine has a single figure to stand for it, there would still be here and there an empty column (as is purposely left in [Fig. 77]) which would throw the whole into confusion. To us now it seems a very simple thing to put a sign to show an empty column, as we have learned to do with the zero or 0, so that the number expressed in the picture of the abacus can be written down without any columns, 241093. This invention of a sign for nothing, was practically one of the greatest moves ever made in science. It is the use of the zero which makes the difference between the old arithmetic and our easy ciphering. We give the credit of the invention to the Arabs by using the term Arabic numerals, while the Arabs call them Indian, and there is truth in both acknowledgments of the nations having been scholars in arithmetic one to the other. But this does not go to the root of the matter, and it is still unsettled whether ciphering was first devised in Asia, or may be traced further back in Europe to the arithmeticians of the school of Pythagoras. As to the main point, however, there is no doubt, that modern arithmetic comes out of ancient counting on the columns of the abacus, improved by writing a dot or a round 0 to show the empty column, and by this means young children now work calculations which would have been serious labour to the arithmeticians of the ancient world.
Next as to the art of measuring. Here it may be fairly guessed that man first measured, as he first counted, on his own body. When barbarians tried by finger-breadths how much one spear was longer than another, or when in building huts they saw how to put one foot before the other to get the distance right between two stakes, they had brought mensuration to its first stage. We sometimes use this method still for rough work, as in taking a horse’s height by hands, or stepping out the size of a carpet. If care is taken to choose men of average size as measurers, some approach may be made to fair measurement in this way. That it was the primitive way can hardly be doubted, for civilized nations who have more exact means still use the names of the body-measures. Besides the cubit, hand, foot, span, nail, already mentioned in [p. 17], we have in English the ell, (of which the early meaning of arm or fore-arm is seen in el-bow, the arm-bend), also the fathom or cord stretched by the outspread arms in sailors’ fashion, and the pace or double step (Latin passus) of which a thousand (mille) made the mile. But though these names keep up the recollection of early measurement by men’s limbs, they are now only used as convenient names for standard measures which they happen to come tolerably near to, as for instance one may go a long way to find a man’s foot a foot long by the rule. Our modern measurements are made by standard lengths, which we have inherited with more or less change from the ancients. It was a great step in civilization when nations such as the Egyptians and Babylonians made pieces of wood or metal of exact lengths to serve as standards. The Egyptian cubit-rules with their divisions may still be seen, and the King’s Chamber in the Great Pyramid measures very exactly 20 cubits by 10, the cubit being 20·63 of our inches. Our foot has scarcely altered for some centuries, and is not very different from the ancient Greek and Roman feet. The French at the first Revolution made a bold attempt to cast off the old traditional standards and go straight to nature, so they established the metre, which was to be a ten-millionth of the distance from the pole to the equator. The calculation however proved inexact, so that the metre is now really a standard measure of the old sort, but so great is the convenience of using the same measures, that the metre and its fractions are coming more and more into use for scientific work all over the world. The use of scales and weights, and of wet and dry measures, had already begun among the civilized nations in the earliest known times. Our modern standards can even to some extent be traced back to those of the old world, as for instance the pound and ounce, gallon and pint, come from the ancient Roman weights and measures.
Fig. 78.—Rudimentary practical Geometry. 1, scalene triangle; 2, folded right angle; 3, folded triangle; 4, rectangle folded in circle.
From measuring feet in length, men would soon come to reckoning the contents, say of an oblong floor, in square feet. But to calculate the contents of less simple figures required more difficult geometrical rules. The Greeks acknowledged the Egyptians as having invented geometry, that is, “land-measuring,” and there may be truth in the old story that the art was invented in order to parcel out the plots of fertile mud on the banks of the Nile. There is in the British Museum an ancient Egyptian manual of mensuration (the Rhind papyrus), one of the oldest books in the world, originally written more than 1,000 years before Euklid’s time, and which shows what the Egyptians then knew and did not know about geometry. From its figures and examples it appears that they used square measure, but reckoned it roughly; for instance, to get the area of the triangular field ABC [Fig. 78] (1) they multiplied half AC by AB, which would only be correct when BAC is a right angle. When the Egyptians wanted the area of a circular field, they subtracted one-ninth from the diameter and squared; thus if the diameter were 9 perches, they estimated that the circle contained 64 square perches, which the reader will find on trial is a good approximation. All this was admirable for the beginnings of geometry, and the record may well be believed that Greek philosophers such as Thales and Pythagoras, when they came to Egypt, gained wisdom from the geometer-priests of the land. But these Egyptian mathematicians, being a priestly order, had come to regard their rules as sacred, and therefore not to be improved on, while their Greek disciples, bound by no such scientific orthodoxy, were free to go on further to more perfect methods. Greek geometry thus reached results which have come down to us in the great work of Euklid, who used the theorems known to his predecessors, adding new ones and proving the whole in a logical series. It must be clearly understood that elementary geometry was not actually invented by means of definitions, axioms, and demonstrations like Euklid’s. Its beginnings really arose out of the daily practical work of land-measurers, masons, carpenters, tailors. This may be seen in the geometrical rules of the altar-builders of ancient India, which do not tell the bricklayer to draw a plan of such and such lines, but to set up poles at certain distances, and stretch cords between them. It is instructive to see that our term straight line still shows traces of such an early practical meaning; line is linen thread, and straight is the participle of the old verb to stretch. If we stretch a thread tight between two pegs, we see that the stretched thread must be the shortest possible; which suggests how the straight line came to be defined as the shortest distance between two points. Also, every carpenter knows the nature of a right angle, and he is accustomed to parallel lines, or such as keep the same distance from one another. To the tailor, the right angle presents itself in another way. Suppose him cutting a doubled piece of cloth to open out into the gore or wedge-shaped piece BAC in [Fig. 78] (2). He must cut ADB a right angle, or his piece when he opens it will have a projection or a recess, as seen in the figure. When he has cut it right, so that BDC opens in a straight line, then he cannot but see that the sides AB, AC, and the angles ABC, ACB must exactly match, having in fact been cut out on one another. Thus he arrives, by what may be called tailor’s geometry, at the result of Euklid I. 5, which now often goes by the name of the “asses’ bridge.” Such easy properties of figures must have been practically known very early. But it is also true that the ancients were long ignorant of some of the problems which now belong to elementary teaching. Thus it has just been mentioned how the Egyptian land-surveyors failed to make out an exact rule to measure a triangular field. Yet had it occurred to them to cut out the diagram of a triangle from a sheet of papyrus, as we may do with the triangle ABC in [Fig. 78] (3), and double it up as shown in the figure, then they would have found that it folds into the rectangle EFHG, and, therefore, its area is the product of the height by half the base. It would be seen that this is no accident, but a property of all triangles, while at the same time it would appear that the three angles at A, B, C, all folding together at D, make up two right angles. Though the more ancient Egyptian geometers do not seem to have got at either of these properties of the triangle, the Greek geometers had in some way become well aware of them before Euklid’s time. The old historians who tell the origin of mathematical discoveries do not always seem to have understood what they were talking of. Thus it is said of Thales that he was the first to inscribe the right-angled triangle in the circle, and thereupon sacrificed a bull. But a mathematician of such eminence could hardly have been ignorant of what any intelligent carpenter has reason to know, how an oblong board fits into a circle symmetrically; the problem of the right-angled triangle in the semicircle is involved in this, as is seen by (4) in the present figure. Perhaps the story really meant that Thales was the first to work out a strict geometrical demonstration of the problem. The tale is also told of Pythagoras, and another version is that he sacrificed a hekatomb on discovering that the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the other two sides (Euklid I. 47). The story is not a likely one of a philosopher who forbad the sacrifice of any animal. As for the proposition, it is one which may present itself practically to masons working with square paving-stones or tiles; thus, when the base is 3 tiles long, and the perpendicular 4, the hypothenuse will be 5, and the tiles which form a square on it will just be as many as together form squares on the other two sides. Whether Pythagoras got a hint from such practical rules, or whether he was led by studying arithmetical squares, at any rate he may have been the first to establish as a general law this property of the right-angled triangle, on which the whole systems of trigonometry and analytical geometry depend.
The early history of mathematics seems so far clear, that its founders were the Egyptians with their practical surveying, and the Babylonians whose skill in arithmetic is plain from the tables of square and cube numbers drawn up by them, which are still to be seen. Then the Greek philosophers, beginning as disciples of these older schools, soon left their teachers behind, and raised mathematics to be, as its name implies, the “learning” or “discipline” of the human mind in strict and exact thought. In its first stages, mathematics chiefly consisted of arithmetic and geometry, and so had to do with known numbers and quantities. But in ancient times the Egyptians and Greeks had already begun methods of dealing with a number without as yet knowing what it was, and the Hindu mathematicians, going further in the same direction, introduced the method now called algebra. It is to be noticed that the use of letters as symbols in algebra was not reached all at once by a happy thought, but grew out of an earlier and clumsier device. It appears from a Sanskrit book that the venerable teachers began by expressing unknown quantities by the term “so-much-as,” or by the names of colours, as “black,” “blue,” “yellow,” and then the first syllables of these words came to be used for shortness. Thus if we had to express twice the square of an unknown quantity, and called it “so much squared twice,” and then abbreviated this to so sq 2, this would be very much as the Hindus did in working out the following problem, given in Colebrooke’s Hindu Algebra: “The square root of half the number of a swarm of bees is gone to a shrub of jasmin: and so are eight-ninths of the whole swarm: a female is buzzing to one remaining male, that is humming within a lotus, in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees.” This Hindu equation is worked out clumsily from the want of the convenient set of signs = + -, which were invented later in Europe, but the minus numbers are marked, and the solution is in principle an ordinary quadratic. The Arab mathematicians learnt from India this admirable method, and through them it became known in Europe in the middle ages. The Arabic name given to it is al-jabr wa-l-mukabalah, that is, “consolidation and opposition,” this meaning what is now done by transposing quantities on the two sides of an equation; thence comes the present word algebra. It was not till about the 17th century in Europe that the higher mathematics were thoroughly established, when Descartes worked into a system the application of algebra to geometry, and Galileo’s researches on the path of a ball or flung stone brought in the ideas which led up to Newton’s fluxions and Leibnitz’s differential calculus, with the aid of which mathematics have risen to their modern range and power. Mathematical symbols have not lost the traces of their first beginnings as abbreviated words, as where n still stands for number and r for radius, while √, which is a running-hand r, does duty for root (radix), and ∫, which is an old-fashioned s, stands for the sum (summa) in integration.