GEOMETRY 



2445 



GEOMETRY 



ing amounts to; the earliest photographs were 

 hazy and uncertain of line, because the men 

 who made the first cameras did not understand 

 well enough the projection of the sitter's face 

 on the sensitive plate. There are also many 

 other very important and practical applications 

 of projective geometry. Of course it is not 

 necessary for the schoolboy who toils over the 

 map of Europe, or for the girl who delightedly 

 "snaps" baby sister's picture with her kodak 

 to know the underlying principles or even to 

 know that there is such a thing as projective . 

 geometry; but the proper fulfilling of their 

 tasks is possible just because somewhere, some 

 time, wise men worked out those principles. 



Terms Used. The student beginning ele- 

 mentary geometry finds himself brought face 

 to face with many new and strange terms. In 

 algebra, as in arithmetic, the things he was 

 called on to solve were called problems. Here 

 he hears talk of propositions and theorems, 

 corollaries, axioms and hypotheses, or sees at 

 the bottom of a solved "problem" the myste- 

 rious letters Q. E. D. or Q. E. F. A few defi- 

 nitions, however, will make all of these plain: 



A proposition is the statement of some truth 

 that is to be proved or of some operation 

 that is to be performed. A proposition of the 

 former type, which states some truth that may 

 be logically demonstrated, is a theorem; one 

 of the latter type, which proposes a question 

 for solution, is a problem. Thus, "The sum of 

 the three angles of a triangle is equal to two 

 right angles" and "Construct a square equiva- 

 lent to the sum of two given squares" are 

 both propositions, but the former is a theorem, 

 the latter a problem. 



A corollary is a truth easily deduced from 

 one or more propositions already proved. 

 Thus, if it is true that the sum of the three 

 angles of a triangle is equal to two right 

 angles, it follows as a self-evident corollary 

 that a triangle can have but one right angle 

 or one obtuse angle. 



A demonstration is the proof of a theorem; 

 a solution is the process of solving a problem. 



The hypothesis is the "if" part of a propo- 

 sition the part which states the thing or 

 things taken for granted; the conclusion sets 

 forth the things which are to be proved true. 

 In the proposition "// two sides oj a quadri- 

 lateral are equal and parallel, the figure is a 

 parallelogram," the part in italics is the hy- 

 pothesis, the rest the conclusion. 



Q. E. D. is the abbreviation for the Latin 

 words Quod erat demonstrandum, meaning 



"which was to be proved," and is placed at the 

 close of the demonstration of every satisfac- 

 torily proved theorem. 



Q. E. F. means Quod erat faciendum, or 

 "which was to be done," and is placed at the 

 close of the solution of any geometric problem. 



One of the terms used very frequently in 

 geometry is axiom, which means a self-evident 

 truth. Geometry seeks to prove most things 

 about the properties -and relations of lines, 

 surfaces and solids; it takes a few for granted, 

 calling them true in the very nature of things. 

 Who, for instance, would think of questioning 

 the statement that "Things equal to the same 

 thing are equal to each other"? That is an 

 axiom, and there are many others, of which 

 the following are perhaps most important: 



1. If equals be added to equals, the results will 

 be equal. 



2. If equals be taken from equals, the re- 

 mainders will be equal. 



3. The doubles of equals are equal. 



4. The halves of equals are equal. 



5. The whole is greater than any of its parts. 



6. The whole is equal to the sum of all its parts. 



7. Between two points only one straight line 

 can be drawn. 



8. A straight line is the shortest distance be- 

 tween two points. 



Then there are postulates not quite so self- 

 evident as axioms, but very clearly possible; 

 and these, like the axioms, were set forth by 

 Euclid and have helped to form the basis of 

 geometric reasoning ever since. Some of the 

 more important ones are as follows: 



1. Any magnitute can be bisected. 



2. A straight line can be continued indefinitely 

 in either direction. 



3. A circle can be drawn with any radius, and 

 from any point as center. 



4. A straight line can be drawn from any point 

 to any extent in any direction. 



How Geometry Proves Things. Given a 

 theorem to be proved, geometry goes 'about it 

 in a very definite way. In the first place, 

 nothing must be assumed. There must be 

 proper authority for every step in the process; 

 and proper authority lies only in an axiom, a 

 postulate, a definition, or a proposition already 

 proved. But all theorems are not proved in 

 exactly the same way; there is a direct and an 

 indirect method. The former consists either 

 in superimposing one figure upon another, or 

 in starting out with some unquestionable fact 

 and proceeding step by step by means of ax- 

 ioms, postulates or theorems already proved 

 to a conclusion which must be correct because 

 every step in it has been taken with full au- 

 thority. The following is a direct proof: 



