354 Dr. Everett on a new Proportion-Table. 



is printed. It is not found that any serious difference exists be- 

 tween different parts of the same card; but it is found that some 

 cards stretch or shrink more than others. This difficulty has 

 thus far been completely overcome by selecting and pairing 

 together those cards which agree best. 



It is worthy of remark that the breaking up of the scale into 

 parallel columns goes far to diminish the injurious effect of ex- 

 pansion or contraction ; for in a straight slide-rule this cause 

 operates on the whole length, whereas in the present arrange- 

 ment it only operates upon a twentieth part of the length. 



I may mention, as matter of curiosity, that in the case of a cir- 

 cular slide-rule it may be shown that, when expansion or con- 

 traction operates in one direction only and affects both circles 

 alike, its injurious effect is, in the most unfavourable position, 

 the same as if it operated tangentially along an arc equal to 

 the diameter. 



A somewhat singular property which incidentally attaches to 

 the new arrangement is, that if any straight line be drawn across 

 either of the two cards, the numbers through which it passes 

 form a geometrical progression. Hence, if two numbers have an 

 odd number of columns between them, a mean proportional can 

 be found by observing where the joining line cuts the central 

 intermediate column. As a particular case of this rule, the 

 square root of any number can be extracted by taking a line 

 from it to one or other of four fixed points, according to a simple 

 rule which is of universal application. 



The accuracy of working by the Proportion-Table is practi- 

 cally found to be about the same as in the use of four-figure 

 logarithms ; that is to say, it may be depended on to about one 

 part in four thousand ; and it has the advantage of greater sim- 

 plicity in use and less liability to mistake. To perform an ope- 

 ration in multiplication or division by logarithms, three refer- 

 ences to Tables are required besides the addition or subtraction 

 of the logarithms. In performing the same operation by the 

 Proportion-Table, the arithmetical work of adding or subtracting 

 is altogether dispensed with, and question and answer are read 

 off at one view. The saving of labour is especially great when 

 several numbers are to be multiplied or divided by the same fac- 

 tor, as one setting of the cards suffices for all. 



The facility with which multiplication and division can thus 

 be performed is of important service in simplifying certain tri- 

 gonometrical processes. There are certain formulae both in plane 

 and spherical trigonometry which, though particularly simple 

 and easily remembered, are not adapted to logarithmic computa- 

 tion, by reason of the number of references to the Tables which 

 their use would involve. These formulae are found to be conve- 



