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.

• There are very obtuse-angled [triangles] such as 10, 17, 26.

• And less [obtuse-angled triangles] such as 10, 23, 26.

• [There are very acute-angled triangles] such as 10, 24, 26.

• And less acute-angled [triangles] such as 10 25, 26.

Therefore, we cannot create an obtuse-angled or an acute-angled [triangle] if the sides are very far apart.

• Know that according to the arithmetic method, there are 79 scalene triangles in the first rank, of which only 33 are valid.

• Also the isosceles triangles are not all valid either.

• The four is the beginning of all the secondary numbers.

Therefore, every even number is a secondary number that is a composite number, except for two, because of the power of the one, for it is the source of its actions.

Every number multiplied by one does not increases, because it is the essence of every number.

• Hence, the beginning of the squares is the four.

• Every square, multiplied by a square, is a square.

• Likewise, [every square], divided by a square, is a square.

• The ratio of a square to a square is a square.

Therefore, the measures of the wise are in squares.

Five

• The five is a circular number, and this is because the end of the first rank is 9; therefore, it contains itself, as I will explain.

• When you add the one that is like a square to the square of the first even number, it is five.

• If you add it to the square of the first non-even number, it is its double.

• When you add its square to the square of its double, the sum is the same as its cube.

• If you double this number, then [the sum of] its square with the square of its double is half its cube.

• If you want to know the ratio of [the sum of] the square of any number you wish and the square of its double to its cube, it is the same as the ratio of the number to 5, if it is smaller, or the ratio of five to it, if it is greater.

Such as 3, because [the sum of its square and] the square of its double is, the cube is 27, and its ratio to the sum of the squares is as the ratio of 3 to 5.

For 6, its square and double [its square] are 180, the cube is 216, and the number is five-sixths of the cube.

Likewise for all numbers.

• This number consists of the first even and the first non-even [= odd] number.

• Some say that the five senses correspond to this number.

• They say that the sight of the eye is of the nature of the stars, therefore the eye sees timelessly.

• The smell of the nature of fire, because it is like the blood.

• The taste is of the nature of the water and the essence is the moisture.

• The hearing is of the nature of the air.

• the touch is of the nature of the earth.

Six

• The six consists of 3, because the boundary of the body is threefold:

• From the length: the front and the back.

• From the breadth: the right and left.

• From the depth: the top and bottom.

So from three excellent directions:

For the movement of all the livings is forward, as well as to the right, because this is the movement of the great sphere, as well as upwards, because the movement of all vegetation is upwards.

And I already mentioned that it is straight.

Seven

• The seven consists of the first non-even number and the second even number.

• It consists also of the first even number and the second non-even number.

Because of this the philosophers called it a perfect number.

The author of Sefer Yetzirah [the Book of Creation] says that it is established in the midst, and this is the secret of the One that is not a body with the body that has six directions.

• It resembles the one, because it is followed by a number that consists only of two, and then by 9, that consists of three.

So it is like the beginning of the numbers.

• It resembles the two, because the scales of their squares are equal.

• It resembles the 3, because its distance from the analogous is 3, therefore there is a 9 in its the nine squares.

• It resembles the four, because the scales of a square is a square or 7.

• It resembles the five, because the sum of the the square of one and the squares of the first even and of the first non-even number is double of 7, which is 14.

And [the sum of all squares] up to the square of 7 is 140, which is analogous to 14.

• It resembles the six, because the sum of the numbers preceding it with 7, is a perfect number that equal to [the sum of] its part.

• Also it has a secret in the calculation of the conjunctions of the three planets:

• For if they were 4, [the number of the combinations of] two is not as the number [4].

• Likewise, if they are 5, then [the number of the combinations of] the two [out of five] is ten and from five upwards there is no even number that is opposite to its nature.

• The same for six and for every number after seven.

• For the [number of the combinations of] two out of seven is 21, which is the product of 3 that is non-even number by 7.

Likewise are the quadruple conjunctions

Eight

• The eight is known for the length is between two points; the surface is [between] four points, and the body [between] eight, which is the thickness.

• Therefore, the word eight is derived from As for Asher, his bread shall be fat [Genesis 49,20].

• It is also a cubic number.

• There are eight greate circles that are the spheres.

• The scales of the 8 are as that of the one.

The scales of the 7 are as that of the two.

The scales of 6 are as the scales of 3.

The scales of 4 are as the scales of 5.

• Corresponding to the 8 are the fire, the air, the water and the earth, because each of them has two natures, that are consist of four elements.

Nine

• The nine is the end of the first rank.

• Therefore, the scale of any number that is multiplied by itself or by another is the 9.

• It is the beginning of the squares of the non-even numbers.

• If you draw a circle and you write the 9 digits on it, the 9 rolls over all the numbers preceding it:

• because 9 by 9 is 81: the 1 that is the units is on the left, and 80 that is the tens corresponding to the 8 is on the right.

• Then, to the left, 9 by 8 is 72: 2 on the left, and 7 on the right for the tens.

• Then, 3 is on the left and 6 is on the right, which is 9 by 7 that is 63.

• Then, 4 is on the left for the units and 5 is on the right for the tens, which is the product of 9 by 6 that is 54.

• Then, 5 is on the left and the number 40 returns corresponding to the 4 that are the units, because the circle turns over and the fifty becomes five.

• Likewise, 9 by 4 are 36 - the units on the right and the tens on the left.

• The same for 27.

• Also 18.

• We consider that there are two spheres, corresponding to the upper spheres: one to east, the other to west. We also say that the 9 is at the head of the dragon in the celestial circle and one [of the spheres] goes in one direction and the other in the opposite direction.

There is 1 in square of 9, as well as all numbers, hence the number 5 is in the middle as if it were in the tail of the dragon, and is therefore a circular number.

• We wish to know how much is the square of 19.

The closest to it is 400. Our number precedes it, therefore we subtract one from the complete twenty. We double it and subtract it, the result is the number 360. We add the square of the 1 that we had subtracted. The total result is 361 and this is the square of 19.