Arithmetic, Geometry, and Music

A circular diagram contains the word "quadruple" quadruplus. Below it are two connected semicircular arches labeled "triple" triplus and "sesquitertius" sesquitertius a ratio of 4:3. Beneath these is a horizontal grid with three cells containing the numbers 3, 9, and 12.

The quadruple and the sesquiquartus ratio of 5:4 accumulate together: a quintuple quincuplus ratio of 5:1 will immediately result.

A circular diagram contains the word "quintuple" quincuplus. Below it are two connected semicircular arches labeled "quadruple" quadruplus and "sesquiquartus" sesquiquartus. Beneath these is a horizontal grid with three cells containing the numbers 4, 16, and 20.

If the quintuple is joined with the sesquiquintus ratio of 6:5, the proportion of a sextuple sexcuplus ratio of 6:1 will be coupled.

A circular diagram contains the word "sextuple" sexcuplus. Below it are two connected semicircular arches labeled "quintuple" quincuplus and "sesquiquintus" sesquiquintus. Beneath these is a horizontal grid with three cells containing the numbers 5, 25, and 30.

And so, according to this progression, all types of multiplicity will arise without any change in the order of the ratio. This happens in such a way that the double duplus combined with the sesqualter ratio of 3:2 creates a triple, the triple combined with the sesquitertius creates a quadruple, the quadruple combined with the sesquiquartus creates a quintuple, and the others in the same manner, so that no end hinders this continuation.

Regarding quantity standing by itself, which is considered in geometric figures; the common ratio of all magnitudes. Chapter 4.

b

And indeed, let what has been said suffice for the present regarding the quantity that we observe in relation to something else. Now, however, in this following part, I shall explain certain things concerning that quantity which exists by itself and does not refer to anything else, which can be useful to us for those things that we will treat later regarding the quantity related to something else. For in a certain way, the investigation of mathematics is established by the alternate reasoning of proofs. Now, however, we are to speak about those numbers which revolve around geometric figures and their spaces and dimensions, that is: linear numbers, triangular or square numbers, and others which only a plane dimension reveals, as well as those joined by an unequal exposition of sides. Also about solids, that is: cubes, spheres, or pyramids, as well as bricks, beams, and wedges, all of which belong to the proper consideration of geometry. But just as the science of geometry itself is produced from arithmetic as if from a certain root and mother, so too do we find the seeds of its figures in the primary numbers. We have indeed made it plain that this destroyed would consume all disciplines, which, once established, it would weaken. But it must be known that these placed signs of numbers, which even now people describe in the designation of sums, were not formed by natural institution, so that even if the small note placed below the quinarius five signifies 5, or that of the denarius ten which we have described signifies 10, and others of this kind, nature did not place them, but custom attached them. For they wished to note five or ten or any others with those symbols for the sake of brevity, lest every time someone wished to show units, lines would have to be drawn that many times. But we, whenever we wish to show something, especially in these formulas, are not burdened to place a multitude of ordered lines. For we wish to demonstrate the number five: we make five lines, and we draw them in this way, IIIII, and with 7, the same amount, and with 10, nonetheless. It is more natural to designate any number by as many units as it contains in itself than by symbols.

Therefore, the unit holds the place of a point, the beginning of an interval and length, yet it is itself capable of neither interval nor length, just as a point is indeed the beginning of a line and of an interval, but is itself neither an interval nor a line. For a point placed upon a point does not create any interval, just as if you join nothing to nothing. For there is nothing that is born from the procreation of nothings. The same proportion indeed remains regarding equalities. For if there are any number of equal terms, the distance from the first to the second is as much as from the second to the third. But between the first and the second, or the second and the third, there is no length or space of an interval. For if you place three sixes in this way: 6, 6, 6; just as the first is to the second, so is the second to the third. But there is no difference between the first and the second. For 6 and 6 are not separated by any intervals of space. Thus also, the unit multiplied by itself creates nothing. For once multiplied by one, it produces nothing else from itself than what it is. For that which lacks an interval does not receive the power of generating intervals, which does not appear to happen in other numbers. For every number multiplied by itself makes another that is greater than itself, because the multiplied intervals extend themselves with a greater equality of space. That, however, which is without an interval does not have the power of bringing forth more than itself. Therefore, from this principle, that is, from the unit, the length of all things increases, which unfolds itself into all numbers from the beginning of the binary number, since the first interval is a line, but two intervals are length and breadth, that is, a line and a surface. Therefore, there are three intervals: length, breadth, and height, that is, line, surface, and solidity. Beyond these, no other intervals can be found. For it will either be one interval, which is length, or something exposed by two intervals, as if something had length and breadth, or it is extended by a triple dimension of an interval, if it is considered by length, height, and breadth. Above which nothing can be found, so that the forms of the six motions are composed according to the nature and number of the intervals. For one interval contains two motions in itself, so that in three intervals, the sum of six motions is completed in this way. For there is forward and backward in length, left and right in breadth, upward and downward in height. It is necessary, however, that whatever a solid body may be, it has length, breadth, and height, and whatever contains these three in itself, that is called a solid by its own name. For these three revolve around every body with inseparable conjunction, and are established in the nature of bodies. Therefore, whatever lacks one interval, that body is not solid. For that which retains only two intervals is called a surface. For every surface is contained only by length and breadth. And here the same conversion remains. For everything that is a surface has length...