as an axiom that the parallelogram will contain 5 × 3 = 15 square units. Hence, to find the areas of right-angled parallelograms, mul tiply the base by the altitude. EXPLANATION OF TERMS. 55. An Axiom is a self-evident truth, not only too simple to require, but too simple to admit of, demonstration. 56. A Proposition is something which is either pro posed to be done, or to be demonstrated, and is either a problem or a theorem. 57. A Problem is something proposed to be done. 58. A Theorem is something proposed to be demonstrated. 59. A Hypothesis is a supposition made with a view to draw from it some consequence which establishes the truth or falsehood of a proposition, or solves a problem. 60. A Lemma is something which is premised, or demonstrated, in order to render what follows more easy. 61. A Corollary is a consequent truth derived iminediately from some preceding truth or demonstration. 62. A Scholium is a remark or observation made upon something going before it. 63. A Postulate is a problem, the solution of which is self-evident. I. That a straight line can be drawn from any one poir t to any other point; Il. That a straight line can be produced to any distance, or terminated at any point; III. That the circumference of a circle can be described about any center, at any distance from that center. AXIOMS. 1. Things which are equal to the same thing are equal to each other. 2. When equals are added to equals the wholes are equal. 3. When equals are taken from equals the remainders are equal. 4. When equals are added to unequals the wholes are unequal. 5. When equals are taken from unequals the remainders are unequal. 6. Things which are double of the same thing, or equal things, are equal to each other. 7. Things which are halves of the same thing, or of equal things, are equal to each other. 8. The whole is greater than any of its parts. 9. Every whole is equal to all its parts taken together. 10. Things which coincide, or fill the same space, are identical, or mutually equal in all their parts. 11. All right angles are equal to one another. 12. A straight line is the shortest distance between two points. 18. Two straight lines cannot inclose a space. ABBREVIATIONS. The common algebraic signs are used in this work, and demonstrations are sometimes made through the medium of equations; and it is so necessary that the student in geometry should understand some of the more simple operations of algebra, that we assume that he is acquainted with the use of the signs. As the terms circle, angle, triangle, hypothesis, axiom, theorem, corollary, and definition, are constantly occurring in a course of geometry, we shall abbreviate them as shown in the following list: Equality and Equivalency are expressed by Thus: B is greater than A, is written B>A B<A L R. L Degrees, minutes, and seconds, are expressed by A triangle is expressed by The term Hypothesis is expressed by ト Perpendicular is expressed by The difference of two quantities, when it is not known which is the greater, is expressed by the symbol ~ Thus, the difference between A and B is written AN B. BOOK I. OF STRAIGHT LINES, ANGLES, AND POLYGONS. THEOREM I. When one straight line meets another, not at its extremity, the two angles thus formed are two right angles, or they are together equal to two right angles. Let AB meet CD, and if AB is perpendicular to CD, it does not incline to either extremity of CD. In that case, the angle ABD is equal to the angle ABC, and is C a right angle, by Definition 15. E B D But if these angles are unequal, we are to show that their sum is equal to two right angles. Conceive the line BE to be drawn from the point B, so as not to incline toward either extremity of CD; then, by Def. 15, the angles CBE and EBD are right angles; but the angles CBA and ABD make the same sum, or fill the same angular space, as the two angles CBE and EBD, and are, consequently, equal to two right angles. Hence the theorem; when one straight line meets another, not at its extremity, the sum of the two angles is equal to two right angles. Cor. Hence, the two angles ABC and ABD are supplementary to each other, (Def. 21). THEOREM II. From any point in a straight line, not at its extremity, the sum of all the angles that can be formed on the same side of the line is equal to two right angles. Let CD be any line, and B any point in it. H We are to show that the sum of all the angles which can be formed at B, on one d side of CD, will be equal to two right angles By Th. 1, any two supplementary angles, as ABD, ABC, are together equal to two right angles. And since the angular space about the point B is neither increased nor diminished by the number of lines drawn from that point, the sum of all the angles DBA, ABE, EBH, HBC, fills the same spaces as any two angles HBD, HBC. Hence the theorem; from any point in a line, the sum of all the angles that can be formed on the same side of the line is equal to two right angles. Cor. 1. And, as the sum of all the angles that can be formed on the other side of the line, CD, is also equal to two right angles; therefore, all the angles that can be formed quite round a point, B, by any number of lines, are together equal to four right angles. Cor. 2. Hence, also, the whole circumference of a circle, being the sum of the measures of all the angles that can be made about the center F, (Def. 53), is the measure of four right angles; consequently, a semicircumference, is the mea F sure of two right angles; and a quadrant, or 90°, is the measure of one right angle. THEOREM III. If one straight line meets two other straight lines at a common point, forming two angles, which together are equal to two right angles the two straight lines are one and the same line. Let the line AB meet the lines BD and BE at the common point B, making the sum of the two angles ABD, ABE, equal to two right angles; we are to prove that DB and BE are one straight line. т E B |