P l a n e G e o m e t r y
An Adventure in Language and Logic
Book I. Propositions 11 and 12
WE ARE NOW GOING TO CONSTRUCT right angles. Not only will we show our geometrical skill, but we satisfy a requirement of logic: We will prove that these "right angles" that we have defined actually exist. For a definition is required only to be understood. We may not assume that what we have defined exists.
Thus, to draw a straight line at right angles to the straight line AB from the given point C, place the point of the compass at C, and mark off equal distances at D and E. With D as center and radius DE, draw an arc. With E as center and with the same radius, draw an arc, and call their point of intersection F. Then CF will be perpendicular to AB at C.
Do you see why?
Here is the proof.
PROPOSITION 11. PROBLEM
Using reductio ad absurdam we can prove:
Only one line can be drawn perpendicular to a given straight line
For if we suppose that CF and CG are both perpendicular to AB, then angle ACF is equal to angle ACG (Postulate 4),
the smaller to the larger; which is absurd.
Dropping a perpendicular
Propostion 12 is the problem of "dropping" a perpendicular:
To a given straight line that may be made as long as we please,
Can you solve this problem? Can you draw a perpendicular line from C to AB?
To see the answer, pass your mouse over the colored area.
Place the point of the compass at C, and with a radius greater than the distance of C to AB, draw arcs at F and G. Now bisect the straight line FG at H. That is, with F as center and radius FG, draw arcs. With G as center and that same radius, draw arcs intersecting those at K and L. Then the straight line KCHL will be the perpendicular bisector of FG.
Here is the proof.
PROPOSITION 12. PROBLEM
This Proposition shows that it is possible to draw a line that satisfies the definition of a perpendicular line, and therefore what we have called a "perpendicular line" actually exists. (See the Commentary on the Definitions.)
A word about what we mean by a given line or a given point, because those expressions occur in many propositions. For something to be given in this science, we must in some way be able to recognize it or know it. "That one." A line will be given if we can reproduce it.
For a point to be given, we must be able to reproduce its position. "There." This is analogous to the idea of a given number, which is a number we can recognize by its name; if it is not rational, then we must be able to approximate it by a rational number as closely as we please.
The requirement of given points, lines, and numbers insures that we do not lose touch with what we actually know.
Please "turn" the page and do some Problems.
Continue on to the next proposition.
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