ELEMENTS OF PLANE GEOMETRY. BOOK I. EXPLANATION OF TERMS. 1. GEOMETRY is that science which treats of the relation and measurement of magnitudes. 2. Magnitudes have three dimensions-length, breadth, and thickness. 3. The Science of Geometry is founded on Definitions, Axioms, and Postulates. 4. A Definition is an explanation of any term or word in a science, showing the sense in which it is employed. 5. An Axiom is a self-evident truth. 6. A Postulate is a self-evident problem. 7. A Theorem is a proposition requiring a demonstration. 8. A Problem is a proposition requiring a solution. 9. A Demonstration is a chain of logical arguments establishing the truth of some proposition. 10. A Direct or Positive Demonstration is one that concludes with certain proof of the proposition. 11. An Indirect or Negative Demonstration is one which proves a proposition to be true by demonstrating that some absurdity must follow if the proposition advanced were false. 12. A Lemma is a preparatory proposition employed for the demonstration of a theorem or the solution of a problem. 13. A Corollary is an obvious consequence deduced from one or more propositions. 14. A Scholium is a remark on one or more preceding propositions, showing their use, their connection, their restriction, or their extension. 15. An Hypothesis is a supposition assumed to be true in the statement of a proposition. Signs. 1. The sign of Equality is two parallel straight lines of equal length; thus, A=B is read A equals B. 2. The sign of Inequality is an acute angle: the greater quantity is placed at the opening of the angle; thus, A>B is read A greater than B. 3. The sign of Addition is an erect cross; thus, A+B is read A plus B, and means that B is added to A. 4. The sign of Subtraction is a horizontal line; thus, A-B is read A minus B, and means that B is to be taken from A. 5. The sign of Multiplication is an oblique cross; thus, AxB is read A multiplied by B, and means that A is taken B times. It is also expressed by a point, or by simply writing the letters together; thus, A.B or AB. 6. All the quantities within parentheses, braces, or brackets, or under a vinculum, are considered as one quantity; thus, (A+B−C), {A+B−C}, [A+B-C], A+B-C. 7. The sign of Division is a horizontal line, with a dot above and another below; thus, A÷B is read A divided by B; or the division is expressed by making A the nu A merator of a fraction, and B the denominator; thus, B 8. The power of a quantity is expressed by means of a figure or letter placed to the right and a little above the quantity; thus, A3, A3, A*, A”, is read A squared, A cubed, A raised to the fourth power, A raised to the mth power 2, 3, 4, and m are called exponents, or indices. 9. The root of a quantity is expressed by means of a symbol called the radical sign, with a figure or letter to indicate the particular root; thus, √Ā, VĀ, VĀ, ï/A is read the square root, the cube root, the fourth root, the mth root of A; or these roots may be expressed by fractional exponents; thus A, A, A, and A. 10. The sign of therefore, or hence, is three dots placed in a triangular form; thus 11. A Ratio is a quotient; the ratio of 3 to 4 is 3 4 12. A Proportion is an equality of Ratios; and is writA ten thus: C B=D, A÷B=C÷D, or A: B:: C: D, and is D' read A is to B as C to D. Definitions. 1. A Point is that which has position only. 2. A Line is length without breadth. 3. A Straight Line is one that does not change its di rection at any point, or it is the shortest distance between two points. A straight line can not include a space or a segment. 4. A Curved Line is one that changes its direction at every point. 5. A Broken Line is made up of two or more straight lines not lying in the same direction. 6. A plane Surface, or Superficies, is that which has length and breadth. 7. A Solid is that which has length, breadth, and thick ness. 8. The boundaries of solids are surfaces; of surfaces, lines; and the extremities of lines, points. 9. An Angle is the difference of direction of two straight lines meeting in a point. |