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Classical Arithmetic, Book I, Chapter 1

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1. Study the lesson for mastery.
2. Earn a perfect score on the lesson examination.

King David playing the Lyre
A statue of King David playing the ancient lyre. Notice the different lengths of the strings.


In this first lesson, we seek to prove that Arithmetic is the first of the four mathematical sciences to be studied.

A “paradigm” is a pattern according to which something is formed.  “Intellection” refers to understanding or wisdom. This wisdom is “distributed” when it is applied to produce works of creation, which make that wisdom known.  Thus, when we read that Arithmetic is the “paradign of the Creator’s distributed intellection”, this means that classical Arithmetic is the source of the design according to which God created the world.1. Arithmetic is to be learned the first of the mathematical sciences , because it has the relation of a principle and mother to all the rest. For it is prior to all of them, not only because the Creator of the universe employed this as the first paradigm of his distributed intellection , and constituted all things according to number; but the priority of arithmetic is also evinced by this, that whenever that which is prior by nature is subverted , that which is posterior is at the same time subverted; but when that which is posterior perishes, that which is prior suffers no essential mutation of its former condition. Thus, if you take away “animal”, the nature of “man” is immediately destroyed; but by taking away “man”, “animal” will not perish.

2. And on the contrary, those things are always posterior which together with themselves introduce something else; but those have a priority of subsistence, which when they are enunciated, co-introduce with themselves nothing of a posterior nature. Thus, if you speak of “man”, you will at the same time introduce “animal”; for “man” is an “animal”. But if you speak of “animal”, you will not at the same time introduce the species “man”; for “animal” is not the same as “man”. The same thing is seen to take place in Geometry and Arithmetic. For if you take away “number”, whence will “triangle” or “quadrangle”, or whatever else is the subject of Geometry subsist? All which are denominative of numbers. But if you take away “triangle” and “quadrangle”, and the whole of Geometry is subverted, “three” and “four”, and the names of other numbers will not perish. Again, when we speak of any geometrical figure, it is at the same time connected with some numerical name; but when we speak of numbers, we do not at the same time introduce geometrical figures.

3. The priority likewise of numbers to Music may from hence be especially demonstrated, that not only those things which subsist by themselves are naturally prior to those which are referred to something else; but musical modulation itself is stamped with numerical names. And the same thing may take place in Music, which has been already noticed in Geometry. For diatessarondiapente, and diapason, are denominated by the antecedent names of numbers . The proportion likewise of sounds to each other is found in numbers alone. For the sound which subsists in the symphony diapason, is in a double ratio 2:1 (two to one). The modulation diatessaron consists in the ratio of 4:3. That which is called the symphony diapente is conjoined by the ratio of 3:2. That which in numbers is sesquioctave, is a tone in music. And in short, the priority of Arithmetic to music will be indubitably demonstrated in the course of this work.

It’s helpful to understand that, in the ancient world, musical sounds were produced by pipes and strings. Different sounds were produced by different lengths of string–the longer the string, the lower the tone.  In pipes, the longer the length of the pipe (which is controlled by opening and closing holes made in the pipe with the fingers), the lower the tone.  When sounds were played at the same time, it was noted that sounds played with certain mathematical relationships (ratios) made pleasant harmony, while others did not.  The art of pleasant music-making, was then understood to be a mathematical science.

4. But since Geometry and Music are prior to Astronomy, it follows that Astronomy is in a still greater degree posterior to Arithmetic. For in this science, the circle, the sphere, the center, parallel circles and the axis are considered, all which pertain to Geometry. Hence also, the senior power of Geometry may from this be shown, that all motion is after rest, and that permanency is always naturally prior to mobility. But Astronomy is the doctrine of moveable, and Geometry of immoveable natures. The motion of the stars likewise is celebrated as being accompanied with harmonic modulations. Whence also it appears that the power of Music precedes in antiquity the course of the stars. And it cannot be doubted that Arithmetic naturally surpasses Astronomy, since it appears to be more ancient than Geometry and Music which are prior to it. For by numbers we collect the rising and setting of the stars, the swiftness and slowness of the planets, and the eclipses and manifold variations of the moon.

In modern times, it is common to assume that the sun and stars are at rest, and that the earth is in motion, but in the ancient world, it was the other way around.  The earth was assumed to be at  rest and the  heavenly objects moved around the earth in fixed  mathematical spheres.  While modern scientists criticize the ancient view, Einstein explained in the book “The Evolution of Physics” that both models may be true, so there is no problem with the ancient view of the the earth being at rest.


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Classical Arithmetic, Lesson 01 Examination