A Short Account of the History of Mathematics

"There appeared in December 1921, just before this reprint was struck off, Sir T. L. Heath's work in 2 volumes on the History of Greek Mathematics. This may now be taken as the standard authority for this [first] period."

"The subject-matter of this book... primarily it is intended to give a short and popular account of those leading facts in the history of mathematics which many who are unwilling, or have not the time, to study it systematically may yet desire to know."

"The first edition was substantially a transcript of some lectures which I delivered in the year 1888 with the object of giving a sketch of the history, previous to the nineteenth century, that should be intelligible to any one acquainted with the elements of mathematics."

"Doubtless an exaggerated view of the discoveries of those mathematicians who are mentioned may be caused by the non-allusion to minor writers who preceded and prepared the way for them, but in all historical sketches this is to some extent inevitable, and I have done my best to guard against it by interpolating remarks on the progress of the science at different times."

"Generally I have not referred to the results obtained by practical astronomers and physicists unless there was some mathematical interest in them."

"In quoting results I have commonly made use of modern notation; the reader must therefore recollect that, while the matter is the same as that of any writer to whom allusion is made, his proof is sometimes translated into a more convenient and familiar language."

"For the history previous to 1758, I need only refer, once for all, to the closely printed pages of M. Cantor's monumental Vorlesungen über die Geschichte der Mathematik (hereafter alluded to as Cantor), which may be regarded as the standard treatise on the subject."

"Although the history of mathematics commences with that of the Ionian schools, there is no doubt that those Greeks who first paid attention to the subject were largely indebted to the previous investigations of the Egyptians and Phoenicians. Our knowledge of the mathematical attainments of those races is imperfect and partly conjectural..."

"The history of mathematics cannot with certainty be traced back to any school or period before that of the…Greeks…."

"Though all early races which have left records behind them knew something of numeration and mechanics, and though the majority were also acquainted with the elements of land-surveying, yet the rules which they possessed were in general founded only on the results of observation and experiment, and were neither deduced from nor did they form part of any science."

"The fact... that various nations in the vicinity of Greece had reached a high state of civilisation does not justify us in assuming that they had studied mathematics."

"Greek tradition uniformly assigned the special development of geometry to the Egyptians, and that of the science of numbers either to the Egyptians or to the Phoenicians."

"The magnitude of the commercial transactions of Tyre and Sidon necessitated a considerable development of arithmetic, to which it is probable the name of science might be properly applied."

"A Babylonian table of the numerical value of the squares of a series of consecutive integers has been found, and this would seem to indicate that properties of numbers were studied."

"According to Strabo the Tyrians paid particular attention to the sciences of numbers, navigation, and astronomy; they had, we know, considerable commerce with their neighbours and kinsmen the Chaldaeans."

"Whatever was the extent of their [the Chaldaeans] attainments in arithmetic, it is almost certain that the Phoenicians were equally proficient."

"It seems probable that the early Greeks were largely indebted to the Phoenicians for their knowledge of practical arithmetic or the art of calculation, and perhaps also learnt from them a few properties of numbers. It may be worthy of note that Pythagoras was a Phoenician; and according to Herodotus, but this is more doubtful, Thales was also of that race."

"The almost universal use of the abacus or swanpan rendered it easy for the ancients to add and subtract without any knowledge of theoretical arithmetic. These instruments... afford a concrete way of representing a number in the decimal scale, and enable the results of addition and subtraction to be obtained by a merely mechanical process."

"About forty years ago a hieratic papyrus, forming part of the Rhind collection in the British Museum, was deciphered... The manuscript was written by a scribe named Ahmes... The work is called "directions for knowing all dark things," and consists of a collection of problems in arithmetic and geometry; the answers are given, but in general not the processes by which they are obtained. It appears to be a summary of rules and questions familiar to the priests."

"The first part [of the Rhind Papyrus] deals with the reduction of fractions of the form 2/(2n + 1) to a sum of fractions each of whose numerators is unity... Probably he had no rule for forming the component fractions, and the answers given represent the accumulated experiences of previous writers: in one solitary case, however, he has indicated his method, for, after having asserted that 2/3 is the sum of 1/2 and 1/6, he adds that therefore two-thirds of one-fifth is equal to the sum of a half of a fifth and a sixth of a fifth, that is, to 1/10 + 1/30."

"That so much attention was paid to fractions is explained by the fact that in early times their treatment was found difficult. The Egyptians and Greeks simplified the problem by reducing a fraction to the sum of several fractions, in each of which the numerator was unity, the sole exception to this rule being the fraction 2/3. This remained the Greek practice until the sixth century of our era. The Romans, on the other hand, generally kept the denominator constant and equal to twelve, expressing the fraction (approximately) as so many twelfths. The Babylonians did the same in astronomy, except that they used sixty as the constant denominator; and from them through the Greeks the modern division of a degree into sixty equal parts is derived. Thus in one way or the other the difficulty of having to consider changes in both numerator and denominator was evaded. To-day when using decimals we often keep a fixed denominator, thus reverting to the Roman practice."

"In multiplication he [Ahmes] seems to have relied on repeated additions. Thus in one numerical example, where he requires to multiply a certain number, say a, by 13, he first multiplies by 2 and gets 2a, then he doubles the results and gets 4a, then he again doubles the result and gets 8a, and lastly he adds together a, 4a, and 8a. Probably division was also performed by repeated subtractions, but, as he rarely explains the process by which he arrived at a result, this is not certain."

"Ahmes goes on to the solution of some simple numerical equations. For example, he says "heap, its seventh, its whole, it makes nineteen," by which he means that the object is to find a number such that the sum of it and one-seventh of it shall be together equal to 19; and he gives as the answer 16 + 1/2 + 1/8, which is correct."

"The arithmetical part of the [Rhind] papyrus indicates that he had some idea of algebraic symbols. The unknown quantity is always represented by the symbol which means a heap; addition is sometimes represented by a pair of legs walking forwards, subtraction by a pair of legs walking backwards or by a flight of arrows; and equality..."

"He [Ahmes] concludes the work with some arithmetico-algebraical questions, two of which deal with arithmetical progressions and seem to indicate that he knew how to sum such series."

"Some methods of land-surveying must have been practised from very early times, but the universal tradition of antiquity asserted that the origin of geometry was to be sought in Egypt. ...Herodotus states that the periodical inundations of the Nile (which swept away the landmarks in the valley of the river, and by altering its course increased or decreased the taxable value of the adjoining lands) rendered a tolerably accurate system of surveying indispensable, and thus led to a systematic study of the subject by the priests."

"We have no reason to think that any special attention was paid to geometry by the Phoenicians, or other neighbours of the Egyptians. A small piece of evidence which tends to show that the Jews had not paid much attention to it is to be found in the mistake made in their sacred books, where it is stated that the circumference of a circle is three times its diameter: the Babylonians also reckoned that was equal to 3."

"That some geometrical results were known at a date anterior to Ahmes's work seems clear if we admit... that, centuries before it was written, the following method of obtaining a right angle was used in laying out the ground-plan of certain buildings. The Egyptians were very particular about the exact orientation of their temples; and they had therefore to obtain with accuracy a north and south line, as also an east and west line. By observing the points on the horizon where a star rose and set, and taking a plane midway between them, they could obtain a north and south line. To get an east and west line, which had to be drawn at right angles to this, certain professional "rope-fasteners" were employed. These men used a rope... divided by knots or marks... in the ratio 3 : 4 : 5. ...A similar method is constantly used at the present time by practical engineers for measuring a right angle. ...But though these are interesting facts in the history of the Egyptian arts we must not press them too far as showing that geometry was then studied as a science. Our real knowledge of the nature of Egyptian geometry depends mainly on the Rhind papyrus."

"Ahmes then goes on to find the area of a circular field … and gives the result as (d - 1/9d)2, where d is the diameter of the circle: this is equivalent to taking 3.1604 as the value of π, the actual value being very approximately 3.1416."

"Ahmes gives some problems on pyramids. ...Ahmes was attempting to find, by means of data obtained from the measurement of the external dimensions of a building, the ratio of certain other dimensions which could not be directly measured: his process is equivalent to determining the trigonometrical ratios of certain angles. The data and the results given agree closely with the dimensions of some of the existing pyramids. Perhaps all Ahmes's geometrical results were intended only as approximations correct enough for practical purposes."

"All the specimens of Egyptian geometry which we possess deal only with particular numerical problems and not with general theorems; and even if a result be stated as universally true, it was probably proved to be so only by a wide induction. ...Greek geometry was from its commencement deductive. There are reasons for thinking that Egyptian geometry and arithmetic made little or no progress subsequent to the date of Ahmes's work; and though for nearly two hundred years after the time of Thales Egypt was recognised by the Greeks as an important school of mathematics, it would seem that, almost from the foundation of the Ionian school, the Greeks outstripped their former teachers."

"Ahmes's book gives us much that idea of Egyptian mathematics which we should have gathered from statements about it by various Greek and Latin authors, who lived centuries later. Previous to its translation it was commonly thought that these statements exaggerated the acquirements of the Egyptians, and its discovery must increase the weight to be attached to the testimony of these authorities."

"We know nothing of the applied mathematics (if there were any) of the Egyptians or Phoenicians. The astronomical attainments of the Egyptians and Chaldaeans were no doubt considerable, though they were chiefly the results of observation: the Phoenicians are said to have confined themselves to studying what was required for navigation."

"At a very early period the Chinese were acquainted with several geometrical or rather architectural implements, such as the rule, square, compasses, and level; with a few mechanical machines, such as the wheel and axle; that they knew of the characteristic property of the magnetic needle; and were aware that astronomical events occurred in cycles. But the careful investigations of L. A. Sédillot have shown that the Chinese made no serious attempt to classify or extend the few rules of arithmetic or geometry with which they were acquainted, or to explain the causes of the phenomena which they observed."

"The idea that the Chinese had made considerable progress in theoretical mathematics seems to have been due to a misapprehension of the Jesuit missionaries who went to China in the sixteenth century. ...they failed to distinguish between the original science of the Chinese and the views which they found prevalent on their arrival|the latter being founded on the work and teaching of Arab or Hindoo missionaries who had come to China in the course of the thirteenth century or later, and while there introduced a knowledge of spherical trigonometry."

"The only geometrical theorem with which we can be certain that the ancient Chinese were acquainted is that in certain cases (namely, when the ratio of the sides is 3 : 4 : 5, or 1 : 1 : √2) the area of the square described on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares described on the sides. It is barely possible that a few geometrical theorems which can be demonstrated in the quasi-experimental way of superposition were also known to them."

"Their [the ancient Chinese] arithmetic was decimal in notation, but their knowledge seems to have been confined to the art of calculation by means of the swan-pan."

"Our acquaintance with the early attainments of the Chinese... serves to illustrate the fact that a nation may possess considerable skill in the applied arts while they are ignorant of the sciences on which those arts are founded."

"Our knowledge of the mathematical attainments of those who preceded the Greeks is very limited; but... the early Greeks learned the use of the abacus for practical calculations, symbols for recording the results, and as much mathematics as is contained or implied in the Rhind papyrus. It is probable that this sums up their indebtedness..."

"Thales... must have had considerable reputation as a man of affairs and as a good engineer, since he was employed to construct an embankment so as to divert the river Halys in such a way as to permit of the construction of a ford."

"We cannot form any exact idea as to how Thales presented his geometrical teaching. We infer, however, from Proclus that it consisted of a number of isolated propositions which were not arranged in a logical sequence, but that the proofs were deductive, so that the theorems were not a mere statement of an induction from a large number of special instances, as probably was the case with the Egyptian geometricians. The deductive character which he thus gave to the science is his chief claim to distinction."

"The following comprise the chief propositions that can now with reasonable probability be attributed to him [Thales]...(i) The angles at the base of an isosceles triangle are equal (Euc. I, 5). Proclus seems to imply that this was proved by taking another exactly equal isosceles triangle, turning it over, and then superposing it on the first—a sort of experimental demonstration. (ii) If two straight lines cut one another, the vertically opposite angles are equal (Euc. I, 15). Thales may have regarded this as obvious, for Proclus adds that Euclid was the first to give a strict proof of it. (iii) A triangle is determined if its base and base angles be given (cf. Euc. I, 26). Apparently this was applied to find the distance of a ship at sea—the base being a tower, and the base angles being obtained by observation. (iv) The sides of equiangular triangles are proportionals (Euc. VI, 4, or perhaps rather Euc. VI, 2). This is said to have been used by Thales when in Egypt to find the height of a pyramid. "...the pyramid [height] was to the stick [height] as the shadow of the pyramid to the shadow of the stick." …we are told that the king Amasis, who was present, was astonished at this application of abstract science. (v) A circle is bisected by any diameter. This may have been enunciated by Thales, but it must have been recognised as an obvious fact from the earliest times. (vi) The angle subtended by a diameter of a circle at any point in the circumference is a right angle (Euc. III, 31). This appears to have been regarded as the most remarkable of the geometrical achievements of Thales... It has been conjectured that he may have come to this conclusion by noting that the diagonals of a rectangle are equal and bisect one another, and that therefore a rectangle can be inscribed in a circle. If so, and if he went on to apply proposition (i), he would have discovered that the sum of the angles of a right-angled triangle is equal to two right angles, a fact with which it is believed that he was acquainted. It has been remarked that the shape of the tiles used in paving floors may have suggested these results."