Then we may, ignoring the units G and H, speak of ON and NP as being equal to x and u respectively.

We take a fixed line OX, usually drawn horizontally; for each value of X we measure a length or abscissa ON equal to x.L, and draw an ordinate NP at right angles to OX and equal to the corresponding value of y .

On any line OX take a length ON equal to xG, and from N draw NP at right angles to OX and equal to uH; G and H being convenient units of length.

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Iii., written apparently after 146, he explains that he thought it desirable to add some account of the manner in which the Romans exercised the power they had won, of their temperament and policy and of the final catastrophe which destroyed Carthage and for ever broke np the Achaean League (iii.

Hence if N is the point of contact, NP must be normal to the traced curve.

Let KA and LB be the positions of NP corresponding to the extreme values of x.

Hence, finally, the resultant is expressed in terms of the coefficients of the three equations, and since it is at once seen to be of degree mn in the coefficient of the third equation, by symmetry it must be of degrees np and pm in the coefficients of the first and second equations respectively.

The assemblage of ordinates NP is then the graph of Y.