MAES IN THE AUTUMN OF 1877. 
237 
paucity of 7’s in the number representing the ratio of a circle’s 
circumference to its diameter.) The corresponding fractions 
arP 1 2 15 32 47 7_0 284 36 3 .6 4 7. 489 2 5 5 3 9 • aTir l 
aTe T’ T’ 85 TV’ 2~5? 42? 15 1? 19 3? 3 4 4? 2 6 01? 2 9 45? ? dJAU 
the corresponding relations, which chiefly concern us, are these: 
1 sid, per. of Mars — 1 sid. year 
2 sid. years — 1 sid. per. of Mars 
8 sid. per. of Mars — 15 sid. years 
32 sid. years — 17 sid. per. of Mars 
25 sid. per. of Mars — 47 sid. years 
79 sid. years — 42 sid. per. of Mars 
151 sid. per. of Mars — 284 sid. years 
363 sid. years — 193 sid. per. of Mars 
344 sid. per. of Mars — 647 sid. years 
4892 sid. years — 2601 sid. per. of Mars 
2945 sid. per. of Mars — 5539 sid. years 
321-7333 days 
43-5331 „ 
16-9916 „ 
95499 „ 
7-4417 „ 
2-1082 „ 
1-1171 „ 
0-9911 „ 
0-1260 „ 
0*1091 „ 
00169 „ 
So that between the time of the opposition of Mars which 
occurred in the year 3662 b.c. until our earth came to the same 
part of her orbit (not the same part of the year of seasons, but 
of the sidereal year) Mars will have made 2945 sidereal circuits 
plus his motion in 0*0169 days, or in 24 min. 20*16 seconds. 
This is the nearest periodic approach (I refer to the near equality 
of 2945 periods of Mars and 5539 years) during what is com- 
monly regarded as historical time ; since the next fraction after 
derived from the continual fraction above given, has for 
its numerator 5539 x 6 + 4892 or 38,126, which is the number 
of sidereal years of the corresponding relation between the 
periods of the earth and Mars : 20,271 sidereal periods of Mars 
fall short of this long interval by *0077 days, or 11 min. 5*28 sec. 
Beyond this we need not care to proceed. 
The use of the relations above tabulated will be very obvious. 
It shows first the number of years separating similar oppositions 
of Mars, the closeness of the similarity depending on the small- 
ness of the number of days representing the difference between 
so many years and the corresponding number of periods of Mars. 
Thus we see that in fifteen years from any given opposition, 
Mars is removed seventeen days’ journey from the prolongation 
of the line extending from the sun to the earth. That line, or 
the earth’s radius vector produced, has passed Mars recently 
therefore, and we have to carry it back at the earth’s daily rate 
of angular motion round the sun, carrying Mars back by his 
less rate of motion, until the earth’s radius vector passes through 
Mars, to get the place of opposition, which of necessity precedes 
the place of opposition fifteen years before. Since the earth’s 
mean daily rate is 3548 // *193, and Mars’s 1886"*518, the dif- 
ference, 1661 "*675, is the earth’s daily gain ; and we have only 
to find out how often this is contained in 17 (or more exactly 
