A History of Mathematics

“The contemplation of the various steps by which mankind has come into possession of the vast stock of mathematical knowledge can hardly fail to interest the mathematician. He takes pride in the fact that his science, more than any other, is an exact science and that hardly anything ever done in mathematics has proved to be useless.”

“The chemist smiles at the childish efforts of alchemists but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindoos as useful and admirable as any research of today.”

“Another reason for the desirability of historical study is the value of historical knowledge to the teacher of mathematics.”

“The interest which pupils take in their studies may be greatly increased if the solution of problems and the cold logic of geometrical demonstrations are interspersed with historical remarks and anecdotes.”

“A class in arithmetic will be pleased to hear about the Hindoos and their invention of the "Arabic notation;" they will marvel at the thousands of years which elapsed before people had even thought of introducing into the numeral notation that Columbus-egg -- the zero; they will find it astounding that it should have taken so long to invent a notation which they themselves can now learn in a month.”

“After the pupils have learned how to bisect a given angle, surprise them by telling of the many futile attempts which have been made to solve, by elementary geometry, the apparently very simple problem of the trisection of an angle.”

“When they [students] know how to construct a square whose area is double the area of a given square, tell them about the duplication of the cube -- how the wrath of Apollo could be appeased only by the construction of a cubical altar double the given altar, and how mathematicians long wrestled with this problem.”

“After the class have exhausted their energies on the theorem of the right triangle, tell them something about its discoverer -- how Pythagoras, jubilant over his great accomplishment, sacrificed a hecatomb to the Muses who inspired him.”

“When the value of mathematical training is called in question, quote the inscription over the entrance into the academy of Plato, the philosopher: "Let no one who is unacquainted with geometry enter here."”

“Students in analytical geometry should know something of Descartes, and, after taking up the differential and integral calculus, they should become familiar with the parts that Newton, Leibniz, and Lagrange played in creating that science.”

“In his historical talk it is possible for the teacher to make it plain to the student that mathematics is not a dead science, but a living one in which steady progress is made.”

“The history of mathematics is important also as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development.”

“Of the largest numbers written in cuneiform symbols, which have hitherto been found, none go as high as a million.”

“The second [Babylonian] tablet records the magnitude of the illuminated portion of the moon's disc for every day from new to full moon, the whole disc being assumed to consist of 240 parts. ...This table not only exhibits the use of the sexagesimal system but also indicates the acquaintance of the Babylonians with [ geometric and arithmetic ] progressions.”

“Not to be overlooked is the fact that in the [Babylonian] sexagesimal notation of integers the "principle of position" was employed. Thus in 1.4 (64)... The introduction of this principle at so early a date is the more remarkable, because in the decimal notation it was not introduced till about the fifth or sixth century after Christ.”

“The principle of position, in its general and systematic application, requires a symbol for zero. We ask, Did the Babylonians possess one? Neither of the above tables answers this question for they... contain no number in which there was occasion to use a zero.”

“The division of the day into 24 hours, and of the hour into minutes and seconds on the scale of 60, is due to the Babylonians.”

“Though there is great difference of opinion regarding the antiquity of Egyptian civilisation, yet all authorities agree in the statement that, however far back they go, they find no uncivilised state of society.”

“All Greek writers are unanimous in ascribing, without envy, to Egypt the priority of invention in the mathematical sciences. Geometry, in particular, is said by Herodotus, Diodorus, Diogenes Laertius, Iamblichus, and other ancient writers to have originated in Egypt.”

“Some problems in this [Rhind] papyrus seem to imply a rudimentary knowledge of proportion.”

“On the walls of the celebrated temple of Horus at Edfu have been found hieroglyphics, written about 100 B.C., which enumerate the pieces of land owned by the priesthood, and give their areas. The area of any quadrilateral, however irregular, is there found by the formula (a+b)/2*(c+d)/2. ...The incorrect formulae of Ahmes of 3000 years B.C. yield generally closer approximations than those of the Edfu inscriptions, written 200 years after Euclid!”

“An insight into Egyptian methods of numeration was obtained through the ingenious deciphering of the hieroglyphics by Champollion, Young, and their successors. ...The symbol for 1 represents a vertical staff, that for 10,000 a pointing finger, that for 100,000 a burbot, that for 1,000,000 a man in astonishment.”

“Fractions were a subject of very great difficulty with the ancients. Simultaneous changes in both numerator and denominator were usually avoided. In manipulating fractions the Babylonians kept the denominators (60) constant. The Romans likewise kept them constant, but equal to 12. The Egyptians and Greeks, on the other hand, kept the numerators constant, and dealt with variable denominators.”

“Having finished the subject of fractions, Ahmes proceeds to the solution of equations of one unknown quantity. The unknown quantity is called 'hau' or heap. ...It thus appears that the beginnings of algebra are as ancient as those of geometry.”

“The principal defect of Egyptian arithmetic was the lack of a simple comprehensive symbolism, a defect which not even the Greeks were able to remove.”

“About the seventh century B.C. an active commercial intercourse sprang up between Greece and Egypt. Naturally there arose an interchange of ideas as well as of merchandise. Greeks, thirsting for knowledge, sought the Egyptian priests for instruction. Thales, Pythagoras, Œnopides, Plato, Democritus, Eudoxus, all visited the land of the pyramids.”

“The Egyptians carried geometry no further than was absolutely necessary for their practical wants. The Greeks, on the other hand, had within them a strong speculative tendency. They felt a craving to discover the reasons for things. They found pleasure in the contemplation of science.”

“To Thales of Miletus (640-546 B.C.), one of the "seven wise men," and the founder of the Ionic school, falls the honour of having introduced the study of geometry into Greece. During middle life he engaged in commercial pursuits which took him to Egypt. He is said to have resided there and to have studied the physical sciences and mathematics with the Egyptian priests.”

“The theorem that all angles inscribed in a semicircle are right angles is attributed by some ancient writers to Thales, by others to Pythagoras.”

“Thales was doubtless familiar with other theorems, not recorded by the ancients. It has been inferred that he knew the sum of the three angles of a triangle to be equal to two right angles, and the sides of equiangular triangles to be proportional.”

“The Egyptians must have made use of the above theorems on the straight line, in some of their constructions found in the Ahmes papyrus, but it was left for the Greek philosopher to give these truths, which others saw, but did not formulate into words, an explicit abstract expression, and to put into scientific language and subject to proof that which others merely felt to be true.”

“Thales may be said to have created the geometry of lines, essentially abstract in its character, while the Egyptians studied only the geometry of surfaces and the rudiments of solid geometry, empirical in their character.”

“With Thales begins also the study of scientific astronomy. He acquired great celebrity by the prediction of a solar eclipse in 585 B.C. Whether he predicted the day of the occurrence, or simply the year, is not known.”

“It is told of him [Thales] that while contemplating the stars during an evening walk, he fell into a ditch. The good old woman attending him exclaimed, "How canst thou know what is doing in the heavens when thou seest not what is at thy feet?"”

“The two most prominent pupils of Thales were Anaximander (b. 611 B.C.) and Anaximenes (b. 570 B.C.). They studied chiefly astronomy and physical philosophy.”

“The Ionic school lasted over one hundred years. The progress of mathematics during that period was slow, as compared with its growth in a later epoch of Greek history. A new impetus to its progress was given by Pythagoras.”

“Pythagoras (580?-500? B.C.). was one of those figures which impressed the imagination of succeeding times to such an extent that their real histories have become difficult to be discerned through the mythical haze that envelops them. The following account of Pythagoras excludes the most doubtful statements.”

“We are obliged to speak of the Pythagoreans as a body, and find it difficult to determine to whom each particular discovery is to be ascribed. The Pythagoreans themselves were in the habit of referring every discovery back to the great founder of the sect.”

“Pythagoras raised mathematics to the rank of a science. Arithmetic was courted by him as fervently as geometry. In fact, arithmetic is the foundation of his philosophic system.”

“The Eudemian Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression.”

“Like Egyptian geometry, the geometry of the Pythagoreans is much concerned with areas.”

“What the Pythagorean method of proof was has been a favourite topic for conjecture.”

“The theorem on the sum of the three angles of a triangle, presumably known to Thales, was proved by the Pythagoreans after the manner of Euclid. They demonstrated also that the plane about a point is completely filled by six equilateral triangles, four squares, or three regular hexagons, so that it is possible to divide up a plane into figures of either kind.”

“Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons."”

“The star-shaped pentagram was used as a symbol of recognition by the Pythagoreans, and was called by them Health.”

“Pythagoras called the sphere the most beautiful of all solids, and the circle the most beautiful of all plane figures.”

“The Pythagoreans were... familiar with the construction of a polygon equal in area to a given polygon and similar to another given polygon. This problem depends upon several important and somewhat advanced theorems, and testifies to the fact that the Pythagoreans made no mean progress in geometry.”

“Of the theorems generally ascribed to the Italian school, some cannot be attributed to Pythagoras himself, nor to his earliest successors. The progress from empirical to reasoned solutions must, of necessity, have been slow. It is worth noticing that on the circle no theorem of any importance was discovered by this school.”

“Among the later Pythagoreans, Philolaus and Archytas are the most prominent.”

“Philolaus wrote a book on the Pythagorean doctrines. By him were first given to the world the teachings of the Italian school, which had been kept secret for a whole century.”

“There is every reason to believe that the later Pythagoreans exercised a strong influence on the study and development of mathematics at Athens. The Sophists acquired geometry from Pythagorean sources. Plato bought the works of Philolaus and had a warm friend in Archytas.”

“Athens soon became the headquarters of Grecian men of letters, and of mathematicians in particular. The home of mathematics among the Greeks was first in the Ionian Islands, then in Lower Italy, and during the time now under consideration, at Athens.”

“In his study of the quadrature and duplication-problems, Hippocrates contributed much to the geometry of the circle.”

“Bryson of Heraclea, a contemporary of Antiphon, advanced the problem of the quadrature considerably by circumscribing polygons at the same time that he inscribed polygons. He erred, however, in assuming that the area of a circle was the arithmetical mean between circumscribed and inscribed polygons.”

“Hankel refers this Method of Exhaustion back to Hippocrates of Chios but the reasons for assigning it to this early writer rather than to Eudoxus seem insufficient.”

“During the Peloponnesian War (431-404 B.C.) the progress of geometry was checked. After the war, Athens sank into the background as a minor political power, but advanced more and more to the front as the leader in philosophy, literature, and science.”

“Plato observed that geometry trained the mind for correct and vigorous thinking. Hence it was that the Eudemian Summary says, "He filled his writings with mathematical discoveries, and exhibited on every occasion the remarkable connection between mathematics and philosophy."”

“One of the greatest achievements of Plato and his school is the invention of analysis as a method of proof. To be sure, this method had been used unconsciously by Hippocrates and others; but Plato, like a true philosopher, turned the instinctive logic into a conscious, legitimate method.”

“The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions.”

“It is now generally admitted that the duplication problem, as well as the trisection and quadrature problems, cannot be solved by means of the ruler and compass only.”

“Plato gave a healthful stimulus to the study of stereometry [solid geometry], which until his time had been entirely neglected. The sphere and the regular solids had been studied to some extent, but the prism, pyramid, cylinder, and cone were hardly known to exist. All these solids became the subjects of investigation by the Platonic school.”

“Another great geometer was Dinostratus, the brother of Menæchmus and pupil of Plato. Celebrated is his mechanical solution of the quadrature of the circle, by means of the quadratrix of Hippias.”

“The fame of the academy of Plato is to a large extent due to Eudoxus's pupils of the school at Cyzicus, among whom are Menaechmus, Dinostratus, Athenaeus, and Helicon.”

“Diogenes Laertius describes Eudoxus as astronomer, physician, legislator, as well as geometer.”

“Eudoxus added much to the knowledge of solid geometry. He proved, says Archimedes, that a pyramid is exactly one-third of a prism, and a cone one-third of a cylinder, having equal base and altitude. The proof that spheres are to each other as the cubes of their radii is probably due to him. He made frequent and skilful use of the method of exhaustion, of which he was in all probability the inventor.”

“A scholiast on Euclid, thought to be Proclus, says that Eudoxus practically invented the whole of Euclid's fifth book.”

“A skilful mathematician of whose life and works we have no details is Aristæus, the elder, probably a senior contemporary of Euclid. The fact that he wrote a work on conic sections tends to show that much progress had been made in their study during the time of Menæchmus. Aristaeus wrote also on regular solids and cultivated the analytic method. His works contained probably a summary of the researches of the Platonic school.”

“Aristotle (384-322 B.C.), the systematiser of deductive logic, though not a professed mathematician, promoted the science of geometry by improving some of the most difficult definitions. His Physics contains passages with suggestive hints of the principle of virtual velocities. About this time there appeared a work called Mechanica, of which he is regarded by some as the author. Mechanics was totally neglected by the Platonic school.”

“We have seen the birth of geometry in Egypt, its transference to the Ionian Islands, thence to Lower Italy and to Athens. We have witnessed its growth in Greece from feeble childhood to vigorous manhood, and now we shall see it return to the land of its birth and there derive new vigour.”

“Demetrius Phalereus was invited from Athens to take charge of the Library, and it is probable, says Gow, that Euclid was invited with him to open the mathematical school.”

“Euclid's greatest activity was during the time of the first Ptolemy, who reigned from 306 to 283 B.C. Of the life of Euclid, little is known, except what is added by Proclus to the Eudemian Summary.”

“When Ptolemy once asked Euclid if geometry could not be mastered by an easier process than by studying the Elements, Euclid returned the answer, "There is no royal road to geometry."”

“Pappus states that Euclid was distinguished by the fairness and kindness of his disposition, particularly toward those who could do anything to advance the mathematical sciences. Pappus is evidently making a contrast to Apollonius, of whom he more than insinuates the opposite character.”

“A pretty little story is related by Stobæus: "A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, 'What do I get by learning these things?' So Euclid called his slave and said, 'Give him threepence, since he must make gain out of what he learns.'"”

“It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences.”

“Among the manuscripts sent by Napoleon I. from the Vatican to Paris was found a copy of the Elements believed to be anterior to Theon's recension. Many variations from Theon's version were noticed therein, but they were not at all important, and showed that Theon generally made only verbal changes. The defects in the Elements for which Theon was blamed must, therefore, be due to Euclid himself.”

“The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard.”

“The term 'axiom' was used by Proclus, but not by Euclid. He speaks, instead, of 'common notions'—common either to all men or to all sciences.”

“There has been much controversy among ancient and modern critics on the postulates and axioms. An immense preponderance of manuscripts and the testimony of Proclus place the 'axioms' about right angles and parallels (Axioms 11 and 12) among the postulates. This is indeed their proper place, for they are really assumptions, and not common notions or axioms.”

“The postulate about parallels plays an important role in the history of non-Euclidean geometry.”

“The only postulate which Euclid missed was the one of superposition, according to which figures can be moved about in space without any alteration in form or magnitude.”

“The regular solids were studied so extensively by the Platonists that they received the name of "Platonic figures." The statement of Proclus that the whole aim of Euclid in writing the Elements was to arrive at the construction of the regular solids, is obviously wrong. The fourteenth and fifteenth books, treating of solid geometry, are apocryphal.”

“A remarkable feature of Euclid's, and of all Greek geometry before Archimedes is that it eschews mensuration. Thus the theorem that the area of a triangle equals half the product of its base and its altitude is foreign to Euclid.”

“Another extant book of Euclid is the Data. It seems to have been written for those who, having completed the Elements, wish to acquire the power of solving new problems proposed to them. The Data is a course of practice in analysis. It contains little or nothing that an intelligent student could not pick up from the Elements itself.”

“The following are the other extant works generally attributed to Euclid: Phœnomena, a work on spherical geometry and astronomy; Optics, which develops the hypothesis that light proceeds from the eye, and not from the object seen; Catoptrica, containing propositions on reflections from mirrors; De Divisionibus, a treatise on the division of plane figures into parts having to one another a given ratio; Sectio Canonis, a work on musical intervals.”

“The immediate successors of Euclid in the mathematical school at Alexandria were probably Conon, Dositheus, and Zeuxippus, but little is known of them.”

“The Quadrature of the Parabola contains two solutions to the problem -- one mechanical, the other geometrical. The method of exhaustion is used in both.”

“Archimedes studied also the ellipse and accomplished its quadrature, but to the hyperbola he seems to have paid less attention. It is believed that he wrote a book on conic sections.”

“By the word 'conoid,' in his book on Conoids and Spheroids, is meant the solid produced by the revolution of a parabola or a hyperbola about its axis. Spheroids are produced by the revolution of an ellipse, and are long or flat, according as the ellipse revolves around the major or minor axis. The book leads up to the cubature of these solids.”

“The proof of the property of the lever, given in his Equiponderance of Planes, holds its place in text-books to this day. His [Archimedes'] estimate of the efficiency of the lever is expressed in the saying attributed to him, "Give me a fulcrum on which to rest, and I will move the earth."”

“After examining the writings of Archimedes, one can well understand how, in ancient times, an 'Archimedean problem' came to mean a problem too deep for ordinary minds to solve, and how an 'Archimedean proof' came to be the synonym for unquestionable certainty. Archimedes wrote on a very wide range of subjects, and displayed great profundity in each. He is the Newton of antiquity.”

“The preface of the second book [of Conic Sections] is interesting as showing the mode in which Greek books were 'published' at this time. It reads thus: "I have sent my son Apollonius to bring you (Eudemus) the second book of my Conics. Read it carefully and communicate it to such others as are worthy of it.”

“It is worthy of notice that Apollonius nowhere introduces the notion of directrix for a conic, and that, though he incidentally discovered the of an ellipse and hyperbola, he did not discover the focus of a parabola. Conspicuous in his geometry is also the absence of technical terms and symbols, which renders the proofs long and cumbrous.”

“About the time of Nicomedes, flourished also Diocles, the inventor of the cissoid (ivy-like). This curve he used for finding two mean proportionals between two given straight lines.”

“Geminus of Rhodes (about 70 B.C.) published an astronomical work still extant. He wrote also a book, now lost, on the Arrangement of Mathematics, which contained many valuable notices of the early history of Greek mathematics. Proclus and Eutocius quote it frequently.”

“Dionysodorus of Amisus in Pontus applied the intersection of a parabola and hyperbola to the solution of a problem which Archimedes, in his Sphere and Cylinder, had left incomplete. The problem is "to cut a sphere so that its segments shall be in a given ratio."”