Chapter 1b

# Squares and Numbers Equal to Roots

## Squares and Numbers Equal to Roots

“A square and 21 in numbers are equal to 10 roots of the same square.”

What must be the amount of a square, which, when 21 dirhems are added to it, becomes equal to the equivalent of ten roots of that square?

# Halve the number of the roots; Multiply this by the moiety the product itself; Solution is is

five. twenty-five. Subtract from this the twenty-one which are connected with the square root ; roots, it is is is four. Extract its Subtract this from the moiety of the two. which the remainder ; five ; the remainder is three. This is the root of the square which you required, and the square is nine. roots ; Or you may add the the sum is seven this ; which you sought for, root to the moiety of the is the root of the square and the square itself is forty- nine. When this case, you meet with an instance which refers you to try its solution by addition, and if that do not serve, then subtraction certainly will. both addition and subtraction will For in this case may be employed, which not answer in any other of the three cases in which

• 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*

# you G, which number of seven, represented likewise be equal to ten roots of the by the root of the ori- the moiety of the you add twenty-one If two. you add the number two if and root, represented the moiety of the roots, then the remainder A C R, sum same square. will Here19 ( Demonstration of the Case

) " three Roots Simple Numbers are equal to and four of a Square"* Let the square be represented by a quadrangle, the sides of which are unknown to us, though they are equal among drate This themselves, as also the angles. is the qua- A D, which comprises the three roots and the four of numbers mentioned in drate one of its sides, In every qua- this instance. multiplied by a unit, is its root. We now cut off the quadrangle H D from the quadrate A D, and take one of its sides H C for three, which is the number of the roots. It follows, then, that the The same is quadrangle equal to H B represents Now the four of numbers which are added to the roots. we halve the side G the point H T, which ; C H, which is R D. equal to three roots, at from this division we construct the square is the product of half the roots (or one and (14) a half) multiplied by themselves, that We add is to say, two and G T a piece equal to the line A H, namely, the piece T L accordingly the line G L becomes equal to A G, and the line K N equal to T L. Thus a new quadrangle, with equal sides and angles, arises, namely, the quadrangle G M and we find that the line A G equal to M L, and the same line A G equal to G L. By these means the line C G remains equal to N R, and the line M N a quarter. then to the line ; ; is is equal to T equal to the H B L, and from the quadrangle quadrangle K L is cut a piece off,

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