2. A coaxal system having real points of intersection has imaginary
limiting points;
3. A coaxal system having imaginary points of intersection has real
limiting points;
4. Every circle through the limiting points cuts all circles of the
system orthogonally;
5. The limiting points are inverse points for every circle of the
system.
The theory of centres of similitude and coaxal circles affords elegant
demonstrations of the famous problem: To describe a circle to touch
three given circles. This problem, also termed the "Apollonian problem,"
was demonstrated with the aid of conic sections by Apollonius in his
book on _Contacts_ or _Tangencies_; geometrical solutions involving the
conic sections were also given by Adrianus Romanus, Vieta, Newton and
others. The earliest analytical solution appears to have been given by
the princess Elizabeth, a pupil of Descartes and daughter of Frederick
V. John Casey, professor of mathematics at the Catholic university of
Dublin, has given elementary demonstrations founded on the theory of
similitude and coaxal circles which are reproduced in his _Sequel to
Euclid_; an analytical solution by Gergonne is given in Salmon's _Conic
Sections_. Here we may notice that there are eight circles which solve
the problem.
_Mensuration of the Circle._
All exact relations pertaining to the mensuration of the circle involve
the ratio of the circumference to the diameter. This ratio, invariably
denoted by [pi], is constant for all circles, but it does not admit of
exact arithmetical expression, being of the nature of an incommensurable
number. Very early in the history of geometry it was known that the
circumference and area of a circle of radius r could be expressed in the
forms 2[pi]r and [pi]r squared. The exact geometrical evaluation of the second
quantity, viz. [pi]r squared, which, in reality, is equivalent to determining a
square equal in area to a circle, engaged the attention of
mathematicians for many centuries. The history of these attempts,
together with modern contributions to our knowledge of the value and
nature of the number [pi], is given below (_Squaring of the Circle_).
The following table gives the values of this constant and several
expiessions involving it:--
+--------------+-----------+-----------+
| | Number. | Logarithm.|
+--------------+-----------+-----------+
| [pi] | 3.1415927 | 0.4971499 |
| 2 [pi] | 6.2831858 | 0.7981799 |
| 4 [pi] |12.5663706 | 1.0992099 |
| (1/2) [pi] | 1.5707963 | 0.1961199 |
| (1/3) [pi] | 1.0471976 | 0.0200286 |
| (1/4) [pi] | 0.7853982 | 1.8950899 |
| (1/6) [pi] | 0.5235988 | 1.7189986 |
| (1/8) [pi] | 0.3926991 | 1.5940599 |
| (1/12) [pi] | 0.2617994 | 1.4179686 |
| (4/3) [pi] | 4.1887902 | 0.6220886 |
| | | |
| [pi] | | |
| ------ | 0.0174533 | 2.2418774 |
| 180 | | |
| | | |
| 1 | | |
| ------ | 0.3183099 | 1.5028501 |
| [pi] | | |
| | | |
| 4 | | |
| ------ | 1.2732395 | 0.1049101 |
| [pi] | | |
| | | |
| 1 | | |
| ------ | 0.0795775 | 2.9097901 |
| 4 [pi] | | |
| | | |
| 180 | | |
| ------ |57.2957795 | 1.7581226 |
| [pi] | | |
| | | |
| [pi] squared | 9.8696044 | 0.9942997 |
| | | |
| 1 | | |
| -------- | 0.0168869 | 2.2275490 |
| 6 [pi] squared | | |
| | | |
| _____ | | |
| \/ [pi] | 1.7724539 | 0.2485750 |
| | | |
| _____ | | |
| \ cubed/ [pi] | 1.4645919 | 0.1657166 |
| | | |
| | | |
| 1 | | |
| -------- | | |
| _____ | 0.5641896 | 1.7514251 |
| \/ [pi] | | |
| | | |
| 2 | | |
| -------- | | |
| _____ | 1.1283792 | 0.0524551 |
| \/ [pi] | | |
| | | |
| 1 | | |
| ---------- | | |
| _____ | 0.2820948 | 1.4503951 |
| 2 \/ [pi] | | |
| | | |
| _____ | | |
| / 6 | | |
| \ cubed/ ---- | 1.2407010 | 0.0936671 |
| V [pi] | | |
| | | |
| ______ | | |
| / 3 | | |
| \ cubed/ ------- | 0.6203505 | 1.7926371 |
| V 4 [pi] | | |
| | | |
| log e [pi] | 1.1447299 | 0.0587030 |
+--------------+-----------+-----------+
Useful fractional approximations are 22/7 and 355/113.
A synopsis of the leading formula connected with the circle will now
be given.