Twelve Books on Harmony

A geometric diagram shows an ellipse with focal points E and K. Lines of reflection (ECK, EBK, and EAK) represent the path of sound or light from one focus to another. The major axis is labeled G H, illustrating the constant length of the reflecting paths as described by the "string method" of construction.

G D C B A represents a curve that reflects a sound made at focus or center E into focus K. This occurs through the sonic lines ECK, EBK, and EAK. Each of these lines is equal to the major diameter G H. This is the origin of the method for drawing an ellipse using a string equal to the length of line G H. By fixing the two ends of the string at points K and E, the rest of the string describes the side G D B F G and the opposite side as it is moved around the circumference. Since the parabolic method for creating reflections of sound is quite ingenious, I also wish to discuss it. Let there be a parabola E S T L, which reflects the parallel sounds M E, F S, R T, and R L into the focus O. If another very small mirror A C G I P B is placed opposite it, sharing the same focus and axis, it will reflect the sonic rays C G I P as parallel lines in D H, N Q. This happens as soon as the rays cross at the focus and touch the concave plane of the smaller parabola. Furthermore, these same parabolas suggest the design of telescopes original: "perspicilla". Indeed, distant objects M F R K will be seen clearly by an eye placed at points D or H. Additionally, heat or fire generated near the focus at C G, I P can be transmitted in this way to D, H, and so on. You can then reflect this heat to the right or left wherever you wish with the help of a flat mirror.

A diagram illustrates parabolic reflection. Parallel vertical rays labeled M, F, R, and K enter a large parabola (E, S, T, and L) and converge toward a focal point O. A smaller inverted parabola (A, B, C, G, I, and P) is positioned near the focus to intercept and reflect these rays back into parallel vertical lines (D, H, N, and Q). This setup functions as a reflecting telescope or a sound concentrator.

There is also another way to reflect sounds and rays here and there by means of a parabola. In this method, a very small parabola B C occupies an important place. It shares the same focus as the larger parabola N P M L. By intercepting the rays N P and M L before they meet at the focus O and cross one another, it reflects them as parallel lines in Q R. If these are light rays, this parabola functions as an excellent telescope. If the retina is placed at Q or R, it will show the individual parts of an object as very large, according to the size of the concave part of the larger parabola. If they are sound rays, they will transmit the harmony of voices or instruments produced far away at D E, H I to a listener at Q and R. Other details can be found in the French book On Sounds original: "libro Gallico de Sonis", referring to Mersenne's Harmonie Universelle from proposition 26 to 28, where echoes are discussed at length. Proposition 29 discusses the refraction of sounds and rays and their passage through different media. Furthermore, these opaque telescopes can produce a more noble effect than transparent ones. We have also spoken extensively about the various ways of reflecting rays in the Commentaries on Genesis Mersenne’s Quaestiones celeberrimae in Genesim (1623) from page 498 to 538, and provided some information on telescopes on page 762.

Two stacked diagrams show parabolic reflection. The upper part shows parallel rays (N, P, M, and L) being reflected by a large parabola and then a smaller parabola (B, C) so they remain parallel (Q, R). The lower diagram shows a more complex system where rays (Q, T, M, I, M, N, R, and S) converge toward a focus and are redirected by a small inner curved surface (V, X, F, and P) to be projected outward.

The following parabola A E B is more suitable for constructing an echo. Sounds produced at Q, H, M, R, and so on, namely Q T, M I, M N, and R S, fall upon the surface B S N E C T A. They are then reflected to the common focus of this larger parabola and another smaller one C D E attached to the back of the previous one. The small surface of the second parabola reflects these same rays as parallel lines C V L, X F, O X P, and so on. In this way, the voices at Q H M R are heard very clearly along the lines V X L F P. Furthermore, a simple parabola without the smaller one seems most suitable of all. This is because a single voice will not only be heard at a single point or focus, as happens in the previous ellipse, but in all places located in the region of the concave parabolic surface. For example: if a voice is at the focus original: "foro" of the parabola B a C, its sonic rays e B, e O, eq, er, and so on, will be reflected across the entire surface M, N, L, G, K. In this area, the strength of the voice will be preserved over a very long distance.

This acoustic diagram shows a profile of a human head at point 'a' speaking into a small parabolic curve (B-a-C). The sound rays are captured and reflected by a larger parabolic surface (S, P, and Q) and then projected onto a wide horizontal receiving area (M, N, L, G, H, I, and K) below. This illustrates the principle of a speaking trumpet or an acoustic reflector.

If anyone wishes to have an echo that reflects many syllables, we have experienced a remarkable echo in the countryside. This was at the home of the most noble man, Lord D'Ormesson, Counselor of the Privy Council. On a wall of his house, an echo repeats seven syllables. If you approach within forty paces of the wall, it repeats only six. If you move another forty paces closer, it repeats only five, and so on. Thus, every forty common paces either subtract or add a syllable. Therefore, if someone desires an echo of twelve syllables, he should stand 480 paces from the wall. For twenty-four syllables, he should stand 960 paces away. However, since the human voice is not strong enough for more than seven or eight syllables to return to the ear with enough force, trumpets or other instruments of this kind must be used. Additionally, one can use several parabolas following one another, of which...