Given any three points _A_, _B_, _D_ in one straight line, the quadrilateral construction finds the point _C_ harmonic to _A_ with respect to _B_, _D_ by the following method: Take any point _O_ outside the straight line _ABD_, and join it to _B_ and _D_. Through _A_ draw any straight line cutting _OD_, _OB_ in _P_ and _Q_. Join _DQ_, _BP_, and let them intersect in _R_. Join _OR_, and let _OR_ meet _ABD_ in _C_. Then _C_ is the point required. To prove this, let _DRQ_ meet _OA_ in _T_, and draw _AR_, meeting _OD_ in _S_. Then a projective transformation of _A_, _B_, _C_, _D_ from _R_ on to _OD_ gives the points _S_, _P_, _O_, _D_, which, projected from _A_ on to _DQ_, give _R_, _Q_, _T_, _D_. But these again, projected from _O_ on to _ABD_, give _C_, _B_, _A_, _D_. Hence _A_, _B_, _C_, _D_ can be projectively transformed into _C_, _B_, _A_, _D_, and therefore form a harmonic range. From this point, the proof that the construction is unique and general follows simply[125]. The introduction of numbers, by this construction, offers no difficulties of principle--except, indeed, those which always attend the application of number to continua--and may be studied satisfactorily in Klein's Nicht-Euklid (I. p. 337 ff.). The principle of it is, to assign the numbers 0, 1, ∞ to _A_, _B_, _D_ and therefore the number 2 to _C_, in order that the differences _AB_, _AC_, _AD_ may be in harmonic progression. By taking _B_, _C_, _D_ as a new triad corresponding to _A_, _B_, _D_, we find a point harmonic to _B_ with respect to _C_, _D_ and assign to it the number 3, and so on. In this way, we can obtain any number of points, and we are sure of having no number and no point twice over, so that our coordinates have the essential property of a unique correspondence with the points they denote, and _vice versa_. =114.= The point of importance in the above construction, however, and the reason why I have reproduced it in detail, is that it proceeds entirely by means of the general principles of transformation enunciated above. From this stage onwards, everything is effected by means of the two fundamental ideas we have just discussed, and everything, therefore, depends on our general principle of projective equivalence. This principle, as regards two dimensions, may be stated more simply than in the passage quoted from Cremona. It starts, in two dimensions, from the following definitions: To project the points _A_, _B_, _C_, _D_ ... from a centre _O_, is to construct the straight lines _OA_, _OB_, _OC_, _OD_.... To cut a number of straight lines _a_, _b_, _c_, _d_ ... by a transversal _s_, is to construct the points _sa_, _sb_, _sc_, _sd_....[126]