A History of Mathematics

A History of Mathematics by Florian Cajori was the first popular history of mathematics written in the United States. It was published in 1893.

Introduction

 * 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.
 * p. 1.


 * 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.
 * p. 1.


 * [Mathematics] warns us against hasty conclusions; it points out the importance of a good notation upon the progress of the science; it discourages excessive specialisation on the part of investigators, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure; it teaches that fortifications can be taken in other ways than by direct attack, that when repulsed from a direct assault it is well to reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken.
 * p. 1.


 * An untold amount of intellectual energy has been expended on the quadrature of the circle, yet no conquest has been made by direct assault. The circle-squarers have existed in crowds ever since the period of Archimedes. After innumerable failures to solve the problem at a time, even when investigators possessed that most powerful tool, the differential calculus, persons versed in mathematics dropped the subject, while those who still persisted were completely ignorant of its history and generally misunderstood the conditions of the problem. ...But progress was made on this problem by approaching it from a different direction and by newly discovered paths. Lambert proved in 1761 that the ratio of the circumference of a circle to its diameter is incommensurable. Some years ago, Lindemann demonstrated that this ratio is also transcendental and that the quadrature of the circle, by means of the ruler and compass only, is impossible. He thus showed by actual proof that which keen minded mathematicians had long suspected; namely, that the great army of circle-squarers have, for two thousand years, been assaulting a fortification which is as indestructible as the firmament of heaven.
 * p. 2




 * Another reason for the desirability of historical study is the value of historical knowledge to the teacher of mathematics.
 * p. 3.


 * 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.
 * p. 3.


 * 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.
 * p. 3




 * 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.
 * p. 3.


 * 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.
 * p. 3.


 * 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.
 * p. 3.


 * 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."
 * p. 3.


 * 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.
 * p. 4.


 * 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.
 * p. 4.


 * 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.
 * p. 4.

The Babylonians

 * Of the largest numbers written in cuneiform symbols, which have hitherto been found, none go as high as a million.
 * p. 6.


 * Most surprising... is the fact that Sumerian inscriptions disclose the use, not only of the... decimal system but also of a sexagesimal one. ...We possess two Babylonian tablets which exhibit its use. One... contains a table of square numbers up to 602. The numbers 1, 4, 9, 16, 25, 36, 49, are given as the squares of the first seven integers respectively. We have next 1.4=82 1.21=92 1.40=102 2.1=112, etc. This remains unintelligible unless we assume the sexagesimal scale, which makes 1.4=60+4 1.21=60+21 2.1=2*60+1.
 * p. 6.


 * 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.
 * p. 6.


 * 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.
 * p. 7.


 * 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.
 * p. 7.


 * The sexagesimal system was used also in fractions. Thus, in the Babylonian inscriptions, 1/2 and 1/3 are designated by 30 and 20, the reader being expected, in his mind, to supply the word "sixtieths." The Greek geometer Hypsicles and the Alexandrian astronomer Ptolemæus borrowed the sexagesimal notation of fractions from the Babylonians and introduced it into Greece. From that time sexagesimal fractions held almost full sway in astronomical and mathematical calculations until the sixteenth century, when they finally yielded their place to the decimal fractions.
 * p. 7.


 * It may be asked, What led to the invention of the sexagesimal system? Why was it that 60 parts were selected? ...Cantor offers the following theory: At first the Babylonians reckoned the year at 360 days. This led to the division of the circle into 360 degrees, each degree representing the daily amount of the supposed yearly revolution of the sun around the earth. Now they were, very probably, familiar with the fact that the radius can be applied to its circumference as a chord 6 times, and that each of these chords subtends an arc measuring exactly 60 degrees. Fixing their attention upon these degrees, the division into 60 parts may have suggested itself to them. Thus, when greater precision necessitated a subdivision of the degree, it was partitioned into 60 minutes.
 * p. 7.


 * 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.
 * p. 8.


 * Iamblichus attributes to them [the people in the Tigro-Euphrates basin] also a knowledge of proportion, and even the invention of the so called musical proportion. Though we possess no conclusive proof, we have nevertheless reason to believe that in practical calculation they used the abacus. ...Now, Babylon was once a great commercial centre,—the metropolis of many nations,—and it is therefore not unreasonable to suppose that her merchants employed this most improved aid to calculation.
 * p. 8.


 * In geometry the Babylonians accomplished almost nothing. Besides the division of the circumference [of the circle] into 6 parts by its radius, and into 360 degrees, they had some knowledge of geometrical figures, such as the triangle and quadrangle, which they used in their auguries. Like the Hebrews (1 Kin. 7:23), they took &pi;=3. Of geometrical demonstrations, there is, of course no trace. "As a rule, in the Oriental mind the intuitive powers eclipse the severely rational and logical."
 * p. 8.


 * When Alexander the Great, after the battle of Arbela (331 B.C.), took possession of Babylon, Callisthenes found there on burned brick astronomical records reaching back as far as 2234 B.C. Porphyrius says that these were sent to Aristotle. Ptolemy, the Alexandrian astronomer, possessed a Babylonian record of eclipses going back to 747 BC. Recently, Epping and [Johann Nepomuk] Strassmaier threw considerable light on Babylonian chronology and astronomy by explaining two calendars of the years 123 B.C. and 111 B.C. ...These scholars have succeeded in giving an account of the Babylonian calculation of the new and full moon and have identified by calculations the Babylonian names of the planets and of the twelve zodiacal signs and twenty-eight normal stars which correspond to some extent with the twenty eight nakshatras of the Hindoos.
 * p. 8.

The Egyptians

 * 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.
 * p. 9.


 * 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.
 * p. 10.


 * A hieratic papyrus, included in the Rhind collection of the British Museum, was deciphered by Eisenlohr in 1877, and found to be a mathematical manual containing problems in arithmetic and geometry. It was written by Ahmes some time before 1700 B.C., and was founded on an older work believed by Birch to date back as far as 3400 B.C.! This curious papyrus -- the most ancient mathematical handbook known to us -- puts us at once in contact with the mathematical thought in Egypt of three or five thousand years ago. It is entitled "Directions for obtaining the Knowledge of all Dark Things." We see from it that the Egyptians cared but little for theoretical results. Theorems are not found in it at all. It contains "hardly any general rules of procedure, but chiefly mere statements of results intended possibly to be explained by a teacher to his pupils."
 * p. 10.


 * In geometry the forte of the Egyptians lay in making constructions and determining areas. The area of an isosceles triangle, of which the sides measure 10 ruths and the base 4 ruths, was erroneously given as 20 square ruths, or half the product of the base by one side. The area of an isosceles trapezoid is found, similarly by multiplying half the sum of the parallel sides by one of the non-parallel sides. The area of a circle is found by deducting from the diameter 1/2 of its length and squaring the remainder. Here &pi; is taken=(16/9)2=3.1604..., a very fair approximation. The papyrus explains also such problems as these,—To mark out in the field a right triangle whose sides are 10 and 4 units; or a trapezoid whose parallel sides are 6 and 4, and the non-parallel sides each 20 units.
 * p. 11.


 * Some problems in this [Rhind] papyrus seem to imply a rudimentary knowledge of proportion.
 * p. 11.


 * The base lines of the pyramids run north, and south and east and west, but probably only the lines running north and south were determined by astronomical observations. This, coupled with the fact that the word harpedonaptæ, applied to Egyptian geometers, means "rope-stretchers," would point to the conclusion that the Egyptian, like the Indian and Chinese geometers, constructed a right triangle upon a given line, by stretching around three pegs a rope consisting of three parts in the ratios 3:4:5, and thus forming a right triangle. If this explanation is correct, then the Egyptians were familiar, 2000 years B.C., with the well-known property of the right triangle, for the special case at least when the sides are in the ratio 3:4:5.
 * p. 11.


 * 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!
 * p. 12.


 * The Egyptians failed in two essential points without which a science of geometry, in the true sense of the word, cannot exist. In the first place, they failed to construct a rigorously logical system of geometry, resting upon a few axioms and postulates. A great many of their rules, especially those in solid geometry, had probably not been proved at all, but were known to be true merely from observation or as matters of fact. The second great defect was their inability to bring the numerous special cases under a more general view, and thereby to arrive at broader and more fundamental theorems. Some of the simplest geometrical truths were divided into numberless special cases of which each was supposed to require separate treatment.
 * p. 12.


 * 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.
 * p. 13.


 * 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.
 * p. 14.


 * Ahmes used the term "fraction" in a restricted sense, for he applied it only to unit-fractions, or fractions having unity for the numerator. It was designated by writing the denominator and then placing over it a dot. Fractional values which could not be expressed by any one unit-fraction were expressed as the sum of two or more of them. ...The first important problem naturally arising was, how to represent any fractional value as the sum of unit-fractions. This was solved by aid of a table, given in the [Rhind] papyrus, in which all fractions of the form 2/(2n+1) (where n designates successively all the numbers up to 49) are reduced to the sum of unit fractions.
 * p. 14.


 * 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.
 * p. 15.


 * 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.
 * p. 15.


 * The Ahmes papyrus doubtless represents the most advanced attainments of the Egyptians in arithmetic and geometry. It is remarkable that they should have reached so great proficiency in mathematics at so remote a period of antiquity. But strange, indeed, is the fact that during the next two thousand years, they should have made no progress whatsoever in it. ...All the knowledge of geometry which they possessed when Greek scholars visited them, six centuries B.C., was doubtless known to them two thousand years earlier, when they built those stupendous and gigantic structures—the pyramids. An explanation for this stagnation of learning has been sought in the fact that their early discoveries in mathematics and medicine had the misfortune of being entered upon their sacred books and that, in after ages, it was considered heretical to augment or modify anything therein. Thus the books themselves closed the gates to progress.
 * p. 15.

Greek Geometry

 * 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.
 * p. 16.


 * 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 ideal relations and loved science as science.
 * p. 16.


 * The early mathematicians, Thales and Pythagoras, left behind no written records of their discoveries. A full history of Greek geometry and astronomy during this period, written by Eudemus, a pupil of Aristotle, has been lost. It was well known to Proclus, who, in his commentaries on Euclid, gives a brief account of it. This abstract constitutes our most reliable information. We shall quote it frequently under the name of Eudemian Summary.
 * p. 16.

The Ionic School

 * 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.
 * p. 17.


 * Plutarch declares that Thales soon excelled his masters and amazed King Amasis by measuring the heights of the pyramids from their shadows. ...by considering that the shadow cast by a vertical staff of known length bears the same ratio to the shadow of the pyramid as the height of the staff bears to the height of the pyramid. This solution presupposes a knowledge of proportion, and the Ahmes papyrus actually shows that the rudiments of proportion were known to the Egyptians. According to Diogenes Laertius the pyramids were measured by Thales in a different way; viz. by finding the length of the shadow of the pyramid at the moment when the shadow of a staff was equal to its own length.
 * p. 17.


 * The Eudemian Summary ascribes to Thales the invention of the theorems on the equality of vertical angles, the equality of the angles at the base of an isosceles triangle, the bisection of a circle by any diameter, and the congruence of two triangles having a side and the two adjacent angles equal respectively. The last theorem he applied to the measurement of the distances of ships from the shore. Thus Thales was the first to apply theoretical geometry to practical uses.
 * p. 17.


 * The theorem that all angles inscribed in a semicircle are right angles is attributed by some ancient writers to Thales, by others to Pythagoras.
 * p. 18.


 * 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.
 * p. 18.


 * 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.
 * p. 18.


 * 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.
 * p. 18.


 * 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.
 * p. 18.


 * 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?"
 * p. 18.


 * 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.
 * p. 18.


 * Of Anaxagoras, a pupil of Anaximenes, and the last philosopher of the Ionic school, we know little, except that while in prison, he passed his time attempting to square the circle. This is the first time, in the history of mathematics, that we find mention of the famous problem of the quadrature of the circle, that rock upon which so many reputations have been destroyed. It turns upon the determination of the exact value of &pi;. Approximations to &pi; had been made by the Chinese, Babylonians, Hebrews, and Egyptians. But the invention of a method to find its exact value, is the knotty problem which has engaged the attention of many minds from the time of Anaxagoras down to our own. Anaxagoras did not offer any solution of it, and seems to have luckily escaped paralogisms.
 * p. 18.


 * About the time of Anaxagoras, but isolated from the Ionic school, flourished Œnopides of Chios. Proclus ascribes to him the solution of the following problems: From a point without, to draw a perpendicular to a given line, and to draw an angle on a line equal to a given angle. That a man could gain a reputation by solving problems so elementary as these, indicates that geometry was still in its infancy, and that the Greeks had not yet gotten far beyond the Egyptian constructions.
 * p. 19.


 * 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.
 * p. 19.

The School of 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.
 * p. 19.


 * He [Pythagoras] ...visited the ancient Thales, who incited him to study in Egypt. ...He settled at Croton, and founded the famous Pythagorean school. This was not merely an academy for the teaching of philosophy, mathematics, and natural science, but it was a brotherhood, the members of which were united for life. This brotherhood had observances approaching masonic peculiarity. They were forbidden to divulge the discoveries and doctrines of their school.
 * p. 20.


 * 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.
 * p. 20.


 * 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.
 * p. 20.


 * 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.
 * p. 21.


 * Like Egyptian geometry, the geometry of the Pythagoreans is much concerned with areas.
 * p. 21.


 * To Pythagoras is ascribed the important theorem that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. He had probably learned from the Egyptians the truth of the theorem in the special case when the sides are 3, 4, 5, respectively. The story goes, that Pythagoras was so jubilant over this discovery that he sacrificed a hecatomb. Its authenticity is doubted, because the Pythagoreans believed in the transmigration of the soul and opposed, therefore, the shedding of blood. In the later traditions of the Neo-Pythagoreans this objection is removed by replacing this bloody sacrifice by that of "an ox made of flour."! The proof of the law of three squares, given in Euclid's Elements, I. 47, is due to Euclid himself, and not to the Pythagoreans.
 * p. 21.


 * What the Pythagorean method of proof was has been a favourite topic for conjecture.
 * p. 21.


 * 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.
 * p. 21.


 * From the equilateral triangle and the square arise the solids, namely the tetraedron, octaedron, icosaedron, and the cube. These solids were, in all probability, known to the Egyptians, excepting perhaps the icosaedron. In Pythagorean philosophy, they represent respectively the four elements of the physical world; namely fire, air, water, and earth. Later another regular solid was discovered, namely the dodecaedron, which, in absence of a fifth element, was made to represent the universe itself.
 * p. 21.


 * Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons."
 * p. 22.


 * The star-shaped pentagram was used as a symbol of recognition by the Pythagoreans, and was called by them Health.
 * p. 22.


 * Pythagoras called the sphere the most beautiful of all solids, and the circle the most beautiful of all plane figures.
 * p. 22.


 * According to Eudemus, the Pythagoreans invented the problems concerning the application of areas, including the cases of defect and excess, as in Euclid, VI. 28, 29.
 * p. 22.


 * 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.
 * p. 22.


 * 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.
 * p. 22.


 * Among the later Pythagoreans, Philolaus and Archytas are the most prominent.
 * p. 22.


 * 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.
 * p. 22.


 * The brilliant Archytas of Tarentum (428-347 B.C.), known as a great statesman and general, and universally admired for his virtues, was the only great geometer among the Greeks when Plato opened his school. Archytas was the first to apply geometry to mechanics and to treat the latter subject methodically. He also found a very ingenious mechanical solution to the problem of the duplication of the cube. His solution involves clear notions on the generation of cones and cylinders. This problem reduces itself to finding two mean proportionals between two given lines. These mean proportionals were obtained by Archytas from the section of a half-cylinder. The doctrine of proportion was advanced through him.
 * p. 23.


 * 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.
 * p. 23.

The Sophist School

 * Athens... became the richest and most beautiful city of antiquity. All menial work was performed by slaves. ...The citizen of Athens was well to do and enjoyed a large amount of leisure. The government being purely democratic, every citizen was a politician. To make his influence felt among his fellow-men he must, first of all, be educated. Thus there arose a demand for teachers. The supply came principally from Sicily, where Pythagorean doctrines had spread. These teachers were called Sophists, or "wise men." Unlike the Pythagoreans, they accepted pay for their teaching. Although rhetoric was the principal feature of their instruction, they also taught geometry, astronomy, and philosophy.
 * p. 24.


 * 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.
 * p. 24.


 * The geometry of the circle, which had been entirely neglected by the Pythagoreans, was taken up by the Sophists. Nearly all their discoveries were made in connection with their innumerable attempts to solve the following three famous problems:—(1) To trisect an arc or an angle. (2) To "double the cube," i.e. to find a cube whose volume is double that of a given cube. (3) To "square the circle," i.e. to find a square or some other rectilinear figure exactly equal in area to a given circle. These problems have probably been the subject of more discussion and research than any other problems in mathematics.
 * p. 24




 * The bisection of an angle was one of the easiest problems in geometry. The trisection of an angle, on the other hand, presented unexpected difficulties. A right angle had been divided into three equal parts by the Pythagoreans. But the general problem, though easy in appearance, transcended the power of elementary geometry. Among the first to wrestle with it was Hippias of Elis, a contemporary of Socrates, and born about 460 B.C. Like all the later geometers, he failed in effecting the trisection by means of a ruler and compass only. Proclus mentions a man, Hippias, presumably Hippias of Elis, as the inventor of a transcendental curve which served to divide an angle not only into three, but into any number of equal parts. This same curve was used later by Deinostratus and others for the quadrature of the circle. On this account it is called the quadratrix.
 * p. 24.


 * The Pythagoreans had shown that the diagonal of a square is the side of another square having double the area of the original one. This probably suggested the problem of the duplication of the cube, i.e. to find the edge of a cube having double the volume of a given cube. Eratosthenes ascribes to this problem a different origin. The Delians were once suffering from a pestilence and were ordered by the oracle to double a certain cubical altar. Thoughtless workmen simply constructed a cube with edges twice as long, but this did not pacify the gods. The error being discovered, Plato was consulted on the matter. He and his disciples searched eagerly for a solution to this "Delian Problem."
 * p. 25




 * Hippocrates of Chios (about 430 B.C.), a talented mathematician, but otherwise slow and stupid, was the first to show that the [duplication of the cube] problem could be reduced to finding two mean proportionals between a given line and another twice as long. For, in the proportion a:x=x:y=y:2a, since x2=ay and y2=2ax and x4=a2y2, we have x4=2 a3x and x3=2a3. But he failed to find the two mean proportionals. His attempt to square the circle was also a failure; for though he made himself celebrated by squaring a lune, he committed an error in attempting to apply this result to the squaring of the circle.
 * Note: if you have a line of length a and another line twice as long (of length 2a), then the mean proportionals x and y (between these two lengths) are defined by the relation a/x = x/y = y/2a, and if x3=2a3, then x will be the solution for doubling (duplicating) the cube a3 with side of length a.
 * p. 25.


 * In his study of the quadrature and duplication-problems, Hippocrates contributed much to the geometry of the circle.
 * p. 25.


 * The subject of similar figures was studied and partly developed by Hippocrates. This involved the theory of proportion. Proportion had, thus far, been used by the Greeks only in numbers. They never succeeded in uniting the notions of numbers and magnitudes. The term "number" was used by them in a restricted sense. What we call irrational numbers was not included under this notion. Not even rational fractions were called numbers. They used the word in the same sense as we use "integers." Hence numbers were conceived as discontinuous, while magnitudes were continuous. The two notions appeared, therefore, entirely distinct. The chasm between them is exposed to full view in the statement of Euclid that "incommensurable magnitudes do not have the same ratio as numbers." In Euclid's Elements we find the theory of proportion of magnitudes developed and treated independent of that of numbers. The transfer of the theory of proportion from numbers to magnitudes (and to lengths in particular) was a difficult and important step.
 * p. 26.


 * The Sophist Antiphon, a contemporary of Hippocrates, introduced the process of exhaustion for the purpose of solving the problem of the quadrature. He did himself credit by remarking that by inscribing in a circle a square, and on its sides erecting isosceles triangles with their vertices in the circumference, and on the sides of these triangles erecting new triangles, etc., one could obtain a succession of regular polygons of 8, 16, 32, 64 sides, and so on, of which each approaches nearer to the circle than the previous one, until the circle is finally exhausted. Thus is obtained an inscribed polygon whose sides coincide with the circumference. Since there can be found squares equal in area to any polygon, there also can be found a square equal to the last polygon inscribed, and therefore equal to the circle itself.
 * p. 26.


 * 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.
 * p. 27.


 * Unlike Bryson and the rest of Greek geometers, Antiphon seems to have believed it possible, by continually doubling the sides of an inscribed polygon, to obtain a polygon coinciding with the circle. This question gave rise to lively disputes in Athens. If a polygon can coincide with the circle, then, says Simplicius, we must put aside the notion that magnitudes are divisible ad infinitum. Aristotle always supported the theory of the infinite divisibility, while Zeno, the Stoic, attempted to show its absurdity by proving that if magnitudes are infinitely divisible, motion is impossible. Zeno argues that Achilles could not overtake a tortoise; for while he hastened to the place where the tortoise had been when he started, the tortoise crept some distance ahead, and while Achilles reached that second spot, the tortoise again moved forward a little, and so on. Thus the tortoise was always in advance of Achilles. Such arguments greatly confounded Greek geometers. No wonder they were deterred by such paradoxes from introducing the idea of infinity into their geometry. It did not suit the rigour of their proofs.
 * Note: Henry Mendell writes in Antiphon's Squaring of the Circle that Simplicius' On Aristotle's Physics was the principal source for Antiphon.
 * p. 27.


 * The process of Antiphon and Bryson gave rise to the cumbrous but perfectly rigorous "method of exhaustion." In determining the ratio of the areas between two curvilinear plane figures, say two circles, geometers first inscribed or circumscribed similar polygons, and then by increasing indefinitely the number of sides, nearly exhausted the spaces between the polygons and circumferences. From the theorem that similar polygons inscribed in circles are to each other as the squares on their diameters, geometers may have divined the theorem attributed to Hippocrates of Chios that the circles, which differ but little from the last drawn polygons, must be to each other as the squares on their diameters. But in order to exclude all vagueness and possibility of doubt, later Greek geometers applied reasoning like that in Euclid XII. 2...
 * p. 27.


 * 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.
 * p. 28.

The Platonic School

 * 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.
 * p. 29.


 * Plato was born at Athens in 429 B.C., the year of the great plague, and died in 348. He was a pupil and near friend of Socrates, but it was not from him that he acquired his taste for mathematics. After the death of Socrates, Plato traveled extensively. In Cyrene he studied mathematics under Theodoras. He went to Egypt, then to Lower Italy and Sicily, where he came in contact with the Pythagoreans. Archytas of Tarentum and Timæus of Locri became his intimate friends. On his return to Athens, about 389 B.C., he founded his school in the groves of the Academia, and devoted the remainder of his life to teaching and writing.
 * p. 29.


 * Plato's physical philosophy is partly based on that of the Pythagoreans. Like them, he sought in arithmetic and geometry the key to the universe. When questioned about the occupation of the Deity, Plato answered that "He geometrises continually." Accordingly, a knowledge of geometry is a necessary preparation for the study of philosophy. To show how great a value he put on mathematics and how necessary it is for higher speculation, Plato placed the inscription over his porch, "Let no one who is unacquainted with geometry enter here."
 * p. 29.


 * 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."
 * p. 30.


 * With Plato as the head-master, we need not wonder that the Platonic school produced so large a number of mathematicians. Plato did little real original work, but he made valuable improvements in the logic and methods employed in geometry. It is true that the Sophist geometers of the previous century were rigorous in their proofs, but as a rule they did not reflect on the inward nature of their methods. They used the axioms without giving them explicit expression, and the geometrical concepts, such as the point, line, surface, etc., without assigning to them formal definitions. The Pythagoreans called a point "unity in position," but this is a statement of a philosophical theory rather than a definition. Plato objected to calling a point a "geometrical fiction." He defined a point as "the beginning of a line" or as "an indivisible line," and a line as "length without breadth." He called the point, line, surface, the 'boundaries' of the line, surface, solid, respectively. Many of the definitions in Euclid are to be ascribed to the Platonic school. The same is probably true of Euclid's axioms. Aristotle refers to Plato the axiom that "equals subtracted from equals leave equals."
 * p. 30.


 * 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.
 * p. 30.


 * The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it."
 * p. 30.


 * 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.
 * p. 31.


 * Plato is said to have solved the problem of the duplication of the cube. But the solution is open to the very same objection which he made to the solutions by Archytas, Eudoxus, and Menæchmus. He called their solutions not geometrical, but mechanical for they required the use of other instruments than the ruler and compass. He said that thereby "the good of geometry is set aside and destroyed, for we again reduce it to the world of sense, instead of elevating and imbuing it with the eternal and incorporeal images of thought, even as it is employed by God, for which reason He always is God." These objections indicate either that the solution is wrongly attributed to Plato or that he wished to show how easily non-geometric solutions of that character can be found.
 * p. 31.


 * 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.
 * p. 31.


 * 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.
 * p. 31




 * One result of these inquiries was epoch-making. Menæchmus, an associate of Plato and pupil of Eudoxus, invented the conic sections, which, in course of only a century, raised geometry to the loftiest height which it was destined to reach during antiquity. Menæchmus cut three kinds of cones, the 'right angled,' 'acute angled,' and 'obtuse angled,' by planes at right angles to a side of the cones, and thus obtained the three sections which we now call the parabola, ellipse, and hyperbola. Judging from the two very elegant solutions of the "Delian Problem" by means of intersections of these curves, Menæchmus must have succeeded well in investigating their properties.
 * p. 32.


 * 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.
 * p. 32.


 * Perhaps the most brilliant mathematician of this period was Eudoxus. He was born at Cnidus about 408 B.C., studied under Archytas, and later, for two months, under Plato. He was imbued with a true spirit of scientific inquiry, and has been called the father of scientific astronomical observation. From the fragmentary notices of his astronomical researches, found in later writers, Ideler and Schiaparelli succeeded in reconstructing the system of Eudoxus with its celebrated representation of planetary motions by "concentric spheres." Eudoxus had a school at Cyzicus, went with his pupils to Athens, visiting Plato, and then returned to Cyzicus, where he died 355 B.C.
 * p. 32.


 * 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.
 * p. 32.


 * Diogenes Laertius describes Eudoxus as astronomer, physician, legislator, as well as geometer.
 * p. 32.


 * The Eudemian Summary says that Eudoxus "first increased the number of general theorems, added to the three proportions three more, and raised to a considerable quantity the learning, begun by Plato, on the subject of the section, to which he applied the analytical method." By this 'section' is meant, no doubt, the "golden section" (sectio aurea), which cuts a line in extreme and mean ratio. The first five propositions in Euclid XIII. relate to lines cut by this section, and are generally attributed to Eudoxus.
 * p. 32.


 * 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.
 * p. 33.


 * A scholiast on Euclid, thought to be Proclus, says that Eudoxus practically invented the whole of Euclid's fifth book.
 * p. 33.


 * Plato has been called a maker of mathematicians. Besides the pupils already named, the Eudemian Summary mentions the following: Theaetetus of Athens, a man of great natural gifts, to whom, no doubt, Euclid was greatly indebted in the composition of the 10th book, treating of incommensurables; Leodamas of Thasos; Neocleides and his pupil Leon, who added much to the work of their predecessors, for Leon wrote an Elements carefully designed, both in number and utility of its proofs; Theudius of Magnesia, who composed a very good book of Elements and generalised propositions, which had been confined to particular cases; Hermotimus of Colophon, who discovered many propositions of the Elements and composed some on loci; and finally the names of Amyclas of Heraclea, Cyzicenus of Athens, and Philippus of Mende.
 * p. 33.


 * 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.
 * p. 34.


 * 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.
 * p. 34.

The First Alexandrian School

 * [There are] other works with texts more or less complete and generally attributed to Euclid. ...His treatise on Porisms is lost, but much learning has been expended by Robert Simson and M. Chasles in restoring it from numerous notes found in the writings of Pappus. The term "porism" is vague in meaning. According to Proclus the aim of a porism is not to state some property or truth, like a theorem, nor to effect a construction, like a problem, but to find and bring to view a thing which necessarily exists with given numbers or a given construction, as, to find the centre of a given circle, or to find the G.C.D. of two given numbers. Porisms, according to Chasles, are incomplete theorems, "expressing certain relations between things variable according to a common law."
 * p. 33.


 * 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.
 * p. 34.


 * In 338 B.C., at the battle of Chæronea, Athens was beaten by Philip of Macedon and her power was broken forever. Soon after, Alexander the Great, the son of Philip, started out to conquer the world. In eleven years he built up a great empire which broke to pieces in a day. Egypt fell to the lot of Ptolemy Soter. Alexander had founded the seaport of Alexandria, which soon became "the noblest of all cities." Ptolemy made Alexandria the capital. The history of Egypt during the next three centuries is mainly the history of Alexandria. Literature, philosophy, and art were diligently cultivated. Ptolemy created the university of Alexandria. He founded the great Library and built laboratories, museums, a zoological garden, and promenades. Alexandria soon became the great centre of learning.
 * p. 34.


 * 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.
 * p. 35.


 * 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.
 * p. 35.


 * Euclid, says Proclus, was younger than Plato and older than Eratosthenes and Archimedes, the latter of whom mentions him. He was of the Platonic sect, and well read in its doctrines. He collected the Elements, put in order much that Eudoxus had prepared, completed many things of Theætetus, and was the first who reduced to unobjectionable demonstration the imperfect attempts of his predecessors.
 * p. 35.


 * 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."
 * p. 35.


 * 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.
 * p. 35.


 * 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.'"
 * p. 35.


 * 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.
 * p. 36.


 * Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him. Allman conjectures that the substance of Books I., II., IV. comes from the Pythagoreans, that the substance of Book VI. is due to the Pythagoreans and Eudoxus, the latter contributing the doctrine of proportion as applicable to incommensurables and also the Method of Exhaustions (Book VII.), that Thætetus contributed much toward Books X. and XIII., that the principal part of the original work of Euclid himself is to be found in Book X.
 * p. 36.


 * Euclid was the greatest systematiser of his time. By careful selection from the material before him, and by logical arrangement of the propositions selected, he built up, from a few definitions and axioms, a proud and lofty structure. It would be erroneous to believe that he incorporated into his Elements all the elementary theorems known at his time. Archimedes, Apollonius, and even he himself refer to theorems not included in his Elements, as being well-known truths.
 * p. 37.


 * 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.
 * p. 37.


 * 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.
 * Compare George Bruce Halstead quote under Quotes about Euclid
 * p. 37.


 * 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.
 * p. 38.


 * 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.
 * p. 38.


 * The postulate about parallels plays an important role in the history of non-Euclidean geometry.
 * p. 38.


 * 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.
 * p. 38.


 * The Elements contains thirteen books by Euclid, and two, of which it is supposed that Hypsicles and Damascius are the authors. The first four books are on plane geometry. The fifth book treats of the theory of proportion as applied to magnitudes in general. The sixth book develops the geometry of similar figures. The seventh, eighth, ninth books are on the theory of numbers, or on arithmetic. In the ninth book is found the proof to the theorem that the number of primes is infinite. The tenth book treats of the theory of incommensurables. The next three books are on stereometry. The eleventh contains its more elementary theorems; the twelfth, the metrical relations of the pyramid, prism, cone, cylinder, and sphere. The thirteenth treats of the regular polygons, especially of the triangle and pentagon, and then uses them as faces of the five regular solids; namely the tetraedron, octaedron, icosaedron, cube, and dodecaedron.
 * p. 38.


 * 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.
 * p. 38.


 * 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.
 * p. 39.


 * 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.
 * p. 39.


 * 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.
 * p. 39.


 * His [Euclid's] treatise on Porisms is lost; but much learning has been expended by Robert Simson and M. Chasles in restoring it from numerous notes found in the writings of Pappus. The term porism is vague in meaning. The aim of a porism is not to state some property or truth, like a theorem, nor to effect a construction, like a problem, but to find and bring to view a thing which necessarily exists with given numbers or a given construction, as, to find the centre of a given circle, or to find the G.C.D. of two given numbers. His other lost works are Fallacies, containing exercises in detection of fallacies; Conic Sections, in four books, which are the foundation of a work on the same subject by Apollonius; and Loci on a Surface, the meaning of which title is not understood. Heiberg believes it to mean "loci which are surfaces."
 * p. 39.


 * The immediate successors of Euclid in the mathematical school at Alexandria were probably Conon, Dositheus, and Zeuxippus, but little is known of them.
 * p. 40.


 * Archimedes was admired by his fellow-citizens chiefly for his mechanical inventions; he himself prized far more highly his discoveries in pure science. He declared that "every kind of art which was connected with daily needs was ignoble and vulgar." Some of his works have been lost. The following are the extant books, arranged approximately in chronological order: 1. Two books on Equiponderance of Planes or Centres of Plane Gravities, between which is inserted his treatise on the Quadrature of the Parabola; 2. Two books on the Sphere and Cylinder; 3. The Measurement of the Circle; 4. On Spirals; 5. Conoids and Spheroids; 6. The Sand-Counter; 7. Two books on Floating Bodies; 8. Fifteen Lemmas.
 * p. 41.


 * In the book on the Measurement of the Circle, Archimedes proves first that the area of a circle is equal to that of a right triangle having the length of the circumference for its base, and the radius for its altitude. In this he assumes that there exists a straight line equal in length to the circumference -- an assumption objected to by some ancient critics, on the ground that it is not evident that a straight line can equal a curved one. The finding of such a line was the next problem. He first finds an upper limit to the ratio of the circumference to the diameter, or &pi;. To do this, he starts with an equilateral triangle of which the base is a tangent and the vertex is the centre of the circle. By successively bisecting the angle at the centre, by comparing ratios, and by taking the irrational square roots always a little too small, he finally arrived at the conclusion that &pi; < 3 1/7. Next he finds a lower limit by inscribing in the circle regular polygons of 6, 12, 24, 48, 96 sides, finding for each successive polygon its perimeter, which is, of course, always less than the circumference. Thus he finally concludes that "the circumference of a circle exceeds three times its diameter by a part which is less than 1/7 but more than 10/71 of the diameter." This approximation is exact enough for most purposes.
 * p. 41.


 * The Quadrature of the Parabola contains two solutions to the problem -- one mechanical, the other geometrical. The method of exhaustion is used in both.
 * p. 42.


 * 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.
 * p. 42.


 * Of all his discoveries Archimedes prized most highly those in his Sphere and Cylinder. In it are proved the new theorems, that the surface of a sphere is equal to four times a great circle; that the surface of a segment of a sphere is equal to a circle whose radius is the straight line drawn from the vertex of the segment to the circumference of its basal circle; that the volume and the surface of a sphere are 2/3 of the volume and surface, respectively, of the cylinder circumscribed about the sphere. Archimedes desired that the figure to the last proposition be inscribed on his tomb. This was ordered done by Marcellus.
 * p. 42.


 * The spiral now called the "spiral of Archimedes," and described in the book On Spirals, was discovered by Archimedes, and not, as some believe, by his friend Conon. His treatise thereon is, perhaps the most wonderful of all his works. Nowadays, subjects of this kind are made easy by the use of the infinitesimal calculus. In its stead the ancients used the method of exhaustion. Nowhere is the fertility of his genius more grandly displayed than in his masterly use of this method. With Euclid and his predecessors the method of exhaustion was only the means of proving propositions which must have been seen and believed before they were proved. But in the hands of Archimedes it became an instrument of discovery.
 * p. 42.


 * 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.
 * p. 43.


 * Archimedes is the author of the first sound knowledge on mechanics. Archytas, Aristotle, and others attempted to form the known mechanical truths into a science, but failed. Aristotle knew the property of the lever, but could not establish its true mathematical theory. The radical and fatal defect in the speculations of the Greeks, says Whewell, was "that though they had in their possession facts and ideas, the ideas were not distinct and appropriate to the facts." For instance, Aristotle asserted that when a body at the end of a lever is moving, it may be considered as having two motions; one in the direction of the tangent and one in the direction of the radius; the former motion is, he says, according to nature, the latter contrary to nature. These inappropriate notions of 'natural' and 'unnatural' motions, together with the habits of thought which dictated these speculations, made the perception of the true grounds of mechanical properties impossible. It seems strange that even after Archimedes had entered upon the right path, this science should have remained absolutely stationary till the time of Galileo -- a period of nearly two thousand years.
 * p. 43.


 * 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."
 * p. 43.


 * While the Equiponderance treats of solids, or the equilibrium of solids, the book on Floating Bodies treats of hydrostatics. His [Archimedes'] attention was first drawn to the subject of specific gravity when King Hieron asked him to test whether a crown, professed by the maker to be pure gold, was not alloyed with silver. The story goes that our philosopher was in a bath when the true method of solution flashed on his mind. He immediately ran home, naked, shouting, "I have found it." To solve the problem, he took a piece of gold and a piece of silver, each weighing the same as the crown. According to one author, he determined the volume of water displaced by the gold, silver, and crown respectively, and calculated from that the amount of gold and silver in the crown. According to another writer, he weighed separately the gold, silver, and crown, while immersed in water, thereby determining their loss of weight in water. From these data he easily found the solution. It is possible that Archimedes solved the problem by both methods.
 * p. 44.


 * 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.
 * p. 44.


 * Eratosthenes, eleven years younger than Archimedes, was a native of Cyrene. He was educated in Alexandria under Callimachus the poet, whom he succeeded as custodian of the Alexandrian Library. His many-sided activity may be inferred from his works. He wrote on Good and Evil, Measurement of the Earth, Comedy, Geography, Chronology, Constellations, and the Duplication of the Cube. He was also a philologian and a poet. He measured the obliquity of the ecliptic and invented a device for finding prime numbers. Of his geometrical writings we possess only a letter to Ptolemy Euergetes, giving a history of the duplication problem and also the description of a very ingenious mechanical contrivance of his own to solve it. In his old age he lost his eyesight, and on that account is said to have committed suicide by voluntary starvation.
 * p. 44.


 * About forty years after Archimedes flourished, Apollonius of Perga's genius nearly equalled that of his great predecessor. He incontestably occupies the second place in distinction among ancient mathematicians. Apollonius was born in the reign of Ptolemy Euergetes and died under Ptolemy Philopator, who reigned 222-205 B.C. He studied at Alexandria under the successors of Euclid, and for some time, also, at Pergamum, where he made the acquaintance of that Eudemus to whom he dedicated the first three books of his Conic Sections. The brilliancy of his great work brought him the title of the "Great Geometer." This is all that is known of his life.
 * p. 45.


 * Apollonius' Conic Sections were in eight books, of which the first four only have come down to us in the original Greek. The next three books were unknown in Europe till the middle of the seventeenth century, when an Arabic translation, made about 1250, was discovered. The eighth book has never been found. In 1710 Halley of Oxford published the Greek text of the first four books and a Latin translation of the remaining three, together with his conjectural restoration of the eighth book, founded on the introductory lemmas of Pappus. The first four books contain little more than the substance of what earlier geometers had done.
 * p. 45.


 * Eutocius tells us that Heraclides, in his life of Archimedes, accused Apollonius of having appropriated, in his Conic Sections, the unpublished discoveries of that great mathematician. It is difficult to believe that this charge rests upon good foundation. Eutocius quotes Geminus as replying that neither Archimedes nor Apollonius claimed to have invented the conic sections, but that Apollonius had introduced a real improvement. While the first three or four books were founded on the works of Menæchmus, Aristæus, Euclid, and Archimedes, the remaining ones consisted almost entirely of new matter.
 * p. 45.


 * 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.
 * p. 46.


 * The first book [of Conic Sections], says Apollonius in his preface to it, "contains the mode of producing the three sections and the conjugate hyperbolas and their principal characteristics, more fully and generally worked out than in the writings of other authors." We remember that Menæchmus, and all his successors down to Apollonius, considered only sections of right cones by a plane perpendicular to their sides, and that the three sections were obtained each from a different cone. Apollonius introduced an important generalisation. He produced all the sections from one and the same cone, whether right or scalene, and by sections which may or may not be perpendicular to its sides. The old names for the three curves were now no longer applicable. Instead of calling the three curves, sections of the 'acute angled,' 'right angled,' and 'obtuse angled' cone, he called them ellipse, parabola, and hyperbola, respectively. To be sure, we find the words 'parabola' and 'ellipse' in the works of Archimedes, but they are probably only interpolations. The word 'ellipse' was applied because y2 &lt; px, p being the parameter; the word 'parabola' was introduced because y2 = px, and the term 'hyperbola' because y2 &gt; px.
 * p. 46.


 * The first book of the Conic Sections of Apollonius is almost wholly devoted to the generation of the three principal conic sections. The second book treats mainly of asymptotes, axes, and diameters. The third book treats of the equality or proportionality of triangles, rectangles, or squares, of which the component parts are determined by portions of transversals, chords, asymptotes, or tangents, which are frequently subject to a great number of conditions. It also touches the subject of foci of the ellipse and hyperbola. In the fourth book, Apollonius discusses the harmonic division of straight lines. He also examines a system of two conics, and shows that they cannot cut each other in more than four points. He investigates the various possible relative positions of two conics, as, for instance, when they have one or two points of contact with each other. The fifth book reveals better than any other the giant intellect of its author. Difficult questions of maxima and minima, of which few examples are found in earlier works, are here treated most exhaustively. The subject investigated is, to find the longest and shortest lines that can be drawn from a given point to a conic. Here are also found the germs of the subject of evolutes and centres of osculation. The sixth book is on the similarity of conies. The seventh book is on conjugate diameters. The eighth book, as restored by Halley, continues the subject of conjugate diameters.
 * p. 48.


 * It is worthy of notice that Apollonius nowhere introduces the notion of directrix for a conic, and that, though he incidentally discovered the focus 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.
 * p. 49.


 * The discoveries of Archimedes and Apollonius, says M. Chasles, marked the most brilliant epoch of ancient geometry. Two questions which have occupied geometers of all periods may be regarded as having originated with them. The first of these is the quadrature of curvilinear figures, which gave birth to the infinitesimal calculus. The second is the theory of conic sections, which was the prelude to the theory of geometrical curves of all degrees, and to that portion of geometry which considers only the forms and situations of figures, and uses only the intersection of lines and surfaces and the ratios of rectilineal distances. These two great divisions of geometry may be designated by the names of Geometry of Measurements and Geometry of Forms and Situations, or, Geometry of Archimedes and of Apollonius.
 * p. 49.


 * Besides the Conic Sections, Pappus ascribes to Apollonius the following works: On Contacts, Plane Loci, Inclinations, Section of an Area, Determinate Section, and gives lemmas from which attempts have been made to restore the lost originals. Two books on De Sectione Rationis have been found in the Arabic. The book on Contacts as restored by Vieta, contains the so-called "Apollonian Problem:" Given three circles, to find a fourth which shall touch the three.
 * p. 49.


 * Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent.
 * p. 50.


 * Among the earliest successors of Apollonius was Nicomedes. Nothing definite is known of him, except that he invented the conchoid (mussel-like). He devised a little machine by which the curve could be easily described. With aid of the conchoid he duplicated the cube. The curve can also be used for trisecting angles in a way much resembling that in the eighth lemma of Archimedes. Proclus ascribes this mode of trisection to Nicomedes, but Pappus, on the other hand, claims it as his own. The conchoid was used by Newton in constructing curves of the third degree.
 * p. 50.


 * 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.
 * p. 50.


 * Perseus... lived some time between 200 and 100 B.C. From Heron and Geminus we learn that he wrote a work on the spire, a sort of anchor-ring surface described by Heron as being produced by the revolution of a circle around one of its chords as an axis. The sections of this surface yield peculiar curves called spiral sections, which, according to Geminus, were thought out by Perseus. These curves appear to be the same as the Hippopede of Eudoxus.
 * p. 50.


 * Probably somewhat later than Perseus lived Zenodorus. He wrote an interesting treatise on a new subject; namely, isoperimetrical figures. Fourteen propositions are preserved by Pappus and Theon. Here are a few of them: Of isoperimetrical regular polygons, the one having the largest number of angles has the greatest area; the circle has a greater area than any regular polygon of equal periphery; of all isoperimetrical polygons of n sides, the regular is the greatest; of all solids having surfaces equal in area, the sphere has the greatest volume.
 * p. 51.


 * Hypsicles (between 200 and 100 B.C.) was supposed to be the author of both the fourteenth and fifteenth books of Euclid, but recent critics are of opinion that the fifteenth book was written by an author who lived several centuries after Christ. The fourteenth book contains seven elegant theorems on regular solids. A treatise of Hypsicles on Risings is of interest because it is the first Greek work giving the division of the circumference into 360 degrees after the fashion of the Babylonians.
 * p. 51.


 * Hipparchus of Nicaea in Bithynia was the greatest astronomer of antiquity. He established inductively the famous theory of epicycles and eccentrics. As might be expected, he was interested in mathematics, not per se, but only as an aid to astronomical inquiry. No mathematical writings of his are extant, but Theon of Alexandria informs us that Hipparchus originated the science of trigonometry, and that he calculated a "table of chords" in twelve books. Such calculations must have required a ready knowledge of arithmetical and algebraical operations.
 * p. 51.


 * About 155 B.C. flourished Heron the Elder of Alexandria. He was the pupil of Ctesibius, who was celebrated for his ingenious mechanical inventions, such as the hydraulic organ, the water clock, and catapult. It is believed by some that Heron was a son of Ctesibius. He exhibited talent of the same order as did his master by the invention of the eolipile and a curious mechanism known as "Heron's fountain." Great uncertainty exists concerning his writings. Most authorities believe him to be the author of an important Treatise on the Dioptra, of which there exist three manuscript copies, quite dissimilar. But M. Marie thinks that the Dioptra is the work of Heron the Younger, who lived in the seventh or eighth century after Christ, and that Geodesy, another book supposed to be by Heron, is only a corrupt and defective copy of the former work. Dioptra contains the important formula for finding the area of a triangle expressed in terms of its sides; its derivation is quite laborious and yet exceedingly ingenious. "It seems to me difficult to believe," says Chasles, "that so beautiful a theorem should be found in a work so ancient as that of Heron the Elder, without that some Greek geometer should have thought to cite it." Marie lays great stress on this silence of the ancient writers, and argues from it that the true author must be Heron the Younger or some writer much more recent than Heron the Elder. But no reliable evidence has been found that there actually existed a second mathematician by the name of Heron.
 * p. 52.


 * "Dioptra," says Venturi, were instruments which had great resemblance to our modern theodolites. The book Dioptra is a treatise on geodesy containing solutions, with aid of these instruments, of a large number of questions in geometry, such as to find the distance between two points, of which one only is accessible, or between two points, which are visible but both inaccessible; from a given point to draw a perpendicular to a line which cannot be approached; to find the difference of level between two points; to measure the area of a field without entering it.
 * p. 52.


 * Heron was a practical surveyor. This may account for the fact that his writings bear so little resemblance to those of the Greek authors, who considered it degrading the science to apply geometry to surveying. The character of his geometry is not Grecian but decidedly Egyptian. ...There are ...points of resemblance between Heron's writings and the ancient Ahmes papyrus. Thus Ahmes used unit-fractions exclusively; Heron uses them oftener than other fractions. Like Ahmes and the priests at Edfu, Heron divides complicated figures into simpler ones by drawing auxiliary lines; like them, he shows, throughout, a special fondness for the isosceles trapezoid. The writings of Heron satisfied a practical want, and for that reason were borrowed extensively by other peoples. We find traces of them in Rome, in the Occident during the Middle Ages, and even in India.
 * p. 53.


 * 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.
 * p. 53.


 * 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."
 * p. 54.

The Second Alexandrian School

 * The close of the dynasty of the Lagides which ruled Egypt from the time of Ptolemy Soter, the builder of Alexandria, for 300 years; the absorption of Egypt into the Roman Empire; the closer commercial relations between peoples of the East and of the West; the gradual decline of paganism and spread of Christianity,—these events were of far-reaching influence on the progress of the sciences, which then had their home in Alexandria. Alexandria became a commercial and intellectual emporium. Traders of all nations met in her busy streets, and in her magnificent Library, museums, lecture halls, scholars from the East mingled with those of the West; Greeks began to study older literatures and to compare them with their own. In consequence of this interchange of ideas the Greek philosophy became fused with Oriental philosophy. Neo-Pythagoreanism and Neo-Platonism were the names of the modified systems. These stood, for a time, in opposition to Christianity. The study of Platonism and Pythagorean mysticism led to the revival of the theory of numbers. Perhaps the dispersion of the Jews and their introduction to Greek learning helped in bringing about this revival. The theory of numbers became a favourite study. This new line of mathematical inquiry ushered in what we may call a new school. There is no doubt that even now geometry continued to be one of the most important studies in the Alexandrian course. This Second Alexandrian School may be said to begin with the Christian era. It was made famous by the names of Claudius Ptolemæus, Diophantus, Pappus, Theon of Smyrna, Theon of Alexandria, Iamblichus, Porphyrius, and others.
 * p. 54.


 * Serenus of Antissa was connected more or less with this [Second Alexandrian] school. He wrote on sections of the cone and cylinder, in two books, one of which treated only of the triangular section of the cone through the apex. He solved the problem, "given a cone (cylinder), to find a cylinder (cone), so that the section of both by the same plane gives similar ellipses." Of particular interest is the following theorem [show figure], which is the foundation of the modern theory of harmonics: If from D we draw DF, cutting the triangle ABC, and choose H on it, so that DE:DF=EH:HF, and if we draw the line AH, then every transversal through D, such as DG will be divided by AH, so that DK:DG=KJ:JO.
 * p. 55.

Newton to Euler

 * In letters which went between me and that most excellent geometer. G.G. Leibniz, ten years ago, when I signified that I was in the knowledge of a method of determining maxima and minima, of drawing tangents, and the like, and when I concealed it in transposed letters involving this sentence (Data æquatione, etc., above cited) that most distinguished man wrote back that he had also fallen upon a method of the same kind, and communicated his method, which hardly differed from mine, except in his forms of words and symbols.
 * Isaac Newton, Principia (1687) 1st edition, Book II. Prop.7, scholium

Euler, Lagrange, and Laplace

 * [ Joseph Fourier] carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series $$\sum_{n=0}^{n=\infty} (a_n\sin nx+b_n\cos nx)$$ represents the function $$\phi(x)$$ for every value of $$x$$ if the coefficients $$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}\phi(x) \sin nx\,dx$$, and $$b_n$$ be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function.