The Elements of Euclid for the Use of Schools and Colleges; Comprising the First Six Books and Portions of the Eleventh and Twelfth Books

by Euclid

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1882 ...and the diameter of the circle described about the triangle. Let ABC be a triangle, and let AD be the perpendicular from the angle A to the base the rectangle BA, AC shall be equal to the rectangle contained by AD and the diameter of the circle described about the triangle. Describe the circle ACB about the triangle; IV. 5. draw the diameter AE, and join EC. Then, because the right angle BDA is equal to the angle ECA in a semicircle; III. 81. and the angle ABD is equal to the angle AEC, for they are in the same segment of the circle; PH. 21. therefore the triangle ABD ia equiangular to the triangle AEC. Therefore BA is to AD as EA is to AC; i. therefore the rectangle BA, AC is equal to the rectangle EA, AD VI. 16. "Wherefore, if from the vertical angle &c Q.e.d. PROPOSITION D. THEOREM, The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles contained by its opposite sides..Let ABCD be any quadrilateral figure inscribed in a circle, and join AC, the rectangle contained by A C, BD shall be equal to the two rectangles contained by AB, CD and by AD, BC. Make the angle ABE equal to the angle DBC; I. 23. add to each of these equals the angle EBD, then the angle ABD is equal to the angle EBC. Axiom 2. And the angle BDA is equal to the angle B CE, for they are in the same segment of the circle; 111.21. therefore the triangle A BD is equiangular to the triangle EBG. Therefore A D is to DB as EC is to CB; therefore the rectangle AD, CB is equal to the rectangle DB, EC. VI. 16. Again, because the angle ABE is equal to the angle DBC, Construction. and the angle BAE is equal to the angle BDC, for they are in the same segment of the circle; III. 21. therefore the triangle ABE is equiangu...

108 pages, Paperback

First published January 1, 1957

About the author

Euclid (Ancient Greek: Εὐκλείδης Eukleidēs -- "Good Glory", ca. 365-275 BC) also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I (323–283 BC). His Stoicheia (Elements) is a 13-volume exploration all corners of mathematics, based on the works of, inter alia, Aristotle, Eudoxus of Cnidus, Plato, Pythagoras. It is one of the most influential works in the history of mathematics, presenting the mathematical theorems and problems with great clarity, and showing their solutions concisely and logically. Thus, it came to serve as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor. He is sometimes credited with one original theory, a method of exhaustion through which the area of a circle and volume of a sphere can be calculated, but he left a much greater mark as a teacher.