BOOK I RECTILINEAR FIGURES A SYSTEM OF PROPOSITIONS 125. In the preceding pages many simple geometric principles were presented and illustrated. There remain, however, many other facts of geometry which are not so apparent, or which, even though they seem quite easy of comprehension, are not easily proved. The aim of demonstrative geometry is to point out methods of discovery and of proof, as well as a logical arrangement, of those geometric principles which are most important in the development of the subject, and which have the widest application in other fields of investigation. Such principles are called propositions, and, except for those which are taken for granted at the outset, are established by means of formal demonstrations. 126. A complete demonstration demands a reason for every statement which is included in it. It is obvious that such a justification of the particular steps of a proof is greatly facilitated by arranging the propositions in a definite progressive order. 127. The most fundamental propositions, which are first considered, are those which are most often in demand as the work progresses; much as the multiplication table is needed in the study of arithmetic. Other propositions are shown to depend more or less directly upon these, and thus gradually arises a system of propositions, each of which depends upon some or all of those preceding, in the order of consideration. For this reason the study of geometry has been compared to the building of a house or other structure; each stage of the work is supported by what has already been completed. ARRANGEMENT OF A DEMONSTRATION 128. The formal demonstration of a theorem in geometry involves three steps: 1. The statement of what is given (the hypothesis). 2. The statement of what is to be proved (the conclusion). 3. The proof, consisting of several steps, each based upon the hypothesis, or upon the authority of definitions, axioms, preliminary theorems, or preceding propositions. These three steps are indicated by the words Given, To prove, Proof. 129. In order to avoid a mechanical repetition of the text, and to gain the ability to reason independently in the field of geometry, it is advisable for the student to bear in mind the following RULES 1. Read the proposition carefully, noting the meaning of each term employed. 2. Draw a figure which represents the conditions demanded by the proposition, and no other special conditions. 3. State the hypothesis and the conclusion as applied to that particular figure. 4. Recall the geometric processes which generally lead to this conclusion, and examine the hypothesis and its immediate consequences in order to discover whether such processes may be applied in the particular instance. 5. Try to formulate a proof before examining the one given in the text. 6. Write out the proof, arranging the steps in order and numbering the main divisions. The final step should agree with the conclusion. After each step give the authority which supports it. Proper authorities are the hypothesis and any definitions, axioms, preliminary theorems, and propositions previously established. Use freely such symbols and abbreviations as are suggested below. 7. At a later period in the development of the work it is often sufficient, and even preferable, to write merely a brief summary of the demonstration. 8. As a review exercise, practice reciting the proof from a mental figure only. a. s. a., having a side and the two adjoining angles of one equal respectively to a side and the two adjoining angles of the other. s. a. s., having two sides and the included angle of one equal respectively to two sides and the included angle of the other. s. s. s., having three sides of one equal respectively to three sides of the other. rt. A, h. 1., being right triangles and having the hypotenuse and a leg of one equal respectively to the hypotenuse and a leg of the other. rt. A, h. a., being right triangles and having the hypotenuse and an adjoining angle of one equal respectively to the hypotenuse and an adjoining angle of the other. AXIOMS. PRELIMINARY PROPOSITIONS 130. All demonstration finally depends on taking some truths for granted. Such truths are sometimes called axioms. They are assumed to be so clear in themselves that every one will accept them without proof. Geometry is no exception to this general rule. It is, however, essential in elementary geometry that the facts taken for granted and without demonstration should be of a simple nature, so simple, in fact, as to be obvious if merely plainly stated. The geometrical facts which are assumed in Book I will now be tabulated under the heading Preliminary Assumptions and Propositions, and it will be observed that we are thus giving a recapitulation of some of the main results of the informal discussion of the Preliminary Course. 131. Preliminary Assumptions and Propositions. 1. Through a given point an indefinite number of straight lines may be drawn (§ 11). 2. Two straight lines can intersect in but one point (§ 11). 3. One and only one straight line can be drawn through two given points (§ 11). 4. Two straight lines that have two points in common coincide throughout and form but one line (§ 11). 5. Two straight lines do not inclose a space (§ 11). 6. Two line-segments whose extremities can be made to coincide must coincide throughout (§ 13). 7. Two line-segments a and b must be in one of three relations to each other, namely, a>b, a = b, or a<b (§ 13). 8. Line-segments may be added together. Of two unequal line-segments the smaller may be subtracted from the larger. Line-segments may be multiplied by a given number (§16). 9. All round angles are equal (§ 22). 10. All straight angles are equal (§ 24). 11. All right angles are equal (§ 28). 12. Two angles A and B must be in one of three relations to each other: ∠A > ∠B, ∠A = ∠B, or ∠ A <∠B (§ 32). 13. At a given point in a given line only one perpendicular can be drawn to that line (in the same plane) (§ 41). 14. The complements of the same angle, or of equal angles, are equal (§ 41). 15. The supplements of the same angle, or of equal angles, are equal (§ 41). 16. Vertical angles are equal (§ 41). 17. If two adjacent angles have their exterior sides in a straight line, they are supplementary (§ 41). 18. If two adjacent angles are supplementary, their exterior sides lie in a straight line (§ 41). 19. Radii of the same circle are equal (§ 60). 20. At a given point in a given line a line may be drawn making with the given line an angle equal to a given angle (§ 89). 21. Figures congruent to the same figure are congruent to each other (§ 113). 132. General Axioms. Further fundamental truths involving magnitude are as follows: 1. Magnitudes which are equal to the same magnitude, or to equal magnitudes, are equal to each other. In other words, A magnitude may be substituted for its equal. 2. If equals are added to equals, the sums are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. If equals are multiplied by equals, the products are equal. 5. If equals are divided by equals, the quotients are equal. The divisor must not be zero. 6. Like powers or like positive roots of equals are equal. parts. 8. The whole of any magnitude is greater than any part of it. |