It is readily seen that, if four circles touch a fifth circle, there are three distinct possible types of configuration of the four circles relatively to the circle which they touch; (1) the four circles may lie on the same side of the fifth circle, (2) three of the four circles may lie on one side and the fourth circle on the other side, (3) two of the four circles may lie on one side and the other two on the other side. The converse of Casey's Theorem may be stated in the following manner: if an equation of the form of the equation in Casey's Theorem exist between the common tangents of four circles taken in pairs, the common tangents being chosen in accordance with one of the three possible types of configuration, then the four circles touch a fifth circle. The truth of this converse theorem which is often assumed without any attempt at proof can be proved, but the proof of it is thought to be beyond the scope of this work. CONTINUITY. Let us consider a variable point P on a given straight line, on which A, B are two fixed points. It is seen at once that, (1) if P be outside AB beyond A, then the excess of PB over PA is equal to AB; and (2) if P be in AB, then the sum of AP and PB is equal to AB, (3) if P be outside AB beyond B, then the excess of AP over BP is equal to AB. Here we observe that, while P changes from one side of A to the other, the distance PA, which vanishes when P coincides with A, changes sides in the equation, which otherwise remains unchanged; and again that while P changes from one side of B to the other, the distance PB similarly changes sides in the equation. A geometrical theorem consists, in many cases, of a proof that a certain equation exists between a number of geometrical magnitudes, such equation remaining unchanged in form for variations in the geometrical magnitudes involved, consistent with the conditions to which they are subject. It is found in many of such theorems, as in the illustration which we have just given, that, if subject to continuous variation of some chosen geometrical magnitude some other magnitude continuously diminish and vanish, then in the equation which applies to the configuration determined by the next succeeding values of the chosen variable magnitude, the magnitude which has vanished appears on the opposite side of the equation. This fact is due to the absence of any sudden changes in the magnitudes under consideration. The general law that no sudden change occurs is often spoken of as THE PRINCIPLE OF CONTINUITY. Let us consider a variable point P on a given straight line, on which A is a fixed point; and let us consider any equation between variable geometrical magnitudes, one of which is PA the distance between P and A. The principle of continuity leads us to expect that, if P in the variation of its position pass from one side to the other of A, the sign of PA in the equation will change. In other words, we may consider the equation to remain unchanged in form, if we resolve to represent by the expression PA not only the distance between P and A, but also the fact that the distance is measured from P towards A. This result is at once obtained by resolving that PA shall represent a distance equal to PA and measured in the opposite direction; in other words, that AP= −PA. Let us return to the consideration of a variable point P on a given straight line, on which A, B are two fixed points. It appears that the equation which exists between the distances between the points takes different forms according as P is (1) in BA produced, (2) in AB, or (3) in AB produced. If we allow the use of the minus sign, we may write these equations, (1) AB+PA-PB=0, (2) AB-AP-PB=0, (3) AB - AP+BP=0, where each symbol such as AB represents merely the length of a line measured in the same direction as AB. It is at once seen that, if we adopt the convention that PQ = - QP, all these equations are the same; each may be written The first form expresses that the operation of passing from A to B is the same as passing from A to P and then from P to B; the second form expresses that the aggregate result of the operations of passing from A to B, and then from B to P and then from P to A is to arrive at the point A of starting; both of which facts are true for all combinations of three points A, B, P on a straight line. The results of the theorems contained in Propositions 5 and 6 of Book II. become the same, if we take into account the fact that the distance BD is measured in opposite directions in the two figures: and similarly the results of Propositions 12 and 13 of Book II. become the same, if we take into account the sign of CD. As a further illustration of the Principle of Continuity we v take Ptolemy's Theorem. Let us consider a variable point P on a circle, on which A, B, ‹ are three fixed points. It is proved in III. Prop. 37 B, that and (1) if P be in the arc AB, AB. PC BC.PA+ CA. PB; BC.PA CA. PB+AB.PC; (2) if P be in the arc BC, (3) if P be in the arc CA, CA.PB AB.PC+BC. PA. These equations may be written (1) AB. PC-BC.PA-CA.PB=0, Hence, while P passes along the arc from one side of B to the other, the sign of PB, which vanishes when P coincides with B, changes sign in the equation, which otherwise remains unchanged, and so on for passage through C or A. |