CHAPTER III

The community has found out that such schemes may be well fitted to give the children a good time in school, but lead them to a bad time afterward. Life is hard work, and if they have never learned in school to give their concentrated attention to that which does not appeal to them and which does not interest them immediately, they have missed the most valuable lesson of their school years. The little practical information they could have learned at any time; the energy of attention and concentration can no longer be learned if the early years are wasted. However narrow and commercial the standpoint which is chosen may be, it can always be found that it is the general education which pays best, and the more the period of cultural work can be expanded the more efficient will be the services of the school for the practical services of the nation.[15]

Of course no one should construe these remarks as opposing in the slightest degree the laudable efforts that are constantly being put forth to make geometry more

interesting and to vitalize it by establishing as strong motives as possible for its study. Let the home, the workshop, physics, art, play,—all contribute their quota of motive to geometry as to all mathematics and all other branches. But let us never forget that geometry has araison d'êtrebeyond all this, and that these applications are sought primarily for the sake of geometry, and that geometry is not taught primarily for the sake of these applications.

When we consider how often geometry is attacked by those who profess to be its friends, and how teachers who have been trained in mathematics occasionally seem to make of the subject little besides a mongrel course in drawing and measuring, all the time insisting that they are progressive while the champions of real geometry are reactionary, it is well to read some of the opinions of the masters. The following quotations may be given occasionally in geometry classes as showing the esteem in which the subject has been held in various ages, and at any rate they should serve to inspire the teacher to greater love for his subject.

The enemies of geometry, those who know it only imperfectly, look upon the theoretical problems, which constitute the most difficult part of the subject, as mental games which consume time and energy that might better be employed in other ways. Such a belief is false, and it would block the progress of science if it were credible. But aside from the fact that the speculative problems, which at first sight seem barren, can often be applied to useful purposes, they always stand as among the best means to develop and to express all the forces of the human intelligence.—Abbé Bossut.

The sailor whom an exact observation of longitude saves from shipwreck owes his life to a theory developed two thousand years ago by men who had in mind merely the speculations of abstract geometry.—Condorcet.

If mathematical heights are hard to climb, the fundamental principles lie at every threshold, and this fact allows them to be comprehended by that common sense which Descartes declared was "apportioned equally among all men."—Collet.

It may seem strange that geometry is unable to define the terms which it uses most frequently, since it defines neither movement, nor number, nor space,—-the three things with which it is chiefly concerned. But we shall not be surprised if we stop to consider that this admirable science concerns only the most simple things, and the very quality that renders these things worthy of study renders them incapable of being defined. Thus the very lack of definition is rather an evidence of perfection than a defect, since it comes not from the obscurity of the terms, but from the fact that they are so very well known.—Pascal.

God eternally geometrizes.—Plato.

God is a circle of which the center is everywhere and the circumference nowhere.—Rabelais.

Without mathematics no one can fathom the depths of philosophy. Without philosophy no one can fathom the depths of mathematics. Without the two no one can fathom the depths of anything.—Bordas-Demoulin.

We may look upon geometry as a practical logic, for the truths which it studies, being the most simple and most clearly understood of all truths, are on this account the most susceptible of ready application in reasoning.—D'Alembert.

The advance and the perfecting of mathematics are closely joined to the prosperity of the nation.—Napoleon.

Hold nothing as certain save what can be demonstrated.—Newton.

To measure is to know.—Kepler.

The method of making no mistake is sought by every one. The logicians profess to show the way, but the geometers alone ever reach it, and aside from their science there is no genuine demonstration.—Pascal.

The taste for exactness, the impossibility of contenting one's self with vague notions or of leaning upon mere hypotheses, the necessity for perceiving clearly the connection between certain propositions and the object in view,—these are the most precious fruits of the study of mathematics.—Lacroix.

Bibliography.Smith, The Teaching of Elementary Mathematics, p. 234, New York, 1900; Henrici, Presidential Address before the British Association,Nature, Vol. XXVIII, p. 497; Hill, Educational Value of Mathematics,Educational Review, Vol. IX, p. 349; Young, The Teaching of Mathematics, p. 9, New York, 1907. The closing quotations are from Rebière, Mathématiques et Mathématiciens, Paris, 1893.

The geometry of very ancient peoples was largely the mensuration of simple areas and solids, such as is taught to children in elementary arithmetic to-day. They early learned how to find the area of a rectangle, and in the oldest mathematical records that have come down to us there is some discussion of the area of triangles and the volume of solids.

The earliest documents that we have relating to geometry come to us from Babylon and Egypt. Those from Babylon are written on small clay tablets, some of them about the size of the hand, these tablets afterwards having been baked in the sun. They show that the Babylonians of that period knew something of land measures, and perhaps had advanced far enough to compute the area of a trapezoid. For the mensuration of the circle they later used, as did the early Hebrews, the value π = 3. A tablet in the British Museum shows that they also used such geometric forms as triangles and circular segments in astrology or as talismans.

The Egyptians must have had a fair knowledge of practical geometry long before the date of any mathematical treatise that has come down to us, for the building of the pyramids, between 3000 and 2400B.C., required the application of several geometric principles. Some knowledge of surveying must also have been necessaryto carry out the extensive plans for irrigation that were executed under Amenemhat III, about 2200B.C.

The first definite knowledge that we have of Egyptian mathematics comes to us from a manuscript copied on papyrus, a kind of paper used about the Mediterranean in early times. This copy was made by one Aah-mesu (The Moon-born), commonly called Ahmes, who probably flourished about 1700B.C.The original from which he copied, written about 2300B.C., has been lost, but the papyrus of Ahmes, written nearly four thousand years ago, is still preserved, and is now in the British Museum. In this manuscript, which is devoted chiefly to fractions and to a crude algebra, is found some work on mensuration. Among the curious rules are the incorrect ones that the area of an isosceles triangle equals half the product of the base and one of the equal sides; and that the area of a trapezoid having basesb,b', and the nonparallel sides each equal toa, is ½a(b+b'). One noteworthy advance appears, however. Ahmes gives a rule for finding the area of a circle, substantially as follows: Multiply the square on the radius by (16/9)2, which is equivalent to taking for π the value 3.1605. This papyrus also contains some treatment of the mensuration of solids, particularly with reference to the capacity of granaries. There is also some slight mention of similar figures, and an extensive treatment of unit fractions,—fractions that were quite universal among the ancients. In the line of algebra it contains a brief treatment of the equation of the first degree with one unknown, and of progressions.[16]

Herodotus tells us that Sesostris, king of Egypt,[17]divided the land among his people and marked out the boundaries after the overflow of the Nile, so that surveying must have been well known in his day. Indeed, theharpedonaptæ, or rope stretchers, acquired their name because they stretched cords, in which were knots, so as to make the right triangle 3, 4, 5, when they wished to erect a perpendicular. This is a plan occasionally used by surveyors to-day, and it shows that the practical application of the Pythagorean Theorem was known long before Pythagoras gave what seems to have been the first general proof of the proposition.

From Egypt, and possibly from Babylon, geometry passed to the shores of Asia Minor and Greece. The scientific study of the subject begins with Thales, one of the Seven Wise Men of the Grecian civilization. Born at Miletus, not far from Smyrna and Ephesus, about 640B.C., he died at Athens in 548B.C.He spent his early manhood as a merchant, accumulating the wealth that enabled him to spend his later years in study. He visited Egypt, and is said to have learned such elements of geometry as were known there. He founded a school of mathematics and philosophy at Miletus, known from the country as the Ionic School. How elementary the knowledge of geometry then was may be understood from the fact that tradition attributes only about four propositions to Thales,—(1) that vertical angles are equal, (2) that equal angles lie opposite the equal sides of an isosceles triangle, (3) that a triangle is determined by two angles and the included side, (4) that a diameter bisects the circle, and possibly the propositions about the

angle-sum of a triangle for special cases, and the angle inscribed in a semicircle.[18]

The greatest pupil of Thales, and one of the most remarkable men of antiquity, was Pythagoras. Born probably on the island of Samos, just off the coast of Asia Minor, about the year 580B.C., Pythagoras set forth as a young man to travel. He went to Miletus and studied under Thales, probably spent several years in Egypt, very likely went to Babylon, and possibly went even to India, since tradition asserts this and the nature of his work in mathematics suggests it. In later life he went to a Greek colony in southern Italy, and at Crotona, in the southeastern part of the peninsula, he founded a school and established a secret society to propagate his doctrines. In geometry he is said to have been the first to demonstrate the proposition that the square on the hypotenuse is equal to the sum of the squares upon the other two sides of a right triangle. The proposition was known in India and Egypt before his time, at any rate for special cases, but he seems to have been the first to prove it. To him or to his school seems also to have been due the construction of the regular pentagon and of the five regular polyhedrons. The construction of the regular pentagon requires the dividing of a line into extreme and mean ratio, and this problem is commonly assigned to the Pythagoreans, although it played an important part in Plato's school. Pythagoras is also said to have known that six equilateral triangles, threeregular hexagons, or four squares, can be placed about a point so as just to fill the 360°, but that no other regular polygons can be so placed. To his school is also due the proof for the general case that the sum of the angles of a triangle equals two right angles, the first knowledge of the size of each angle of a regular polygon, and the construction of at least one star-polygon, the star-pentagon, which became the badge of his fraternity. The brotherhood founded by Pythagoras proved so offensive to the government that it was dispersed before the death of the master. Pythagoras fled toMegapontum, a seaport lying to the north of Crotona, and there he died about 501B.C.[19]

Fanciful Portrait of Pythagoras Calandri's Arithmetic, 1491Fanciful Portrait of Pythagoras Calandri's Arithmetic, 1491

For two centuries after Pythagoras geometry passed through a period of discovery of propositions. The stateof the science may be seen from the fact that Œnopides of Chios, who flourished about 465B.C., and who had studied in Egypt, was celebrated because he showed how to let fall a perpendicular to a line, and how to make an angle equal to a given angle. A few years later, about 440B.C., Hippocrates of Chios wrote the first Greek textbook on mathematics. He knew that the areas of circles are proportional to the squares on their radii, but was ignorant of the fact that equal central angles or equal inscribed angles intercept equal arcs.

Antiphon and Bryson, two Greek scholars, flourished about 430B.C.The former attempted to find the area of a circle by doubling the number of sides of a regular inscribed polygon, and the latter by doing the same for both inscribed and circumscribed polygons. They thus approximately exhausted the area between the polygon and the circle, and hence this method is known as the method of exhaustions.

About 420B.C.Hippias of Elis invented a certain curve called the quadratrix, by means of which he could square the circle and trisect any angle. This curve cannot be constructed by the unmarked straightedge and the compasses, and when we say that it is impossible to square the circle or to trisect any angle, we mean that it is impossible by the help of these two instruments alone.

During this period the great philosophic school of Plato (429-348B.C.) flourished at Athens, and to this school is due the first systematic attempt to create exact definitions, axioms, and postulates, and to distinguish between elementary and higher geometry. It was at this time that elementary geometry became limited to the use of the compasses and the unmarked straightedge,which took from this domain the possibility of constructing a square equivalent to a given circle ("squaring the circle"), of trisecting any given angle, and of constructing a cube that should have twice the volume of a given cube ("duplicating the cube"), these being the three famous problems of antiquity. Plato and his school interested themselves with the so-called Pythagorean numbers, that is, with numbers that would represent the three sides of a right triangle and hence fulfill the condition thata2+b2=c2. Pythagoras had already given a rule that would be expressed in modern form, as ¼(m2+ 1)2=m2+ ¼(m2- 1)2. The school of Plato found that ((½m)2+ 1)2=m2+ ((½m)2- 1)2. By giving various values tom, different Pythagorean numbers may be found. Plato's nephew, Speusippus (about 350B.C.), wrote upon this subject. Such numbers were known, however, both in India and in Egypt, long before this time.

One of Plato's pupils was Philippus of Mende, in Egypt, who flourished about 380B.C.It is said that he discovered the proposition relating to the exterior angle of a triangle. His interest, however, was chiefly in astronomy.

Another of Plato's pupils was Eudoxus of Cnidus (408-355B.C.). He elaborated the theory of proportion, placing it upon a thoroughly scientific foundation. It is probable that Book V of Euclid, which is devoted to proportion, is essentially the work of Eudoxus. By means of the method of exhaustions of Antiphon and Bryson he proved that the pyramid is one third of a prism, and the cone is one third of a cylinder, each of the same base and the same altitude. He wrote the first textbook known on solid geometry.

The subject of conic sections starts with another pupil of Plato's, Menæchmus, who lived about 350B.C.He cut the three forms of conics (the ellipse, parabola, and hyperbola) out of three different forms of cone,—the acute-angled, right-angled, and obtuse-angled,—not noticing that he could have obtained all three from any form of right circular cone. It is interesting to see the far-reaching influence of Plato. While primarily interested in philosophy, he laid the first scientific foundations for a system of mathematics, and his pupils were the leaders in this science in the generation following his greatest activity.

The great successor of Plato at Athens was Aristotle, the teacher of Alexander the Great. He also was more interested in philosophy than in mathematics, but in natural rather than mental philosophy. With him comes the first application of mathematics to physics in the hands of a great man, and with noteworthy results. He seems to have been the first to represent an unknown quantity by letters. He set forth the theory of the parallelogram of forces, using only rectangular components, however. To one of his pupils, Eudemus of Rhodes, we are indebted for a history of ancient geometry, some fragments of which have come down to us.

The first great textbook on geometry, and the greatest one that has ever appeared, was written by Euclid, who taught mathematics in the great university at Alexandria, Egypt, about 300B.C.Alexandria was then practically a Greek city, having been named in honor of Alexander the Great, and being ruled by the Greeks.

In his work Euclid placed all of the leading propositions of plane geometry then known, and arranged themin a logical order. Most geometries of any importance written since his time have been based upon Euclid, improving the sequence, symbols, and wording as occasion demanded. He also wrote upon other branches of mathematics besides elementary geometry, including a work on optics. He was not a great creator of mathematics, but was rather a compiler of the work of others, an office quite as difficult to fill and quite as honorable.

Euclid did not give much solid geometry because not much was known then. It was to Archimedes (287-212B.C.), a famous mathematician of Syracuse, on the island of Sicily, that some of the most important propositions of solid geometry are due, particularly those relating to the sphere and cylinder. He also showed how to find the approximate value of π by a method similar to the one we teach to-day, proving that the real value lay between 3-1/7 and 3-10/71. The story goes that the sphere and cylinder were engraved upon his tomb, and Cicero, visiting Syracuse many years after his death, found the tomb by looking for these symbols. Archimedes was the greatest mathematical physicist of ancient times.

The Greeks contributed little more to elementary geometry, although Apollonius of Perga, who taught at Alexandria between 250 and 200B.C., wrote extensively on conic sections, and Hypsicles of Alexandria, about 190B.C., wrote on regular polyhedrons. Hypsicles was the first Greek writer who is known to have used sexagesimal fractions,—the degrees, minutes, and seconds of our angle measure. Zenodorus (180B.C.) wrote on isoperimetric figures, and his contemporary, Nicomedes of Gerasa, invented a curve known as the conchoid, by means of which he could trisect any angle. Another contemporary, Diocles, invented the cissoid, or ivy-shapedcurve, by means of which he solved the famous problem of duplicating the cube, that is, constructing a cube that should have twice the volume of a given cube.

The greatest of the Greek astronomers, Hipparchus (180-125B.C.), lived about this period, and with him begins spherical trigonometry as a definite science. A kind of plane trigonometry had been known to the ancient Egyptians. The Greeks usually employed the chord of an angle instead of the half chord (sine), the latter having been preferred by the later Arab writers.

The most celebrated of the later Greek physicists was Heron of Alexandria, formerly supposed to have lived about 100B.C., but now assigned to the first century A.D. His contribution to geometry was the formula for the area of a triangle in terms of its sides a, b, and c, with s standing for the semiperimeter ½(a+b+c). The formula is

sqrt{s(s-a)(s-b)(s-c)}

Probably nearly contemporary with Heron was Menelaus of Alexandria, who wrote a spherical trigonometry. He gave an interesting proposition relating to plane and spherical triangles, their sides being cut by a transversal. For the plane triangleABC, the sidesa,b, andcbeing cut respectively inX,Y, andZ, the theorem asserts substantially that

(AZ/BZ) · (BX/CX) · (CY/AY) = 1.

The most popular writer on astronomy among the Greeks was Ptolemy (Claudius Ptolemaeus, 87-165A.D.), who lived at Alexandria. He wrote a work entitled "Megale Syntaxis" (The Great Collection), which his followers designated asMegistos(greatest), on which account the Arab translators gave it the name "Almagest"(almeaning "the"). He advanced the science of trigonometry, but did not contribute to geometry.

At the close of the third century Pappus of Alexandria (295A.D.) wrote on geometry, and one of his theorems, a generalized form of the Pythagorean proposition, is mentioned inChapter XVIof this work. Only two other Greek writers on geometry need be mentioned. Theon of Alexandria (370A.D.), the father of the Hypatia who is the heroine of Charles Kingsley's well-known novel, wrote a commentary on Euclid to which we are indebted for some historical information. Proclus (410-485A.D.) also wrote a commentary on Euclid, and much of our information concerning the first Book of Euclid is due to him.

The East did little for geometry, although contributing considerably to algebra. The first great Hindu writer was Aryabhatta, who was born in 476A.D.He gave the very close approximation for π, expressed in modern notation as 3.1416. He also gave rules for finding the volume of the pyramid and sphere, but they were incorrect, showing that the Greek mathematics had not yet reached the Ganges. Another Hindu writer, Brahmagupta (born in 598A.D.), wrote an encyclopedia of mathematics. He gave a rule for finding Pythagorean numbers, expressed in modern symbols as follows:

1/4(p^2/q)+q)^2=1/4((p^2/q)-q)^2+p^2

He also generalized Heron's formula by asserting that the area of an inscribed quadrilateral of sidesa,b,c,d, and semiperimeters, is

sqrt{(s-a)(s-b)(s-c)(s-d)}

The Arabs, about the time of the "Arabian Nights Tales" (800A.D.), did much for mathematics, translatingthe Greek authors into their language and also bringing learning from India. Indeed, it is to them that modern Europe owed its first knowledge of Euclid. They contributed nothing of importance to elementary geometry, however.

The greatest of the Arab writers was Mohammed ibn Musa al-Khowarazmi (820A.D.). He lived at Bagdad and Damascus. Although chiefly interested in astronomy, he wrote the first book bearing the name "algebra" ("Al-jabr wa'l-muqābalah," Restoration and Equation), composed an arithmetic using the Hindu numerals,[20]and paid much attention to geometry and trigonometry.

Euclid was translated from the Arabic into Latin in the twelfth century, Greek manuscripts not being then at hand, or being neglected because of ignorance of the language. The leading translators were Athelhard of Bath (1120), an English monk; Gherard of Cremona (1160), an Italian monk; and Johannes Campanus (1250), chaplain to Pope Urban IV.

The greatest European mathematician of the Middle Ages was Leonardo of Pisa[21](ca.1170-1250). He was very influential in making the Hindu-Arabic numerals known in Europe, wrote extensively on algebra, and was the author of one book on geometry. He contributed nothing to the elementary theory, however. The first edition of Euclid was printed in Latin in 1482, the first one in English appearing in 1570.

Our symbols are modern, + and - first appearing in a German work in 1489; = in Recorde's "Whetstone of Witte" in 1557; > and < in the works of Harriot (1560-1621); and × in a publication by Oughtred (1574-1660).

The most noteworthy advance in geometry in modern times was made by the great French philosopher Descartes, who published a small work entitled "La Géométrie" in 1637. From this springs the modern analytic geometry, a subject that has revolutionized the methods of all mathematics. Most of the subsequent discoveries in mathematics have been in higher branches. To the great Swiss mathematician Euler (1707-1783) is due, however, one proposition that has found its way into elementary geometry, the one showing the relation between the number of edges, vertices, and faces of a polyhedron.

There has of late arisen a modern elementary geometry devoted chiefly to special points and lines relating to the triangle and the circle, and many interesting propositions have been discovered. The subject is so extensive that it cannot find any place in our crowded curriculum, and must necessarily be left to the specialist.[22]Some idea of the nature of the work may be obtained from a mention of a few propositions:

The medians of a triangle are concurrent in the centroid, or center of gravity of the triangle.

The bisectors of the various interior and exterior angles of a triangle are concurrent by threes in the incenter or in one of the three excenters of the triangle.

The common chord of two intersecting circles is a special case of their radical axis, and tangents to the circles from any point on the radical axis are equal.

IfOis the orthocenter of the triangleABC, andX,Y,Zare the feet of the perpendiculars fromA,B,Crespectively, andP,Q,Rare the mid-points ofa,b,crespectively, andL,M,Nare the mid-points ofOA,OB,OCrespectively; then the pointsL,M,N;P,Q,R;X,Y,Zall lie on a circle, the "nine points circle."

In the teaching of geometry it adds a human interest to the subject to mention occasionally some of the historical facts connected with it. For this reason this brief sketch will be supplemented by many notes upon the various important propositions as they occur in the several books described in the later chapters of this work.

We know little of the teaching of geometry in very ancient times, but we can infer its nature from the teaching that is still seen in the native schools of the East. Here a man, learned in any science, will have a group of voluntary students sitting about him, and to them he will expound the truth. Such schools may still be seen in India, Persia, and China, the master sitting on a mat placed on the ground or on the floor of a veranda, and the pupils reading aloud or listening to his words of exposition.

In Egypt geometry seems to have been in early times mere mensuration, confined largely to the priestly caste. It was taught to novices who gave promise of success in this subject, and not to others, the idea of general culture, of training in logic, of the cultivation of exact expression, and of coming in contact with truth being wholly wanting.

In Greece it was taught in the schools of philosophy, often as a general preparation for philosophic study. Thus Thales introduced it into his Ionic school, Pythagoras made it very prominent in his great school at Crotona in southern Italy (Magna Græcia), and Plato placed above the door of hisAcademiathe words, "Let no one ignorant of geometry enter here,"—a kind of entrance examination for his school of philosophy. Inthese gatherings of students it is probable that geometry was taught in much the way already mentioned for the schools of the East, a small group of students being instructed by a master. Printing was unknown, papyrus was dear, parchment was only in process of invention. Paper such as we know had not yet appeared, so that instruction was largely oral, and geometric figures were drawn by a pointed stick on a board covered with fine sand, or on a tablet of wax.

But with these crude materials there went an abundance of time, so that a number of great results were accomplished in spite of the difficulties attending the study of the subject. It is said that Hippocrates of Chios (ca.440B.C.) wrote the first elementary textbook on mathematics and invented the method of geometric reduction, the replacing of a proposition to be proved by another which, when proved, allows the first one to be demonstrated. A little later Eudoxus of Cnidus (ca.375B.C.), a pupil of Plato's, used thereductio ad absurdum, and Plato is said to have invented the method of proof by analysis, an elaboration of the plan used by Hippocrates. Thus these early philosophers taught their pupils not facts alone, but methods of proof, giving them power as well as knowledge. Furthermore, they taught them how to discuss their problems, investigating the conditions under which they are capable of solution. This feature of the work they called thediorismus, and it seems to have started with Leon, a follower of Plato.

Between the time of Plato (ca.400B.C.) and Euclid (ca.300B.C.) several attempts were made to arrange the accumulated material of elementary geometry in a textbook. Plato had laid the foundations for the science, in the form of axioms, postulates, and definitions, and hehad limited the instruments to the straightedge and the compasses. Aristotle (ca.350B.C.) had paid special attention to the history of the subject, thus finding out what had already been accomplished, and had also made much of the applications of geometry. The world was therefore ready for a good teacher who should gather the material and arrange it scientifically. After several attempts to find the man for such a task, he was discovered in Euclid, and to his work the next chapter is devoted.

After Euclid, Archimedes (ca.250B.C.) made his great contributions. He was not a teacher like his illustrious predecessor, but he was a great discoverer. He has left us, however, a statement of his methods of investigation which is helpful to those who teach. These methods were largely experimental, even extending to the weighing of geometric forms to discover certain relations, the proof being given later. Here was born, perhaps, what has been called the laboratory method of the present.

Of the other Greek teachers we have but little information as to methods of imparting instruction. It is not until the Middle Ages that there is much known in this line. Whatever of geometry was taught seems to have been imparted by word of mouth in the way of expounding Euclid, and this was done in the ancient fashion.

The early Church leaders usually paid no attention to geometry, but as time progressed thequadrivium, or four sciences of arithmetic, music, geometry, and astronomy, came to rank with thetrivium(grammar, rhetoric, dialectics), the two making up the "seven liberal arts." All that there was of geometry in the first thousand years of Christianity, however, at least in the greatmajority of Church schools, was summed up in a few definitions and rules of mensuration. Gerbert, who became Pope Sylvester II in 999A.D., gave a new impetus to geometry by discovering a manuscript of the old Roman surveyors and a copy of the geometry of Boethius, who paraphrased Euclid about 500A.D.He thereupon wrote a brief geometry, and his elevation to the papal chair tended to bring the study of mathematics again into prominence.

Geometry now began to have some place in the Church schools, naturally the only schools of high rank in the Middle Ages. The study of the subject, however, seems to have been merely a matter of memorizing. Geometry received another impetus in the book written by Leonardo of Pisa in 1220, the "Practica Geometriae." Euclid was also translated into Latin about this time (strangely enough, as already stated, from the Arabic instead of the Greek), and thus the treasury of elementary geometry was opened to scholars in Europe. From now on, until the invention of printing (ca.1450), numerous writers on geometry appear, but, so far as we know, the method of instruction remained much as it had always been. The universities began to appear about the thirteenth century, and Sacrobosco, a well-known medieval mathematician, taught mathematics about 1250 in the University of Paris. In 1336 this university decreed that mathematics should be required for a degree. In the thirteenth century Oxford required six books of Euclid for one who was to teach, but this amount of work seems to have been merely nominal, for in 1450 only two books were actually read. The universities of Prague (founded in 1350) and Vienna (statutes of 1389) required most of plane geometry for the teacher'slicense, although Vienna demanded but one book for the bachelor's degree. So, in general, the universities of the thirteenth, fourteenth, and fifteenth centuries required less for the degree of master of arts than we now require from a pupil in our American high schools. On the other hand, the university students were younger than now, and were really doing only high school work.

The invention of printing made possible the study of geometry in a new fashion. It now became possible for any one to study from a book, whereas before this time instruction was chiefly by word of mouth, consisting of an explanation of Euclid. The first Euclid was printed in 1482, at Venice, and new editions and variations of this text came out frequently in the next century. Practical geometries became very popular, and the reaction against the idea of mental discipline threatened to abolish the old style of text. It was argued that geometry was uninteresting, that it was not sufficient in itself, that boys needed to see the practical uses of the subject, that only those propositions that were capable of application should be retained, that there must be a fusion between the demands of culture and the demands of business, and that every man who stood for mathematical ideals represented an obsolete type. Such writers as Finæus (1556), Bartoli (1589), Belli (1569), and Cataneo (1567), in the sixteenth century, and Capra (1678), Gargiolli (1655), and many others in the seventeenth century, either directly or inferentially, took this attitude towards the subject,—exactly the attitude that is being taken at the present time by a number of teachers in the United States. As is always the case, to such an extreme did this movement lead that there was a reaction that brought the Euclid type of bookagain to the front, and it has maintained its prominence even to the present.

The study of geometry in the high schools is relatively recent. The Gymnasium (classical school preparatory to the university) at Nürnberg, founded in 1526, and the Cathedral school at Württemberg (as shown by the curriculum of 1556) seem to have had no geometry before 1600, although the Gymnasium at Strassburg included some of this branch of mathematics in 1578, and an elective course in geometry was offered at Zwickau, in Saxony, in 1521. In the seventeenth century geometry is found in a considerable number of secondary schools, as at Coburg (1605), Kurfalz (1615, elective), Erfurt (1643), Gotha (1605), Giessen (1605), and numerous other places in Germany, although it appeared but rarely in the secondary schools of France before the eighteenth century. In Germany the Realschulen—schools with more science and less classics than are found in the Gymnasium—came into being in the eighteenth century, and considerable effort was made to construct a course in geometry that should be more practical than that of the modified Euclid. At the opening of the nineteenth century the Prussian schools were reorganized, and from that time on geometry has had a firm position in the secondary schools of all Germany. In the eighteenth century some excellent textbooks on geometry appeared in France, among the best being that of Legendre (1794), which influenced in such a marked degree the geometries of America. Soon after the opening of the nineteenth century thelycéesof France became strong institutions, and geometry, chiefly based on Legendre, was well taught in the mathematical divisions. A worthy rival of Legendre's geometry was thework of Lacroix, who called attention continually to the analogy between the theorems of plane and solid geometry, and even went so far as to suggest treating the related propositions together in certain cases.

In England the preparatory schools, such as Rugby, Harrow, and Eton, did not commonly teach geometry until quite recently, leaving this work for the universities. In Christ's Hospital, London, however, geometry was taught as early as 1681, from a work written by several teachers of prominence. The highest class at Harrow studied "Euclid and vulgar fractions" one period a week in 1829, but geometry was not seriously studied before 1837. In the Edinburgh Academy as early as 1885, and in Rugby by 1839, plane geometry was completed.

Not until 1844 did Harvard require any plane geometry for entrance. In 1855 Yale required only two books of Euclid. It was therefore from 1850 to 1875 that plane geometry took a definite place in the American high school. Solid geometry has not been generally required for entrance to any eastern college, although in the West this is not the case. The East teaches plane geometry more thoroughly, but allows a pupil to enter college or to go into business with no solid geometry. Given a year to the subject, it is possible to do little more than cover plane geometry; with a year and a half the solid geometry ought easily to be covered also.

Bibliography.Stamper, A History of the Teaching of Elementary Geometry, New York, 1909, with a very full bibliography of the subject; Cajori, The Teaching of Mathematics in the United States, Washington, 1890; Cantor, Geschichte der Mathematik, Vol. IV, p. 321, Leipzig, 1908; Schotten, Inhalt und Methode des planimetrischen Unterrichts, Leipzig, 1890.

It is fitting that a chapter in a book upon the teaching of this subject should be devoted to the life and labors of the greatest of all textbook writers, Euclid,—a man whose name has been, for more than two thousand years, a synonym for elementary plane geometry wherever the subject has been studied. And yet when an effort is made to pick up the scattered fragments of his biography, we are surprised to find how little is known of one whose fame is so universal. Although more editions of his work have been printed than of any other book save the Bible,[23]we do not know when he was born, or in what city, or even in what country, nor do we know his race, his parentage, or the time of his death. We should not feel that we knew much of the life of a man who lived when the Magna Charta was wrested from King John, if our first and only source of information was a paragraph in the works of some historian of to-day; and yet this is about the situation in respect to Euclid. Proclus of Alexandria, philosopher, teacher, and mathematician, lived from 410 to 485 A.D., and wrote a commentary on the works of Euclid. In his writings, which seem to set forth in amplified form his lectures to the students in the Neoplatonist School

of Alexandria, Proclus makes this statement, and of Euclid's life we have little else:

Not much younger than these[24]is Euclid, who put together the "Elements," collecting many of the theorems of Eudoxus, perfecting many of those of Theætetus, and also demonstrating with perfect certainty what his predecessors had but insufficiently proved. He flourished in the time of the first Ptolemy, for Archimedes, who closely followed this ruler,[25]speaks of Euclid. Furthermore it is related that Ptolemy one time demanded of him if there was in geometry no shorter way than that of the "Elements," to whom he replied that there was no royal road to geometry.[26]He was therefore younger than the pupils of Plato, but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.[27]

Thus we have in a few lines, from one who lived perhaps seven or eight hundred years after Euclid, nearly all that is known of the most famous teacher of geometry that ever lived. Nevertheless, even this little tells us about when he flourished, for Hermotimus and Philippus were pupils of Plato, who died in 347B.C., whereas Archimedes was born about 287B.C.and was writing about 250B.C.Furthermore, since Ptolemy I reigned from 306 to 283B.C., Euclid must have been teaching about 300B.C., and this is the date that is generally assigned to him.

Euclid probably studied at Athens, for until he himself assisted in transferring the center of mathematical

culture to Alexandria, it had long been in the Grecian capital, indeed since the time of Pythagoras. Moreover, numerous attempts had been made at Athens to do exactly what Euclid succeeded in doing,—to construct a logical sequence of propositions; in other words, to write a textbook on plane geometry. It was at Athens, therefore, that he could best have received the inspiration to compose his "Elements."[28]After finishing his education at Athens it is quite probable that he, like other savants of the period, was called to Alexandria by Ptolemy Soter, the king, to assist in establishing the great school which made that city the center of the world's learning for several centuries. In this school he taught, and here he wrote the "Elements" and numerous other works, perhaps ten in all.

Although the Greek writers who may have known something of the life of Euclid have little to say of him, the Arab writers, who could have known nothing save from Greek sources, have allowed their imaginations the usual latitude in speaking of him and of his labors. Thus Al-Qifṫī, who wrote in the thirteenth century, has this to say in his biographical treatise "Ta'rīkh al-Ḥukamā":

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