The Order of the Classical Quadrivium

The following is taken from Thomas Taylor’s “Theoretic Arithmetic” (1818)1

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

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 fabricator2 of the universe employed this as the first paradigm of his distributed intellection3, 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.

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 hot the same as man.

The same thing is seen to take place in Geometry and Arithmetic. For if you take away numbers, 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 appellations of other numbers will not perish. Again, when we speak of any geometrical figure, it is at the same time connected with some numerical appellation; but when we speak of numbers, we do not at the same time introduce geometrical figure.

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 appellations. And the same thing may take place in this, which has been already noticed in Geometry. For diatessaron, diapente, 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. The modulation diatessaron consists in the ratio of 4 to 3. And that which is called the symphony diapente is conjoined by the ratio of 3 to 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.

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 the geometric discipline. 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.

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  1. Taylor, Thomas. Theoretic Arithmetic. (1818)
  2. See:
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