The Infinite in the Finite

By Alistair Macintosh Wilson | Go to book overview
Save to active project

12
PROPORTION

THE GEOMETRICAL SOLUTION OF AHA PROBLEMS

Although the Greeks far surpassed their teachers, the Egyptians, in the science of geometry, they never succeeded in equalling the ancient civilizations in the treatment of aha problems. The Greeks tried to solve aha problems by using their powerful methods of geometrical reasoning. But geometrical methods are not really well suited for these problems, and they were not successful. In this chapter we shall describe what they did achieve, since this will form the basis of our later discussions of geometry.

The methods of solving aha problems described in the first part of this book passed via the Greeks to the Hindu mathematicians of India, and from them to the Islamic scholars at Baghdad, where they were finally drawn together into a new science. We shall describe how this happened in Chapters 17 and 18.

The Greek geometrical treatment of aha problems is described in the fifth and sixth books of Euclid Elements. Euclid began in a natural way by describing the theory of ratio and proportion.


THE THEORY OF PROPORTION

At the time of Euclid, the Greeks were familiar with three different kinds of proportion. The first they called arithmetical proportion. A set of line segments are in arithmetical proportion when their lengths increase by an equal amount as we go from one to another (Fig. 12.la).

The fifth book of the Elements is concerned with a different kind of proportion, which we now know as geometrical proportion. This is the kind of proportion we meet with in similar triangles. Consider Fig. 12.1b, in which lines BD and CE are parallel. We see that triangles ADB and AEC are similar, so that
AB/AD = AC/AE.

-252-

Notes for this page

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
Loading One moment ...
Project items
Notes
Cite this page

Cited page

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited page

Bookmark this page
The Infinite in the Finite
Settings

Settings

Typeface
Text size Smaller Larger
Search within

Search within this book

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

While we understand printed pages are helpful to our users, this limitation is necessary to help protect our publishers' copyrighted material and prevent its unlawful distribution. We are sorry for any inconvenience.
Full screen
/ 532

matching results for page

Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

Cited passage

Welcome to the new Questia Reader

The Questia Reader has been updated to provide you with an even better online reading experience.  It is now 100% Responsive, which means you can read our books and articles on any sized device you wish.  All of your favorite tools like notes, highlights, and citations are still here, but the way you select text has been updated to be easier to use, especially on touchscreen devices.  Here's how:

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select.

OK, got it!

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.

Are you sure you want to delete this highlight?