By Parshin Shafarevich
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Numbers which cannot be expressed as the ratio of integers are called irrational numbers. These numbers have nonrepeating decimal equivalents. , have a difficult time dissociating the concept of number from the symbol for number. However, in ancient Greece no symbols for numbers, as we know them, existed. The symbols that had been used previously by the Babylonians and Egyptians for the purpose of surveying or keeping records had long since been forgotten. Instead of representing numbers by symbols, Greek philosophers conceived of number as being the ratio of lengths.
Illustration by Tom Prentiss. ) Proportion in Architecture 31 It can be verified that the lengths of Series 1 are the arithmetic means of the lengths from Series 2. 4, the lengths of Series 2 supply the harmonic means between adjacent pairs of lengths from Series 1. 618 for the red and blue series). 14). 21) are each Pell's series with ratios closely approximating V 2 . 20) also possess the following additive properties: 1. , 1 + 2 = 3. 2. , 2 + 3 = 5. 3. , 7 - 3 = 2 - 2 . Using these additive properties and beginning with the two lengths 1, V 2 , all other lengths of Series 1 and 2 can be constructed with compass and straightedge.
1 Ratio 1:1 4:3 3:2 16:9 2:1 9:4 8:3 3:1 4:1 Musical interval Unison Fourth (diatesseronl Fifth (diapente) Octave (diapasonl Eleventh (fourth above octave) Twelfth (fifth above octave) Fifteenth (next octave) All were consonant (or pleasant sounding) except for 9:4 and 16:9, which were compound ratios composed of successive fifths and fourths. To understand how these ratios are all related by a common system, we must first consider the series upon which all systems of proportion are built, the geometric series.
Algebraic Geometry 3 Curves Jaobians by Parshin Shafarevich