Arithmetica

From this, it is easy to gather information about evenly even numbers that can be divided by two repeatedly until reaching one which are also diminished. For, as will be shown later, a diminished number is one whose total parts, when heaped together, do not equal the whole. Now, each part of an evenly even number is found in the sequence of doubles starting from unity. This is according to the second property. Furthermore, the preceding parts joined together restore a sum that is one less than the following number, according to the rule just established. Therefore, every evenly even number is diminished. From this place, something is taken away from perfection completeness which is established as the opposite of unity. But you will not easily recognize that unity which does not have it nearby, especially since existence itself is entirely derived from action.

Inanimate things are more limited than plants, for plants not only exist but also live. Plants are more limited than animals, by one degree, insofar as animals possess sense perception in addition to being and living. Animals are likewise more limited than humans, as humans are capable of reason. Finally, humans are more limited than angels by one degree, for angels possess intellect, which is more fully illuminated by the torch of divinity. These things should be understood in reverse: as one approaches the assigned unity, one is understood to tend more toward plurality and thus toward imperfection. For matter is by far the most imperfect of creatures, while an angel is the most perfect. By this law, in the nature of evenly even numbers, as if in a symbol, the first and highest plurality and imperfection is uncovered in matter. The second is in inanimate things. The third is in plants. The fourth is in animals. The last is in angels.

There is also another way of philosophizing from the opposite, while preserving the simplicity of unity but contracting it into excellence. In this way, it becomes known that the "super-immense" being holds the peak of perfection. From His eternal dimension, angels are distant by one degree, namely by the initiating limit. Furthermore, angels surpass humans in intelligence. Humans surpass beasts in reason. Beasts surpass plants in sense perception. Plants surpass inanimate things in life. Finally, inanimate things surpass matter by the fact of being, for matter is almost non-being. If you wish to follow a direct analogy, you will easily see that in the divine nature there is no division at all. In angels, there is one division. In the heavens, two. In the compositions of nature, three. Thus, the super-celestial world is shortened by one division from the celestial. The celestial world is likewise shortened from the elementary world. But these things will be discussed more fully in their proper place.

On the occasion of the above, it should be noted that some things in numbers correspond to things in reality by direct analogy, while others correspond by opposite analogy. The famous author Dionysius Dionysius the Areopagite, a Christian Neoplatonist philosopher mentions that the same is observed in things. In his work On the Celestial Hierarchy, he states that earthly images express those divine minds. Sometimes this happens by the law of direct analogy, but sometimes by the opposite. For example, when the oracles of the prophets express the Seraphic spirits through fire, or the Thrones by the name of seats, he asserts that this is done because of properties that correspond directly. But when they express them by names like anger, lust, incontinence, irrationality, and insensibility, it is done by opposite analogy. And this is done by a certain divine and more secret intelligence. Surely, those "unlike images" serve a negative theology defining God by what He is not. This contributed significantly to bringing out the prophetic meanings.

Furthermore, he believes the same is done so that divine meanings do not become accessible to everyone. Also, if they were expressed by high images, men might remain stuck or attached to them. They might falsely believe that those spirits possess earthly and bodily forms. It is otherwise with unlike images, especially since people would not dare to ascribe such things to them, but rather revere the hidden and veiled intelligence within them.

What he adds in the last place as a property is this. If evenly even numbers are arranged in an even series, the product of the means (for there are two means when the series is even) is equal to the product of the extremes, until the series is complete. Now, a product is what arises from the multiplication and mutual leading of numbers. For example, in this series: 1, 2, 4, 8, 16, 32. This is bounded by six discrete marks and is even. Therefore, it obtains two means, namely 8 and 4. By their mutual leading and multiplication, the sum of 32 arises. For 8 taken four times is 32. Conversely, the same sum arises if you lead the nearby extremes, 2 and 16, into each other. For both twice 16 and sixteen times two make 32. The same is also contained under 1 and 32.

Furthermore, it is appropriate for this series of evenly even numbers to be continuous and uninterrupted, so that no intermediate number is omitted. Otherwise, if any intermediate is left out and not taken, this will not happen. This is clear in a series arranged like this: 1, 2, 4, 16. In this series,
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one number, the eight, is left out between 4 and 16. For under the means, namely 2 and 4, only 8 is contained. But under unity and 16, the double is contained, namely sixteen. Nor should it be omitted that those are called the "means" in a series which are equally distant, primarily from the extremes. One should not look at the excess of units, but at the plurality of intercepted numbers. For example, if two are skipped on each side, or three, or any other number. It is also necessary that those which are on the outside

be taken correctly. This happens if they are separated from those means by the same interval on both sides, with either no numbers or the same number of numbers intercepted on each side. Furthermore, the numbers which are led into each other are parts corresponding to each other, namely the denominator and the denominated.

If the series is odd, and there is a single mean equally distant from the extremes on both sides, then the product of that mean multiplied by itself is equal to the product of the extremes. This continues until the series is filled. For example, in this series: 1, 2, 4, 8, 16. This is completed by five numbers and is odd. Therefore, there is one mean, namely four. If this is led into itself, it restores the sum of 16. The numbers 2 and 8, placed around it, show the same through mutual leading into each other. For twice eight, or eight times two, is 16. The same is true for 1 and 16. This will happen no matter how far the series is extended. Furthermore, this property belongs not only to evenly even numbers, but to all numbers that maintain a geometric relationship. This will be shown in its proper place. From this property, one can grasp the rarity, density, resistance, activity, weight, and lightness of the elements, and the alternate excess according to these things. But for now, it is enough to have pointed this out.

ON THE EVENLY ODD NUMBER AND ITS PROPERTIES. CHAPTER VII.

20 Every mean of an evenly odd number is odd

A large decorative initial P featuring floral patterns and a small putto figure at its base.
The evenly odd an even number whose half is odd, such as 6, 10, or 14 number is one that has been allotted the nature and substance of evenness, but in its contrary division, it is opposed to the nature of the evenly even number. It will be taught by how much a different reasoning this number is divided. For, since it is even, it receives a cut into equal parts. But its parts will immediately remain indivisible and inseparable.
21 such as 6, 10, 14, 18, 22, and those similar to them. For if you divide these numbers into twin parts, you run into an odd number, which you cannot cut.

Its half is evenly odd

Therefore it is a property

For the divisible ??? evenly odd is found first

Whenever a part of an evenly odd number has a divided denominator, the quantity of parts of this even... odd... contrary

It happens to these numbers that all parts have contrary denominations to the quantities of the parts being named. It can never happen that any part of this number receives a denomination and quantity of the same genre. For always, if the denomination is even, the quantity of the part will be odd. If the denomination is odd, the quantity will be even. For example, in 18: the second part is its half, which is an even name, and is 9, which is an odd quantity. The third part, which is an odd denomination, is six, which is an even plurality. Again, if you convert it, the sixth part, which is an even denomination, is three, but three is odd. And the ninth part, which is an odd word, is 2, which is an even number. The same is found in all others that are evenly odd. It can never happen that
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any part has a name and number of the same genre.

The production of these numbers happens if you arrange numbers that differ by two starting from one, that is, all the odd numbers set in their natural sequence and order. For if these are multiplied by two, a rightly measured plurality will produce all the evenly odd numbers. Let the first be unity, 1, and after this the one that differs from it by two, which is 3, and after this the one that again differs from the previous by two, which is 5, and so on to infinity. Let the arrangement be like this: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. These are the naturally following odd numbers, with no even number distinguishing them in the middle. If you multiply these by two, you will produce them in this way: twice one is two, which is indeed divided, but its parts are found to be indivisible because of