ositions carefully selected and graded, the object of which is to secure original investigation on the part of the pupil. Students will thus learn not merely to “ follow old trails" of thought as presented in the text-book, but will also learn to reason for themselves, without which thought-studies are of but little value in education. A general acknowledgment of indebtness is due to those who have previously written upon the subject of geometry, especially to American and English authors, most of whose works have been read with interest and profit, and whose views have done much to give shape to the conception of a course in the elements of geometry here presented. The work is respectfully submitted to the judgment of teachers of geometry with the hope that it may contribute to awaken a deeper interest in the beautiful science of form, and through it aid in the development of the power of logical thinking in our country. EDWARD BROOKS. PHILADELPHIA, Aug. 30, 1889. NOTE TO REVISED EDITION. This revision consists in the modification and simplification of a number of theorems and demonstrations for the purpose of making more easy the pathway of the student, and of bringing the work up to the latest and most advanced ideas of geometrical instruction. EDWARD BROOKS. PHILADELPHIA, Jan. 16, 1901. CONTENTS. PLANE GEOMETRY. PAGE . . 33 Subject Matter of Geometry . 13 Theorems for Demonstration . 126 Problems of Construction . 129 Geometry, etc. 23 Abbreviations 31 Theorems : Angles and Perpendiculars Squares on Lines 154 72 Theorems for Demonstration . 186 Loci of Straight Lines 76 Practical Examples . 77 Theorems for Demonstration . 192 Theorems for Demonstration . 78 Geometrical Constructions . 193 Problems for Construction. . 210 RATIO AND PROPORTION. Theorems for Demonstration . 98 Regular Polygons . 99 Theorems for Demonstration . 233 Axioms and Postulates . 101 ) Problems for Construction . . . 235 · 187 MAXIMA AND MINIMA. MISCELLANEOUS PROPOSITIONS. Theorems . Theorems and Problems . 243 Theorems in Book III. SECTION II. Supplementary to Book I. 244 | Theorems in Book V. Supplementary to Book III. 246 Problems in Book V. Supplementary to Book IV. 249 Derived General Expressions 263 HISTORY OF GEOMETRY. GEOMETRY is generally supposed to have had its origin in Egypt, where the annual overflowing of the Nile obliterated the landmarks and rendered it necessary to have recourse to mathematical measurement to reestablish them. This origin is indicated by the term itself, Geometry being from two Greek words, yn, earth, and Metpov, measure, signifying, literally, the measurement of the earth. But, whatever may have been the origin of the term, the natural tendency of the human mind to compare things in respect to their forms and magnitudes is so universal that a geometry more or less perfect must have existed since the first dawn of civilization. Geometry is supposed to have been introduced into Greece by Thales, who lived about the year 650 B. C. He is said to have measured the height of the Pyramids by means of their shadow, and also determined the distance of vessels from the shore by the principles of geometry. On his return to Greece he founded the Iouian school, named from Ionia, his native country. He is supposed to have made various discoveries concerning the circle and the comparison of triangles, and it is known that he discovered that all the angles in a semicircle are right angles. One of the disciples of Thales composed an elementary treatise on Geometry—the earliest on record-and he is said to have invented the gnomon, geographical charts, and sun-dials. Anaxagoras, another disciple of the Ionian school, having been imprisoned for his opinions relating to Astronomy, employed his time in attempting to square the circle. Pythagoras, who lived about 570 B. C., was one of the earliest Greek geometers. He studied under Thales, traveled in Egypt and India, and on his return settled in Italy and founded one of the most celebrated schools of antiquity. He is supposed to have discovered the following principles : 1. Only three plane figures can fill up the space around a point; 2. The sum of the angles of a plane triangle equals two right angles; 3. The circle is greater than any other plane figure of equal perimeter; 4. The celebrated proposition of the square on the hypotenuse. It was said that in honor of the last discovery he sacrificed one hundred oxen. Plutarch, however, says but one ox; and Cicero doubts even that, as it was in opposition to his doctrines to offer bloody sacrifices, and suggests that they may have been images made of clay or flour. The next geometer of eminence was Hippocrates, who lived about 400 B.C. He was the first to effect the quadrature of a curvilinear space by finding a rectilinear one equal to it. He showed that a crescent formed by half the circumference of one circle and one-fourth the circumference of another is equal to an isosceles right triangle whose hypotenuse is the common chord of the arcs. He also showed that the duplication of the cube depends on finding two mean proportionals between two given lines, Plato, the “poetical philosopher,” delighted in the science and cultivated it with great success. He made mathematics the basis of his instruction, placing over the door of his school the inscription, “Let no one ignorant of geometry enter here.” He is reputed to have invented geometrical analysis; and the conic sections were first studied in his school. He gave a simple and elegant solution of the duplication of the cube; and the problem of the trisection of an angle, which occupied much attention in his school, led to many valuable discoveries in respect to the conic sections and other branches of geometry. Eudoxus, a contemporary of Plato, discovered the measure of the pyramid and cone. Euclid, the most celebrated geometer of antiquity, lived about 300 B. C. He studied in Athens under the disciples of Plato, and became connected with the celebrated school at Alexandria. It is related of him that when Ptolemy asked if there was not some easier way of learning geometry, he replied, “There is no royal road to geometry.” Euclid collected the propositions which had been discovered by his predecessors, constituting his famous “ Elements "-a work of such eminent excellence that it is still used in England in various editions, and regarded as the best textbook on elementary geometry. It consists of fifteen books, thirteen of which are known to have been written by Euclid, but the fourteenth and fifteenth are supposed to have been added about two centuries later by Hypsicles of Alexandria. The first four books treat of plane figures; the seventh, eighth, ninth, and tenth relate to arithmetic and incommensurables; the eleventh and twelfth contain the elements of solid geometry; the thirteenth treats of the five regular solids; and the fourteenth and fifteenth treat of regular solids. It is only the first six books and the eleventh and twelfth that are now much used in schools. After Euclid comes Archimedes, born at Syracuse about 287 B. C., who is distinguished by the discovery of the beautiful theorem that the sphere is two-thirds of the circumscribing cylinder, and that their surfaces bear the same relation. He is also distinguished for his works on conoids and spheroids, his discovery of the exact quadrature of the parabola, and his very ingenious approximation to that of the circle. In his work on conoids he compares the area of an ellipse with that of a circle, and in his work on the circle he shows that when the diameter is unity the circumference will be between 318 and 341. Archimedes was followed by Apollonius of Perga, born about 250 B. C. He studied in the Alexandrian school under the successors of Euclid, and so great was his genius that he acquired the name of the Great Geometer. His principal treatise was a work on conic sections in eight books. He is said to have given these curves their names, parabola, ellipse, and hyperbola. Other geometers followed, among whom the most illustrious were Pappus and Diophantus; but the Greek geometry, though it was afterward enriched by many new theorems, may be said to have reached its limits |