J 20 On Allan's Dividing Instrument. 



slightly touch on a few points as a corroboralion of my 

 former opinion, 



VVhfn the wooden model of the wheel and ring; was first 

 put into mv hands al the Society's house, I immediatelv saw 

 thccreat utihty ot the ring. I am still in the same opinion, 

 that the principle, which is not attended with any great de- 

 gree of difficulty in formation, will be found to produce that 

 accuracy v\hich 1 think is not likely soon to be excelled. 

 The great improycnient which has teen made in lathes for 

 turninii large work would render that part of the opcratioit 

 quite easv, eyen it it was carried to six feet or more in dia- 

 nu-ter. A partial expansion at tho tinte of racking, seems 

 to me one of the greatest objections that can be iTiadc 

 to a \ar\rc size, and the bad effects of that I believe might 

 be overcome. 



-To that extraordinary remark in the Retrospect, saying, 

 " We can conceive cases in which the effect of the shifting 

 would br/ to diminish th.e size of one or more of the teeth 

 each time below that of the rest, instead of equalizing 

 them, and finallv, to cut some of them awav entirely," 

 I sliall make no rcjily ; only, that it cannot take place 

 without an error of at least one whole tooth, beibre the 

 shifting of the ring. Any man that knows the prmeiplc of 

 a wheelbarrow will see the absurdity of this remark in a 

 moment. 



T hope it is mentioned in the Transactions of the Society 

 of Arts, that when under the operation of racking or form- 

 ing the teeth, the ring must, after being reversed or moved 

 1 S0° once or twice, which will produce correctly all the way 

 roimd opposite teeth, although unequal in themselves, be 

 shifted onequarter or yO" ; and then proceed with the racking 

 and revcrsiuir as before, which will in the end, I have no 

 doubt, provided all the work and apparatus concerned 

 is good, produce the number of equal teeth required 

 round a circle, for the purpose of dividino; mathematical 

 iustrumenlB probably not yet equalled. 1 have had some 

 experience in these things, which 1 hope will allow me to 

 speak with that confidence I do on this point. The in- 

 strument alluded to in the I^etrospect is the first essay on 

 the art of dividing by the rack and screw, brouijht forth by 

 Dr. Hook in the year 1674, was merely a quadrant for 

 the purpose of astronomy, with teeth cut with a screw all 

 the way on the outer edge ; which was not into degrees 

 and minutes, but their true value was to be found after- 

 wards : therefore it was not intended as a machine to divide 



others 



