b = ar c = br = arr d = cr = arrr
and so on; whence the series
| a | b | c | d | &c. | ||||
|---|---|---|---|---|---|---|---|---|
| is | a | ar | arr | arrr | &c. | |||
| Hence | a | : | c | ∷ | a | : | arr | |
| (186) | ∷ | aa | : | aarr | ||||
| ∷ | aa | : | bb | |||||
because, b being ar, bb is arar or aarr. Again,
| a | : | d | ∷ | a | : | arrr | ||
| (186) | ∷ | aaa | : | aaarrr | ||||
| ∷ | aaa | : | bbb | |||||
| a | : | e | ∷ | aaaa | : | bbbb | , and so on; | |
that is, the first bears to the nᵗʰ term from the first the same proportion as the nᵗʰ power of the first to the nᵗʰ power of the second.
193. A short rule may be found for adding together any number of terms of a continued proportion. Let it be first required to add together the terms 1, r, rr, &c. where r is greater than unity. It is evident that we do not alter any expression by adding or subtracting any numbers, provided we afterwards subtract or add the same. For example,
p = p - q + q - r + r - s + s
Let us take four terms of the series, 1, r, rr, &c. or,
1 + r + rr + rrr