Difference between revisions of "ספר האחד"

One

• The one counts itself and no other counts it, rather it counts every number.

• It is a square root, a cube root, a square and a cube.

• It resembles the substance of the thing that bears the accidents.

• Every number is in its potentiality and it is in every number in practice.

• It is the existence and every number exists for it.

• It is eternal and every number is created.

• It is the cause of every even and non-even number.

• It is not a number.

• It is not multiplied or divided, but is the cause of all multiplication and division.

• It has no similarity, no opposite and no alteration.

• It does with one side what every number does with two sides.

• It encompasses every whole and every part, because it is the first and the last; it has no fraction, only in the thought of dividing the whole so that it will become one.

• It is the beginning of all the odd numbers that generate the square numbers when summed successively.

• It is also the beginning of all the even numbers, where the products of the extremes are equal to one another

and if the number is odd, the product is as the square of the mean.

and the difference between each square and square is as one part and half of the part.

• Since the one is not a fraction, the two are parts in the right sense, and any [kind of] parts is like the two.

• For every triangular shape that is the beginning of the shapes and every shape is divided into it:

If one side [of the shape] is one, the triangle can only be equilateral or isosceles acute triangle, because then the base have to be one.

• And the number ten is analogous to one, as are the hundred and thousand successively.

• And the arithmeticians, when they want to multiply a number by a number in one or two ranks, they subtract one for the foundation; and they gave many reasons for this.

But, the principle is that the analogy begins in the second rank.

And to make it easier for the students, they calculated in the first rank, which are the units and added for the two numbers. Therefore, they had to subtract one.

Because the rank of the tens is really one, the hundreds two, the thousands three, so the reason is the same.

Two

• The two is the beginning of all numbers.

• It is called so, because it receives a difference, as the sentence: And their laws differ from [those of] every people [Esther 3, 8].

• Its sum is as its product by itself.

• Every being is substance and form, but the One is not so.

• Every triangle, whose one side is two, can only be either equilateral or isosceles or obtuse.

• Know that all triangles are divided into two different categories:

• The one is uniform and it is in two ways: equilateral or isosceles, all angles of which are always acute.

• The second category: whose sides are unequal and it is in three ways:

• The one whose one angle is obtuse, because two angles are always acute.

The square of the largest side exceeds the sum of the squares of the two smaller sides by double the product of one [of the smaller] sides multiplied by its own extension up to its height that falls on the outside.

The beginning of the obtuse-angled triangles is by the succession: 2, 3, 4.

• The second type is where one angle is a right angle and the [other] two are also acute.

The [sum of the] squares of the two smaller [sides] is as the square of the hypotenuse, which is the largest side.

• The third type is called acute-angled, because all of its angles are acute.

Three

• The three is opposite to the one, because the sum of one with one is greater than its product by itself, the calculation of two is equal, and the sum of three with three is smaller than its product by itself. The way of three applies to every [other] number.

• It is the beginning of the triangular numbers.

• Its square is the number that is the beginning of the [scalene] triangles.

• Its cube is the number of sides of the equilateral triangles that are in the first rank.

• The number three is the beginning of every number that is not even [= odd numbers], whose product by itself is greater than itself.

• There is no number other than it, such that the sum of the preceding numbers is the same as itself.

Therefore, the number six that consists of it is equal [to the sum of] its parts.

The number that consists of the six is ​​the beginning of the numbers whose parts exceed over them.

• If you sum it with its parts, the sum is a perfect number.

• Know that the one is as the length, which is the line.

Two is as the surface

And the three is as the body, for it has length, breadth and depth.

• The theologians say that the time is divided into three.

• The vegetative soul, the vital soul and the rational soul are three.

• It is the beginning of the right triangle.

Four

• The four is the first visible square.

• It is the beginning of the even-times-even numbers.

• It is the beginning of the composite numbers.

For the first rank are 9 [numbers], of which the one is the foundation of all numbers, the four prime numbers remain: 2, 3, 5, 7; and the other four are composed.

• With the sum of the preceding numbers they are ten, which is the beginning of the analogue decade, and four is the principle.

Since the first of the units is one that is indivisible, therefore [four] is the opposite.

Likewise 2 with 5, 3 with 6, 4 with 7 and so on endlessly.

• Therefore the fourth zodiac sign is opposite to the first: for the warm and cold are the active, and the moisture and dryness are the passive.

So every fourth sign of the zodiac is the opposite of the first in terms of the activity, and the fiery signs alone are opposite in the activity and the passivity.

Accordingly the astrologers said that the aspect of the quarter of the celestial circle is hatred and the aspect of the sixth is half love.

For, the third of the signs is, in the nature, same in activity and opposite in passivity with the first, so they said that it is half love.

The aspect of the third is total love, because the aspect of the third of the celestial circle, which is the fifth, is in the nature of the first, hence the aspect of total love in the activity and the passivity.

So are 1 with 5, because both preserve themselves.

So also the 2 with 6, because both are even-times-odd numbers.

As are 3 with 7, because both are odd numbers and ?.

Not so for: 1 with 3; and 2 with 4; and 3 with 5.

The aspect of half the celestial circle and the aspekt of its quarter is that of its half, which is an aspect of hostility in passivity.

Do not be surprised that the seventh house also indicates to the companions, as the author of the Book of Creation mentioned: A.M.Š for male and A.Š.M. for female.

• The circle that is one is divided by the diameter.

The diameter is also divided and its aspect is the quarter.

If you puncture it in the quarter, the circle is divided into three equal parts each is equal to a triangle; and its half is the sixth of the circle.

There is no other number that is divided by nature into several fractions.

• The four is also the beginning of the scalene acute-angled triangles.

Every number follows it successively in all its paths.

As for the first obtuse-angled triangle: the excess of the longest [side] over [the sum of] the other two [sides] is as the middle [side] and the shortest is less than the longest by the number of the middle [side].

I shall give you a rule: if there are three successive numbers, from four upwards, and you wish to know the excess of the two over the greatest, always subtract four from the mean and multiply the remainder by the mean number.

For example: 10, 11, 12.

We subtract 4 from 11; 7 remain. We multiply this by 11; the result is 77 and this is the deficit of the square of the greatest.

כפלנוהו על י"א עלה ע"ז והוא חסרון מרבע היתר

Know that there are those angles that are very obtuse and those that are very close to the right [angle].

Such as [the ratio of] 4, 8, 9.

So there are acute angles that are half of the right [angle] or its third or less.

Do not think that you can make triangles out of any number.

• Because in the acute-angled triangle there can be no smaller number than four.

• There is no triangle whose side is longer than [the sum of the other] two.

• Also, [you cannot] make a right-angled [triangle] if the numbers of the sides are far apart.

Only in a combination such as 3, 4, 5, or its multiple.

Or so that the shorter sides are successive [numbers], such as 20, 21, 29.

Or the longer ones are successive [numbers], such as 5, 12, 13.

Five

Six

Seven

Eight

Nine