Now just as a slowly moving boat produces no bow-wave, but the bow-wave is seen only when the boat moves with a speed which is greater than the velocity of propagation of surface-waves in water, so, in like manner, no wave of compression is visible in front of a projectile so long as the speed of the projectile is less than the velocity of sound. But if the speed of the projectile reaches and exceeds the velocity of sound, then the head-wave, as we shall call it, augments noticeably in power, and is more and more extended, that is, the angle made by the contours of the wave with the direction of flight is more and more diminished, just as when the speed of a boat is increased a similar phenomenon is noticed in connexion with the bow-wave. In fact, we can from an instantaneous photograph so taken approximately estimate the speed with which the projectile is travelling. The explanation of the bow-wave of a ship and that of the head-wave of a body travelling in atmospheric space both repose upon the same principle, long ago employed by Huygens. Conceive a number of pebbles to be cast into a pond of water at regular intervals in such wise that all the spots struck are situate in the same straight line, and that every spot subsequently struck lies a short space farther to the right. The spots first struck will furnish then the wave-circles which are widest, and all of them together will, at the points where they are thickest, form a sort of cornucopia closely resembling the bow-wave. (Fig. 54.) The resemblance is greater the smaller the pebbles are, and the more quickly they succeed each other. If a rod be dipped into the water and quickly carried along its surface, the falling of the pebbles will then take place, so to speak, uninterruptedly, and we shall have a real bow-wave. If we put the compressed air-wave in the place of the surface-waves of the water, we shall have the head-wave of the projectile. [Illustration: Fig. 54.] You may be disposed to say now, it is all very pretty and interesting to observe a projectile in its flight, but of what practical use is it? It is true, I reply, one cannot _wage war_ with photographed projectiles. And I have likewise often had to say to medical students attending my lectures on physics, when they inquired for the practical value of some physical observation, "You cannot, gentlemen, cure diseases with it." I had also once to give my opinion regarding how much physics should be taught at a school for millers, supposing the instruction there to be confined _exactly_ to what was necessary for a miller. I was obliged to reply: "A miller always _needs_ exactly as much physics as he _knows_." Knowledge which one does not possess one cannot use.