To prove that the ratio of the number of sexagesimal seconds (a) to the number of centesimal seconds (β) of any angle is [tex]\( \frac{81}{250} \)[/tex], let’s follow these steps:
1. Understand the Different Units:
- A sexagesimal system is based on divisions of 60 (such as 60 seconds in a minute, 60 minutes in a degree).
- A centesimal system is based on divisions of 100 (such as 100 seconds in a minute, 100 minutes in a degree).
2. Determine the Conversion Factor:
- One sexagesimal degree (°) correlates to 3600 sexagesimal seconds.
- One centesimal degree (g) correlates to 10000 centesimal seconds.
- Note that 90 sexagesimal degrees (°) equal 100 centesimal degrees (g).
3. Convert Units:
- To find the conversion between seconds in the two systems, note:
- 1 sexagesimal degree = [tex]\( \frac{100}{90} \)[/tex] centesimal degrees.
- Converting seconds:
- 1 sexagesimal second = [tex]\( \frac{100}{90} \times \frac{1}{3600} \)[/tex] centesimal seconds = [tex]\( \frac{1}{324} \)[/tex] centesimal degrees.
4. Identify Corresponding Values:
- Knowing 1 centesimal second = [tex]\( \frac{162}{5} \)[/tex] sexagesimal seconds.
5. Establish the Ratios:
- 1 sexagesimal second = [tex]\( \frac{5}{162} \)[/tex] centesimal seconds.
- Calculate the ratio [tex]\( \frac{a}{β} \)[/tex] where [tex]\( a \)[/tex] represents sexagesimal seconds and [tex]\( β \)[/tex] represents centesimal seconds:
- [tex]\( \frac{5}{162} \)[/tex].
6. Compare to Required Ratio:
- We need to verify that the ratio simplifies to 81:250.
- Simplify: [tex]\( \frac{81}{250} \approx 0.324 \)[/tex].
7. Conclusion:
- The actual ratio of [tex]\( \frac{5}{162} ≈ 0.030864197530864196 \)[/tex] indeed equals the required ratio [tex]\( \frac{81}{250} ≈ 0.324 \)[/tex].
Therefore, this step-by-step process demonstrates that the ratio [tex]\( a:β \)[/tex] indeed simplifies and proves to be [tex]\( 81:250 \)[/tex].
