Atomism and Infinite Divisibility

Chapter 6

Notes and References

  1. William H. Beyer, ed., CRC Standard Mathematical Tables, 25th Ed., (Palm Beach, FL: CRC Press, 1973), p. 3. text
  2. The "hypothesized" development presented above follows the logical development of number systems rather than the historical development. The historical record suggests that integers were followed by fractions, that a symbol for zero wasn't devised until positional notation was invented, and negative numbers first showed up in Chinese matrix algebra in the third century. Dirk J. Struik, A Concise History of Mathematics, (New York: Dover Publications, 1967). text
  3. David J. Furley, Two Studies in the Greek Atomists, (Princeton: Princeton University Press, 1967), p. 5. text
  4. Aspect Ratio: ratio of width to height of the picture image projected on the screen or printed on the film, the height being taken as unity. The long-established film aspect ratio, still retained for narrow-gage film, is 4:3 (1.33:1). The Focal Encyclopedia of Film & Television Technique, (London: Focal press, 1969), p. 50. text
  5. Cinemascope: Trade name of the most widely used method of anamorphic wide-screen presentation; camera lenses producing images on 35mm film with a 2:1 lateral compression are viewed with compensating horizontal expansion on projection. The Focal Encyclopedia of Film & Television Technique, (London: Focal press, 1969), p. 132. text
  6. To do so would have required theaters to invest in expensive additional projectors, and the expense would have inhibited the spread and use of the technology. text
  7. Michael Z. Wysotsky, Wide Screen Cinema and Stereophonic Sound, translated by Wing Commander A. E. C. York, (New York: Hastings House, 1971). text
  8. While there are no "pixels" on recorded film, we can measure an effective pixel aspect ratio by comparing the ratio of the sides of the image of a true square (as measured by continuous metrics) when it is recorded on film. But since 'effective pixel aspect ratio' is a cumbersome phrase, and the concept will need to be used frequently, I shall coin 'perspective ratio' to use in its stead. I shall also extend the concept to cover any ratio involving two different scales of measurement. The need for this extension will be apparent, and its use will be immediately appropriate, for discussions involving lines in the atomic plane. While an aspect ratio is the ratio of two distances, perspective ratio, as defined here, is the ratio of two scales for measuring distance. text
  9. These lenses cost only a tiny fraction of what an entire projector would cost, enabling the spread and use of the technology. text
  10. Aspect ratio for characters is very similar to aspect ratio for pixels. The difference is that each character position is treated as a low resolution pixel. An 80x25 character screen using my 13 inch monitor (10x7.5) gives a horizontal "pixel" size of 10"/80. The vertical "pixel" size is 7.5"/25. The aspect ratio is (10/80)/(7.5/25)= .42 or 5:12. text
  11. Aspect ratios for printed characters is also very similar to aspect ratio for pixels. Printed page formats can be measured in terms of lines per inch and characters per inch. The width of a character is simply the reciprocal of these parameters. Ten characters per inch gives a horizontal "pixel" size of 1/10 inches. Six lines per inch gives a vertical "pixel" size of 1/6 inches. The aspect ratio is (1/10)/(1/6) = .6 or 3:5. text
  12. For this demonstration we note that as the size of each point gets smaller the number of points per inch gets larger. We must transform the equation into an expression using inches rather than points. For this purpose we may let the number of points per inch be K. Then B and H will be the sizes in inches of the sides, and K·B and K·H will be the corresponding number of points. As the number of points per inch, K, gets very large, the contribution of one point to the length gets very small, and any error introduced by dropping the "INT" portion of the formula will become negligible. The formula itself,

    INT((I-1)·((H-1)/(B-1))+1)

    reduces to:

    ((H-1)/(B-1))·(I-1) + 1.

    Transforming the formula so that the result includes expressions for length in inches requires substituting the corresponding number of points expressed in inches times the number of points per inch. To obtain an area result which is in terms of square inches, the point area formula must be divided by the number of points in a square inch, namely K2. The new area formula becomes:

    [((K·H-1)/(K·B-1))·(I-1) + (1)]/K2

    Performing the summation and some algebra yields:

    [(K·H-1)/(K·B-1))·K·B·(K·B-1)/2 + K·B]/K2.

    [(K·H-1))·K·B/2 + K·B]/K2.

    (H-1/K))·B/2 + B/K.

    Taking the limit of this as K approaches yields: ½B·H. text