Chapter 1

Three kinds of numbers in calculations

People always want a number when calculating. Every number is composed of units. Any number can be divided into units.

Every number, which may be expressed from one to ten, surpasses the preceding by one unit. Afterwards, the ten is doubled or tripled, just as before the units were. Thus arise 20, 30, until 100. The 100 is doubled and tripled in the same way up to 1,000, and so on without limit.

There are three kinds of numbers in calculations=

  1. roots - any quantity multiplied by itself
  2. squares - the result of the roots when multiplied
  3. simple numbers that are not root nor square

A number belonging to one of these three may be equal to a number of another class. For example, a square equal to a root, or a root equal to a square.

An example of squares equal to roots:

  • A square is equal to 5 roots.
  • The root of the square is 5
  • The square is 25, which is 5 times its root

If you say “1/3 of the square is equal to 4 roots”=

  • The whole square is equal to 12 roots
  • 144 is the square and its root is 12

If you say “5 squares are equal to 10 roots” then=

  • 1 square is equal to 2 roots
  • The root of the square is 2
  • Its square is 4

In this way, whether the squares be many or few, multiplied or divided by any number), they are reduced to a single square and the same is done with the roots, which are their equivalents i.e. they are reduced in the same proportion as the squares.

When Squares are Equal to Numbers

We say “a square is equal to 9”.

  • This means its root is 3

“5 squares are equal to 80”

  • This means 1 square is equal to 1/5 of 80
  • This leads to 16

“Half of the square is equal to 18”

  • This means the square is 36
  • Its root is 6

Thus, all squares, multiples, and sub-multiples of them, are reduced to a single square.

  • If there be only part of a square, you add to it until there is a whole square.
  • You do the same with the equivalent in numbers.

When Roots are Equal to Numbers

“One root equals 3 in number”

  • This means the root is 3
  • Its square is 9

“4 roots are equal to 20”

  • This means one root is 5
  • The square is 25

“Half the root is equal to 10”

  • This means the root is 20
  • The square is 400

The following 3 kinds can be combined to create new compounds:

  • roots
  • squares
  • numbers

These new compounds are:

  • square and roots equal to numbers
  • squares and numbers equal to roots
  • roots and numbers equal to squares

Roots and square equal to numbers

“One square and 10 roots of the same amount to 39 dirhems.”

  • This means what is that square which, when increased by 10 of its own roots amount to 39?

The solution is that:

  • you halve the number of the roots, which now yields 5.
  • multiply this by itself
  • the product is 25
  • add 39 to this to get 64
  • take its root which is 8
  • subtract from it half the number of roots which is 5
  • the remainder is 3 – this is the root of the square you were looking for
  • its square is 9

The solution is the same when 2 or 3 squares are specified.

  • You reduce them to one single square
  • In the same proportion you reducethe roots and simple numbers which are connected with them.

“2 squares and 10 roots are equal to 48 dirhems”

  • What must be the amount of 2 squares which, when added and then added to 10 times the root of one of them, lead to 48 dirhems?

The solution is:

  • reduce the 2 squares to one square
  • this one square is the moiety of both
  • reduce everything mentioned in the statement to its half

This reorganizes the question: “A square and 5 roots of it are equal to 24 dirhems”.

  • What must be the square which when added to 5 times its root, is equal to 24 dirhems?

The new solution is:

  • Half the number of roots
  • The moiety is 2.5
  • Multiply that by itself to come up with 6.25
  • add it to 24 leading to 30.25
  • deduct from this moiety 2.5 (the number of roots) which leads to 3
  • the square is 9

The same goes for the question: “Half of a square of 5 roots are equal to 28 dirhems”

  • What is the square which, when added to the five of its roots, is equal to 28 dirhems?

You must:

  • first complete your square by doubling the whole question
  • this will give you 1 square, 5 roots, 56 dirhems
  • halve the roots to get 5
  • multiply this by itself to get 25
  • add 25 to 56 to get 81
  • the root of 81 is 9
  • subtract from 9 the number of roots which is 5
  • you get 4 as the root
  • the square is 16

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