The members of the Central Council have received the following as part of the Methods Committee’s report this year. In the hope that the question of extension can be finished with at the coming meeting of the Council, the members of the Committee are anxious that the Exercise as a whole should know what is proposed and they invite comment:-

Since presenting our last report to the Central Council at Whitsun, 1951, we have further considered the subject of Extension in the light of comments and suggestions made by various people prompted by our invitation published in *The Ringing World*. In particular, we have directed our attention to the problem of the movement of the bells in the coursing order from lead to lead.

It will be appreciated that, if the structure of a lead of an extension be satisfactory, the “likeness” of the blue-line of the extension to that of the parent will be governed by the order in which the leads are joined together. This order is determined by the sequence of pivot bells from parent to extension. Each pivot will move all the bells two distinct numbers of places through the coursing order, dependant upon whether the lead-end place is 2nds or ultimate; e.g., if in a Major method the pivot bell be the 3rd followed by an ultimate place, the lead head will be 3527486, each bell having moved one place back in the coursing order, but if followed by a 2nds place, the lead head will be 5738264, each bell having moved two places back in the coursing order. Moreover, each movement may be regarded in two different ways, backward or forward. In the example given above, the first movement may be regarded as one place backward or six forward, and the second movement as two backward or five forward. Thus if it is desired to extend an ultimate place Major method having 3rd as pivot, and the movement be regarded as one place backward, the pivot sequence in Major, Royal and Maximus wil be 3, 3, 3; but if the movement be regarded as six forward, the required sequence will be 3, 5, 7. In the first case the lead heads will be 3527486, 352749608 and 3527496E8T0; but in the second case 3527486, 795038264 and ET907856342 respectively. It will be readily seen that while in the first case the blue line of the extensions will closely follow that of the parent, in the second place it will be so badly cut up that it will bear very little resemblance to that of the parent.

Similarly, if the method be 2nds place and the 3rd the pivot and the movement be regarded as two backward, the pivot sequence will again be 3, 3, 3, and for five forward again, 3, 5, 7, and once again the first will piece together satisfactorily and the second will not.

The following table shows the results for Major methods having the 5th as pivot. In these the movement may be regarded as four forward (ult.) and three forward (2nds) or three backward (ult.) and four backward (2nds).

Pivot Sequence | Lead-head | Pivot Sequence | Lead-head | |||

Ult. | 2nds | Ult. | 2nds | |||

Major | 5 | 7856342 | 8674523 | 5 | 7856342 | 8674523 |

Royal | 7 | 089674523 | 860492735 | 5 | 795038264 | 907856342 |

Maximus | 9 | 08T6E492735 | 8604T2E3957 | 5 | 795E3T20486 | 9E7T5038264 |

None of these will piece the leads together satisfactorily.

Investigation has shown that very few sequences permit the leads of an extension to be joined together in an order that produces a blue line bearing a close resemblance to that of the parent, and so it is our desire to amend the table of pivot sequences recommended to the Council in 1951 so that it reads as follows:-

Ultimate Place Methods only | 2nds Place Methods only | All Methods | ||||

Minor | 2 | 4 | 4 | 6 | 3 | 5 |

Major | 2 | 6 | 4 | 8 | 3 | 7 |

Royal | 2 | 8 | 4 | 0 | 3 | 9 |

Maximus | 2 | 0 | 4 | T | 3 | E |

The above sequences produce the lead heads of Plain Bob on all numbers and their reverse order, and those of Cambridge Surprise on all numbers and their reverse order, and no others. Whilst recognising that many true mathematical extensions give other sequences than those in the above table, we recommend that none but those above be recognised.

Our last year’s definition must now be amended as follows:-

“The pivot bell should be the same treble or the same tenor throughout the series of extensions but that the third treble and the third tenor be the limits. In the case of extensions from minor where the pivot may be regarded as either the third treble or the third tenor, second’s place methods should be regarded as the third treble and ultimate place methods as the third tenor.”

K. W. H. FELSTEAD

St. Michael’s Vicarage,

Southampton.

*The Ringing World*, May 30, 1952, page 345