Chapter 1

Three kinds of numbers in calculations

January 31, 2020

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.

As to the case in for instance, then this you which squares are equal " a is squares are equal to eighty thirty-six, Thus, is and all ; root three. is Or " “f then one square which to one-fifth of eighty, of the square its numbers; equal to nine ;”* square say, a square, and is to is is equal Or “the sixteen. five half equal to eighteen ;”{ then the square its root is is six. and sub-multiples of squares, multiples, them, are reduced to a single square. If there be only part of a square, you add thereto, until there a whole is square; you do the same with the equivalent in numbers. As to the case in which roots are equal to numbers ; " " one root for three in number then instance, the root is equals three, and are equal to twenty square nine. its ;" ; then one root || Or " four roots is equal to five, and the square to be formed of it is twenty-five. Or “half the root is equal to ten; “f then the * 2 .r' 2 t 5*

t || ^[ = 9 #=3 = 8o/. 0= 2=^ = 36 x=3 18.-. 2 4 =20 -f=10 .-. /. = 16 …

  • = 6 #=5

= 20

(5)8 ( whole root formed of I is and the square which equal to twenty, it is is four hundred. three kinds found that these squares, ) ; roots, namely, and numbers, may be combined together, and " arise that ;* is, squares compound species " and numbers numbers to ;” and roots equal squares to roots;” “roots and numbers equal to squares.” thus three equal Roots and Squares are equal " one stance, and ten square, to thirty-nine dirhems the square which, roots, amounts ;" when Numbers to ;f for in- amount roots of the same, that to say, is what must be increased by ten of The solution to thirty-nine? is own its this

you halve the number= )= of the roots, which in the present instance yields the product the is sum eight, roots, is is This you multiply by five. Add twenty-five. sixty- four. Now and subtract from which is five ; take the root of half the it itself; this to thirty-nine; the remainder is this, which number of three. This the root of the square which you sought for; square itself is nine.

  • The three cases considered are, cx 2 cx~+ a 3d. cx* f +bxa ist. 2d. bx = 1st case= Example x 2 -f 10^=39
  • 2d case, ex 1 --a-bx Example. #‘4-21 S = N " 25 ~ 21the that, when And know, the roots must be halved. number of in a question belonging to this case you have halved the number of the roots and multiplied the moiety by itself, if the product be less than the the square, then the number of dirhems connected with instance (8) is impossible;* but the product be equal to if the dirhems by themselves, then the root of the square is equal to the moiety of the roots alone, without either addition or subtraction. In every instance where you have two squares, or more or less, reduce them to one entire square, f as I have explained under the first case. Roots and Numbers are equal to Squares;^, for instance, " three roots and four of simple numbers are equal to a square." is is one and a Solution = Halve the roots the moiety ; the product this half. by itself; Multiply two and a quarter. Add this to the four the sum ;
  • If in an equation, of the form the case in supposed the 2 x’ +a=bx, 2 (|) = (|)2=a, then* 2 f cx +a=bx is to be reduced \ 3d case z. equation cannot happen. ex to 2 x 2 --^~x bx + a 2 Example x = +4 = V'6f = 2j -f ij is a, If13 ( and a quarter. six Add half. Extract this to the square, and the square four. is is These are the two and a to This is the root of the a multiple or sub-multiple one entire square. which six cases I mentioned in the They have now been introduction to this book. plained. it is sixteen. Whenever you meet with it root ; its moiety of the roots, which was one and a half; the sum of a square, reduce ) ex- have shown that three among them do not I require that the roots be halved, and I have taught how they must be resolved. As for the other three, in which halving the roots is necessary, I think it expe- dient, more accurately, to explain them by separate chapters, in which a figure be given for each will case, to point out the reasons for halving. Demonstration of the Case = " a Square and ten Roots are equal The which to thirty -nine Dirhems"* figure to explain this a quadrate, the sides of are unknown. It represents the square, the which, or the root of which, you wish to know. the figure as one of sides A B, its each side of which roots ; and if may be This is considered you multiply one of these by any number, then the amount of that number may be looked upon as the are added to the square. number of Each the roots which side of the quadrate represents the root of the square; and, as in the instance,
  • Geometrical illustration of the case, x 2 + io# = 39 (9)14 ( ) the roots were connected with the square, one-fourth of ten, that combine Thus with two and a to say, take half, and of the four sides of the figure. with each it is we may A the original quadrate new paral- B, four each having a side of the qua- lelograms are combined, drate as its length, and the number of two and a half as and they are the parallelograms C, G, T, its breadth K. We have now a quadrate of equal, though sides ; ; unknown but in each of the four corners of which a square two and a half piece of two and a half multiplied by wanting. In order to compensate for this want and to complete the quadrate, we must add (to that have already) four times the square of two and a is, twenty-five. figure, which we half, that We know (by the statement) that the first namely, the quadrate representing the square, together with the four parallelograms around represent the ten roots, bers. If to this we is it, which equal to thirty-nine of num- add twenty-five, which is the equivalent A B, completed, then we of the four quadrates at the corners of the figure by which the great figure D H know makes that this together of this great quadrate is its is sixty-four. root, that subtract twice a fourth of ten, that as is is, One side If eight. is five, from we eight, from the two extremities of the side of the great quadrate D H, then the remainder of such a side will be three, and that is the root of the square, or the side of the original figure we have halved A B. It must be observed, that the number of the roots, product of the moiety multiplied by and added the itself to the numberthirty-nine, in order to complete the great figure in four corners ; because the fourth of any plied by itself, and then by is four, number of the moiety of that number its multi- equal to the product multiplied by itself.* Accordingly, we multiplied only the moiety of the roots by instead of multiplying itself, This then by four. its the figure is fourth by itself, and = o. The same may also be explained We proceed from the the square. It is on two it becomes by another figure. B, which represents our next business to add to roots of the same. so that quadrate A it the ten We halve for this purpose the ten, five, and construct two quadrangles A B, namely, G and D, sides of the quadrate the length of each of them being five, as the moiety of the ten roots, whilst the breadth of each A B. side of the quadrate Then number of the by five = roots which two sides of the first this five A B. This is equal being half of the we have added quadrate. equal to a a quadrate remains opposite the corner of the quadrate to five multiplied is to each of the Thus we know that (10)the quadrate, which first quadrangles on its sides, the square, and the two is which are the ten roots, make In order to complete the great together thirty-nine. quadrate, there wants only a square of five multiplied (11) by five, or twenty-five. This we add to thirty-nine, in We extract its root, sixty-four. The sum S H. order to complete the great square eight, which is is one of By subtracting from the sides of the great quadrangle. same quantity which we have before added, namely five, we obtain three as the remainder. This is this the the side of the quadrangle square; it is itself is nine. A B, which represents the and the square the root of this square, This is the figure = H Demonstration of the Case = " a Square and twenty-one Dirhems are equal We to ten Roots"* represent the square by a quadrate length of whose side we do not know. This paralellogram is A D, such H B. The D, the To this we join a parallelogram, the breadth of which the sides of the quadrate A is equal to one of as the side H N. length of the two
  • Geometrical illustration of the case, x*
  • Geometrical illustration of the 3d case, 2 x' 30= -f- 4But we know that the quadrangle A R represents the four of numbers which are added to the three roots. The quadrangle A N and K L the quadrangle A gether equal to the quadrangle are to- R, which represents the four of numbers. We have seen, prises the and a also, that the quadrangle G M com- product of the moiety of the roots, or of one half, multiplied by itself; that is to say two and a quarter, together with the four of numbers, which are AN represented by the quadrangles and K L. There remains now from the side of the great original quadrate A D, which represents the whole square, only the moiety of the roots, that line G If C. is to say, we add G M, which is is the line A This it B which is C G, or the moiety of half, makes four, C, or the root to a square, which A D. Here follows was which we were desirous to explain. (15) A G, and a represented by the quadrate the figure. namely, the being equal to two and together with this, the three roots, namely, one half, the line this to the root of the quadrate a half; then one and a M JLWe have observed that every question which requires equation or reduction for one of the book. I now have its solution, will refer which cases six you have proposed in I to this also explained their arguments. Bear them, therefore, in mind.

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