(188.) The product of the numerical expressions for the mass of the body and the square of the radius of gyration is a quantity much used in mechanical science, and has been called the _moment of inertia_. The moments of inertia, therefore, for different axes in the same body are proportional to the squares of the corresponding radii of gyration; and consequently increase as the distances of the axes from the centre of gravity increase. (187.) (189.) From what has been explained in (187.), it follows, that the moment of inertia of any axis may be computed by common arithmetic, if the moment of inertia of a parallel axis through the centre of gravity be previously known. To determine this last, however, would require analytical processes altogether unsuitable to the nature and objects of the present treatise. The velocity of rotation which a body receives from a given impulse is great in exactly the same proportion as the moment of inertia is small. Thus the moment of inertia may be considered in rotatory motion analogous to the mass of the body in rectilinear motion. From what has been explained in (187.) it follows that a given impulse at a given distance from the axis will communicate the greatest angular velocity when the axis passes through the centre of gravity, and that the velocity which it will communicate round other axes will be diminished in the same proportion as the squares of their distances from the centre of gravity added to the square of the radius of gyration for a parallel axis through the centre of gravity are augmented. (190.) If any point whatever be assumed in a body, and right lines be conceived to diverge in all directions from that point, there are generally two of these lines, which being taken as axes of rotation, one has a greater and the other a less moment of inertia than any of the others. It is a remarkable circumstance, that, whatever be the nature of the body, whatever be its shape, and whatever be the position of the point assumed, these two axes of greatest and least moment will always be at right angles to each other. These axes and a third through the same point, and at right angles to both of them, are called the _principal axes_ of that point from which they diverge. To form a distinct notion of their relative position, let the axis of greatest moment be imagined to lie horizontally from north to south, and the axis of least moment from east to west; then the third principal axis will be presented perpendicularly upwards and downwards. The first two being called the principal axes of greatest and least moment, the third may be called the _intermediate principal axis_.