[This proposition, which is deduced directly from props. 12, 13, is not given by Euclid, but it seems to deserve a place in the second Book, not only on account of its own interest, but because also it is the key to a large class of problems, which involve by the enunciations the bisection of some line. Some examples of such problems will now be given.] 1. If straight lines be drawn from each angle of a triangle to bisect the opposite sides, four times the sum of the squares on these lines is equal to three times the sum of the squares on the sides. 2. If AB, one of the equal sides of an isosceles triangle ABC, be produced beyond the base to D, so that BD = AB, show that the square on CD is equal to the square on AB together with twice the square on BC. 3. From any point P lines are drawn to the angles of a rectangle ABCD; show that the squares on PA, PC are together equal to the squares on PB, PD. 4. A square BDEC is described on the hypotenuse of a right-angled triangle ABC ; show that the squares on DA, AC are together equal to the squares on EA, AB. 5. The squares on the diagonals of a parallelogram are together equal to the squares on the four sides. 6. If two points be taken in the diameter of a circle equally distant from the centre, the sum of the squares of two lines drawn from these points to any point in the circumference will be constant. 7. The hypotenuse of a right-angled triangle ABC is trisected at D, E, and CD, CE are joined ; show that the sum of the squares on the sides of the triangle CDE is equal to two-thirds of the square on AB. 8. In any quadrilateral, the squares on the diagonals are together double of the sum of the squares on the two lines joining the bisections of the opposite sides. G PROPOSITION XIV. PROBLEM. To describe a square that shall be equal to a given rectilineal figure. Let A be the given rectil. figure. It is required to describe a square that shall be equal to A. EF = Constr. Describe the rectangular O BCDE = the rectil. figure A. (i. 45) Then if BE = ED, it is a sq., and what was required is done. But if BE be not = ED, produce BE to F, and make ED, bisect BF in G, (i. 10) from centre G, at distance GB or GF, describe semicircle BHF, and produce DE to meet the circumference in H. The sq. on EH shall equal the given figure A. Dem. :: BF is divided into two equal parts at G and into two unequal parts at E, .-, rect. BE, EF with sq. on EG = sq. on GF; (ii. 5) but GF=GH, and .. sq. on GF = sq. on GH, and .. = sqs. on GE, EH ; (i. 47) :, rect. BE, EF, with sq. on EG = sqs. on EG, EH ; take away from each the sq. on EG ; and rect. BE, EF = sq. on EH. But rect. BE, EF is BD : EF = ED, (constr.) .: BD= sq. on EH ; and BD = A; (constr.) :: sq. on EH rectil. fig. A. Therefore, a square has been described as required. Q. E. F. THE PUPIL-TEACHER'S COURSE OF MATHEMATICS. PART 11.-ALGEBRA. THE SECOND PART of 'The Pupil-Teacher's Course of Mathematics' will contain all the rules of Algebra which pupil-teachers are required to study-viz. in the fourth year, the first four rules, greatest common measure, least common multiple, fractions, square root, and simple equations involving one unknown quantity, with easy problems ; and in the fifth year, the same, with cube root, simple equations with two unknown quantities, quadratics involving one unknown quantity, and easy problems that lead to them. In most works on Algebra the student is too soon made to face the difficulties of Symbolical Algebra ; it is far better to defer this until Arithmetical Algebra has been mastered as far as quadratic equations. Pupil-teachers are not required to determine what the symbols denote in order that the rules of Arithmetical Algebra may hold in all cases. Accordingly in the present treatise the symbols will denote the numbers and operations of arithmetic only, which will greatly simplify our proofs of the rules. The Author is convinced, by long experience, that this is by far the best way of teaching this subject. The theory of indices and sums will thus be left for the student's more advanced reading hereafter. Many examples and hints will be given, and special regard will be paid to the solution of equations. The very best way to become an accomplished and expert analyst is to solve thousands of equations ; until at last the proper method of attacking a difficult equation is discovered almost at a glance. As in the case of the First Part, it is hoped that all beginners may find this little book useful. NATIONAL SOCIETY'S DEPOSITORY, SANCTUARY, WESTMINSTER. THE TEACHER'S COMPLETE MANUAL OF THE SCIENCE AND ART OF TEACHING. INCLUDING TWO SERIES OF INDEPENDENT MANUALS. The First Series, which relates to the SCIENCE of Teaching, consists of 1. THE CULTIVATION OF THE SENSES. Price 8d. 2. THE CULTIVATION OF THE MEMORY. Price 6d. 3. ON THE USE OF WORDS. Price 8d. 4. ON DISCIPLINE. Price 8d. 5. ON CLASS TEACHING. Price 6d. The Second Series, which relates to the ART of Teaching, consists of 1. HOW TO TEACH GRAMMAR. Price 8d. 2. ARITHMETIC AND HOW TO TEACH IT. Price 8d. 3. HOW TO TEACH GEOGRAPHY. Price 6d. 4. HOW TO TEACH HISTORY. Price 8d. 5. HOW TO TEACH READING AND WRITING. Price 8d. 6. SCHOOL FURNITURE AND APPARATUS, AND THEIR USES. Price 6d. THE PUPIL-TEACHER'S COURSE OF MATHEMATICS. Part 1.-EUCLID, Books I. and II. WITH NOTES, EXAMPLES, AND EXPLANATIONS. By a late Fellow and Senior Mathematical Lecturer ; Examiner for the Oxford and Cambridge Board, the Cambridge Syndicate, &c. Price Is. 6d. NATIONAL SOCIETY'S DEPOSITORY, WESTMINSTER; And through all Booksellers. |