And for an analogous reason, wise men know that in the ’Varsity boat-race it is judicious and prudent to bet on the heavier crew. Leaving aside the question of the supply of energy, and keeping to that of the mechanical efficiency of the machine, we may find endless biological illustrations of the principle of similitude. In the case of the flying bird (apart from the initial difficulty of raising itself into the air, which involves another problem) it may be shewn that the bigger it gets (all its proportions remaining the same) the more difficult it is for it to maintain itself aloft in flight. The argument is as follows: In order to keep aloft, the bird must communicate to the air a downward momentum equivalent to its own weight, and therefore proportional to _the cube of its own linear dimensions_. But the momentum so communicated is proportional to the mass of air driven downwards, and to the rate at which it is driven: the mass being proportional to the bird’s wing-area, and also (with any given slope of wing) to the speed of the bird, and the rate being again proportional to the bird’s speed; accordingly the whole momentum varies as the wing-area, i.e. as _the square of the linear dimensions, and also as the square of the speed_. Therefore, in order that the bird may maintain level flight, its speed must be proportional to _the square root of its linear dimensions_. Now the rate at which the bird, in steady flight, has to work in order to drive itself forward, is the rate at which it communicates energy to the air; and this is proportional to _m_ _V_^2, i.e. to the mass and to the square of the velocity of the air displaced. But the mass of air displaced per second is proportional to the wing-area and to the speed of the bird’s motion, and therefore to the power 2½ of the linear dimensions; and the speed at which it is displaced is proportional to the bird’s speed, and therefore to the square root of the linear dimensions. Therefore the energy communicated per second (being proportional to the mass and to the square of the speed) is jointly proportional to the power 2½ of the linear dimensions, as above, and to the first power thereof: {25} that is to say, it increases in proportion _to the power_ 3½ _of the linear dimensions_, and therefore faster than the weight of the bird increases. Put in mathematical form, the equations are as follows: (_m_ = the mass of air thrust downwards; _V_ its velocity, proportional to that of the bird; _M_ its momentum; _l_ a linear dimension of the bird; _w_ its weight; _W_ the work done in moving itself forward.) _M_ = _w_ = _l_^3. But _M_ = _m_ _V_, and _m_ = _l_^2 _V_. Therefore _M_ = _l_^2 _V_^2, and _l_^2 _V_^2 = _l_^3, or _V_ = √_l_. But, again, _W_ = _m_ _V_^2 = _l_^2 _V_ × _V_^2 = _l_^2 × √_l_ × _l_ = _l_^{3½}.