Table of Contents  Introduction  Home P l a n e G e o m e t r y An Adventure in Language and Logic based on CONSTRUCTION

1.  From a given point to draw a straight line equal to a given straight line.  
2.  Let A be the given point, and BC the given straight line;  
3.  we are required to draw from the point A a straight line equal  
4.  to BC.  
(The construction begins by drawing the equilateral triangle ABD; then extending DA, DB, to E and F; drawing a circle with radius BC, thus making BG equal to BC; then making DG, DL radii of a circle, so that on subtracting DB, DA respectively, AL will equal BG and therefore BC.) 

5.  From the point A to the point B, draw the straight line AB;  
(Postulate 1) 
6.  and on it draw the equilateral triangle ABD;  (I. 1)  
7.  and extend the straight lines DA, DB to E, F.  (Postulate 2)  
8.  With B as center and BC as radius, draw the circle CHG,  
9.  meeting DF at G.  (Postulate 3)  
10.  With D as center and DG as radius, draw the circle GKL,  
11.  meeting DE at L.  (Postulate 3)  
12.  Then AL will equal BC. 
13.  For, since B is the center of the circle CHG,  
14.  BC is equal to BG.  (Definition 16)  
15.  And since D is the center of circle GKL,  
16.  DL is equal to DG.  
17.  But in those lines, DA is equal to DB;  (Definition 9)  
18.  therefore the remainder AL is equal to the remainder BG.  
(Axiom 3)  
19.  And we have shown that BC is equal to BG;  
20.  therefore each of the straight lines AL, BC is equal to BG.  
21.  And things equal to the same thing are equal to one another.  
(Axiom 1)  
22.  Therefore AL is equal to BC.  
23.  Therefore from the given point A we have drawn a straight line  
24.  AL equal to the straight line BC.  
25.  Which is what we wanted to do. 
This proposition leads directly to the next one, where we will be required to cut off from the longer of two straight lines a length equal to the shorter line. The solution is obvious  but notice how we must rely on Proposition 2; line 6 below. In fact, the next proposition is the only one that requires Proposition 2. The "given" point is the endpoint of a line.
The student should now begin to see how each proposition depends on previous propositions. That is the nature of any logical theory. That is the axiomatic method.
1.  Given two unequal straight lines, to cut off from the longer line  
2.  a straight line equal to the shorter line.  
3.  Let AB and C be the two given straight lines, and let AB be  
4.  longer;  
5.  we are required to cut off from AB a straight line equal to C.  
6.  From the point A draw AD equal to C;  (Proposition 2) 
7.  and with A as center and radius AD, draw the circle DEF.  
(Postulate 3)  
8.  Then, since the point A is the center of circle DEF,  
9.  AE is equal to AD.  (Definition 16) 
10.  But C is also equal to AD.  (Construction) 
11.  Therefore each of the lines AE, C is equal to AD.  
12.  Therefore AE is equal to C.  (Axiom 1) 
13.  Therefore from AB the longer of two straight lines  
14.  we have cut off a straight line AE equal to C, the shorter line.  
15.  Which is what we wanted to do. 
Please "turn" the page and do some Problems.
or
Continue on to the next proposition.
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