Let us, at first, consider the mathematical relationships perceived by the sense of sight. Magnitudes sensible to the eye may form amongst each other aggregates of parts connected by mathematical laws. For instance, a piece of wood or stone may have geometrical form, that of a cube, a cone, a cylinder, or a sphere, which establishes regular relationships of distance between the different points of its outline. Furthermore, its dimensions may be quantities mutually related in simple proportions which the eye can seize readily; height, may be two, three, or four times greater than thickness or breadth: this constitutes a second series of mathematical relationships. Finally, many of these pieces of wood or stone may be placed symmetrically on the top or by the side of each other, according to distances and angles mathematically combined. Architecture is established on this aggregate of connected parts. An architect conceiving some dominant character, either serenity, simplicity, strength, or elegance, as formerly in Greece or Rome, or the strange, the varied, the infinite, the fantastic, as in Gothic times, may select and combine connections, proportions, dimensions, forms, and positions--in short, the relationships of materials, that is to say, certain visible magnitudes in such a way as to display the character aimed at. By the side of magnitudes perceived by sight there are magnitudes perceived by the hearing,--I mean the velocities of sonorous vibrations; and these vibrations being magnitudes may also form aggregates of parts connected by mathematical laws. In the first place, as you are aware, a musical sound is composed of continuous vibrations of equal velocity, and this equality already places between them a mathematical relationship; in the second place, two sounds being given, the second may be composed of vibrations, two, three, or four times the rapidity of the first; accordingly, there is between these two sounds a mathematical relationship, which is figured by placing them at an equal distance from each other on the musical stave. If, consequently, instead of taking two, we take a number of sounds, and place them at equal distances,--we form a scale, which scale is the gamut, all the sounds being thus bound together according to their relative position on the gamut. You can now establish these connections either between successive or simultaneous sounds, the first order of sounds constituting melody, and the second harmony. This is music: it has two essential parts, based, like architecture, on mathematical relationships, which the artist is free to combine and modify.