We called on Mr. Macalister when in Dunedin, who, when he heard that we were interested in the important question to which he had devoted so much attention, gave us a pressing invitation to hear the children at their rabbit exercises.
‘Noo, sir,’ said he, ‘ye wad aiblins like to hear the laddie bairns dae their Bunnyian Calculus; it’s jist wonnerfu.’
We said we ‘aiblins would.’
‘Well, ye maun ken then,’ continued he, addressing himself, chalk in hand, to his blackboard, ‘that a guid deal depends on the assoumptions even in the exawct scieences. Ane is that the doe rabbit litters aucht times in the twalmonth; and anither, that her feemly consists o’ twa he and four she anes. Monnie mae assoumptions maun be made that it wadna jist a’ thegither dae to explain to the callants. Ho’someever, we’ll reckon the term o’ life to be sax years, and start wi ae bonnie winsome doe rabbit. It is evident there will be—
(1 + 4) she rabbits at the end of the first term.
(1 + 4)² ” ” ” ” second term.
(1 + 4)⁴⁸ ” ” ” ” sax years.
Here we should soobtrawct ane, for the auld doe will e’en noo dee, while the first four o’ her offspring’ll be hirplin, and maun be deducted at the end o’ the next term, belike. But as we maun ca’ cannie wi’ the bairns, an’ no ding them doited a’ thegither, we’ll just pay no kind of attention whatever to the deed anes; for weel I wat, it maks sma’ difference in the result, those that dee by natural means, and won’t affect the first few significant figures. Nae doot ye see, that it’s jist compownd interest payable aucht times in the twalmonth, and if we further aloo’ for the deed anes, that is exactly as if a brokerage, as it were, were soobtrawcted at the end o’ each payment after the aucht and twa score. I’ll just show ye the exact formula wi’ r for the number o’ she rabbits and R: r the ratio of total rabbits to she ones at a litter, n being the number o’ years that elawpsed.’
And consulting his book, this is what he wrote on another board:
R
N = - (1 + r)⁸ⁿ⁻¹ [(1 + r){(1 + r)⁴⁸ - 8n - 1} + 8n]
r